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Preface
A fundamental problem with quantum theories of gravity, as opposed to
the other forces of nature, is that in Einsteins theory of gravity, general
relativity, there is no background geometry to work with the geometry
of spacetime itself becomes a dynamical variable. This is loosely sum
marized by saying that the theory is generally covariant. The standard
model of the strong, weak, and electromagnetic forces is dominated by the
presence of a nitedimensional group, the Poincare group, acting as the
symmetries of a spacetime with a xed geometrical structure. In general
relativity, on the other hand, there is an innitedimensional group, the
group of dieomorphisms of spacetime, which permutes dierent mathe
matical descriptions of any given spacetime geometry satisfying Einsteins
equations. This causes two dicult problems. On the one hand, one can no
longer apply the usual criterion for xing the inner product in the Hilbert
space of states, namely that it be invariant under the geometrical symmetry
group. This is the socalled inner product problem. On the other hand,
dynamics is no longer encoded in the action of the geometrical symmetry
group on the space of states. This is the socalled problem of time.
The prospects for developing a mathematical framework for quantizing
gravity have been considerably changed by a number of developments in
the s that at rst seemed unrelated. On the one hand, Jones discov
ered a new polynomial invariant of knots and links, prompting an intensive
search for generalizations and a unifying framework. It soon became clear
that the new knot polynomials were intimately related to the physics of
dimensional systems in many ways. Atiyah, however, raised the challenge
of nding a manifestly dimensional denition of these polynomials. This
challenge was met by Witten, who showed that they occur naturally in
ChernSimons theory. ChernSimons theory is a quantum eld theory in
dimensions that has the distinction of being generally covariant and also
a gauge theory, that is a theory involving connections on a bundle over
spacetime. Any knot thus determines a quantity called a Wilson loop,
the trace of the holonomy of the connection around the knot. The knot
polynomials are simply the expectation values of Wilson loops in the vac
uum state of ChernSimons theory, and their dieomorphism invariance
is a consequence of the general covariance of the theory. In the explosion
of work that followed, it became clear that a useful framework for under
standing this situation is Atiyahs axiomatic description of a topological
quantum eld theory, or TQFT.
On the other hand, at about the same time as Jones initial discov
ery, Ashtekar discovered a reformulation of general relativity in terms of
v
vi Preface
what are now called the new variables. This made the theory much more
closely resemble a gauge theory. The work of Ashtekar proceeds from the
canonical approach to quantization, meaning that solutions to Einsteins
equations are identied with their initial data on a given spacelike hyper
surface. In this approach the action of the dieomorphism group gives rise
to two constraints on initial data the dieomorphism constraint, which
generates dieomorphisms preserving the spacelike surface, and the Hamil
tonian constraint, which generates dieomorphisms that move the surface
in a timelike direction. Following the procedure invented by Dirac, one
expects the physical states to be annihilated by certain operators corre
sponding to the constraints.
In an eort to nd such states, Rovelli and Smolin turned to a descrip
tion of quantum general relativity in terms of Wilson loops. In this loop
representation, they saw that at least formally a large space of states
annihilated by the constraints can be described in terms of invariants of
knots and links. The fact that link invariants should be annihilated by the
dieomorphism constraint is very natural, since a link invariant is nothing
other than a function from links to the complex numbers that is invariant
under dieomorphisms of space. In this sense, the relation between knot
theory and quantum gravity is a natural one. But the fact that link in
variants should also be annihilated by the Hamiltonian constraint is deeper
and more intriguing, since it hints at a relationship between knot theory
and dimensional mathematics.
More evidence for such a relationship was found by Ashtekar and Ko
dama, who discovered a strong connection between ChernSimons theory
in dimensions and quantum gravity in dimensions. Namely, in terms
of the loop representation, the same link invariant that arises from Chern
Simons theorya certain normalization of the Jones polynomial called the
Kauman bracketalso represents a state of quantum gravity with cosmo
logical constant, essentially a quantization of antideSitter space. The full
ramications of this fact have yet to be explored.
The present volume is the proceedings of a workshop on knots and
quantum gravity held on May thth, at the University of Cali
fornia at Riverside, under the auspices of the Departments of Mathematics
and Physics. The goal of the workshop was to bring together researchers
in quantum gravity and knot theory and pursue a dialog between the two
subjects.
On Friday the th, Dana Fine began with a talk on ChernSimons
theory and the WessZuminoWitten model. Wittens original work deriv
ing the Jones invariant of links or, more precisely, the Kauman bracket
from ChernSimons theory used conformal eld theory in dimensions as
a key tool. By now the relationship between conformal eld theory and
topological quantum eld theories in dimensions has been explored from
a number of viewpoints. The path integral approach, however, has not yet
Preface vii
been worked out in full mathematical rigor. In this talk Fine described
work in progress on reducing the ChernSimons path integral on S
to the
path integral for the WessZuminoWitten model.
Oleg Viro spoke on Simplicial topological quantum eld theories. Re
cently there has been increasing interest in formulating TQFTs in a man
ner that relies upon triangulating spacetime. In a sense this is an old idea,
going back to the ReggePonzano model of Euclidean quantum gravity.
However, this idea was given new life by Turaev and Viro, who rigorously
constructed the ReggePonzano model of d quantum gravity as a TQFT
based on the j symbols for the quantum group SU
q
. Viro discussed a
variety of approaches of presenting manifolds as simplicial complexes, cell
complexes, etc., and methods for constructing TQFTs in terms of these
data.
Saturday began with a talk by Renate Loll on The loop formulation
of gauge theory and gravity, introduced the loop representation and var
ious mathematical issues associated with it. She also discussed her work
on applications of the loop representation to lattice gauge theory. Abhay
Ashtekars talk, Loop transforms, largely concerned the paper with Jerzy
Lewandowski appearing in this volume. The goal here is to make the loop
representation into rigorous mathematics. The notion of measures on the
space // of connections modulo gauge transformations has long been a
key concept in gauge theory, which, however, has been notoriously di
cult to make precise. A key notion developed by Ashtekar and Isham for
this purpose is that of the holonomy Calgebra, an algebra of observ
ables generated by Wilson loops. Formalizing the notion of a measure on
// as a state on this algebra, Ashtekar and Lewandowski have been able
to construct such a state with remarkable symmetry properties, in some
sense analogous to Haar measure on a compact Lie group. Ashtekar also
discussed the implications of this work for the study of quantum gravity.
Pullin spoke on The quantum Einstein equations and the Jones poly
nomial, describing his work with Bernd Br ugmann and Rodolfo Gambini
on this subject. He also sketched the proof of a new result, that the coef
cient of the nd term of the AlexanderConway polynomial represents a
state of quantum gravity with zero cosmological constant. His paper with
Gambini in this volume is a more detailed proof of this fact.
On Saturday afternoon, Louis Kauman spoke on Vassiliev invari
ants and the loop states in quantum gravity. One aspect of the work
of Br uegmann, Gambini, and Pullin that is especially of interest to knot
theorists is that it involves extending the bracket invariant to generalized
links admitting certain kinds of selfintersections. This concept also plays a
major role in the study of Vassiliev invariants of knots. Curiously, however,
the extensions occurring in the two cases are dierent. Kauman explained
their relationship from the path integral viewpoint.
Sunday morning began with a talk by Gerald Johnson, Introduction
viii Preface
to the Feynman integral and Feynmans operational calculus, on his work
with Michel Lapidus on rigorous path integral methods. Viktor Ginzburg
then spoke on Vassiliev invariants of knots, and in the afternoon, Paolo
CottaRamusino spoke on d quantum gravity and knot theory. This talk
dealt with his work in progress with Maurizio Martellini. Just as Chern
Simons theory gives a great deal of information on knots in dimensions,
there appears to be a relationship between a certain class of dimensional
theories and socalled knots, that is, embedded surfaces in dimensions.
This class, called BF theories, includes quantum gravity in the Ashtekar
formulation, as well as Donaldson theory. CottaRamusino and Martellini
have described a way to construct observables associated to knots, and
are endeavoring to prove at a perturbative level that they give knot in
variants.
Louis Crane spoke on Quantum gravity, spin geometry, and categor
ical physics. This was a review of his work on the relation between
ChernSimons theory and dimensional quantum gravity, his construc
tion with David Yetter of a dimensional TQFT based on the j symbols
for SU
q
, and his work with Igor Frenkel on certain braided tensor
categories. In the nal talk of the workshop, John Baez spoke on Strings,
loops, knots and gauge elds. He attempted to clarify the similarity be
tween the loop representation of quantum gravity and string theory. At
a xed time both involve loops or knots in space, but the stringtheoretic
approach is also related to the study of knots.
As some of the speakers did not submit papers for the proceedings, pa
pers were also solicited from Steve Carlip and also from J. Scott Carter and
Masahico Saito. Carlips paper, Geometric structures and loop variables,
treats the thorny issue of relating the loop representation of quantum grav
ity to the traditional formulation of gravity in terms of a metric. He treats
quantum gravity in dimensions, which is an exactly soluble test case. The
paper by Carter and Saito, Knotted surfaces, braid movies, and beyond,
contains a review of their work on knots as well as a number of new re
sults on braids. They also discuss the role in dimensional topology of
new algebraic structures such as braided tensor categories.
The editor would like to thank many people for making the workshop
a success. First and foremost, the speakers and other participants are to
be congratulated for making it such a lively and interesting event. The
workshop was funded by the Departments of Mathematics and Physics
of U. C. Riverside, and the chairs of these departments, Albert Stralka
and Benjamin Shen, were crucial in bringing this about. Michel Lapidus
deserves warm thanks for his help in planning the workshop. Invaluable
help in organizing the workshop and setting things up was provided by the
sta of the Department of Mathematics, and particularly Susan Spranger,
Linda Terry, and Chris Truett. Arthur Greenspoon kindly volunteered to
help edit the proceedings. Lastly, thanks go to all the participants in the
Preface ix
Knots and Quantum Gravity Seminar at U. C. Riverside, and especially
Jim Gilliam, Javier Muniain, and Mou Roy, who helped set things up.
x Preface
Contents
The Loop Formulation of Gauge Theory and Gravity
Renate Loll
Introduction
YangMills theory and general relativity as
dynamics on connection space
Introducing loops
Equivalence between the connection and loop
formulations
Some lattice results on loops
Quantization in the loop approach
Acknowledgements
Bibliography
Representation Theory of Analytic Holonomy C
algebras
Abhay Ashtekar and Jerzy Lewandowski
Introduction
Preliminaries
The spectrum
. Loop decomposition
. Characterization of //
. Examples of
A
Integration on //
. Cylindrical functions
. A natural measure
Discussion
Appendix A C
loops and U connections
Appendix B C
loops, UN and SUN connections
Acknowledgements
Bibliography
The Gauss Linking Number in Quantum Gravity
Rodolfo Gambini and Jorge Pullin
Introduction
The WheelerDeWitt equation in terms of loops
The Gauss selflinking number as a solution
Discussion
Acknowledgements
xi
xii Contents
Bibliography
Vassiliev Invariants and the Loop States in Quantum
Gravity
Louis H. Kauman
Introduction
Quantum mechanics and topology
Links and the Wilson loop
Graph invariants and Vassiliev invariants
Vassiliev invariants from the functional integral
Quantum gravityloop states
Acknowledgements
Bibliography
Geometric Structures and Loop Variables in
dimensional Gravity
Steven Carlip
Introduction
From geometry to holonomies
Geometric structures from holonomies
to geometry
Quantization and geometrical observables
Acknowledgements
Bibliography
From ChernSimons to WZW via Path
Integrals
Dana S. Fine
Introduction
The main result
//
n
as a bundle over
G
An explicit reduction from ChernSimons to WZW
Acknowledgements
Bibliography
Topological Field Theory as the Key to Quantum
Gravity
Louis Crane
Introduction
Quantum mechanics of the universe. Categorical physics
Ideas from TQFT
Hilbert space is deadlong live Hilbert space
or the observer gas approximation
The problem of time
Contents xiii
Matter and symmetry
Acknowledgements
Bibliography
Strings, Loops, Knots, and Gauge Fields
John C. Baez
Introduction
String eld/gauge eld duality
YangMills theory in dimensions
Quantum gravity in dimensions
Quantum gravity in dimensions
Acknowledgements
Bibliography
BF Theories and knots
Paolo CottaRamusino and Maurizio Martellini
Introduction
BF theories and their geometrical signicance
knots and their quantum observables
Gauss constraints
Path integrals and the Alexander invariant of a knot
Firstorder perturbative calculations and
higherdimensional linking numbers
Wilson channels
Integration by parts
Acknowledgements
Bibliography
Knotted Surfaces, Braid Movies, and Beyond
J. Scott Carter and Masahico Saito
Introduction
Movies, surface braids, charts, and isotopies
. Movie moves
. Surface braids
. Chart descriptions
. Braid movie moves
. Denition locality equivalence
The permutohedral and tetrahedral equations
. The tetrahedral equation
. The permutohedral equation
. Planar versions
categories and the movie moves
. Overview of categories
. The category of braid movies
xiv Contents
Algebraic interpretations of braid movie moves
. Identities among relations and Peier elements
. Moves on surface braids and identities among re
lations
. Symmetric equivalence and moves on surface braids
. Concluding remarks
Acknowledgements
Bibliography
The Loop Formulation of Gauge Theory and
Gravity
Renate Loll
Center for Gravitational Physics and Geometry,
Pennsylvania State University,
University Park, Pennsylvania , USA
email lollphys.psu.edu
Abstract
This chapter contains an overview of the loop formulation of Yang
Mills theory and dimensional gravity in the Ashtekar form.
Since the conguration spaces of these theories are spaces of gauge
potentials, their classical and quantum descriptions may be given in
terms of gaugeinvariant Wilson loops. I discuss some mathematical
problems that arise in the loop formulation and illustrate parallels
and dierences between the gauge theoretic and gravitational appli
cations. Some remarks are made on the discretized lattice approach.
Introduction
This chapter summarizes the main ideas and basic mathematical structures
that are necessary to understand the loop formulation of both gravity and
gauge theory. It is also meant as a guide to the literature, where all the
relevant details may be found. Many of the mathematical problems aris
ing in the loop approach are described in a related review article . I
have tried to treat YangMills theory and gravity in parallel, but also to
highlight important dierences. Section summarizes the classical and
quantum features of gauge theory and gravity, formulated as theories of
connections. In Section , the Wilson loops and their properties are intro
duced, and Section discusses the classical equivalence between the loop
and the connection formulation. Section contains a summary of progress
in a corresponding lattice loop approach, and in Section some of the main
ingredients of quantum loop representations are discussed.
YangMills theory and general relativity as
dynamics on connection space
In this section I will briey recall the Lagrangian and Hamiltonian for
mulation of both pure YangMills gauge theory and gravity, dened on a
dimensional manifold M
R of Lorentzian signature. The gravi
tational theory will be described in the Ashtekar form, in which it most
Renate Loll
closely resembles a gauge theory.
The classical actions dening these theories are up to overall constants
S
YM
A
M
d
x Tr F
F
M
TrF F .
and
S
GR
A, e
M
d
xee
I
e
J
F
IJ
M
IJKL
e
I
e
J
F
KL
.
for YangMills theory and general relativity respectively. In ., the basic
variable A is a LieGvalued connection form on M, where G denotes
the compact and semisimple Liestructure group of the YangMills theory
typically G SUN. In ., which is a rstorder form of the gravi
tational action, A is a selfdual, so,
C
valued connection form, and e
a vierbein, dening an isomorphism of vector spaces between the tangent
space of M and the xed internal space with the Minkowski metric
IJ
.
The connection A is selfdual in the internal space, A
MN
i
MN
IJ
A
IJ
,
with the totally antisymmetric tensor. A mathematical characterization
of the action . is contained in Baez chapter . As usual, F denotes
in both cases the curvature associated with the dimensional connection
A.
Note that the action . has the unusual feature of being complex. I
will not explain here why this still leads to a welldened theory equivalent
to the standard formulation of general relativity. Details may be found in
,.
Our main interest will be the Hamiltonian formulations associated with
. and .. Assuming the underlying principal bre bundles P, G to
be trivial, we can identify the relevant phase spaces with cotangent bundles
over ane spaces / of LieGvalued connections on the dimensional
manifold assumed compact and orientable. The cotangent bundle T
/
is coordinatized by canonically conjugate pairs A,
E, where A is a LieG
valued, pseudotensorial form gauge potential and
E a LieGvalued
vector density generalized electric eld on . We have G SUN, say,
for gauge theory, and G SO
C
for gravity. The standard symplectic
structure on T
/ is expressed by the fundamental Poisson bracket relations
A
i
a
x,
E
b
j
y
i
j
b
a
xy.
for gauge theory, and
A
i
a
x,
E
b
j
yi
i
j
b
a
xy.
for gravity, where the factor of i arises because the connection A is complex
valued. In . and ., a and b are dimensional spatial indices and i
The Loop Formulation of Gauge Theory and Gravity
and j internal gauge algebra indices.
However, points in T
/ do not yet correspond to physical congura
tions, because both theories possess gauge symmetries, which are related
to invariances of the original Lagrangians under certain transformations
on the fundamental variables. At the Hamiltonian level, this manifests it
self in the existence of socalled rstclass constraints. These are sets of
functions C on phase space, which are required to vanish, C , for phys
ical congurations and at the same time generate transformations between
physically indistinguishable congurations. In our case, the rstclass con
straints for the two theories are
YM
D
a
E
ai
.
and
GR
D
a
E
ai
,C
a
F
ab
i
E
b
i
,C
ijk
F
ab i
E
a
j
E
b
k
..
The physical interpretation of these constraints is as follows the so
called Gauss law constraints D
a
E
ai
, which are common to both theo
ries, restrict the allowed congurations to those whose covariant divergence
vanishes, and at the same time generate local gauge transformations in the
internal space. The wellknown YangMills transformation law correspond
ing to a group element g , the set of Gvalued functions on , is
A,
Eg
Ag g
dg, g
Eg. .
The second set, C
a
, of constraints in . generates dimensional
dieomorphisms of and the last expression, the socalled scalar or Hamil
tonian constraint C, generates phase space transformations that are inter
preted as corresponding to the time evolution of the spatial slice in M in
a spacetime picture. Comparing . and ., we see that in the Hamil
tonian formulation, pure gravity may be interpreted as a YangMills theory
with gauge group G SO
C
, subject to four additional constraints in
each point of . However, the dynamics of the two theories is dierent we
have
H
YM
A,
E
d
xg
ab
g
E
ai
E
b
i
gB
ai
B
b
i
.
as the Hamiltonian for YangMills theory where for convenience we have
introduced the generalized magnetic eld B
a
i
abc
F
bc i
. Note the
explicit appearance of a Riemannian background metric g on , to contract
indices and ensure the integrand in . is a density. On the other hand, no
such additional background structure is necessary to make the gravitational
Hamiltonian well dened. The Hamiltonian H
GR
for gravity is a sum of a
Renate Loll
subset of the constraints .,
H
GR
A,
Ei
d
x
N
a
F
ab
i
E
b
i
i
N
ijk
F
ab k
E
a
i
E
b
j
,.
and therefore vanishes on physical congurations.
N and N
a
in .
are Lagrange multipliers, the socalled lapse and shift functions. This
latter feature is peculiar to generally covariant theories, whose gauge group
contains the dieomorphism group of the underlying manifold. A further
dierence from the gaugetheoretic application is the need for a set of reality
conditions for the gravitational theory, because the connection A is a priori
a complex coordinate on phase space.
The space of physical congurations of YangMills theory is the quotient
space // of gauge potentials modulo local gauge transformations. It
is nonlinear and has nontrivial topology. After excluding the reducible
connections from /, this quotient can be given the structure of an innite
dimensional manifold for compact and semisimple G . The corresponding
physical phase space is then its cotangent bundle T
//. Elements of
T
// are called classical observables of the YangMills theory.
The nonlinearities of both the equations of motion and the physical
conguration/phase spaces, and the presence of gauge symmetries for these
theories, lead to numerous diculties in their classical and quantum de
scription. For example, in the path integral quantization of YangMills
theory, a gaugexing term has to be introduced in the sum over all con
gurations
Z
,
dA e
iSY M
,.
to ensure that the integration is only taken over one member
A of each
gauge equivalence class. However, because the principal bundle / //
is nontrivial, we know that a unique and attainable gauge choice does not
exist this is the assertion of the socalled Gribov ambiguity. In any case,
since the expression . can be made meaningful only in a weakeld
approximation where one splits S
YM
S
free
S
I
, with S
free
being just
quadratic in A, this approach has not yielded enough information about
other sectors of the theory, where the elds cannot be assumed weak.
Similarly, in the canonical quantization, since no good explicit descrip
tion of T
// is available, one quantizes the theory a la Dirac. This
means that one rst quantizes on the unphysical phase space T
/, as
if there were no constraints. That is, one uses a formal operator repre
sentation of the canonical variable pairs A,
E, satisfying the canonical
commutation relations
A
i
a
x,
E
b
j
y ih
i
j
b
a
x y, .
The Loop Formulation of Gauge Theory and Gravity
the quantum analogues of the Poisson brackets .. Then a subset of
physical wave functions T
YM
phy
phy
A is projected out from the space
of all wave functions, T A, according to
T
YM
phy
AT
TEA , .
i.e. T
YM
phy
consists of all functions that lie in the kernel of the quantized
Gauss constraint,
TE. For a critical appraisal of the use of such Dirac
conditions, see . The notations T and T
YM
phy
indicate that these are
just spaces of complexvalued functions on / and do not yet carry any
Hilbert space structure. Whereas this is not required on the unphysical
space T, one does need such a structure on T
YM
phy
in order to make physical
statements, for instance about the spectrum of the quantum Hamiltonian
H. Unfortunately, we lack an appropriate scalar product, i.e. an appropri
ate gaugeinvariant measure dA
in
phy
,
t
phy
,/
dA
phy
A
t
phy
A. .
More precisely, we expect the integral to range over an extension of the
space // which includes also distributional elements . Given such a
measure, the space of physical wave functions would be the Hilbert space
H
YM
T
YM
phy
of functions squareintegrable with respect to the inner
product ..
Unfortunately, the situation for gravity is not much better. Here the
space of physical wave functions is the subspace of T projected out accord
ing to
T
GR
phy
AT
TEA ,
C
a
A,
CA . .
Again, one now needs an inner product on the space of such states. As
suming that the states
phy
of quantum gravity can be described as the
set of complexvalued functions on some still innitedimensional space
/, the analogue of . reads
phy
,
t
phy
,
dA
phy
A
t
phy
A, .
where the measure dA is now both gauge and dieomorphism invariant
and projects in an appropriate way to /. If we had such a measure again,
we do not, the Hilbert space H
GR
of states for quantum gravity would con
sist of all those elements of T
GR
phy
which are squareintegrable with respect
to that measure. In theup to nowabsence of explicit measures to make
., . well dened, our condence in the procedure outlined here
stems from the fact that it can be made meaningful for lowerdimensional
model systems such as YangMills theory in and gravity in di
Renate Loll
mensions, where the analogues of the space / are nite dimensional cf.
also .
Note that there is a very important dierence in the role played by
the quantum Hamiltonians in gauge theory and gravity. In YangMills
theory, one searches for solutions in H
YM
of the eigenvector equation
H
YM
phy
AE
phy
A, whereas in gravity the quantum Hamiltonian
C is one of the constraints and therefore a physical state of quantum gravity
must lie in its kernel,
C
phy
A.
After all that has been said about the diculties of nding a canon
ical quantization for gauge theory and general relativity, the reader may
wonder whether one can make any statements at all about their quantum
theories. For the case of YangMills theory, the answer is of course in the
armative. Many qualitative and quantitative statements about quantum
gauge theory can be derived in a regularized version of the theory, where
the at background spacetime is approximated by a hypercubic lattice.
Using rened computational methods, one then tries to extract for either
weak or strong coupling properties of the continuum theory, for exam
ple an approximate spectrum of the Hamiltonian
H
YM
, in the limit as
the lattice spacing tends to zero. Still there are nonperturbative features,
the most important being the connement property of YangMills theory,
which also in this framework are not well understood.
Similar attempts to discretize the Hamiltonian theory have so far been
not very fruitful in the treatment of gravity, because, unlike in YangMills
theory, there is no xed background metric with respect to which one could
build a hypercubic lattice, without violating the dieomorphism invariance
of the theory. However, there are more subtle ways in which discrete struc
tures arise in canonical quantum gravity. An example of this is given by the
latticelike structures used in the construction of dieomorphisminvariant
measures on the space // , . One important ingredient we will be
focusing on are the nonlocal loop variables on the space / of connections,
which will be the subject of the next section.
Introducing loops
A loop in is a continuous map from the unit interval into ,
,
s
a
s, .
s. t. .
The set of all such maps will be denoted by , the loop space of .
Given such a loop element , and a space / of connections, we can dene
The Loop Formulation of Gauge Theory and Gravity
a complex function on /, the socalled Wilson loop,
T
A
N
Tr
R
P exp
A. .
The pathordered exponential of the connection A along the loop , U
P exp
A, is also known as the holonomy of A along . The holonomy
measures the change undergone by an internal vector when parallel trans
ported along . The trace is taken in the representation R of G, and N is
the dimensionality of this linear representation. The quantity T
A
was
rst introduced by K. Wilson as an indicator of conning behaviour in the
lattice gauge theory . It measures the curvature or eld strength in a
gaugeinvariant way the components of the tensor F
ab
itself are not mea
surable, as can easily be seen by expanding T
A
for an innitesimal loop
. The Wilson loop has a number of interesting properties.
. It is invariant under the local gauge transformations ., and there
fore an observable for YangMills theory.
. It is invariant under orientationpreserving reparametrizations of the
loop .
. It is independent of the basepoint chosen on .
It follows from and that the Wilson loop is a function on a crossproduct
of the quotient spaces, T
A
T// /Di
S
.
In gravity, the Wilson loops are not observables, since they are not in
variant under the full set of gauge transformations of the theory. However,
there is now one more reason to consider variables that depend on closed
curves in , since observables in gravity also have to be invariant under dif
feomorphisms of . This means going one step further in the construction
of such observables, by considering Wilson loops that are constant along
the orbits of the Diaction,
T T, Di. .
Such a loop variable will just depend on what is called the generalized
knot class K of generalized since may possess selfintersections and
selfoverlaps, which is not usually considered in knot theory. Note that
a similar construction for taking care of the dieomorphism invariance is
not very contentful if we start from a local scalar function, x, say. The
analogue of . would then tell us that had to be constant on any
connected component of . The set of such variables is not big enough
to account for all the degrees of freedom of gravity. In contrast, loops
can carry dieomorphisminvariant information such as their number of
selfintersections, number of points of nondierentiability, etc.
The use of nonlocal holonomy variables in gauge eld theory was pi
oneered by Mandelstam in his paper on Quantum electrodynamics
Renate Loll
without potentials , and later generalized to the nonAbelian theory
by BialynickiBirula and again Mandelstam . Another upsurge in
interest in pathdependent formulations of YangMills theory took place
at the end of the s, following the observation of the close formal re
semblance of a particular form of the YangMills equations in terms of
holonomy variables with the equations of motion of the dimensional non
linear sigma model , . Another set of equations for the Wilson loop
was derived by Makeenko and Migdal see, for example, the review .
The hope underlying such approaches has always been that one may be
able to nd a set of basic variables for YangMills theory which are better
suited to its quantization than the local gauge potentials Ax. Owing to
their simple behaviour under local gauge transformations, the holonomy
or the Wilson loop seem like ideal candidates. The aim has therefore been
to derive suitable dierential equations in loop space for the holonomy
U
, its trace, T, or for their vacuum expectation values, which are to
be thought of as the analogues of the usual local YangMills equations of
motion. The fact that none of these attempts have been very successful
can be attributed to a number of reasons.
To make sense of dierential equations on loop space, one rst has to
introduce a suitable topology, and then set up a dierential calculus
on . Even if we give locally the structure of a topological vector
space, and are able to dene dierentiation, this in general will ensure
the existence of neither inverse and implicit function theorems nor
theorems on the existence and uniqueness of solutions of dierential
equations. Such issues have only received scant attention in the past.
Since the space of all loops in is so much larger than the set of
all points in , one expects that not all loop variables will be in
dependent. This expectation is indeed correct, as will be explained
in the next section. Still, the ensuing redundancy is hard to control
in the continuum theory, and has therefore obscured many attempts
of establishing an equivalence between the connection and the loop
formulation of gauge theory. For example, there is no action princi
ple in terms of loop variables, which leaves considerable freedom for
deriving loop equations of motion. As another consequence, pertur
bation theory in the loop approach always had to fall back on the
perturbative expansion in terms of the connection variable A.
Equivalence between the connection and loop
formulations
In this section I will discuss the motivation for adopting a pure loop for
mulation for any theory whose conguration space comes from a space
// of connection forms modulo local gauge transformations. For the
YangMills theory itself, there is a strong physical motivation to choose
The Loop Formulation of Gauge Theory and Gravity
the variables T as a set of basic variables on physical grounds, one
requires only physical observables O i.e. in our case the Wilson loops, and
not the local elds A to be represented by selfadjoint operators
O in the
quantum theory, with the classical Poisson relations O, O
t
going over to
the quantum commutators ih
O,
O
t
.
There is also a mathematical justication for reexpressing the classical
theory in terms of loop variables. It is a wellknown result that one can
reconstruct the gauge potential A up to gauge transformations from the
knowledge of all holonomies U
, for any structure group G GLN, C.
Therefore all that remains to be shown is that one can reconstruct the
holonomies U
from the set of Wilson loops T. It turns out that,
at least for compact G, there is indeed an equivalence between the two
quotient spaces
/
T, /Di
S
Mandelstam constraints
..
By Mandelstam constraints I mean a set of necessary and sucient con
ditions on generic complex functions f on /Di
S
that ensure
they are the traces of holonomies of a group G in a given representation
R. They are typically algebraic, nonlinear constraints among the T.
To obtain a complete set of such conditions without making reference to
the connection space / is a nontrivial problem that for general G to my
knowledge has not been solved.
For example, for G SL, C in the fundamental representation by
matrices, the Mandelstam constraints are
a Tpoint loop
bTT
cT
T
.
dTT
t
,
t
eT
T
T
T
.
The loop
t
in eqn d is the loop with an appendix
attached,
where is any path not necessarily closed starting at a point of see
Fig. . In e, is a path starting at a point of
and ending at
see
Fig. . The circle denotes the usual path composition. It is believed
that the set . and all conditions that can be derived from . exhaust
the Mandelstam constraints for this particular gauge group and represen
tation, although I am not aware of a formal proof. Still, owing to the
noncompactness of G, it can be shown that not every element of // can
be reconstructed from the Wilson loops. This is the case for holonomies
that take their values in the socalled subgroup of null rotations . How
ever, this leads only to an incompleteness of measure zero of the Wilson
Renate Loll
Fig. . Adding an appendix
to the loop does not change the
Wilson loop
Fig. . The loop congurations related by the Mandelstam constraint
T
T
T
T
loops on //.
If one wants to restrict the group G to a subgroup of SL, C, there
are further conditions on the loop variables, which now take the form of
inequalities . For SU SL, C, one nds the two conditions
T,
T
T
T
T
,.
whereas for SU, SL, C one derives
T
T
T
T
T
T
,.
and no restrictions for other values of T
and T
. Unfortunately,
The Loop Formulation of Gauge Theory and Gravity
. and . do not exhaust all inequalities there are for SU and
SU, respectively. There are more inequalities corresponding to more
complicated congurations of more than two intersecting loops, which are
critically dependent on the geometry of the loop conguration.
Let me nish this section with some further remarks on the Mandelstam
constraints
In cases where the Wilson loops form a complete set of variables,
they are actually overcomplete, due to the existence of the Mandel
stam constraints i.e. implicitly the nontrivial topological structure
of //. This is the overcompleteness I referred to in the previous
section.
Note that one may interpret some of the constraints . as con
straints on the underlying loop space, i.e. the Wilson loops are eec
tively not functions on /Di
S
, but on some appropriate quo
tient with respect to equivalence relations such as
. The
existence of such relations makes it dicult to apply directly math
ematical results on the loop space or the space /Di
S
of oriented, unparametrized loops such as , , , to the
present case.
One has to decide how the Mandelstam constraints are to be carried
over to the quantum theory. One may try to eliminate as many of
them as possible already in the classical theory, but real progress
in this direction has only been made in the discretized lattice gauge
theory see Section below. Another possibility will be mentioned
in Section .
Finally, let me emphasize that the equivalence of the connection and
the loop description at the classical level does not necessarily mean
that quantization with the Wilson loops as a set of basic variables
will be equivalent to a quantization in which the connections A are
turned into fundamental quantum operators. One may indeed hope
that at some level, those two possibilities lead to dierent results,
with the loop representation being better suited for the description
of the nonperturbative structure of the theory.
Some lattice results on loops
This section describes the idea of taking the Wilson loops as the set of basic
variables, as applied to gauge theory, although some of the techniques and
conclusions may be relevant for gravitational theories too. The lattice
discretization is particularly suited to the loop formulation, because in it
the gauge elds are taken to be concentrated on the dimensional links l
i
of the lattice. The Wilson loops are easily formed by multiplying together
the holonomies U
l
...U
ln
of a set of contiguous, oriented links that form
Renate Loll
Fig. . is a loop on the hypercubic lattice N
d
,N,d.
a closed chain on the lattice see Fig. ,
T Tr
R
U
l
U
l
...U
ln
..
We take the lattice to be the hypercubic lattice N
d
, with periodic
boundary conditions N is the number of links in each direction, and
the gauge group G to be SU, again in the fundamental representation.
Although the lattice is nite, there are an innite number of closed con
tours one can form on it, and hence an innite number of Wilson loops. At
the same time, the number of Mandelstam constraints . is innite. The
only constraints that cause problems are those of type e, because of their
nonlinear and highly coupled character. Nevertheless, as I showed in ,
it is possible to solve this coupled set of equations and obtain an explicit
local description of the nitedimensional physical conguration space
phy
/
lattice
T, a lattice loop
Mandelstam constraints
.
in terms of an independent set of Wilson loops. The nal solution for
the independent degrees of freedom is surprisingly simple it suces to
use the Wilson loops of small lattice loops, consisting of no more than six
lattice links. At this point it may be noted that a similar problem arises
in gravity in the connection formulation on a manifold
g
R, with
a compact, dimensional Riemann surface
g
of genus g. One also has a
discrete set of loops, namely, the elements of the homotopy group of
g
,
and can introduce the corresponding Wilson loops. Again, one is interested
in nding a maximally reduced set of such Wilson loops, in terms of which
all the others can be expressed, using the Mandelstam constraints see, for
example, , .
The Loop Formulation of Gauge Theory and Gravity
Of course it does not suce to establish a set of independent loop
variables on the lattice one also has to show that the dynamics can be
expressed in a manageable form in terms of these variables. Some important
problems are the following.
. Since the space
phy
// is topologically nontrivial, the set of
independent lattice Wilson loops cannot provide a good global chart
for
phy
. It would therefore be desirable to nd a minimal embedding
into a space of loop variables that is topologically trivial together
with a nite number of nonlinear constraints that dene the space
phy
.
. What is the explicit form of the Jacobian of the transformation from
the holonomy link variables to the Wilson loops From this one could
derive an eective action/Hamiltonian for gauge theory, and maybe
derive a prescription of how to translate local quantities into their
loop space analogues.
. Is it possible to dene a perturbation theory in loop space, once one
understands which Wilson loop congurations contribute most in the
path integral, say This would be an interesting alternative to the
usual perturbation theory in A or the
N
expansion, and may lead
to new and more ecient ways of performing calculations in lattice
gauge theory.
. Lastly, in nding a solution to the Mandelstam constraints it was
crucial to have the concept of a smallest loop size on the lattice the
loops of link length going around a single plaquette. What are the
consequences of this result for the continuum theory, where there is
no obstruction to shrinking loops down to points
Some progress on points and has been made for small lattices , and
it is clear that all of the aboveposed problems are rather nontrivial, even
in the discretized lattice theory where the number of degrees of freedom is
eectively nite. One may be able to apply some of the techniques used
here to dimensional gravity, where discrete structures appear too the
generalized knot classes after factoring out by the spatial dieomorphisms.
Quantization in the loop approach
I will now return to the discussion of the continuum theory of both gravity
and gauge theory, and review a few concepts that have been used in their
quantization in a loop formulation. The contributions by Ashtekar and
Lewandowski and Pullin in this volume contain more details about
some of the points raised here.
The main idea is to base the canonical quantization of a theory with
conguration space / on the nonlocal, gaugeinvariant phase space vari
ables T and not on some gaugecovariant local eld variables. This
Renate Loll
is a somewhat unusual procedure in the quantization of a eld theory, but
takes directly into account the nonlinearities of the theory.
You may have noted that so far we have only talked about congura
tion space variables, the Wilson loops. For a canonical quantization we
obviously need some momentum variables that depend on the generalized
electric eld
E. One choice of a generalized Wilson loop that also depends
on
E is
T
a
A,
E
, s Tr U
s
E
a
s. .
This variable depends now on both a loop and a marked point, and it is still
gauge invariant under the transformations .. Also it is strictly speaking
not a function on phase space, but a vector density. This latter fact may be
remedied by integrating T
a
over a dimensional ribbon or strip, i.e. a
nondegenerate parameter congruence of curves
t
s Rs, t, t ,
. In any case, the momentum Wilson loop depends not just on a loop,
but needs an additional geometric ingredient. I will use here the unsmeared
version since it does not aect what I am going to say.
The crucial ingredient in the canonical quantization is the fact that the
loop variables T, T
a
, s form a closing algebra with respect to the
canonical symplectic structure on T
/. This fact was rst established by
Gambini and Trias for the gauge group SUN and later rediscovered
by Rovelli and Smolin in the context of gravity . For the special case of
G SL, C or any of its subgroups in the fundamental representation,
the Poisson algebra of the loop variable s is
T
,T
,
T
a
, s, T
a
,
s
T
s
T
s
,
T
a
, s, T
b
,t.
a
,
s
T
b
s
, ut T
b
s
, ut
b
,
t
T
a
t
, vs T
a
t
, vs
.
In the derivation of this semidirect product algebra, the Mandelstam con
straints .e have been used to bring the righthand sides into a form
linear in T. The structure constants are distributional and depend just
on the geometry of loops and intersection points,
a
,x
dt
t, x
a
t. .
The Poisson bracket of two loop variables is nonvanishing only when
an insertion of an electric eld
E in a T
a
variable coincides in with the
holonomy of another loop. For gauge group SUN, the righthand sides
of the Poisson brackets can no longer be written as linear expressions in T.
The Loop Formulation of Gauge Theory and Gravity
By a quantization of the theory in the loop representation we will mean
a representation of the loop algebra . as the commutator algebra of a
set of selfadjoint operators
T,
T
a
, s or an appropriately smeared
version of the momentum operator
T
a
, s. Secondly, the Mandelstam
constraints must be incorporated in the quantum theory, for example by
demanding that relations like . hold also for the corresponding quan
tum operators
T. For the momentum Wilson loops there are analogous
Mandelstam constraints.
The completion of this quantization program is dierent for gauge the
ory and gravity, in line with the remarks made in Section above. For
YangMills theory, one has to nd a Hilbert space of wave functionals that
depend on either the Wilson loops T or directly on the elements of
some appropriate quotient of a loop space. The quantized Wilson loops
must be selfadjoint with respect to the inner product on this Hilbert space,
since they correspond to genuine observables in the classical theory. Then
the YangMills Hamiltonian . has to be reexpressed as a function of
Wilson loops, and nally, one has to solve the eigenvector equation
H
YM
T, T
a
E, .
to obtain the spectrum of the theory. Because of the lack of an appropriate
scalar product in the continuum theory, the only progress that has been
made along these lines is in the latticeregularized version of the gauge
theory see, for example, , .
For the case of general relativity one does not require the loop vari
ables T, T
a
, s to be represented selfadjointly, because they do not
constitute observables on the classical phase space. At least one such repre
sentation where the generalized Wilson loops are represented as dierential
operators on a space of loop functionals
is known, and has been ex
tensively used in quantum gravity. The completion of the loop quantization
program for gravity requires an appropriate inclusion of the dimensional
dieomorphism constraints C
a
and the Hamiltonian constraint C, cf. .,
which again means that they have to be reexpressed in terms of loop vari
ables. Then, solutions to these quantum constraints have to be found, i.e.
loop functionals that lie in their kernel,
C
a
T, T
a
CT, T
a
..
Finally, the space of these solutions or an appropriate subspace has
to be given a Hilbert space structure to make a physical interpretation
possible. No suitable scalar product is known at the moment, but there
exist a large number of solutions to the quantum constraint equations ..
These solutions and their relation to knot invariants are the subject of
Pullins chapter .
There is one more interesting mathematical structure in the loop ap
Renate Loll
proach to which I would like to draw attention, and which tries to establish
a relation between the quantum connection and the quantum loop rep
resentations. This is the socalled loop transform, rst introduced by
Rovelli and Smolin , which is supposed to intertwine the two types of
representations according to
,/
dA
T
A
phy
A.
and
T
,/
dA
T
A
T
phy
A
,.
and similarly for the action of the
T
a
. Indeed, it was by the heuristic
use of . and . that Rovelli and Smolin arrived at the quantum
representation of the loop algebra . mentioned above. The relation
. is to be thought of as a nonlinear analogue of the Fourier transform
p
dx e
ixp
x.
on L
R, dx, but does in general not possess an inverse. Again, for Yang
Mills theory we do not know how to construct a concrete measure dA
that
would allow us to proceed with a loop quantization program, although in
this case the loop transform can be given a rigorous mathematical meaning.
For the application to gravity, the domain space of the integration will
be a smaller space /, as explained in Section above, but for this case we
know even less about the welldenedness of expressions like . and ..
There is, however, a variety of simpler examples such as dimensional
gravity, electrodynamics, etc. where the loop transform is a very useful
and mathematically welldened tool.
The following diagram Fig. summarizes the relations between con
nection and loop dynamics, both classically and quantummechanically, as
outlined in Sections and .
The Loop Formulation of Gauge Theory and Gravity
Fig. .
Renate Loll
Acknowledgements
Warm thanks to John Baez for organizing an inspiring workshop.
Bibliography
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integrability, Lett. Math. Phys.
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Renate Loll
Representation Theory of Analytic Holonomy
Calgebras
Abhay Ashtekar
Center for Gravitational Physics and Geometry,
Pennsylvania State University,
University Park, Pennsylvania , USA
email ashtekarphys.psu.edu
Jerzy Lewandowski
Department of Physics, University of Florida,
Gainesville, Florida , USA
email lewandfuw.edu.pl
Abstract
Integral calculus on the space A/G of gaugeequivalent connections
is developed. Loops, knots, links, and graphs feature prominently
in this description. The framework is well suited for quantization of
dieomorphisminvariant theories of connections.
The general setting is provided by the Abelian Calgebra of
functions on A/G generated by Wilson loops i.e. by the traces of
holonomies of connections around closed loops. The representation
theory of this algebra leads to an interesting and powerful dual
ity between gauge equivalence classes of connections and certain
equivalence classes of closed loops. In particular, regular measures
on a suitable completion of A/G are in correspondence with
certain functions of loops and dieomorphism invariant measures
correspond to generalized knot and link invariants. By carrying
out a nonlinear extension of the theory of cylindrical measures on
topological vector spaces, a faithful, dieomorphisminvariant mea
sure is introduced. This measure can be used to dene the Hilbert
space of quantum states in theories of connections. The Wilson loop
functionals then serve as the conguration operators in the quantum
theory.
Introduction
The space // of connections modulo gauge transformations plays an im
portant role in gauge theories, including certain topological theories and
general relativity . In a typical setup, // is the classical congu
ration space. In the quantum theory, then, the Hilbert space of states
On leave from Instytut Fizyki Teoretycznej, Warsaw University ul. Hoza ,
Warszawa, Poland
Abhay Ashtekar and Jerzy Lewandowski
would consist of all squareintegrable functions on // with respect to
some measure. Thus, the theory of integration over // would lie at
the heart of the quantization procedure. Unfortunately, since // is an
innitedimensional nonlinear space, the wellknown techniques to dene
integrals do not go through and, at the outset, the problem appears to be
rather dicult. Even if this problem could be overcome, one would still
face the issue of constructing selfadjoint operators corresponding to phys
ically interesting observables. In a Hamiltonian approach, the Wilson loop
functions provide a natural set of manifestly gaugeinvariant congura
tion observables. The problem of constructing the analogous, manifestly
gaugeinvariant momentum observables is dicult already in the classical
theory these observables would correspond to vector elds on // and
dierential calculus on this space is not well developed.
Recently, Ashtekar and Isham hereafter referred to as AI devel
oped an algebraic approach to tackle these problems. The AI approach
is in the setting of canonical quantization and is based on the ideas in
troduced by Gambini and Trias in the context of YangMills theories
and by Rovelli and Smolin in the context of quantum general relativity
. Fix an nmanifold on which the connections are to be dened and
a compact Lie group G which will be the gauge group of the theory un
der consideration
. The rst step is the construction of a Calgebra of
conguration observablesa suciently large set of gaugeinvariant func
tions of connections on . A natural strategy is to use the Wilson loop
functionsthe traces of holonomies of connections around closed loops on
to generate this Calgebra. Since these are conguration observables,
they commute even in the quantum theory. The Calgebra is therefore
Abelian. The next step is to construct representations of this algebra.
For this, the Gelfand spectral theory provides a natural setting since any
given Abelian Calgebra with identity is naturally isomorphic with the
Calgebra of continuous functions on a compact, Hausdor space, the
Gelfand spectrum spC
of that algebra. As the notation suggests,
spC
can be constructed directly from the given Calgebra its ele
ments are homomorphisms from the given Calgebra to the algebra of
complex numbers. Consequently, every continuous cyclic representation
of the AI Calgebra is of the following type the carrier Hilbert space is
L
spC
, d for some regular measure d and the Wilson loop operators
act on squareintegrable functions on spC
simply by multiplication.
AI pointed out that, since the elements of the Calgebra are labelled by
loops, there is a correspondence between regular measures on spC
In typical applications, will be a Cauchy surface in an n dimensional space
time in the Lorentzian regime or an ndimensional spacetime in the Euclidean regime.
In the main body of this paper, n will be taken to be and G will be taken to be
SU both for concreteness and simplicity. These choices correspond to the simplest
nontrivial applications of the framework to physics.
Holonomy Calgebras
and certain functions on the space of loops. Dieomorphisminvariant mea
sures correspond to knot and link invariants. Thus, there is a natural inter
play between connections and loops, a point which will play an important
role in the present paper.
Note that, while the classical conguration space is //, the domain
space of quantum states is spC
. AI showed that // is naturally
embedded in spC
and Rendall showed that the embedding is in fact
dense. Elements of spC
can be therefore thought of as generalized gauge
equivalent connections. To emphasize this point and to simplify the nota
tion, let us denote the spectrum of the AI Calgebra by //. The fact
that the domain space of quantum states is a completion ofand hence
larger thanthe classical conguration space may seem surprising at rst
since in ordinary i.e. nonrelativistic quantum mechanics both spaces are
the same. The enlargement is, however, characteristic of quantum eld
theory, i.e. of systems with an innite number of degrees of freedom. AI
further explored the structure of // but did not arrive at a complete
characterization of its elements. They also indicated how one could make
certain heuristic considerations of Rovelli and Smolin precise and associate
momentum operators with closed strips i.e. ndimensional ribbons
on . However, without further information on the measure d on //,
one cannot decide if these operators can be made selfadjoint, or indeed
if they are even densely dened. AI did introduce certain measures with
which the strip operators interact properly. However, as they pointed out,
these measures have support only on nitedimensional subspaces of //
and are therefore too simple to be interesting in cases where the theory
has an innite number of degrees of freedom.
In broad terms, the purpose of this paper is to complete the AI program
in several directions. More specically, we will present the following results
. We will obtain a complete characterization of elements of the Gelfand
spectrum. This will provide a degree of control on the domain space of
quantum states which is highly desirable since // is a nontrivial,
innitedimensional space which is not even locally compact, while
// is a compact Hausdor space, the completion procedure is quite
nontrivial. Indeed, it is the lack of control on the precise content
of the Gelfand spectrum of physically interesting Calgebras that
has prevented the Gelfand theory from playing a prominent role in
quantum eld theory so far.
. We will present a faithful, dieomorphisminvariant measure on //.
The theory underlying this construction suggests how one might in
troduce other dieomorphisminvariant measures. Recently, Baez
has exploited this general strategy to introduce new measures. These
developments are interesting from a strictly mathematical standpoint
because the issue of existence of such measures was a subject of
Abhay Ashtekar and Jerzy Lewandowski
controversy until recently
. They are also of interest from a physical
viewpoint since one can, for example, use these measures to regulate
physically interesting operators such as the constraints of general
relativity and investigate their properties in a systematic fashion.
. Together, these results enable us to dene a RovelliSmolin loop
transform rigorously. This is a map from the Hilbert space
L
//, d to the space of suitably regular functions of loops on
. Thus, quantum states can be represented by suitable functions
of loops. This loop representation is particularly well suited to
dieomorphisminvariant theories of connections, including general
relativity.
We have also developed dierential calculus on // and shown that the
strip i.e. momentum operators are in fact densely dened, symmetric op
erators on L
//, d i.e. that they interact with our measure correctly.
However, since the treatment of the strip operators requires a number of
new ideas from physics as well as certain techniques from graph theory,
these results will be presented in a separate work.
The methods we use can be summarized as follows.
First, we make the nonlinear duality between connections and loops
explicit. On the connection side, the appropriate space to consider is the
Gelfand spectrum // of the AI Calgebra. On the loop side, the ap
propriate object is the space of hoopsi.e. holonomically equivalent loops.
More precisely, let us consider based, piecewise analytic loops and regard
two as being equivalent if they give rise to the same holonomy evaluated
at the basepoint for any Gconnection. Call each holonomic equivalence
class a Ghoop. It is straightforward to verify that the set of hoops has
the structure of a group. It also carries a natural topology , which,
however, will not play a role in this paper. We will denote it by H and
call it the hoop group. It turns out that H and the spectrum // can be
regarded as being dual to each other in the following sense
. Every element of H denes a continuous, complexvalued function
on // and these functions generate the algebra of all continuous
functions on //.
. Every element of // denes a homomorphism fromH to the gauge
group G which is unique modulo an automorphism of G and every
homomorphism denes an element of //.
In the case of topological vector spaces, the duality mapping respects
the topology and the linear structure. In the present case, however, H has
the structure only of a group and // of a topological space. The duality
mappings can therefore respect only these structures. Property above
We should add, however, that in most of these debates, it was A/Grather than its
completion A/Gthat was the center of attention.
Holonomy Calgebras
provides us with a complete algebraic characterization of the elements
of Gelfand spectrum // while property species the topology of the
spectrum.
The second set of techniques involves a family of projections from
the topological space // to certain nitedimensional manifolds. For
each subgroup of the hoop group H, generated by n independent hoops,
we dene a projection from // to the quotient, G
n
/Ad, of the nth
power of the gauge group by the adjoint action. An element of G
n
/Ad
is thus an equivalence class of ntuples, g
,...,g
n
, with g
i
G, where
g
,...,g
n
g
g
g
,...,g
g
n
g
for any g
G. The lifts of func
tions on G
n
/Ad are then the nonlinear analogs of the cylindrical functions
on topological vector spaces. Finally, since each G
n
/Ad is a compact topo
logical space, we can equip it with measures to integrate functions in the
standard fashion. This in turn enables us to dene cylinder measures on
// and integrate cylindrical functions thereon. Using the Haar measure
on G, we then construct a natural, dieomorphisminvariant, faithful mea
sure on //.
Finally, for completeness, we will summarize the strategy we have adop
ted to introduce and analyze the momentum operators, although, as men
tioned above, these results will be presented elsewhere. The rst observa
tion is that we can dene vector elds on // as derivations on the ring of
cylinder functions. Then, to dene the specic vector elds corresponding
to the momentum or strip operators we introduce certain notions from
graph theory. These specic vector elds are divergencefree with respect
to our cylindrical measure on //, i.e. they leave the cylindrical measure
invariant. Consequently, the momentum operators are densely dened and
symmetric on the Hilbert space of squareintegrable functions on //.
The paper is organized as follows. Section is devoted to preliminaries.
In Section , we obtain the characterization of elements of // and also
present some examples of those elements which do not belong to //, i.e.
which represent genuine, generalized gauge equivalence classes of connec
tions. Using the machinery introduced in this characterization, in Section
we introduce the notion of cylindrical measures on // and then dene
a natural, faithful, dieomorphisminvariant, cylindrical measure. Section
contains concluding remarks.
In all these considerations, the piecewise analyticity of loops on plays
an important role. That is, the specic constructions and proofs presented
here would not go through if the loops were allowed to be just piecewise
smooth. However, it is far from being obvious that the nal results would
not go through in the smooth category. To illustrate this point, in Appendix
A we restrict ourselves to U connections and, by exploiting the Abelian
character of this group, show how one can obtain the main results of this
paper using piecewise C
loops. Whether similar constructions are possible
in the nonAbelian case is, however, an open question. Finally, in Appendix
Abhay Ashtekar and Jerzy Lewandowski
B we consider another extension. In the main body of the paper, is a
manifold and the gauge group G is taken to be SU. In Appendix B,
we indicate the modications required to incorporate SUN and UN
connections on an nmanifold keeping, however, piecewise analytic loops.
Preliminaries
In this section, we will introduce the basic notions, x the notation, and
recall those results from the AI framework which we will need in the
subsequent discussion.
Fix a dimensional, real, analytic, connected, orientable manifold .
Denote by P, SU a principal bre bundle P over the base space
and the structure group SU. Since any SU bundle over a manifold
is trivial we take P to be the product bundle. Therefore, SU connections
on P can be identied with suvalued form elds on . Denote by /
the space of smooth say C
SU connections, equipped with one of the
standard Sobolev topologies see, e.g., . Every SUvalued function
g on denes a gauge transformation in /,
Agg
Ag g
dg. .
Let us denote the quotient of / under the action of this group of
local gauge transformations by //. Note that the ane space structure
of / is lost in this projection // is a genuinely nonlinear space with a
rather complicated topological structure.
The next notion we need is that of closed loops in . Let us begin by
considering continuous, piecewise analytic C
parametrized paths, i.e.
maps
p,s
...s
n
,.
which are continuous on the whole domain and C
on the closed intervals
s
k
,s
k
. Given two paths p
, and p
, such that
p
p
, we denote by p
p
the natural composition
p
p
s
p
s, for s ,
p
s , for s
,.
.
The inverse of a path p , is a path
p
s p s. .
A path which starts and ends at the same point is called a loop. We will
be interested in based loops. Let us therefore x, once and for all, a point
x
. Denote by L
x
the set of piecewise C
loops which are based at
x
, i.e. which start and end at x
.
Given a connection A on P, SU, a loop L
x
, and a xed point
x
in the ber over the point x
, we denote the corresponding element of the
Holonomy Calgebras
holonomy group by H, A. Since P is a product bundle, we will make
the obvious choice x
x
, e, where e is the identity in SU. Using
the space of connections /, we now introduce a key equivalence relation
on L
x
Two loops , L
x
will be said to be holonomically equivalent, , i
H, A H, A, for every SU connection A in /.
Each holonomic equivalence class will be called a hoop. Thus, for example,
two loops which have the same orientation and which dier only in their
parametrization dene the same hoop. Similarly, if two loops dier just
by a retraced segment, they belong to the same hoop. More precisely,
if paths p
and p
are such that p
p
and p
p
x
,
so that p
p
is a loop in L
x
, and if is a path starting at the point
p
p
, then p
p
and p
p
dene the same hoop. Note
that in general the equivalence relation depends on the choice of the gauge
group, whence the hoop should in fact be called an SUhoop. However,
since the group is xed to be SU in the main body of this paper, we
drop this sux. The hoop to which a loop belongs will be denoted by .
The collection of all hoops will be denoted by H thus H L
x
/.
Note that, because of the properties of the holonomy map, the space of
hoops, H, has a natural group structure
. We will call it the
hoop group
.
With this machinery at hand, we can now introduce the AI Calgebra.
We begin by assigning to every H a realvalued function T
on //,
bounded between and
T
A
Tr H, A, .
where
A // A is any connection in the gauge equivalence class
A
any loop in the hoop and where the trace is taken in the fundamental
representation of SU. T
is called the Wilson loop function in the
physics literature. Owing to SU trace identities, products of these
functions can be expressed as sums
T
T
T
T
,.
where, on the right, we have used the composition of hoops in H. There
fore, the complex vector space spanned by the T
functions is closed under
the product it has the structure of a algebra. Denote it by H/ and
call it the holonomy algebra. Since each T
is a bounded continuous func
The hoop equivalence relation seems to have been introduced independently by a
number of authors in dierent contexts e.g. by Gambini and collaborators in their
investigation of YangMills theory and by Ashtekar in the context of gravity.
Recently, the group structure of HG has been systematically exploited and extended
by Di Bartolo, Gambini, and Griego to develop a new and potentially powerful
framework for gauge theories.
Abhay Ashtekar and Jerzy Lewandowski
tion on //, H/ is a subalgebra of the Calgebra of all complexvalued,
bounded, continuous functions thereon. The completion H/ of H/ under
the sup norm has therefore the structure of a Calgebra. This is the AI
Calgebra.
As in , one can make the structure of H/ explicit by constructing it,
step by step, from the space of loops L
x
. Denote by TL
x
the free vector
space over complexes generated by L
x
. Let K be the subspace of TL
x
dened as follows
i
a
i
i
Ki
i
a
i
T
i
A,
A //. .
It is then easy to verify that H/ TL
x
/K. Note that the Kequivalence
relation subsumes the hoop equivalence. Thus, elements of H/can be rep
resented either as
a
i
T
i
or as
a
i
i
K
. The operation, the product,
and the norm on H/ can be specied directly on TL
x
/K as follows
i
a
i
i
K
i
a
i
i
K
,
i
a
i
i
K
j
b
j
j
K
ij
a
i
b
j
i
j
K
i
j
K
,
i
a
i
i
K
sup
A,/
i
a
i
T
i
A, .
where a
i
is the complex conjugate of a
i
. The Calgebra H/ is obtained
by completing H/ in the norm topology.
By construction, H/ is an Abelian Calgebra. Furthermore, it is
equipped with the identity element T
o
o
K
, where o is the trivial i.e.
zero loop. Therefore, we can directly apply the Gelfand representation
theory. Let us summarize the main results that follow . First, we know
that H/ is naturally isomorphic to the Calgebra of complexvalued, con
tinuous functions on a compact Hausdor space, the spectrum spH/.
Furthermore, one can show that the space // of connections mod
ulo gauge transformations is densely embedded in spH/. Therefore, we
can denote the spectrum as //. Second, every continuous, cyclic rep
resentation of H/ by bounded operators on a Hilbert space has the fol
lowing form the representation space is the Hilbert space L
//, d for
some regular measure d on // and elements of H/, regarded as func
tions on //, act simply by pointwise multiplication. Finally, each of
these representations is uniquely determined via the GelfandNaimark
Segal construction by a positive linear functional on the Calgebra
H/
a
i
T
i
a
i
T
i
a
i
T
i
.
The functionals naturally dene functions on the space L
x
of
loopswhich in turn serve as the generating functionals for the representa
Holonomy Calgebras
tionvia T
. Thus, there is a canonical correspondence between
regular measures on // and generating functionals . The question
naturally arises can we write down necessary and sucient conditions for
a function on the loop space L
x
to qualify as a generating functional
Using the structure of the algebra H/ outlined above, one can answer this
question armatively
A function on L
x
serves as a generating functional for the GNS
construction if and only if it satises the following two conditions
i
a
i
T
i
i
a
i
i
and
i,j
a
i
a
j
i
j
i
j
..
The rst condition ensures that the functional on H/, obtained by
extending by linearity, is well dened while the second condition by
. ensures that it satises the positivity property, B
B,B
H/. Thus, together, they provide a positive linear functional on the
algebra H/. Finally, since H/ is dense in H/ and contains the identity, it
follows that the positive linear functional extends to the full Calgebra
H/.
Thus, a loop function satisfying . determines a cyclic represen
tation of H/ and therefore a regular measure on //. Conversely, every
regular measure on // provides through vacuum expectation values a
loop functional T
satisfying .. Note, nally, that the rst
equation in . ensures that the generating function factors through the
hoop equivalence relation .
We thus have an interesting interplay between connections and loops,
or, more precisely, generalized gaugeequivalent connections elements of
// and hoops elements of H. The generators T
of the holonomy
algebra i.e. conguration operators are labelled by elements of H. The
elements of the representation space i.e. quantum states are L
functions
on //. Regular measures d on // are, in turn, determined by functions
on H satisfying .. The group of dieomorphisms on has a natural
action on the algebra H/, its spectrum //, and the space of hoops,
H. The measure d is invariant under this induced action on // i
is invariant under the induced action on H. Now, a dieomorphism
invariant function of hoops is a function of generalized knotsgeneralized,
because our loops are allowed to have kinks, intersections, and segments
that may be traced more than once.
Hence, there is a correspondence
If the gauge group were more general, we could not have expressed products of the
generators T
as sums eqn .. For SU, for example, one cannot get rid of double
products triple and higher products can, however, be reduced to linear combinations
Abhay Ashtekar and Jerzy Lewandowski
between generalized knot invariants which in addition satisfy . and
regular dieomorphisminvariant measures on generalized gaugeequivalent
connections.
The spectrum
The constructions discussed in Section have several appealing features. In
particular, they open up an algebraic approach to the integration theory
on the space of gaugeequivalent connections and bring to the forefront
the duality between loops and connections. However, in practice, a good
control on the precise content and structure of the Gelfand spectrum //
of H/ is needed to make further progress. Even for simple algebras which
arise in nonrelativistic quantum mechanicssuch as the algebra of almost
periodic functions on R
n
the spectrum is rather complicated while R
n
is
densely embedded in the spectrum, one does not have as much control as
one would like on the points that lie in its closure see, e.g., . In the
case of a Calgebra of all continuous, bounded functions on a completely
regular topological space, the spectrum is the StoneCech compactication
of that space, whence the situation is again rather unruly. In the case of
the AI algebra H/, the situation seems to be even more complicated
since the algebra H/ is generated only by certain functions on //, at
least at rst sight, H/ appears to be a proper subalgebra of the Calgebra
of all continuous functions on //, and therefore outside the range of
applicability of standard theorems.
However, the holonomy Calgebra is also rather special it is con
structed from natural geometrical objectsconnections and loopson the
manifold . Therefore, one might hope that its spectrum can be charac
terized through appropriate geometric constructions. We shall see in this
section that this is indeed the case. The specic techniques we use rest
heavily on the fact that the loops involved are all piecewise analytic.
This section is divided into three parts. In the rst, we introduce the
key tools that are needed also in subsequent sections, in the second, we
present the main result, and in the third we give examples of generalized
gaugeequivalent connections, i.e. of elements of // //. To keep the
discussion simple, generally we will not explicitly distinguish between paths
and loops and their images in .
. Loop decomposition
A key technique that we will use in various constructions is the decom
position of a nite number of hoops,
,...,
k
, into a nite number of
independent hoops,
,...,
n
. The main point here is that a given set
of double products of T
and the T
themselves. In this case, the generating functional is
dened on single and double loops, whence, in addition to knot invariants, link invariants
can also feature in the denition of the generating function.
Holonomy Calgebras
of loops need not be holonomically independent every open segment in
a given loop may be shared by other loops in the given set, whence, for
any connection A in /, the holonomies around these loops could be inter
related. However, using the fact that the loops are piecewise analytic, we
will be able to show that any nite set of loops can be decomposed into
a nite number of independent segments and this in turn leads us to the
desired independent hoops. The availability of this decomposition will be
used in the next subsection to obtain a characterization of elements of //
and in Section to dene cylindrical functions on //.
Our aim then is to show that, given a nite number of hoops,
,...,
k
,
with
i
, identity in H for any i, there exists a nite set of loops,
,...,
n
L
x
, which satises the following properties
. If we denote by H
,...,
m
the subgroup of the hoop group H
generated by the hoops
,...,
m
, then we have
H
,...,
k
H
,...,
n
,.
where, as before,
i
denotes the hoop to which
i
belongs.
. Each of the new loops
i
contains an open segment i.e. an embedding
of an interval which is traced exactly once and which intersects any
of the remaining loops
j
with i , j at most at a nite number of
points.
The rst condition makes the sense in which each hoop
i
can be decom
posed in terms of
j
precise while the second condition species the sense
in which the new hoops
,...,
n
are independent.
Let us begin by choosing a loop
i
L
x
in each hoop
i
, for all
i , . . . , k, such that none of the
i
has a piece which is immediately
retraced i.e. none of the loops contains a path of the type
. Now,
since the loops are piecewise analytic, any two which overlap do so either
on a nite number of nite intervals and/or intersect in a nite number
of points. Ignore the isolated intersection points and mark the end points
of all the overlapping intervals including the end points of selfoverlaps of
a loop. Let us call these marked points vertices. Next, divide each loop
i
into paths which join consecutive vertices. Thus, each of these paths
is piecewise analytic, is part of at least one of the loops
,...,
k
, and
has a nonzero parameter measure along any loop to which it belongs.
Two distinct paths intersect at a nite set of points. Let us call these
oriented paths edges. Denote by n the number of edges that result. The
edges and vertices form a piecewise analytically embedded graph. By
construction, each loop in the list is contained in the graph and, ignoring
parametrization, can be expressed as a composition of a nite number of
its n edges. Finally, this decomposition is minimal in the sense that,
if the initial set
,...,
k
of loops is extended to include p additional
loops,
k
,...,
kp
, each edge in the original decomposition of k loops
Abhay Ashtekar and Jerzy Lewandowski
is expressible as a product of the edges which feature in the decomposition
of the k p loops.
The next step is to convert this edge decomposition of loops into a
decomposition in terms of elementary loops. This is easy to achieve. Con
nect each vertex v to the basepoint x
by an oriented, piecewise analytic
curve qv such that these curves overlap with any edge e
i
at most at a
nite number of isolated points. Consider the closed curves
i
, starting
and ending at the basepoint x
,
i
qv
i
e
i
qv
i
,.
where v
i
are the end points of the edge e
i
, and denote by o the set of these
n loops. Loops
i
are not unique because of the freedom in choosing the
curves qv. However, we will show that they provide us with the required
set of independent loops associated with the given set
,...,
k
.
Let us rst note that the decomposition satises certain conditions
which follow immediately from the properties of the segments noted above
. o is a nite set and every
i
o is a nontrivial loop in the sense
that the hoop
i
it denes is not the identity element of H
. every loop
i
contains an open interval which is traversed exactly
once and no nite segment of which is shared by any other loop in o
. every hoop
,...,
k
in our initial set can be expressed as a nite
composition of hoops dened by elements of o and their inverses
where a loop
i
and its inverse
i
may occur more than once
and,
. if the initial set
,...,
k
of loops is extended to include p additional
loops,
k
,...,
kp
, and if o
t
is the set of n
t
loops
t
,...,
t
n
corresponding to
,...,
kp
, such that the paths q
t
v
t
agree with
the paths qv whenever v v
t
, then each hoop
i
is a nite product
of the hoops
t
j
.
These properties will play an important role throughout the paper. For
the moment we only note that we have achieved our goal the property
above ensures that the set
,...,
n
of loops is independent in the sense
we specied in the point just below eqn . and that above implies
that the hoop group generated by
,...,
k
is contained in the hoop group
generated by
,...,
n
, i.e. that our decomposition satises ..
We will conclude this subsection by showing that the independence
of hoops
,...,
n
has an interesting consequence which will be used
repeatedly
Lemma . For every g
,...,g
n
SU
n
, there exists a connection
A
/ such that
H
i
,A
g
i
i , . . . , n. .
Holonomy Calgebras
Proof Fix a sequence of elements g
,...,g
n
of SU, i.e. a point in
SU
n
. Next, for each loop
i
, pick an Abelian connection A
t
i
a
t
i
w
t
where a
t
i
is a realvalued form on and w
t
a xed element of the Lie
algebra of SU. Let furthermore the support of a
t
i
intersect the ith
segment s
i
in a nite interval, and, if j , i, intersect the jth segment s
j
only on a set of measure zero as measured along any loop
m
containing
that segment. Finally, let us suppose that the connection is generic in the
sense that the holonomy that the holonomy H
i
,A
t
i
is not the identity
element of SU. Then, it is of the form
H
i
,A
t
i
exp w, w su, w ,
with w a multiple of w
t
. Now, consider a connection
A
i
tg
A
t
i
g, g SU, t R
Then, we have
H
i
,A
i
exp tg
wg.
Hence, by choosing g and t appropriately, we can make H
i
,A
i
coincide
with any element of SU, in particular g
i
. Because of the independence
of the loops
i
noted above, we can choose connections A
i
independently
for every
i
. Then, the connection
A
A
A
n
.
satises ..
We conclude this subsection with two remarks.
. The result . we just proved can be taken as a statement of the
independence of the SUhoops
,...,
n
it captures the idea
that the loops
,...,
n
we constructed above are holonomically in
dependent as far as SU connections are concerned. This notion
of independence is, however, weaker than the one contained in the
statement above. Indeed, that notion does not refer to connections
at all and is therefore not tied to any specic gauge group. More
precisely, we have the following. We just showed that if loops are
independent in the sense of , they are necessarily independent in the
sense of .. The converse is not true. Let and be two loops in
L
x
which do not share any point other than x
and which have no
selfintersections or selfoverlaps. Set
and
. Then, it
is easy to check that
i
, i , are independent in the sense of .
although they are obviously not independent in the sense of . Simi
larly, given , we can set
. Then, is independent in the
sense of . since the square root is well dened in SU but
not in the sense of . From now on, we will say that loops satisfying
Abhay Ashtekar and Jerzy Lewandowski
are independent and those satisfying . are weakly independent.
This denition extends naturally to hoops. Although we will not
need them explicitly, it is instructive to spell out some algebraic con
sequences of these two notions. Algebraically, weak independence of
hoops
i
ensures that
i
freely generate a subgroup of H. On the
other hand, if
i
are independent, then every homomorphism from
H
,...,
n
to any group G can be extended to every nitely
generated subgroup of H which contains H
,...,
n
. Weak in
dependence will suce for all considerations in this section. However,
to ensure that the measure introduced in Section is well dened,
we will need hoops which are independent.
. The availability of the loop decomposition sheds considerable light
on the hoop equivalence one can show that two loops in L
x
dene
the same SUhoop if and only if they dier by a combination of
reparametrization and immediate retracings. This second charac
terization is useful because it involves only the loops no mention is
made of connections and holonomies. This characterization continues
to hold if the gauge group is replaced by SUN or indeed any group
which has the property that, for every nonnegative integer n, the
group contains a subgroup which is freely generated by n elements.
. Characterization of //
We will now use the availability of this hoop decomposition to obtain a
complete characterization of all elements of the spectrum //.
The characterization is based on results due to Giles . Recall rst
that elements of the Gelfand spectrum// are in correspondence with
multiplicative, preserving homomorphisms from H/ to the algebra of
complex numbers. Thus, in particular, every
A // acts on T
H/
and hence on any hoop to produce a complex number
A . Now, using
certain constructions due to Giles, AI were able to show that
Lemma . Every element
A of // denes a homomorphism
H
A
from
the hoop group H to SU such that, H,
A
Tr
H
A
. .a
They also exhibited the homomorphism explicitly. Here, we have para
phrased the AI result somewhat because they did not use the notion of
hoops. However, the step involved is completely straightforward.
Our aim now is to establish the converse of this result. Let us make a
small digression to illustrate why the converse cannot be entirely straight
forward. Given a homomorphism
H from H to SU we can simply
dene
A via
A
Tr H . The question is whether this
A can qualify
Holonomy Calgebras
as an element of the spectrum. From the denition, it is clear that
AT
sup
A,/
T
A .a
for all hoops . On the other hand, to qualify as an element of the spectrum
//, the homomorphism
A must be continuous, i.e. should satisfy
Af f .b
for every f
a
i
i
K
H/. Since, a priori, .b appears to be a
stronger requirement than .a, one would expect that there are more
homomorphisms
H from the hoop group H to SU than the
H
A
, which
arise from elements of //. That is, one might expect that the converse of
the AI result would not be true. It turns out, however, that if the holon
omy algebra is constructed from piecewise analytic loops, the apparently
weaker condition .a is in fact equivalent to .b, and the converse
does hold. Whether it would continue to hold if one used piecewise smooth
loopsas was the case in the AI analysisis not clear see Appendix A.
We begin our detailed analysis with an immediate application of Lemma
.
Lemma . For every homomorphism
H from H to SU, and every
nite set of hoops
,...,
k
, there exists an SU connection A
such
that for every
i
in the set,
H
i
H
i
,A
,.
where, as before, H
i
,A
is the holonomy of the connection A
around
any loop in the hoop class .
Proof Let
,...,
n
be, as in Section ., a set of independent loops
corresponding to the given set of hoops. Denote by g
,...,g
n
the image
of
,...,
n
under
H. Use the construction of Section . to obtain a
connection A
such that g
i
H
i
,A
for all i. It then follows from
the denition of the hoop group that for any hoop in the subgroup
H
,...,
n
generated by
,...,
n
, we have
HH,A
. Since
each of the given
i
belongs to H
,...,
n
, we have, in particular,
..
We now use this lemma to prove the main result. Recall that each
A // is a homomorphism from H/ to complex numbers and as before
denote by
A the number assigned to T
H/ by
A. Then, we have
Lemma . Given a homomorphism
H from the hoop group H to SU
there exists an element
A
H
of the spectrum // such that
A
H
Tr
H . .b
Two homomorphisms
H and
H
t
dene the same element of the spectrum
Abhay Ashtekar and Jerzy Lewandowski
if and only if
H
t
g
H g, H, for some independent
g in SU.
Proof The idea is to dene the required
A
H
using .b, i.e. to show
that the right side of .b provides a homomorphism from H/ to the
algebra of complex numbers. Let us begin with the free complex vector
space TH over the hoop group H and dene on it a complexvalued
function h as follows h
a
i
i
a
i
Tr
H
i
. We will rst show
that h passes through the Kequivalence relation of AI noted in Section
and is therefore a welldened complexvalued function on the holonomy
algebra H/ TH/K. Note rst that Lemma . immediately implies
that, given a nite set of hoops,
,...,
k
, there exists a connection A
such that
h
k
i
a
i
i
k
i
a
i
T
i
A
sup
A,/
k
i
a
i
T
i
A
..
Therefore, we have
k
i
a
i
T
i
A
A // h
k
i
a
i
i
.
Since the left side of this implication denes the Kequivalence, it follows
that the function h on TH has a welldened projection to the holonomy
algebra H/. That h is linear is obvious from its denition. That it
respects the operation and is a multiplicative homomorphism follows from
the denitions . of these operations on H/ and the denition of the
hoop group H. Finally, the continuity of this funtion on H/ is obvious
from .. Hence h extends uniquely to a homomorphism from the C
algebra H/ to the algebra of complex numbers, i.e. denes a unique
element
A
H
via
A
H
B hB, B H/.
Next, suppose
H
and
H
give rise to the same element of the spectrum.
In particular, then, Tr
H
Tr
H
for all hoops. Now, there is a
general result given two homomorphisms
H
and
H
from a group G to
SU such that Tr
H
g Tr
H
g, g G, there exists g
SU
such that
H
gg
H
gg
, where g
is independent of g. Using the
hoop group H for G, we obtain the desired uniqueness result.
To summarize, by combining the results of the three lemmas, we obtain
the following characterization of the points of the Gelfand spectrum of the
holonomy Calgebra
Theorem . Every point
A of // gives rise to a homomorphism
H
from the hoop group H to SU and every homomorphism
H denes a
point of //, such that
A
Tr
H . This is a correspondence
modulo the trivial ambiguity that homomorphisms
H and g
H g dene
Holonomy Calgebras
the same point of //.
We conclude this subsection with some remarks
. It is striking that Theorem . does not require the homomorphism
H to be continuous indeed, no reference is made to the topology of
the hoop group anywhere. This purely algebraic characterization of
the elements of // makes it convenient to use it in practice.
. Note that the homomorphism
H determines an element of // and
not of //
A
H
is not, in general, a smooth gaugeequivalent con
nection. Nonetheless, as Lemma . tells us, it can be approximated
by smooth connections in the sense that, given any nite number of
hoops, one can construct a smooth connection which is indistinguish
able from
A
H
as far as these hoops are concerned. This is stronger
than the statement that // is dense in //. Necessary and su
cient conditions for
H to arise from a smooth connection were given
by Barrett see also .
. There are several folk theorems in the literature to the eect that
given a function on the loop space L
x
satisfying certain conditions,
one can reconstruct a connection modulo gauge such that the given
function is the trace of the holonomy of that connection. For a
summary and references, see, e.g., . Results obtained in this sub
section have a similar avor. However, there is a key dierence our
main result shows the existence of a generalized connection mod
ulo gauge, i.e. an element of // rather than a regular connection
in //. A generalized connection can also be given a geometrical
meaning in terms of parallel transport, but in a generalized bundle
.
. Examples of
A
Fix a connection A /. Then, the holonomy H, A denes a ho
momorphism
H
A
H SU. A gaugeequivalent connection, A
t
g
Ag g
dg, gives rise to the homomorphism
H
A
g
x
H
A
gx
.
Therefore, by Theorem ., A and A
t
dene the same element of the
Gelfand spectrum // is naturally embedded in //. Furthermore,
Lemma . now implies that the embedding is in fact dense. This provides
an alternative proof of the AI and Rendall
results quoted in Section .
Had the gauge group been dierent and/or had a dimension greater than
, there would exist nontrivial Gprincipal bundles over . In this case,
even if we begin with a specic bundle in our construction of the holon
omy Calgebra, connections on all possible bundles belong to the Gelfand
spectrum see Appendix B. This is one illustration of the nontriviality of
Note, however, that while this proof is tailored to the holonomy Calgebra HA,
Rendalls proof is applicable in more general contexts.
Abhay Ashtekar and Jerzy Lewandowski
the completion procedure leading from // to //.
In this subsection, we will illustrate another facet of this nontriviality
we will give examples of elements of // which do not arise from //.
These are the generalized gaugeequivalent connections
A which have a
distributional character in the sense that their support is restricted to
sets of measure zero in .
Recall rst that the holonomy of the zero connection around any hoop is
identity. Hence, the homomorphism
H
from H to SU it denes sends
every hoop to the identity element e of SU and the multiplicative
homomorphism it denes from the Calgebra H/ to complex numbers
sends each T
to /Tr e. We now use this information to introduce
the notion of the support of a generalized connection. We will say that
A // has support on a subset S of if for every hoop which has at
least one representative loop which fails to pass through S, we have i
H
A
e , the identity element of SU or, equivalently, ii
A
K
. In ii,
A is regarded as a homomorphism of algebras. While
H
A
has
an ambiguity, noted in Theorem ., conditions i and ii are insensitive
to it.
We are now ready to give the simplest example of a generalized connec
tion
A which has support at a single point x . Note rst that, owing to
piecewise analyticity, if L
x
passes through x, it does so only a nite
number of times, say N. Let us denote the incoming and outgoing tan
gent directions to the curve at its jth passage through x by v
j
,v
j
. The
generalized connections we want to dene now will depend only on these
tangent directions at x. Let be a not necessarily continuous mapping
from the sphere S
of directions in the tangent space at x to SU. Set
j
v
j
v
j
,
so that, if at the jth passage, arrives at x and then simply returns by
retracing its path in a neighborhood of x, we have
j
e, the identity
element of SU. Now, dene
H
H SU via
H
e, if / H
x
,
N
...
, otherwise,
.
where H
x
is the space of hoops every loop in which passes through x.
For each choice of , the mapping
H
is well dened, i.e. is independent of
the choice of the loop in the hoop class used in its construction. In
particular, we used tangent directions rather than vectors to ensure this.
It denes a homomorphism from H to SU which is nontrivial only on
H
x
, and hence a generalized connection with support at x. Furthermore,
one can verify that . is the most general element of // which has
support at x and depends only on the tangent vectors at that point. Using
analyticity of the loops, we can similarly construct elements which depend
Holonomy Calgebras
on the higherorder behavior of loops at x. There is no generalized con
nection with support at x which depends only on the zerothorder behavior
of the curve, i.e. only on whether the curve passes through x or not.
One can proceed in an analogous manner to produce generalized con
nections which have support on or dimensional submanifolds of . We
conclude with an example. Fix a dimensional, oriented, analytic sub
manifold M of and an element g of SU, g , e. Denote by H
M
the subset of H consisting of hoops, each loop of which intersects M
nontangentially, i.e. such that at least one of the incoming or the outgo
ing tangent direction to the curve is transverse to M. As before, owing to
analyticity, any L
x
can intersect M only a nite number of times. De
note as before the incoming and the outgoing tangent directions at the jth
intersection by v
j
and v
j
respectively. Let
j
equal if the orientation
of v
j
, M coincides with the orientation of if the two orientations
are opposite and if v
j
is tangential to M. Dene
H
g,M
H SU
via
H
g,M
e, if / H
M
,
g
N
g
N
...g
g
, otherwise,
.
where g
e, the identity element of SU. Again, one can check that
. is a welldened homomorphism from H to SU, which is non
trivial only on H
M
and therefore denes a generalized connection with
support on M. One can construct more sophisticated examples in which
the homomorphism is sensitive to the higherorder behavior of the loop at
the points of intersection and/or the xed SU element g is replaced by
gx M SU.
Integration on //
In this section we shall develop a general strategy to perform integrals on
// and introduce a natural, faithful, dieomorphisminvariant measure
on this space. The existence of this measure provides considerable con
dence in the ideas involving the loop transform and the loop representation
introduced in , .
The basic strategy is to carry out an appropriate generalization of the
theory of cylindrical measures , on topological vector spaces
. The
key idea in our approach is to replace the standard linear duality on vector
spaces by the duality between connections and loops. In the rst part of
This idea was developed also by Baez using, however, an approach which is
somewhat dierent from the one presented here. Chronologically, the authors of this
paper rst found the faithful measure introduced in this section and reported this result
in the rst draft of this paper. Subsequently, Baez and the authors independently
realized that the theory of cylindrical measures provides the appropriate conceptual
setting for this discussion.
Abhay Ashtekar and Jerzy Lewandowski
this section, we will dene cylindrical functions, and in the second part, we
will present a natural cylindrical measure.
. Cylindrical functions
Let us begin by recalling the situation in the linear case. Let V denote
a topological vector space and V
its dual. Given a nitedimensional
subspace S
of V
, we can dene an equivalence relation on V v
v
if
and only if v
,s
v
,s
for all v
,v
V and s
S
, where v, s
is
the action of s
S
on v V . Denote the quotient V/ by S. Clearly, S
is a nitedimensional vector space and we have a natural projection map
S
V S. A function f on V is said to be cylindrical if it is a pullback
via S
to V of a function
f on S, for some choice of S
. These are the
functions we will be able to integrate. Equip each nitedimensional vector
space S with a measure d and set
V
df
S
d
f. .
For this denition to be meaningful, however, the set of measures d that
we chose on vector spaces S must satisfy certain compatibility conditions.
Thus, if a function f on V is expressible as a pullback via S
j
of functions
f
j
on S
j
, for j , , say, we must have
S
d
f
S
d
f
..
Such compatible measures do exist. Perhaps the most familiar example is
the normalized Gaussian measure.
We want to extend these ideas, using // in place of V . An immediate
problem is that // is a genuinely nonlinear space whence there is no
obvious analog of V
and S
which are central to the above construction
of cylindrical measures. The idea is to use the results of Section to nd
suitable substitutes for these spaces. The appropriate choices turn out to
be the following we will let the hoop group H play the role of V
and its
subgroups, generated by a nite number of independent hoops, the role of
S
. Recall that S
is generated by a nite number of linearly independent
basis vectors.
More precisely, we proceed as follows. From Theorem ., we know
that each
A in // is completely characterized by the homomorphism
H
A
from the hoop group H to SU, which is unique up to the freedom
H
A
g
H
A
g
for some g
SU. Now suppose we are given n
independent loops,
,...,
n
. Denote by o
the subgroup of the hoop
group H they generate. Using this o
, let us introduce the following
equivalence relation on //
A
A
i
H
A
g
H
A
g
for all
o
and some hoopindependent g
SU. Denote by o
the
Holonomy Calgebras
projection from // onto the quotient space /// . The idea is to
consider functions f on // which are pullbacks under o
of functions
f on /// and dene the integral of f on // to be equal to the
integral of
f on /// . However, for this strategy to work, ///
should have a simple structure that one can control. Fortunately, this is
the case
Lemma . Let o
be a subgroup of the hoop group which is generated
by a nite number of independent hoops. Then,
a There is a natural bijection
/// Homo
, SU/Ad
b For each choice of independent generators
,...,
n
of o
, there is
a bijection
/// SU
n
/Ad
c Given o
as in a, the topology induced on /// from SU
n
/Ad
according to b is independent of the choice of free generators of o
.
Proof By denition of the equivalence relation , every
A in ///
is in correspondence with the restriction to o
of an equivalence class
H
A
of homomorphisms from the full hoop group H to SU. Now,
it follows from Lemma . that every homomorphism from o
to SU
extends to a homomorphism from the full hoop group to SU. There
fore, there is a correspondence between equivalence classes A and
equivalence classes of homomorphisms from o
to SU where, as usual,
two are equivalent if they dier by the adjoint action of SU.
The statement b follows automatically from a the map in b is
given by the values taken by an element of Homo
, SU on the chosen
generators. The proof of c is a simple exercise.
Some remarks about this result are in order
. Although we began with independent loops
,...,
n
, the equiv
alence relation we introduced makes direct reference only to the sub
group o
of the hoop group they generate. This is similar to the
situation in the linear case where one may begin with a set of linearly
independent elements s
,...,s
n
of V
, let them generate a subspace
S
and then introduce the equivalence relation on S using S
directly.
. However, a specic basis is often used in subsequent constructions. In
particular, cylindrical measures are generally introduced as follows.
One xes a basis in S
, uses it to dene an isomorphism between
S V/ and R
n
, introduces a measure on R
n
for each n and
uses the isomorphism to dene measures d on spaces S. These, in
turn, dene a cylindrical measure d on V through .. However,
since S
admits other bases, at the end one must make sure that
Abhay Ashtekar and Jerzy Lewandowski
the nal measure d on V is welldened, i.e., is insensitive to the
specic choice of basis made in its denition. In the case now under
consideration, we will follow a similar strategy. As we just showed in
Lemma ., given a set of independent loops which generate o
, we
can identify /// with SU
n
/Ad. Therefore, we can dene
a cylindrical measure on // by rst choosing suitable measures on
SU
n
/Ad for each n, using the isomorphism to dene measures on
/// and showing nally that the nal integrals are insensitive
to the initial choices of generators of o
.
. The equivalence relation of Lemma . says that two generalized
connections are equivalent if their actions on the group o
, generated
by a nite number of independent loops, are indistinguishable. It
follows from Lemma . that each equivalence class
A contains a
regular connection A. Finally, if A and A
t
are two regular connections
whose pullbacks to all
i
are the same for i , . . . , n, then they
are equivalent where
i
is any loop in the hoop class
i
. Thus,
in eect, the projection o
maps the domain space // of the
continuum quantum theory to the domain space of a lattice gauge
theory where the lattice is formed by the n loops
,...,
n
. The
initial choice of the independent hoops is, however, arbitrary and it is
this arbitrariness that is responsible for the richness of the continuum
theory.
We can now dene cylindrical functions on //. Let o
be a subgroup
of the hoop group which is generated by a nite number of independent
loops. Let us set
B
S
Homo
, SU/Ad
Using Lemma . we shall regard o
as a projection map from //
onto B
S
and parametrize B
S
by SU
n
/Ad. For the topology on B
S
we will use the one dened by its identication with SU
n
/Ad. By a
cylindrical function f on //, we shall mean the pullback to // under
o
of a continuous function
f on B
S
, for some subgroup o
of the hoop
group which is generated by a nite number of independent loops. These
are the functions we want to integrate. Let us note an elementary property
of cylindrical functions which will be used repeatedly. Let o
and o
t
be
two nitely generated subgroups of the hoop group and such that o
t
o
.
Then, it is easy to check that if a function f on // is cylindrical with
respect to o
t
, it is also cylindrical with respect to o
. Furthermore, there
is now a natural projection from B
S
onto B
S
given by the projection
Homo
, SU Homo
t
, SU and
f is the pullback of
f
t
via this
projection.
In the remainder of this subsection, we will analyse the structure of the
space of cylindrical functions. First we have
Holonomy Calgebras
Lemma . has the structure of a normed algebra with respect to the
operations of taking linear combinations, multiplication, complex conjuga
tion and sup norm of cylindrical functions.
Proof It is clear by denition of that if f , then any complex multiple
f as well as the complex conjugate
f also belongs to . Next, given two
elements, f
i
with i , , of , we will show that their sum and their
product also belong to . Let o
i
be nitely generated subgroups of the hoop
group H such that f
i
are the pullbacks to // of continuous functions
f
i
on B
S
i
. Consider the sets of n
and n
independent hoops generating
respectively o
and o
. While the rst n
and the last n
hoops are
independent, there may be relations between hoops belonging to the rst
set and those belonging to the second. Using the technique of Section .,
nd the set of independent hoops, say,
,...,
n
which generate the given
n
n
hoops and denote by o
the subgroup of the hoop group generated
by
,...,
n
. Then, since o
i
o
, it follows that f
i
are cylindrical also
with respect to o
. Therefore, f
f
and f
f
are also cylindrical with
respect to o
. Thus has the structure of a algebra. Finally, since
B
S
being homeomorphic to SU
n
/Ad is compact for any o
, and
since elements of are pullbacks to // of continuous functions on B
S
,
it follows that the cylindrical functions are bounded. We can therefore use
the sup norm to endow with the structure of a normed algebra.
We will use the sup norm to complete and obtain a Calgebra . A
natural class of functions one would like to integrate on // is given by
the Gelfand transforms of the traces of holonomies. The question then
is if these functions are contained in . Not only is the answer armative,
but in fact the traces of holonomies generate all of . More precisely, we
have
Theorem . The Calgebra is isomorphic to the Calgebra H/.
Proof Let us begin by showing that H/ is embedded in . Given an
element F with FA
a
i
T
i
A of H/, its Gelfand transform is a
function f on the spectrum //, given by f
A
a
i
A
i
. Consider
the set of hoops
,...,
k
that feature in the sum, decompose them into
independent hoops
,...,
n
as in Section ., and denote by o
the sub
group of the hoop group they generate. Then, it is clear that the function
f is the pullback via o
of a continuous function
f on B
S
. Hence we
have f . Since the norms of F in H/ and f in are given by
F sup
A,
FA and f sup
A,/
f
A,
since the restriction of f to // coincides with F, and since // is em
bedded densely in //, it is clear that the map from F to f is isometric.
Finally, since H/ is obtained by a C
completion of the space of functions
Abhay Ashtekar and Jerzy Lewandowski
of the form F, we have the result that H/ is embedded in the Calgebra
.
Next, we will show that there is also an inclusion in the opposite di
rection. We rst note a fact about SU. Fix n elements g
,...,g
n
of SU. Then, from the knowledge of traces in the fundamental rep
resentation of all elements which belong to the group generated by these
g
i
, i , . . . , n, we can reconstruct g
,...,g
n
modulo an adjoint map.
It therefore follows that the space of functions
f on B
S
considered as
a copy of SU
n
/Ad which come from the functions F on H/ using
all the hoops
i
o
suce to separate points of B
S
. Since this space
is compact, it follows from the Weierstrass theorem that the Calgebra
generated by these projected functions is the entire Calgebra of contin
uous functions on B
S
. Now, suppose we are given a cylindrical function
f
t
on //. Its projection under the appropriate o
is, by the preceed
ing result, contained in the projection of some element of H/. Thus, is
contained in H/.
Combining the two results, we have the desired result the Calgebras
H/ and are isomorphic.
Remark Theorem . suggests that from the very beginning we could
have introduced the Calgebra in place of H/. This is indeed the case.
More precisely, we could have proceeded as follows. Begin with the space
//, and introduce the notion of hoops and hoop decomposition in terms of
independent hoops. Then given a subgroup o
of H generated by a nite
number of independent hoops, we could have introduced an equivalence
relation on // not // which we do not yet have as follows A
A
iH,A
g
H,A
g for all o
and some hoop independent
g SU. It is then again true due to Lemma . that the quotient
/// is isomorphic to B
S
. This is true inspite of the fact that we
are using // rather than // because, as remarked immediately after
Lemma ., each
A /// contains a regular connection. We
can therefore dene cylindrical functions, but now on // rather than
// as the pullbacks of continuous functions on B
S
. These functions
have a natural Calgebra structure. We can use it as the starting point
in place of the AI holonomy algebra. While this strategy seems natural
at rst from an analysis viewpoint, it has two drawbacks, both stemming
from the fact that the Wilson loops are now assigned a secondary role.
For physical applications, this is unsatisfactory since Wilson loops are the
basic observables of gauge theories. From a mathematical viewpoint, the
relation between knot/link invariants and measures on the spectrum of the
algebra would now be obscure. Nonetheless, it is good to keep in mind
that this alternate strategy is available as it may make other issues more
transparent.
Holonomy Calgebras
. A natural measure
In this subsection, we will discuss the issue of integration of cylindrical
functions on //. Our main objective is to introduce a natural, faithful,
dieomorphism invariant measure on //.
As mentioned in the remarks following Lemma ., the idea is similar
to the one used in the case of topological vector spaces. Thus, for each
subgroup o
of the hoop group which is generated by a nite number of
independent hoops, we will equip the space B
S
with a measure d
S
and,
as in ., set
,/
df
B
S
d
S
f.
where f is any cylindrical function compatible with o
. It is clear that
for the integral to be welldened, the set of measures we choose on dif
ferent B
S
should be compatible so that the analog of . holds. We will
now exhibit a natural choice for which the compatibility is automatically
satised.
Denote by d the normalized Haar measure on SU. It naturally in
duces a measure on SU
n
which is invariant under the adjoint action of
SU and therefore projects down unambiguously to a measure dn on
SU
n
/Ad. Next, consider an arbitrary group o
which is generated by
n independent loops. Choose a set of independent generators
,...,
n
and use the corresponding map SU
n
/Ad B
S
to transport the mea
sure dn to a measure d
S
on B
S
. We will show that the measure
d
S
is insensitive to the specic choice of the generators of o
used in its
construction and that the resulting family of measures on the various B
S
satises the appropriate compatibility conditions. We will then explore the
properties of the resulting cylindrical measure on //. Our construction
is natural since SU comes equipped with the Haar measure. In partic
ular, we have not introduced any additional structure on the underlying
manifold on which the initial connections A are dened hence the re
sulting cylindrical measure will be automatically invariant under the action
of Di.
Theorem . a Let f be a cylindrical function on // with respect to
two subgroups o
i
, i , , of the hoop group, each of which is generated
by a nite number of independent hoops. Fix a set of generators in each
subgroup and denote by d
S
i
the corresponding measures on B
S
i
. Then, if
f
i
denote the projections of f to B
S
i
, we have
B
S
d
S
f
B
S
d
S
f
..
In particular, if o
o
, the integral is independent of the choice of gen
erators used in its denition.
Abhay Ashtekar and Jerzy Lewandowski
b The functional v H/ C dened below extends to a linear, strictly
positive and Diinvariant functional on H/
vF
,/
df.
where d is dened by . and where the cylindrical function f on //
is the Gelfand transform of the element F of H/ and,
c The cylindrical measure d dened by . is a genuine, regular and
strictly positive measure on //. Furthermore, d is Di invariant.
Proof Fix n
i
independent hoops that generate o
i
and, as in the proof of
Theorem ., use the construction of Section . to carry out the decompo
sition of these n
n
hoops in terms of independent hoops, say
,...,
n
.
Thus, the original n
n
hoops are contained in the group o
generated
by the
i
. Clearly, o
i
o
. Hence, the given f is cylindrical also with
respect to o
, i.e. is the pullback of a function
f on B
S
. We will now
establish . by showing that each of the two integrals in that equation
equals the analogous integral of
f over B
S
.
Note rst that, since the generators have been xed, we can ignore the
distinction between B
S
i
and SU
ni
/Ad and between B
S
and SU
n
/
Ad. It will also be convenient to regard functions on SU
n
/Ad as
functions on SU
n
which are invariant under the adjoint SU ac
tion. Thus, instead of carrying out integrals over B
S
i
and B
S
of functions
on these spaces, we can integrate the lifts of these functions to SU
ni
and SU
n
respectively.
To deal with quantities with suxes and simultaneously, for no
tational simplicity we will let a prime stand for either the sux or .
Then, in particular, we have nitely generated subgroups o
and o
t
of
the hoop group, with o
t
o
, and a function f which is cylindrical with
respect to both of them. The generators of o
are
,...,
n
while those
of o
t
are
t
,...,
t
n
with n
t
lt n. From the construction of o
Sec
tion . it follows that every hoop in the second set can be expressed as a
composition of the hoops in the rst set and their inverses. Furthermore,
since the hoops in each set are independent, the construction implies that
for each i
t
,...,n
t
, there exists Ki
t
, . . . , n with the following
properties i if i
t
,j
t
, Ki
t
, Kj
t
and, ii in the decomposition of
t
i
in terms of unprimed hoops, the hoop
Ki
or its inverse appears
exactly once and if i
t
,j
t
, the hoop
Kj
does not appear at all. For
simplicity, let us assume that the orientation of the primed hoops is such
that it is
Ki
rather than its inverse that features in the decomposition
of
t
i
. Next, let us denote by g
,...,g
n
a point of the quotient SU
n
which is projected to g
t
,...,g
t
n
in SU
n
. Then g
t
i
is expressed in
terms of g
i
as g
t
i
..g
Ki
.., where .. denotes a product of elements of the
Holonomy Calgebras
type g
m
where m , Kj for any j. Since the given function f on //
is a pullback of a function
f on SU
n
as well as of a function
f
t
on
SU
n
, it follows that
fg
,...,g
n
f
t
g
t
,...,g
t
n
f
t
..g
K
.., . . . , ..g
Kn
...
Hence,
d
d
n
fg
,...,g
n
m,K...Kn
d
m
d
K
d
Kn
f
t
..g
K
.., . . . , ..g
Kn
..
n
mn
d
m
d
d
n
f
t
g
,...,g
n
d
d
n
f
t
g
,...,g
n
,
where, in the rst step, we simply rearranged the order of integration, in
the second step we used the invariance property of the Haar measure and
simplied notation for dummy variables being integrated, and, in the third
step, we used the fact that, since the measure is normalized, the integral
of a constant produces just that constant
. This establishes the required
result ..
Next, consider the vacuum expectation value functional v on H/ de
ned by .. The continuity and linearity of v follow directly from its
denition. That it is positive, i.e. satises vF
F , is equally obvi
ous. In fact, a stronger result holds if F , , then vF
F gt since
the value vF
F of v on F
F is obtained by integrating a nonnegative
function
f
f on SU
n
/Ad with respect to a regular measure. Thus, v
is strictly positive. Since H/ is dense in H/ it follows that the functional
extends to H/ by continuity.
To see that we have a regular measure on //, recall rst that the C
algebra of cylindrical functions is naturally isomorphic with the Gelfand
transform of the Calgebra H/, which, in turn is the Calgebra of all
continuous functions on the compact Hausdor space //. The vacuum
expectation value function v is therefore a continuous linear functional
on the algebra of all continuous functions on // equipped with the L
i.e. sup norm. Hence, by the RieszMarkov theorem, it follows that
In the simplest case, with n and n
, we would for example have
fg
,g
f
g
g
f
g
. Then
dg
dg
fg
,g
dg
dg
f
g
g
dg
fg
, where in the second step we have used the invariance and normalization
properties of the Haar measure.
Abhay Ashtekar and Jerzy Lewandowski
vf
df is in fact the integral of the continuous function f on //
with respect to a regular measure. This is just our cylindrical measure.
Finally, since our construction did not require the introduction of any
extra structure on such as a metric or a volume element, it follows that
the functional v and the measure d are Di invariant.
We will refer to d as the induced Haar measure on //. We now
explore some properties of this measure and of the resulting representation
of the holonomy Calgebra H/.
First, the representation of H/ can be constructed as follows. The
Hilbert space is L
//, d and the elements F of H/ act by multiplica
tion so that we have F f for all L
//, d, where f is
the Gelfand transform of F. This representation is cyclic and the function
dened by
A on // is the vacuum state. Every generator
T
of the holonomy Calgebra is represented by a bounded, selfadjoint
operator on L
//, d.
Second, it follows from the strictness of the positivity of the measure
that the resulting representation of the Calgebra H/ is faithful. To our
knowledge, this is the rst faithful representation of the holonomy C
algebra that has been constructed explicitly. Finally, the dieomorphism
group Di of has an induced action on the Calgebra H/ and hence
also on its spectrum //. Since the measure d on // is invariant under
this action, the Hilbert space L
//, d carries a unitary representation
of Di. Under this action, the vacuum state
is left invariant.
Third, restricting the values of v to the generators T
K
of H/,
we obtain the generating functional of this representation
d
A . This functional on the loop space L
x
is dieomorphism invari
ant since the measure d is. It thus dened a generalized knot invariant.
We will see below that, unlike the standard invariants, vanishes on
all smoothly embedded loops. Its action is nontrivial only when the loop
is retraced, i.e. only on generalized loops.
Let us conclude this section by indicating how one computes the integral
in practice. Fix a loop L
x
and let
i
iN
N
.
be the minimal decomposition of into independent loops
,...,
n
as
in Section ., where
k
, keeps track of the orientation and i
k
, . . . , n. We have used the double sux i
k
because any one independent
loop
,...,
n
can arise more than once in the decomposition. Thus,
N n. The loop denes an element T
of the holonomy Calgebra H/
whose Gelfand transform t
is a function on //. This is the cylindrical
function we wish to integrate. Now, the projection map of Lemma .
assigns to t
the function
t
on SU
n
/Ad given by
t
g
,...,g
n
Tr g
i
...g
N
iN
.
Holonomy Calgebras
Thus, to integrate t
on //, we need to integrate this
t
on SU
n
/Ad.
Now, there exist in the literature useful formulae for integrating polyno
mials of the matrix element functions over SU see, e.g., . These
can be now used to evaluate the required integrals on SU
n
/Ad. In
particular, one can readily show the following result
,/
dt
,
unless every loop
i
is repeated an even number of times in the decompo
sition . of .
Discussion
In this paper, we completed the AI program in several directions. First,
by exploiting the fact that the loops are piecewise analytic, we were able
to obtain a complete characterization of the Gelfand spectrum // of
the holonomy Calgebra H/. AI had shown that every element of //
denes a homomorphism from the hoop group H to SU. We found
that every homomorphism from H to SU denes an element of the
spectrum and two homomorphisms
H
and
H
dene the same element
of // only if they dier by an SU automorphism, i.e. if and only if
H
g
H
g for some g SU. Since this characterization is rather
simple and purely algebraic, it is useful in practice. The second main result
also intertwines the structure of the hoop group with that of the Gelfand
spectrum. Given a subgroup o
of H generated by a nite number, say
n, of independent hoops, we were able to dene a projection o
from
// to the compact space SU
n
/Ad. This family of projections en
ables us to dene cylindrical functions on // these are the pullbacks to
// of the continuous functions on SU
n
/Ad. We analysed the space
of these functions and found that its completion has the structure of
a Calgebra which, moreover, is naturally isomorphic with the Calgebra
of all continuous functions on the compact Hausdor space // and hence,
via the inverse Gelfand transform, also to the holonomy Calgebra H/
with which we began. We then carried out a nonlinear extension of the
theory of cylindrical measures to integrate these cylindrical functions on
//. Since is the Calgebra of all continuous functions on //, the
cylindrical measures in fact turn out to be regular measures on //. Fi
nally, we were able to introduce explicitly such a measure on // which is
natural in the sense that it arises from the Haar measure on SU. The
resulting representation of the holonomy Calgebra has several interesting
properties. In particular, the representation is faithful and dieomorphism
invariant. Hence, its generating function which sends loops based at
x
to real numbers denes a generalized knot invariant generalized
because we have allowed loops to have kinks, selfintersections, and self
Abhay Ashtekar and Jerzy Lewandowski
overlaps. These constructions show that the AI program can be realized
in detail.
Having the induced Haar measure d on // at ones disposal, we
can now make the ideas on the loop transform T of , rigorous. The
transform T sends states in the connection representation to states in the
socalled loop representation. In terms of machinery introduced in this
paper, a state in the connection representation is a squareintegrable
function on // thus L
//, d. The transform T sends it to a
function on the space H of hoops. Sometimes, it is convenient to lift
canonically to the space L
x
and regard it as a function on the loop space.
This is the origin of the term loop representation. We have T
with
,/
dt
A
At
,,.
where t
with t
A
A is the Gelfand transform of the trace of
the holonomy function, T
, on //, associated with the hoop , and ,
denotes the inner product on L
//, d. Thus, in the transform the
traces of holonomies play the role of the integral kernel. Elements of the
holonomy algebra have a natural action on the connection states . Using
T , one can transform this action to the hoop states . It turns out that
the action is surprisingly simple and can be represented directly in terms of
elementary operations on the hoop space. Consider a generator T
of H/.
In the connection representation, we have T
AT
A
A. On
the hoop states, this action translates to
T
.
where is the composition of hoops and in the hoop group. Thus,
one can forgo the connection representation and work directly in terms of
hoop states. We will show elsewhere that the transform also interacts well
with the momentum operators which are associated with closed strips, i.e.
that these operators also have simple action on the hoop states.
The hoop states are especially well suited to deal with dieomorphism
i.e. Di invariant theories. In such theories, physical states are re
quired to be invariant under Di. This condition is awkward to impose
in the connection representation, and it is dicult to control the structure
of the space of resulting states. In the loop representation, by contrast,
the task is rather simple physical states depend only on generalized knot
and link classes of loops. Because of this simplication and because the
action of the basic operators can be represented by simple operations on
hoops, the loop representation has proved to be a powerful tool in quantum
general relativity in and dimensions.
The use of this representation, however, raises several issues which are
Holonomy Calgebras
still open. Perhaps the most basic of these is that, without referring back to
the connection representation, we do not yet have a useful characterization
of the hoop states which are images of connection states, i.e. of elements
of L
//, d. Neither do we have an expression of the inner product be
tween hoop states. It would be extremely interesting to develop integration
theory also over the hoop group H and express the inner product between
hoop states directly in terms of such integrals, without any reference to the
connection representation. This may indeed be possible using again the
idea of cylindrical functions, but now on H, rather than on //, and
exploiting, as before, the duality between these spaces. If this is achieved
and if the integrals over // and H can be related, we would have a non
linear generalization of the Plancherel theorem which establishes that the
Fourier transform is an isomorphism between two spaces of L
functions.
The loop transform would then become an extremely powerful tool.
Appendix A C
loops and U connections
In the main body of the paper, we restricted ourselves to piecewise analytic
loops. This restriction was essential in Section . for our decomposition
of a given set of a nite number of loops into independent loops which in
turn is used in every major result contained in this paper. The restriction
is not as severe as it might rst seem since every smooth manifold admits
a unique analytic structure. Nonetheless, it is important to nd out if
our arguments can be replaced by more sophisticated ones so that the
main results would continue to hold even if the holonomy Calgebra were
constructed from piecewise smooth loops. In this appendix, we consider
U connections rather than SU and show that, in this theory, our
main results do continue to hold for piecewise C
loops. However, the new
arguments make a crucial use of the Abelian character of U and do not
by themselves go over to nonAbelian theories. Nonetheless, this Abelian
example is an indication that analyticity may not be indispensible even in
the nonAbelian case.
The appendix is divided into three parts. The rst is devoted to cer
tain topological considerations which arise because manifolds admit non
trivial Ubundles. The second proves the analog of the spectrum theo
rem of Section . The third introduces a dieomorphisminvariant measure
on the spectrum and discusses some of its properties. Throughout this ap
pendix, by loops we shall mean continuous, piecewise C
loops. We will use
the same notation as in the main text but now those symbols will refer to
the U theory. Thus, for example, H will denote the Uhoop group
and H/, the U holonomy algebra.
A. Topological considerations
Fix a smooth manifold . Denote by / the space of all smooth U con
nections which belong to an appropriate Sobolev space. Now, unlike SU
Abhay Ashtekar and Jerzy Lewandowski
bundles, U bundles over manifolds need not be trivial. The question
therefore arises as to whether we should allow arbitrary U connections
or restrict ourselves only to the trivial bundle in the construction of the
holonomy Calgebra. Fortunately, it turns out that both choices lead to
the same Calgebra. This is the main result of this subsection.
Denote by /
the subspace of / consisting of connections on the trivial
U bundle over . As is common in the physics literature, we will identify
elements A
of /
with realvalued forms. Thus, the holonomies dened
by any A
/
will be denoted by H, A
expi
A
expi,
where takes values in , and depends on both the connection A
and
the loop . We begin with the following result
Lemma A. Given a nite number of loops,
,...,
n
, for every A /
there exists A
/
such that
H
i
,AH
i
,A
, i , . . . , n.
Proof Suppose the connection A is dened on the bundle P. We will show
rst that there exists a local section s of P which contains the given loops
i
. To construct the section we use a map S
which generates the
bundle P, U see Trautman . The map carries each
i
into a
loop
i
in S
. Since the loops
i
cannot form a dense set in S
, we may
remove from the sphere an open ball B which does not intersect any of
i
. Now, since S
B is contractible, there exists a smooth section of
the Hopf bundle U SU S
,
S
B SU.
Hence, there exists a section s of P, U dened on a subset V
S
B which contains all the loops
i
. Having obtained this
section s, we can now look for the connection A
. Let be a globally
dened, real form on such that V s
A. Hence, the connection
A
denes on the given loops
i
the same holonomy elements as A
i.e.
H
i
,A
H
i
, A A.
for all i.
This lemma has several useful implications. We mention two that will
be used immediately in what follows.
. Denition of hoops A priori, there are two equivalence relations on
the space L
x
of based loops that one can introduce. One may say
that two loops are equivalent if the holonomies of all connections in
/ around them are the same, or, one could ask that the holonomies
Holonomy Calgebras
be the same only for connections in /
. Lemma A. implies that the
two notions are in fact equivalent there is only one hoop group H.
As in the main text, we will denote by the hoop to which a loop
L
x
belongs.
. Sup norm Consider functions f on / dened by nite linear combi
nations of holonomies on / around hoops in H. Since the holono
mies themselves are complex numbers, the trace operation is now re
dundant. We can restrict these functions to /
. Lemma A. implies
that
sup
A,
fA sup
A
,
fA
. A.
As a consequence of these implications, we can use either / or /
to
construct the holonomy Calgebra in spite of topological nontrivialities,
there is only one H/. To construct this algebra we proceed as follows. Let
H/ denote the complex vector space of functions f on /
of the form
fA
n
j
a
j
H, A
n
j
a
j
exp
i
A
, A.
where a
j
are complex numbers. Clearly, H/ has the structure of a
algebra with the product law H, A
H, A
H,A
and the
relation given by H, A
H
,A
. Equip it with the sup norm
and take the completion. The result is the required Calgebra H/.
We conclude by noting another consequence of Lemma A. the
Abelian hoop group H has no torsion. More precisely, we have the
following result
Lemma A. Let H. Then if
n
e, the identity in H, for
n Z, then e.
Proof Since
n
e, we have, for every A
/
,
H, A
H
n
,
n
A
,
where is any loop in the hoop . Hence, by Lemma A., it follows that
e.
Although the proof is more complicated, the analogous result holds also
in the SU case treated in the main text. However, in that case, the hoop
group is nonAbelian. As we will see below, it is the Abelian character of
the Uhoop group that makes the result of this lemma useful.
A. The Gelfand spectrum of H/
We now want to obtain a complete characterization of the spectrum of the
U holonomy Calgebra H/ along the lines of Section . The analog of
Lemma . is easy to establish every element
A of the spectrum denes
Abhay Ashtekar and Jerzy Lewandowski
a homomorphism
H
A
from the hoop group H to U now given simply
by
H
A
A . This is not surprising. Indeed, it is clear from the
discussion in Section that this lemma continues to hold for piecewise
smooth loops even in the full SU theory. However, the situation is quite
dierent for Lemma . because there we made a crucial use of the fact
that the loops were continuous and piecewise analytic. Therefore, we
must now modify that argument suitably. An appropriate replacement is
contained in the following lemma.
Lemma A. For every homomorphism
H from the hoop group H to
U, every nite set of hoops
,...,
k
, and every gt , there exists
a connection A
/
such that
H
i
H
i
,A
lt , i , . . . , k. A.
Proof Consider the subgroup H
,...,
n
H that is generated by
,...,
k
. Since H is Abelian and since it has no torsion, H
,...,
k
is nitely and freely generated by some elements, say
,...,
n
. Hence, if
A. is satised by
i
for a suciently small
t
then it will also be satised
by the given
j
for the given . Consequently, without loss of generality,
we can assume that the hoops
i
are weakly Uindependent, i.e. that
they satisfy the following condition
if
k
n
kn
e, then k
i
i.
Now, given such hoops
i
, the homomorphism
H H U denes
a point
H
,...,
H
k
U
k
. On the other hand, there also exists
a map from /
to U
k
, which can be expressed as a composition
/
A
A
,...,
k
A
e
i
A
,...,e
i
k
A
U
k
.
A.
The rst map in A. is linear and its image, V R
k
, is a vector space. Let
,mm
,...,m
k
Z
k
. Denote by V
m
the subspace of V orthogonal
to m. Now if V
m
were to equal V then we would have
m
k
m
k
,
which would contradict the assumption that the given set of hoops is in
dependent. Hence we necessarily have V
m
lt V . Furthermore, since a
countable union of subsets of measure zero has measure zero, it follows
that
mZ
k
V
m
lt V.
This strict inequality implies that there exists a connection B
/
such
Holonomy Calgebras
that
v
B
,...,
n
B
V
mZ
k
V
m
.
In other words, B
is such that every ratio v
i
/v
j
of two dierent components
of v is irrational. Thus, the line in /
dened by B
via A
t tB
is
carried by the map A. into a line which is dense in U
k
. This ensures
that there exists an A
on this line which has the required property A..
Armed with this substitute of Lemma ., it is straightforward to show
the analog of Lemma .. Combining these results, we have the U
spectrum theorem
Theorem A. Every
A in the spectrum of H/ denes a homomorphism
H from the hoop group H to U, and, conversely, every homomorphism
H denes an element
A of the spectrum such that
H
A . This is a
correspondence.
A. A natural measure
Results presented in the previous subsection imply that one can again intro
duce the notion of cylindrical functions and measures. In this subsection,
we will exhibit a natural measure. Rather than going through the same
constructions as in the main text, for the sake of diversity, we will adopt
here a complementary approach. We will present a strictly positive linear
functional on the holonomy Calgebra H/ whose properties suce to en
sure the existence of a dieomorphisminvariant, faithful, regular measure
on the spectrum of H/.
We know from the general theory outlined in Section that, to specify
a positive linear functional on H/, it suces to provide an appropriate
generating functional on L
x
. Let us simply set
, if o
, otherwise,
A.
where o is the identity hoop in H. It is clear that this functional on L
x
is dieomorphism invariant. Indeed, it is the simplest of such functionals
on L
x
. We will now show that it does have all the properties to be a
generating function
.
Theorem A. extends to a continuous positive linear function on the
holonomy Calgebra H/. Furthermore, this function is strictly positive.
Since this functional is so simple and natural, one might imagine using it to dene a
positive linear functional also for the SU holonomy algebra of the main text. However,
this strategy fails because of the SU identities. For example, in the SU holonomy
algebra we have T
T
T
. Evaluation of the generating
functional A. on this identity would lead to the contradiction . We can dene a
Abhay Ashtekar and Jerzy Lewandowski
Proof By its denition, the loop functional admits a welldened
projection on the hoop space H which we will denote again by . Let
TH be the free vector space generated by the hoop group. Extend to
TH by linearity. We will rst establish that this extension satises the
following property given a nite set of hoops
j
, j , . . . , n, such that,
if
i
,
j
if i , j, we have
n
j
a
j
j
n
j
a
j
j
lt sup
A
,
n
j
a
j
H
j
,A
n
j
a
j
H
j
,A
.
A.
To see this, note rst that, by denition of , the left side equals
a
j
.
The right side equals
a
j
expi
j
a
j
exp i
j
where
j
depend on A
and the hoop
j
. Now, by Lemma A., given the n hoops
j
, one can
nd an A
such that the angles
j
can be made as close as one wishes to a
prespecied ntuple
j
. Finally, given the n complex numbers a
i
, one can
always nd
j
such that
a
j
lt
a
j
expi
j
a
j
expi
j
. Hence
we have A..
The inequality A. implies that the functional has a welldened
projection on the holonomy algebra H/ which is obtained by taking
the quotient of TH by the subspace K consisting of
b
i
i
such that
b
i
H
i
,A
for all A
/
. Furthermore, the projection is positive
denite f
f for all f H/, equality holding only if f . Next,
since the norm on this algebra is the sup norm on /
, it follows from
A. that the functional is continuous on H/. Hence it admits a unique
continuous extension to the Calgebra H/. Finally, A. implies that the
functional continues to be strictly positive on H/.
Theorem A. implies that the generating functional provides a contin
uous, faithful representation of H/ and hence a regular, strictly positive
measure on the Gelfand spectrum of this algebra. It is not dicult to
verify by direct calculations that this is precisely the U analog of the
induced Haar measure discussed in Section .
Appendix B C
loops, UN and SUN connections
In this appendix, we will consider another extension of the results presented
in the main text. We will now work with analytic manifolds and piecewise
analytic loops as in the main text. However, we will let the manifold have
positive linear function on the SU holonomy Calgebra via
, if T
AAA
, otherwise,
where we have regarded A
as a subspace of the space of SU connections. However,
on the SU holonomy algebra, this positive linear functional is not strictly positive
the measure on A/G it denes is concentrated just on A
. Consequently, the resulting
representation of the SU holonomy Calgebra fails to be faithful.
Holonomy Calgebras
any dimension and let the gauge group G be either UN or SUN. Con
sequently, there are two types of complications algebraic and topological.
The rst arise because, for example, the products of traces of holonomies
can no longer be expressed as sums while the second arise because the
connections in question may be dened on nontrivial principal bundles.
Nonetheless, we will see that the main results of the paper continue to hold
in these cases as well. Several of the results also hold when the gauge group
is allowed to be any compact Lie group. However, for brevity of presen
tation, we will refrain from making digressions to the general case. Also,
since most of the arguments are rather similar to those given in the main
text, details will be omitted.
Let us begin by xing a principal bre bundle P, G, obtain the
main results, and then show, at the end, that they are independent of
the initial choice of P, G. Denitions of the space L
x
of based loops,
and holonomy maps H, A, are the same as in Section . To keep the
notation simple, we will continue to use the same symbols as in the main
text to denote various spaces which, however, now refer to connections on
P, G. Thus, / will denote the space of suitably regular connections
on P, G, H will denote the P, Ghoop group, and H/ will be the
holonomy Calgebra generated by nite sums of nite products of traces
of holonomies of connections in / around hoops in H.
B. Holonomy Calgebra
We will set T
A
N
TrH, A, where the trace is taken in the fun
damental representation of UN or SUN, and regard T
as a function
on //, the space of gaugeequivalent connections. The holonomy alge
bra H/ is, by denition, generated by all nite linear combinations with
complex coecients of nite products of the T
, with the relation given
by f
f, where the bar stands for complex conjugation. Being traces
of UN and SUN matrices, T
are bounded functions on //. We can
therefore introduce on H/ the sup norm
f sup
A,
fA,
and take the completion of H/ to obtain a Calgebra. We will denote it
by H/. This is the holonomy Calgebra.
The key dierence between the structure of this H/ and the one we
constructed in Section is the following. In Section , the algebra H/
was obtained simply by imposing an equivalence relation K, see eqn .
on the free vector space generated by the loop space L
x
. This was possible
because, owing to SU Mandelstam identities, products of any two T
could be expressed as a linear combination of T
eqn .. For the groups
under consideration, the situation is more involved. Let us summarize the
situation in the slightly more general context of the group GLN. In
Abhay Ashtekar and Jerzy Lewandowski
this case, the Mandelstam identities follow from the contraction of N
matricesH
, A, . . . , H
N
, A in our casewith the identity
j
i
,...,
jN
iN
where
j
i
is the Kronecker delta and the bracket stands for the antisym
metrization. They enable one to express products of N T
functions
as a linear combination of traces of products of , , . . . , N T
functions.
Hence, we have to begin by considering the free algebra generated by the
hoop group and then impose on it all the suitable identities by extend
ing the denition of the kernel .. Consequently, if the gauge group is
GLN, an element of the holonomy algebra H/ is expressible as an equiv
alence class a, b
,...,c
N
, where a, b, c are complex numbers
products involving N and higher loops are redundant. For subgroups
of GLNsuch as SUNfurther reductions arise.
Finally, let us note identities that will be useful in what follows. These
result from the fact that the determinant of N N matrix M can be
expressed in terms of traces of powers of that matrix, namely
det M FTrM, TrM
, . . . , TrM
N
for a certain polynomial F. Hence, if the gauge group G is UN, we have,
for any hoop ,
FT
,...,T
NFT
,...,T
N . B.a
In the SUN case a stronger identity holds
FT
,...,T
N . B.b
B. Loop decomposition and the spectrum theorem
The construction of Section . for decomposition of a nite number of
loops into independent loops makes no direct reference to the gauge group
and therefore carries over as is. Also, it is still true that the map
/AH
, A, . . . , H
n
,AG
n
is onto if the loops
i
are independent. The proof requires some minor
modications which consist of using local sections and a suciently large
number of generators of the gauge group. A direct consequence is that
the analog of Lemma . continues to hold.
As for the Gelfand spectrum, the AI result that there is a natural
embedding of // into the spectrum // of H/ goes through as before
and so does the general argument due to Rendall that the image of //
is dense in //. This is true for any group and representation for which
the traces of holonomies separate the points of //. Finally, using the
analog of Lemma ., it is straightforward to establish the analog of the
Holonomy Calgebras
main conclusion of Lemma ..
The conversethe analog of Lemma .on the other hand requires
further analysis based on Giles results along the lines used by AI in
the SU case. For the gauge group under consideration, given an ele
ment of the spectrum
Ai.e. a continuous homomorphism from H/ to
the algebra of complex numbersGiles theorem provides us with a
homomorphism
H
A
from the hoop group H to GLN.
What we need to show is that H
A
can be so chosen that it takes values
in the given gauge group G. We can establish this by considering the
eigenvalues
,...,
N
of H
A
. After all, the characteristic polynomial
of the matrix H
A
is expressible directly by the values of
A taken on
the hoops ,
,...,
n
. First, we note from eqn B.a that in particular
i
, , for every i. Thus far, we have used only the algebraic properties
of
A. From continuity it follows that for every hoop , TrH
A
N.
Substituting for its powers we conclude that
k
k
N
N
for every integer k. This suces to conclude that
i
.
With this result in hand, we can now use Giles analysis to conclude that
given
A we can nd H
A
which takes values in UN. If G is SUN then,
by the identity B.b, we have
det H
A
,
so that the analog of Lemma . holds.
Combining these results, we have the spectrum theorem every element
A of the spectrum // denes a homomorphism
H from the hoop group
H to the gauge group G, and, conversely, every homomorphism
H from
the hoop group H to G denes an element
A of the spectrum // such
that
A
N
Tr
H for all H. Two homomorphisms
H
and
H
dene the same element of the spectrum if and only if Tr
H
Tr
H
.
B. Cylindrical functions and the induced Haar measure
Using the spectrum theorem, we can again associate with any subgroup
o
H
,...,
n
generated by n independent hoops
,...,
n
, an
equivalence relation on //
A
A
i
A
A
for all o
.
This relation provides us with a family of projections from // onto
the compact manifolds G
n
/Ad since, for groups under consideration, the
traces of all elements of a nitely generated subgroup in the fundamental
representation suce to characterize the subgroup modulo an overall ad
joint map. Therefore, as before, we can dene cylindrical functions on //
as the pullbacks to // of continuous functions on G
n
/Ad. They again
Abhay Ashtekar and Jerzy Lewandowski
form a Calgebra. Using the fact that traces of products of group elements
suce to separate points of G
n
/Ad when G UN or G SUN ,
it again follows that the Calgebra of cylindrical functions is isomorphic
with H/. Finally, the construction of Section . which led us to the deni
tion of the induced Haar measure goes through step by step. The resulting
representation of the Calgebra H/ is again faithful and dieomorphism
invariant.
B. Bundle dependence
In all the constructions above, we xed a principal bundle P, G. To
conclude, we will show that the Calgebra H/ and hence its spectrum are
independent of this choice. First, as noted at the end of Section ., using
the hoop decomposition one can show that two loops in L
x
dene the same
hoop if and only if they dier by a combination of reparametrization and
an immediate retracing of segments for gauge groups under consideration.
Therefore, the hoop groups obtained from any two bundles are the same.
Fix a gauge group G from UN, SUN and let P
, G and P
,
G be two principal bundles. Let the corresponding holonomy Calgebras
be H/
and H/
. Using the spectrum theorem, we know that their
spectra are naturally isomorphic //
HomH, G //
. We
wish to show that the algebras are themselves naturally isomorphic, i.e.
that the map dened by
n
i
a
i
T
i
n
i
a
i
T
i
B.
is an isomorphism from H/
to H/
. Let us decompose the given hoops
,...,
k
into independent hoops
,...,
n
. Let o
be the subgroup of
the hoop group they generate and, as in Section ., let
i
o
,i,,
be the corresponding projection maps from // to G
n
/Ad. From the
denition of the maps it follows that the projections of the two functions
in B. to G
N
/Ad are in fact equal. Hence, we have
n
i
a
i
T
i
n
i
a
i
T
i
,
whence it follows that is an isomorphism of Calgebras. Thus, it does
not matter which bundle we begin with we obtain the same holonomy
Calgebra H/. Now, x a bundle, P
say, and consider a connection A
dened on P
. Note that the holonomy map of A
denes a homomorphism
from the bundleindependent hoop group to G. Hence, A
denes also
a point in the spectrum of H/
. Therefore, given a manifold M and a
gauge group G the spectrum // automatically contains all the connections
Holonomy Calgebras
on all the principal Gbundles over M.
Acknowledgements
The authors are grateful to John Baez for several illuminating comments
and to Roger Penrose for suggesting that the piecewise analytic loops may
be better suited for their purpose than the piecewise smooth ones. JL also
thanks Cliord Taubes for stimulating discussions. This work was sup
ported in part by the National Science Foundation grants PHY
and PHY by the Polish Committee for Scientic Research KBN
through grant no. and by the Eberly research fund of Penn
sylvania State University.
Bibliography
. A. Ashtekar, Nonperturbative canonical quantum gravity Notes pre
pared in collaboration with R.S. Tate, World Scientic, Singapore
. A. Ashtekar and C. Isham, Classical amp Quantum Gravity ,
. R. Gambini and A. Trias, Nucl. Phys. B,
. C. Rovelli and L. Smolin, Nucl. Phys. B,
. A. Rendall, Classical amp Quantum Gravity ,
. J. Baez, In Proceedings of the Conference on Quantum Topology, eds
L. Crane and D. Yetter to appear, hepth/
. J. Lewandowski, Classical amp Quantum Gravity ,
. P.K. Mitter and C. Viallet, Commun. Math. Phys. ,
. C. Di Bartolo, R. Gambini, and J. Griego, preprint IFFI/ gr
qc/
. R. Giles, Phys. Rev. D,
. J.W. Barrett, Int. J. Theor. Phys. ,
. A.N. Kolmogorov, Foundations of the Theory of Probability, Chelsea,
New York
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ysis, Manifolds and Physics, NorthHolland, Amsterdam , pp.
. M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press,
Cambridge Chapter
. A. Trautman, Czech. J. Phys. B,
. C. Procesi, Adv. Math. ,
Abhay Ashtekar and Jerzy Lewandowski
The Gauss Linking Number in Quantum
Gravity
Rodolfo Gambini
Intituto de Fsica, Facultad de Ciencias,
Tristan Narvaja , Montevideo, Uruguay
email rgambinising.edu.uy
Jorge Pullin
Center for Gravitational Physics and Geometry,
Pennsylvania State University,
University Park, Pennsylvania , USA
email pullinphys.psu.edu
Abstract
We show that the exponential of the Gauss selflinking number of
a knot is a solution of the WheelerDeWitt equation in loop space
with a cosmological constant. Using this fact, it is straightforward to
prove that the second coecient of the Jones polynomial is a solution
of the WheelerDeWitt equation in loop space with no cosmological
constant. We perform calculations from scratch, starting from the
connection representation and give details of the proof. Implications
for the possibility of generation of other solutions are also discussed.
Introduction
The introduction of the loop representation for quantum gravity has made
it possible for the rst time to nd solutions to the WheelerDeWitt equa
tion the quantum Hamiltonian constraint and therefore to have possible
candidates to become physical wave functions of the gravitational eld. In
the loop representation the Hamiltonian constraint has nonvanishing ac
tion only on functions of intersecting loops. It was rst argued that by
considering wave functions with support on smooth loops one could solve
the constraint straightforwardly , . However, it was later realized that
such solutions are associated with degenerate metrics metrics with zero
determinant and this posed inconsistencies if one wanted to couple the
theory . For instance, if one considered general relativity with a cosmo
logical constant it turns out that nonintersecting loop states also solve the
WheelerDeWitt equation for arbitrary values of the cosmological constant.
This does not appear as reasonable since dierent values of the cosmological
constant lead to widely dierent behaviors in general relativity.
Therefore the problem of solving the WheelerDeWitt equation in loop
Rodolfo Gambini and Jorge Pullin
Fig. . At least a triple intersection is needed to have a state associated
with a nondegenerate dimensional metric in the loop representation.
And even in this case the metric is nondegenerate only at the point of
intersection.
space is far from solved and has to be tackled by considering the action of
the Hamiltonian constraint in the loop representation on states based on
at least triply intersecting loops, as depicted in Fig. .
Although performing this kind of direct calculations is now possible,
since welldened expressions for the Hamiltonian constraint exist in terms
of the loop derivative see also and actually some solutions have been
found with this approach , another line of reasoning has also proved to
be useful.
This other approach is based on the fact that the exponential of the
ChernSimons term based on Ashtekars connection
CS
A exp
d
xA
i
a
b
A
i
c
A
i
a
A
j
b
A
k
c
ijk
abc
.
is a solution of the WheelerDeWitt equation in the connection represen
tation with a cosmological term
H
ijk
A
i
a
A
j
b
F
k
ab
ijk
abc
A
i
a
A
j
b
A
k
c
.
the rst term is just the usual Hamiltonian constraint with and the
second term is just det g where det g is the determinant of the spatial
part of the metric written in terms of triads.
To prove this fact one simply needs to notice that for the ChernSimons
state introduced above,
A
i
a
CS
A
abc
F
i
bc
CS
A, .
and therefore the rightmost functional derivative in the cosmological con
stant term of the Hamiltonian . produces a term in F
i
ab
that exactly
cancels the contribution from the vacuum Hamiltonian constraint.
The Gauss Linking Number in Quantum Gravity
This result for the ChernSimons state in the connection representa
tion has an immediate counterpart in the loop representation, since the
transform of the ChernSimons state into the loop representation
CS
dA
CS
AW
A.
can be interpreted as the expectation value of the Wilson loop in a Chern
Simons theory,
CS
dA e
SCS
W
AW
.
with coupling constant
, and we know due to the insight of Witten
that this coincides with the Kauman bracket of the loop. Therefore the
state
CS
Kauman bracket
.
should be a solution of the WheelerDeWitt equation in loop space.
This last fact can actually be checked in a direct fashion using the
expressions for the Hamiltonian constraint in loop space of . In order to
do this we make use of the identity
Kauman bracket
e
Gauss
Jones polynomial
.
which relates the Kauman bracket, the Gauss selflinking number, and
the Jones polynomial. The Gauss selflinking number is framing dependent,
and so is the Kauman bracket. The Jones polynomial, however, is framing
independent. This raises the issue of up to what extent statements about
the Kauman bracket being a state of gravity are valid, since one expects
states of quantum gravity to be truly dieomorphisminvariant objects and
framing is always dependent on an external device which should conict
with the dieomorphism invariance of the theory. Unfortunately it is not
clear at present how to settle this issue since it is tied to the regularization
procedures used to dene the constraints. We will return to these issues in
the nal discussion.
We now expand both the Jones polynomial and the exponential of the
Gauss linking number in terms of and get the expression
Kauman bracket
Gauss
Gauss
a
Gauss
Gauss
a
a
....
where a
,a
are the second and third coecient of the innite expansion
of the Jones polynomial in terms of . a
is known to coincide with the
second coecient of the Conway polynomial .
Rodolfo Gambini and Jorge Pullin
One could now apply the Hamiltonian constraint in the loop represen
tation with a cosmological constant to the expansion . and one would
nd that certain conditions have to be satised if the Kauman bracket is
to be a solution. Among them it was noticed that
H
a
.
where
H
is the Hamiltonian constraint without cosmological constant
more recently it has also been shown that
H
a
but we will
not discuss it here.
Summarizing, the fact that the Kauman bracket is a solution of the
WheelerDeWitt equation with cosmological constant seems to have as a di
rect consequence that the coecients of the Jones polynomial are solutions
of the vacuum WheelerDeWitt equation This partially answers
the problem of framing we pointed out above. Even if one is reluctant to
accept the Kauman bracket as a state because of its framing dependence,
it can be viewed as an intermediate step of a framingdependent proof that
the Jones polynomial which is a framingindependent invariant solves the
WheelerDeWitt equation it should be stressed that we only have evidence
that the rst two nontrivial coecients are solutions.
The purpose of this paper is to present a rederivation of these facts from
a dierent, and to our understanding simpler, perspective. We will show
that the framingdependent knot invariant
G
e
Gauss
.
is also a solution of the WheelerDeWitt equation with cosmological con
stant. It can be viewed as an Abelian limit of the Kauman bracket more
on this in the conclusions. Given this fact, one can therefore consider their
dierence divided by
,
D
Kauman bracket
G
,.
which is also a solution of the WheelerDeWitt equation with a cosmolog
ical constant. This dierence is of the form
D
a
a
Gauss
a
.....
Now, this dierence is a state for all values of , in particular for
. This means that a
should be a solution of the WheelerDeWitt
equation. This conrms the proof given in , .
Therefore we see that by noticing that the Gauss linking number is
a state with cosmological constant, it is easy to prove that the second
coecient of the innite expansion of the Jones polynomial which coincides
with the second coecient of the Conway polynomial is a solution of the
WheelerDeWitt equation with .
The Gauss Linking Number in Quantum Gravity
The rest of this paper will be devoted to a detailed proof that the expo
nential of the Gauss linking number solves the WheelerDeWitt equation
with cosmological constant. To this aim we will derive expressions for the
Hamiltonian constraint with cosmological constant in the loop represen
tation. We will perform the calculation explicitly for the case of a triply
selfintersecting loop, the more interesting case for gravity purposes it
should be noticed that all the arguments presented above were indepen
dent of the number and order of intersections of the loops we just present
the explicit proof for a triple intersection since in three spatial dimensions
it represents the most important type of intersection.
Apart from presenting this new state, we think the calculations exhib
ited in this paper should help the reader get into the details of how these
calculations are performed and make an intuitive contact between the ex
pressions in the connection and the loop representation.
In Section we derive the expression of the Hamiltonian constraint
with a cosmological constant in the loop representation for a triply in
tersecting loop in terms of the loop derivative. In Section we write an
explicit analytic expression for the Gauss linking number and prove that
it is a state of the theory. We end in Section with a discussion of the
results.
The WheelerDeWitt equation in terms of loops
Here we derive the explicit form in the loop representation of the Hamil
tonian constraint with a cosmological constant. The derivation proceeds
along the following lines. Suppose one wants to dene the action of an
operator
O
L
on a wave function in the loop representation . Applying
the transform
O
L
dA
O
L
W
AA .
the operator
O in the right member acts on the loop dependence of the
Wilson loop. On the other hand, this denition should agree with
O
L
dAW
A
O
C
A.
where
O
C
is the connection representation version of the operator in ques
tion. Therefore, it is clear that
O
L
W
A
O
C
W
A, .
where means the adjoint operator with respect to the measure of inte
gration dA. If one assumes that the measure is trivial, the only eect of
taking the adjoint is to reverse the factor ordering of the operators.
Concretely, in the case of the Hamiltonian constraint without
Rodolfo Gambini and Jorge Pullin
cosmological constant
H
C
ijk
A
i
a
A
j
b
F
k
ab
.
and therefore
H
L
W
A
ijk
F
k
ab
A
i
a
A
j
b
W
A. .
We now need to compute this quantity explicitly. For that we need the
expression of the functional derivative of the Wilson loop with respect to
the connection
A
i
a
x
W
A
dy
a
y xTr
Pexp
y
o
dz
b
A
b
i
Pexp
o
y
dz
b
A
b
.
where o is the basepoint of the loop. The Wilson loop is therefore broken
at the point of action of the functional derivative and a Pauli matrix
i
is inserted. It is evident that with this action of the functional derivative
the Hamiltonian operator is not well dened. We need to regularize it as
H
L
W
A lim
f
xz
ijk
F
k
ab
x
A
i
a
x
A
j
b
z
W
A.
where f
xz xz when . Caution should be exercised, since
such point splitting breaks the gauge invariance of the Hamiltonian. There
are a number of ways of xing this situation in the language of loops. One of
them is to dene the Hamiltonian inserting pieces of holonomies connecting
the points x and z between the functional derivatives to produce a gauge
invariant quantity . Here we will only study the operator in the limit
in which the regulator is removed therefore we will not be concerned with
these issues. A proper calculation would require their careful study.
It is immediate from the above denitions that the Hamiltonian con
straint vanishes in any regular point of the loop, since it yields a term
dy
a
dy
b
F
i
ab
which vanishes due to the antisymmetry of F
i
ab
and the symme
try of dy
a
dy
b
at points where the loop is smooth. However, at intersections
there can be nontrivial contributions. Here we compute the contribution
at a point of triple selfintersection,
ijk
F
k
ab
x
A
i
a
x
A
j
b
z
W
A
ijk
F
k
ab
x.
a
b
W
j
k
A
a
b
W
j
kA
a
b
W
j
kA,
where we have denoted
where
i
are the petals of the loop
as indicated in Fig. . By W
j
k
A we really mean take the holon
omy from the basepoint along
up to just before the intersection point,
insert a Pauli matrix, continue along
to the intersection, continue along
The Gauss Linking Number in Quantum Gravity
, and just before the intersection insert another Pauli matrix, continue to
the intersection, and complete the loop along
. One could pick after the
intersection instead of before to include the Pauli matrices and it would
make no dierence since we are concentrating on the limit in which the
regulator is removed, in which the insertions are done at the intersection.
By
a
we mean the tangent to the petal number just before the inter
section where the Pauli matrix was inserted. This is just shorthand for
expressions like
dy
a
xy when the point x is close to the intersection,
so strictly speaking
a
really is a distribution that is nonvanishing only at
the point of intersection.
One now uses the following identity for traces of SU matrices,
ijk
W
j
kA
W
i AW
AW
AW
i A, .
which is a natural generalization to the case of loops with insertions of the
SU Mandelstam identities,
W
AW
AW
AW
A, .
where
means the loop opposite to . The result of the application of
these identities to the expression of the Hamiltonian is
HW
A.
F
i
ab
a
b
W
iA
a
b
W
iA
a
b
W
iA
.
This expression can be further rearranged making use of the loop deriva
tive. The loop derivative
ab
x
o
, is the dierentiation operator that
appears in loop space when one considers two loops to be close if they
dier by an innitesimal loop appended through a path
x
o
going from the
basepoint to a point of the manifold x as shown in Fig. . Its denition is
x
o
o
x
ab
ab
x
o
.
where
ab
is the element area of the innitesimal loop and by
x
o
o
x
we mean the loop obtained by traversing the path from the basepoint to
x, the innitesimal loop , the path from x to the basepoint, and then
the loop .
We will not discuss all its properties here. The only one we need is that
the loop derivative of a Wilson loop taken with a path along the loop is
given by
ab
x
o
W
A Tr
F
ab
xPexp
dy
c
A
c
,.
which reects the intuitive notion that a holonomy of an innitesimal loop
is related to the eld tensor. Therefore we can write expressions like
F
i
ab
W
iA.
Rodolfo Gambini and Jorge Pullin
Fig. . The loop dening the loop derivative
as
ab
W
A, .
and the nal expression for the Hamiltonian constraint in the loop repre
sentation can therefore be read o as follows
H
a
b
ab
a
b
ab
a
b
ab
..
This expression could be obtained by particularizing that of to the
case of a triply selfintersecting loop and rearranging terms slightly using
the Mandelstam identities. However, we thought that a direct derivation
for this particular case would be useful for pedagogical purposes.
We now have to nd the loop representation form of the operator corre
sponding to the determinant of the metric in order to represent the second
term in .. We proceed in a similar fashion, rst computing the action
of the operator in the connection representation
det q
ijk
abc
A
i
a
A
j
b
A
k
c
.
on a Wilson loop
ijk
abc
A
i
a
A
j
b
A
k
c
W
A
abc
ijk
a
b
c
W
i
j
kA
..
This latter expression can be rearranged with the following identity between
holonomies with insertions of Pauli matrices
ijk
W
i
j
kA
W
AW
AW
A. .
It is therefore immediate to nd the expression of the determinant of
the metric in the loop representation
detq
abc
a
b
c
..
The Gauss Linking Number in Quantum Gravity
With these elements we are in a position to perform the main calculation
of this paper, to show that the exponential of the Gauss selflinking number
of a loop is a solution of the Hamiltonian constraint with a cosmological
constant.
The Gauss selflinking number as a solution
In order to be able to apply the expressions we derived in the previous
section for the constraints to the Gauss selflinking number we need an ex
pression for it in terms of which it is possible to compute the loop derivative.
This is furnished by the wellknown integral expression
Gauss
dx
a
dy
b
abc
xy
c
xy
.
where x y is the distance between x and y with a ducial metric. This
formula is most well known when the two loop integrals are computed along
dierent loops. In that case the formula gives an integer measuring the
extent to which the loops are linked. In the present case we are considering
the expression of the linking of a curve with itself. This is in general not
well dened without the introduction of a framing .
We will rewrite the above expression in a more convenient fashion
Gauss
d
x
d
yX
a
x, X
b
y, g
ab
x, y .
where the vector densities X are dened as
X
a
x,
dz
a
zx.
and the quantity g
ab
x, y is the propagator of a ChernSimons theory ,
g
ab
x, y
abc
xy
c
xy
..
For calculational convenience it is useful to introduce the notation
Gauss X
ax
X
by
g
ax by
.
where we have promoted the point dependence in x, y to a continuous
index and assumed a generalized Einstein convention which means sum
over repeated indices a, b and integrate over the manifold for repeated
continuous indices. This notation is also faithful to the fact that the index
a behaves as a vector density index at the point x that is, it is natural to
pair a and x together.
The only dependence on the loop of the Gauss selflinking number is
through the X, so we just need to compute the action of the loop derivative
on one of them to be in a position to perform the calculation straightfor
Rodolfo Gambini and Jorge Pullin
Fig. . The loop derivative that appears in the denition of the Hamilto
nian is evaluated along a path that follows the loop
wardly. In order to do this we apply the denition of loop derivative, that
is we consider the change in the X when one appends an innitesimal loop
to the loop as illustrated in Fig. . We partition the integral in a portion
going from the basepoint to the point z where we append the innitesimal
loop, which we characterize as four segments along the integral curves of
two vector elds u
a
and v
b
of associated lengths
and
, and then we
continue from there back to the basepoint along the loop. Therefore,
ab
ab
z
o
X
ax
z
o
dy
a
xy
u
a
xz
v
b
u
c
c
xz
u
a
u
c
v
c
c
xz.
v
a
v
c
c
xz
o
z
dy
a
y z.
The last and rst term combine to give back X
ax
and therefore
one can read o the action of the loop derivative from the other terms.
Rearranging one gets
ab
z
o
X
cx
a
c
b
xz.
where the notation
c
b
x z stands for
c
b
x z, the product of the
Kronecker and Dirac deltas.
This is really all we need to compute the action of the Hamiltonian.
We therefore now consider the action of the vacuum part of the
Hamiltonian on the exponential of the Gauss linking number. The action
of the loop derivative is
ab
x
o
exp
X
cy
X
dz
g
cy dz
.
a
c
b
x yX
dz
g
cy dz
exp
X
ew
X
fw
g
ew fw
.
Now we must integrate by parts. Using the fact that
a
X
ax
and
The Gauss Linking Number in Quantum Gravity
the denition of g
cy dz
., we get
ab
x
o
exp
X
cy
X
dz
g
cy dz
.
abc
X
cx
X
cx
X
cx
exp
X
cy
X
dz
g
cy dz
.
Therefore the action of the Hamiltonian constraint on the Gauss linking
number is
He
Gauss
abc
a
b
c
e
Gauss
.
where we again have replaced the distributional tangents at the point of
intersection by an expression only involving the tangents. The expression
is only formal since in order to do this a divergent factor should be kept
in front. We assume such factors coming from all terms to be similar and
therefore ignore them.
It is straightforward now to check that applying the determinant of the
metric one the Gauss linking number one gets a contribution exactly equal
and opposite by inspection from expression .. This concludes the main
proof of this paper.
Discussion
We showed that the exponential of the Gauss selflinking number is a
solution of the Hamiltonian constraint of quantum gravity with a cosmo
logical constant. This naturally can be viewed as the Abelian limit of the
solution given by the Kauman bracket.
What about the issue of regularization The proof we presented is only
valid in the limit where , that is, when the regulator is removed. If
one does not take the limit the various terms do not cancel. However, the
expression for the Hamiltonian constraint we introduced is also only valid
when the regulator is removed. A regularized form of the Hamiltonian con
straint in the loop representation is more complicated than the expression
we presented. If one is to pointsplit, innitesimal segments of loop should
be used to connect the split points to preserve gauge invariance and a more
careful calculation would be in order.
What does all this tell us about the physical relevance of the solutions
The situation is remarkably similar to the one present in the loop repre
sentation of the free Maxwell eld . In that case, as here, there are two
terms in the Hamiltonian that need to be regularized in a dierent way in
the case of gravity, the determinant of the metric requires splitting three
points whereas the Hamiltonian only needs two. As a consequence of this,
it is not surprising that the wave functions that solve the constraint have
some regularization dependence. In the case of the Maxwell eld the vac
uum in the loop representation needs to be regularized. In fact, its form
is exactly the same as that of the exponential of the Gauss linking number
if one replaces the propagator of the ChernSimons theory present in the
Rodolfo Gambini and Jorge Pullin
latter by the propagator of the Maxwell eld. This similarity is remarkable.
The problem is therefore the same the wave functions inherit regulariza
tion dependence since the regulator does not appear as an overall factor of
the wave equation.
How could these regularization ambiguities be cured In the Maxwell
case they are solved by considering an extended loop representation in
which one allows the quantities X
ax
to become smooth vector densities on
the manifold without reference to any particular loop . In the grav
itational case such construction is being actively pursued , although
it is more complicated. It is in this context that the present solutions
really make sense. If one allows the X to become smooth functions the
framing problem disappears and one is left with a solution that is a func
tion of vector elds and only reduces to the Gauss linking number in a
very special singular limit. It has been proved that the extensions of
the Kauman bracket and Jones polynomials to the case of smooth den
sity elds are solutions of the extended constraints. A similar proof goes
through for the extended Gauss linking number. In the extended repre
sentation, there are additional multivector densities needed in the repre
sentation. The Abelian limit of the Kauman bracket the Gauss linking
number appears as the restriction of the extended Kauman bracket to
the case in which higherorder multivector densities vanish. It would be
interesting to study if such a limit could be pursued in a systematic way or
der by order. It would certainly provide new insights into how to construct
nonperturbative quantum states of the gravitational eld.
Acknowledgements
This paper is based on a talk given by J P at the Riverside conference. J P
wishes to thank John Baez for inviting him to participate in the conference
and hospitality in Riverside. This work was supported by grants NSF
PHY, NSFPHY, and by research funds of the University
of Utah and Pennsylvania State University. Support from PEDECIBA
Uruguay is also acknowledged. R G wishes to thank Karel Kuchar and
Richard Price for hospitality at the University of Utah where part of this
work was accomplished.
Bibliography
. T. Jacobson, L. Smolin, Nucl. Phys. B, .
. C. Rovelli, L. Smolin, Phys. Rev. Lett. , Nucl. Phys.
B, .
. B. Br ugmann, J. Pullin, Nucl. Phys. B, .
. R. Gambini, Phys. Lett. B, .
. B. Br ugmann, R. Gambini, J. Pullin, Phys. Rev. Lett. , .
The Gauss Linking Number in Quantum Gravity
. E. Witten, Commun. Math. Phys. , .
. E. Guadagnini, M. Martellini, M. Mintchev, Nucl. Phys. B,
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. B. Br ugmann, R. Gambini, J. Pullin, General Relativity Gravitation
,.
. J. Griego, The extended loop representation a rst approach,
preprint .
. R. Gambini, A. Trias, Phys. Rev. D, .
. C. Di Bartolo, F. Nori, R. Gambini, L. Leal, A. Trias, Lett. Nuovo.
Cim. , .
. D. ArmandUgon, R. Gambini, J. Griego, L. Setaro, Classical loop
actions of gauge theories, preprint hepth .
. C. Di Bartolo, R. Gambini, J. Griego, J. Pullin, in preparation.
Rodolfo Gambini and Jorge Pullin
Vassiliev Invariants and the Loop States in
Quantum Gravity
Louis H. Kauman
Department of Mathematics, Statistics, and Computer Science,
University of Illinois at Chicago,
Chicago, Illinois , USA
email Uuicvm.uic.edu
Abstract
This paper derives at the physical level of rigor a general switch
ing formula for the link invariants dened via the Witten functional
integral. The formula is then used to evaluate the top row of the cor
responding Vassiliev invariants and it is shown to match the results
of BarNatan obtained via perturbation expansion of the integral. It
is explained how this formalism for Vassiliev invariants can be used
in the context of the loop transform for quantum gravity.
Introduction
The purpose of this paper is to expose properties of Vassiliev invariants by
using the simplest of the approaches to the functional integral denition of
link invariants. These methods are strong enough to give the top row eval
uations of Vassiliev invariants for the classical Lie algebras. They give an
insight into the structure of these invariants without using the full pertur
bation expansion of the integral. One reason for examining the invariants
in this light is the possible applications to the loop variables approach to
quantum gravity. The same level of handling the functional integral is
commonly used in the loop transform for quantum gravity.
The paper is organized as follows. Section is a brief discussion of the
nature of the functional integral. Section details the formalism of the
functional integral that we shall use, and works out a dierence formula
for changing crossings in the link invariant and a formula for the change of
framing. These are applied to the case of an SUN gauge group. Section
denes the Vassiliev invariants and shows how to formulate them in terms
of the functional integral. In particular we derive the specic expressions
for the top rows of Vassiliev invariants corresponding to the fundamental
representation of SUN. This gives a neat point of view on the results of
BarNatan, and also gives a picture of the structure of the graphical vertex
associated with the Vassiliev invariant. We see that this vertex is not just
a transversal intersection of Wilson loops, but rather has the structure
Louis H. Kauman
of Casimir insertion up to rst order of approximation coming from the
dierence formula in the functional integral. This claries an issue raised
by John Baez in . Section is a quick remark about the loop formalism
for quantum gravity and its relationships with the invariants studied in the
previous sections. This marks the beginning of a study that will be carried
out in detail elsewhere.
Quantum mechanics and topology
In Edward Witten proposed a formulation of a class of manifold
invariants as generalized Feynman integrals taking the form ZM where
ZM
dAexp
ik/SM, A
.
Here M denotes a manifold without boundary and A is a gauge eld also
called a gauge potential or gauge connection dened on M. The gauge
eld is a form on M with values in a representation of a Lie algebra.
The group corresponding to this Lie algebra is said to be the gauge group
for this particular eld. In this integral the action SM, A is taken to
be the integral over M of the trace of the ChernSimons threeform CS
AdA/AAA. The product is the wedge product of dierential forms.
Instead of integrating over paths, the integral ZM integrates over
all gauge elds modulo gauge equivalence see for a discussion of the
denition and meaning of gauge equivalence. This generalization from
paths to elds is characteristic of quantum eld theory.
Quantum eld theory was designed in order to accomplish the quan
tization of electromagnetism. In quantum electrodynamics the classical
entity is the electromagnetic eld. The question posed in this domain is to
nd the value of an amplitude for starting with one eld conguration and
ending with another. The analogue of all paths from point a to point b is
all elds from eld A to eld B.
Wittens integral ZM is, in its form, a typical integral in quantum
eld theory. In its content ZM is highly unusual. The formalism of
the integral, and its internal logic, support the existence of a large class
of topological invariants of manifolds and associated invariants of knots
and links in these manifolds.
The invariants associated with this integral have been given rigorous
combinatorial descriptions see , , , , , but questions and con
jectures arising from the integral formulation are still outstanding see .
Links and the Wilson loop
We now look at the formalism of the Witten integral and see how it im
plicates invariants of knots and links corresponding to each classical Lie
algebra. We need the Wilson loop. The Wilson loop is an exponentiated
Vassiliev Invariants and Quantum Gravity
version of integrating the gauge eld along a loop K in space that we take
to be an embedding knot or a curve with transversal selfintersections.
For the purpose of this discussion, the Wilson loop will be denoted by the
notation
KA
to denote the dependence on the loop K and the eld A.
It is usually indicated by the symbolism Tr
P exp
KA
.
Thus
KA
Tr
P exp
KA
. Here the P denotes pathordered
integration. The symbol Tr denotes the trace of the resulting matrix.
With the help of the Wilson loop functional on knots and links, Witten
writes down a functional integral for link invariants in a manifold M
ZM, K
dAexp
ik/SM, A
Tr
P exp
KA
dAexp
ik/S
KA
.
Here SM, A is the ChernSimons action, as in the previous discussion.
We abbreviate SM, A as S. Unless otherwise mentioned, the manifold
M will be the dimensional sphere S
.
An analysis of the formalism of this functional integral reveals quite a
bit about its role in knot theory. This analysis depends upon key facts
relating the curvature of the gauge eld to both the Wilson loop and the
ChernSimons Lagrangian. To this end, let us recall the local coordinate
structure of the gauge eld Ax, where x is a point in space. We can write
Ax A
a
k
xT
a
dx
k
where the index a ranges from to m with the Lie
algebra basis T
,T
,T
,...,T
m
. The index k goes from to . For each
choice of a and k, A
k
a
x is a smooth function dened on space. In Ax
we sum over the values of repeated indices. The Lie algebra generators T
a
are actually matrices corresponding to a given representation of an abstract
Lie algebra. We assume some properties of these matrices as follows
.T
a
,T
b
if
abc
T
c
where x, y xy yx, and f
abc
the matrix of
structure constants is totally antisymmetric. There is summation
over repeated indices.
. TrT
a
T
b
ab
/ where
ab
is the Kronecker delta
ab
if a b
and zero otherwise.
We also assume some facts about curvature. The reader may compare
with the exposition in . But note the dierence in conventions on the
use of i in the Wilson loops and curvature denitions. The rst fact is the
relation of Wilson loops and curvature for small loops
Fact The result of evaluating a Wilson loop about a very small planar
circle around a point x is proportional to the area enclosed by this circle
times the corresponding value of the curvature tensor of the gauge eld
evaluated at x. The curvature tensor is written F
a
rs
xT
a
dx
r
dy
s
. It is the
Louis H. Kauman
local coordinate expression of dA AA.
Application of Fact Consider a given Wilson loop
KA
. Ask how
its value will change if it is deformed innitesimally in the neighborhood
of a point x on the loop. Approximate the change according to Fact ,
and regard the point x as the place of curvature evaluation. Let
KA
denote the change in the value of the loop.
KA
is given by the formula
KA
dx
r
dx
s
F
a
rs
xT
a
KA
. This is the rstorder approximation to
the change in the Wilson loop.
In this formula it is understood that the Lie algebra matrices T
a
are to
be inserted into the Wilson loop at the point x, and that we are summing
over repeated indices. This means that each T
a
KA
is a new Wilson
loop obtained from the original loop
KA
by leaving the form of the loop
unchanged, but inserting the matrix T
a
into the loop at the point x. See
the gure below.
KA
T
a
KA
Remark on insertion The Wilson loop is the limit, over partitions of
the loop K, of products of the matrices
Ax
where x runs over the
partition. Thus one can write symbolically,
KA
xK
Ax
xK
A
a
k
xT
a
dx
k
.
It is understood that a product of matrices around a closed loop connotes
the trace of the product. The ordering is forced by the dimensional
nature of the loop. Insertion of a given matrix into this product at a point
on the loop is then a welldened concept. If T is a given matrix then it is
understood that T
KA
denotes the insertion of T into some point of the
loop. In the case above, it is understood from the context of the formula
ds
r
dx
s
F
a
rs
xT
a
KA
that the insertion is to be performed at the point x
indicated in the argument of the curvature.
Vassiliev Invariants and Quantum Gravity
Remark The previous remark implies the following formula for the varia
tion of the Wilson loop with respect to the gauge eld
KA
A
a
k
x
dx
k
T
a
KA
.
Varying the Wilson loop with respect to the gauge eld results in the
insertion of an innitesimal Lie algebra element into the loop.
Proof
KA
A
a
k
x
A
a
k
x
yK
A
a
k
yT
a
dy
k
yltx
A
a
k
yT
a
dy
k
T
a
dx
k
ygtx
A
a
k
yT
a
dy
k
dx
k
T
a
KA
.
.
Fact The variation of the ChernSimons action S with respect to the
gauge potential at a given point in space is related to the values of the
curvature tensor at that point by the following formula where
abc
is the
epsilon symbol for three indices
F
a
rs
x
rst
S
A
a
t
x
.
With these facts at hand we are prepared to determine how the Witten
integral behaves under a small deformation of the loop K.
Proposition Compare . All statements of equality in this proposi
tion are up to order /k
.
. Let ZK ZK, S
and let ZK denote the change of ZK
under an innitesimal change in the loop K. Then
ZK i/k
dAexp
ik/S
rst
dx
r
dy
s
dz
t
T
a
T
a
KA
.
The sum is taken over repeated indices, and the insertion is taken of
the matrix products T
a
T
a
at the chosen point x on the loop K that
is regarded as the center of the deformation. The volume element
rst
dx
r
dy
s
dz
t
is taken with regard to the innitesimal directions of
the loop deformation from this point on the original loop.
. The same formula applies, with a dierent interpretation, to the case
where x is a double point of transversal selfintersection of a loop
Louis H. Kauman
K, and the deformation consists in shifting one of the crossing seg
ments perpendicularly to the plane of intersection so that the self
intersection point disappears. In this case, one T
a
is inserted into
each of the transversal crossing segments so that T
a
T
a
KA
denotes
a Wilson loop with a selfintersection at x and insertions of T
a
at
x
and x
, where
and
denote small displacements along
the two arcs of K that intersect at x. In this case, the volume form is
nonzero, with two directions coming from the plane of movement of
one arc, and the perpendicular direction is the direction of the other
arc.
Proof
ZK
dAexp
ik/S
KA
dAexp
ik/S
dx
r
dy
s
F
a
rs
xT
a
KA
Fact
dAexp
ik/S
dx
r
dy
s
rst
S
A
a
t
x
T
a
KA
Fact
dAexp
ik/S
S
A
a
t
x
rst
dx
r
dy
s
T
a
KA
i/k
dA
exp
ik/S
A
a
t
x
rst
dx
r
dy
s
T
a
KA
i/k
dAexp
ik/S
rst
dx
r
dy
s
T
a
KA
A
a
t
x
integration by parts
i/k
dAexp
ik/S
rst
dx
r
dy
s
dz
t
T
a
T
a
KA
dierentiating the Wilson loop.
This completes the formalism of the proof. In the case of part , the change
of interpretation occurs at the point in the argument when the Wilson
loop is dierentiated. Dierentiating a selfintersecting Wilson loop at a
point of selfintersection is equivalent to dierentiating the corresponding
product of matrices at a variable that occurs at two points in the product
corresponding to the two places where the loop passes through the point.
One of these derivatives gives rise to a term with volume form equal to
zero, the other term is the one that is described in part . This completes
the proof of the proposition. .
Applying Proposition As the formula of Proposition shows, the in
Vassiliev Invariants and Quantum Gravity
tegral ZK is unchanged if the movement of the loop does not involve
three independent space directions since
rst
dx
r
dy
s
dz
t
computes a vol
ume. This means that ZK ZS
, K is invariant under moves that
slide the knot along a plane. In particular, this means that if the knot K
is given in the nearly planar representation of a knot diagram, then ZK
is invariant under regular isotopy of this diagram. That is, it is invari
ant under the Reidemeister moves II and III. We expect more complicated
behavior under move I since this deformation does involve three spatial
directions. This will be discussed momentarily.
We rst determine the dierence between ZK
and ZK
where K
and K
denote the knots that dier only by switching a single crossing.
We take the given crossing in K
to be the positive type, and the crossing
in K
to be of negative type.
The strategy for computing this dierence is to use K
as an intermediate,
where K
is the link with a transversal selfcrossing replacing the given
crossing in K
or K
. Thus we must consider
ZK
ZK
and
ZK
ZK
. The second part of Proposition applies to each
of these dierences and gives
i/k
dAexp
ik/S
rst
dx
r
dy
s
dz
t
T
a
T
a
K
A
where, by the description in Proposition , this evaluation is taken along
the loop K
with the singularity and the T
a
T
a
insertion occurs along the
two transversal arcs at the singular point. The sign of the volume element
will be opposite for
and consequently we have that
.
The volume element
rst
dx
r
dy
s
dz
t
must be given a conventional value in
our calculations. There is no reason to assign it dierent absolute values
for the cases of
and
since they are symmetric except for the sign.
Therefore ZK
ZK
ZK
ZK
. Hence
ZK
/
ZK
ZK
.
Louis H. Kauman
This result is central to our further calculations. It tells us that the eval
uation of a singular Wilson loop can be replaced with the average of the
results of resolving the singularity in the two possible ways.
Now we are interested in the dierence ZK
ZK
ZK
ZK
i/k
dAexp
ik/S
rst
dx
r
dy
s
dz
t
T
a
T
a
K
A
.
Volume convention It is useful to make a specic convention about the
volume form. We take
rst
dx
r
dy
s
dz
t
for
and
for
.
Thus
ZK
ZK
i/k
dAexp
ik/S
T
a
T
a
K
A
.
Integral notation Let ZT
a
T
a
K
denote the integral
ZT
a
T
a
K
dAexp
ik/S
T
a
T
a
K
A
.
Dierence formula Write the dierence formula in the abbreviated form
ZK
ZK
i/kZT
a
T
a
K
.
This formula is the key to unwrapping many properties of the knot in
variants. For diagrammatic work it is convenient to rewrite the dierence
equation in the form shown below. The crossings denote small parts of
otherwise identical larger diagrams, and the Casimir insertion T
a
T
a
K
is
indicated with crossed lines entering a disk labelled C.
Vassiliev Invariants and Quantum Gravity
The Casimir The element
a
T
a
T
a
of the universal enveloping algebra is
called the Casimir. Its key property is that it is in the center of the algebra.
Note that by our conventions Tr
a
T
a
T
a
a
aa
/ d/ where d is
the dimension of the Lie algebra. This implies that an unknotted loop with
one singularity and a Casimir insertion will have Zvalue d/.
In fact, for the classical semisimple Lie algebras one can choose a basis
so that the Casimir is a diagonal matrix with identical values d/D on
its diagonal. D is the dimension of the representation. We then have the
general formula ZT
a
T
a
K
loc
d/DZK for any knot K. Here K
loc
denotes the singular knot obtained by placing a local selfcrossing loop in
K as shown below
Note that ZK
loc
ZK. Let the at loop shrink to nothing. The
Wilson loop is still dened on a loop with an isolated cusp and it is equal
to the Wilson loop obtained by smoothing that cusp.
Louis H. Kauman
Let K
loc
denote the result of adding a positive local curl to the knot
K, and K
loc
the result of adding a negative local curl to K.
Then by the above discussion and the dierence formula, we have
ZK
loc
ZK
loc
i/kZT
a
T
a
K
loc
ZK i/kd/DZK.
Thus,
ZK
loc
i/kd/D
ZK.
Similarly,
ZK
loc
i/kd/D
ZK.
These calculations show how the dierence equation, the Casimir, and
properties of Wilson loops determine the framing factors for the knot in
variants. In some cases we can use special properties of the Casimir to
obtain skein relations for the knot invariant.
For example, in the fundamental representation of the Lie algebra for
SUN the Casimir obeys the following equation see ,
a
T
a
ij
T
a
kl
il
jk
N
ij
kl
.
Hence
ZT
a
T
a
K
ZK
N
ZK
where K
denotes the result of smoothing a crossing as shown below
Vassiliev Invariants and Quantum Gravity
Using ZK
ZK
ZK
and the dierence identity, we obtain
ZK
ZK
i/k
ZK
/
N
ZK
ZK
.
Hence
i/NkZK
i/NkZK
i/kZK
or
e
N
ZK
e
N
ZK
ee
ZK
where ex expi/kx taken up to O/k
.
Here d N
and D N, so the framing factor is
i/k
N
N
e
N /N
.
Therefore, if PK
wK
ZK denotes the normalized invariant of
ambient isotopy associated with ZK with wK the sum of the crossing
signs of K, then
e/NPK
e/NPK
ee
PK
.
Hence
eNPK
eNPK
ee
PK
.
This last equation shows that PK is a specialization of the Homy poly
nomial for arbitrary N, and that for N
SU
it is a specialization
of the Jones polynomial.
Graph invariants and Vassiliev invariants
We now apply this integral formalism to the structure of rigid vertex graph
invariants that arise naturally in the context of knot polynomials. If V K
is a Laurentpolynomialvalued, or, more generally, commutativering
valued invariant of knots, then it can be naturally extended to an invariant
of rigid vertex graphs by dening the invariant of graphs in terms of the
Louis H. Kauman
knot invariant via an unfolding of the vertex as indicated below
VK
aV K
bV K
cV K
.
Here K
indicates an embedding with a transversal valent vertex . We
use the symbol to distinguish this choice of vertex designation from the
previous one involving a selfcrossing Wilson loop.
Formally, this means that we dene V G for an embedded valent
graph G by taking the sum over a
iS
b
iS
c
iS
V S for all knots S
obtained from G by replacing a node of G either with a crossing of positive
or negative type, or with a smoothing denoted . It is not hard to see
that if V K is an ambient isotopy invariant of knots, then this extension is
a rigid vertex isotopy invariant of graphs. In rigid vertex isotopy the cyclic
order at the vertex is preserved, so that the vertex behaves like a rigid disk
with exible strings attached to it at specic points.
There is a rich class of graph invariants that can be studied in this
manner. The Vassiliev invariants , , constitute the important special
case of these graph invariants where a , b , and c . Thus
V G is a Vassiliev invariant if
VK
VK
VK
.
V G is said to be the nite type k if V G whenever G gt k where
G denotes the number of valent nodes in the graph G. If V is the
nite type k, then V G is independent of the embedding type of the graph
G when G has exactly k nodes. This follows at once from the denition of
nite type. The values of V G on all the graphs of k nodes is called the
top row of the invariant V .
For purposes of enumeration it is convenient to use chord diagrams to
enumerate and indicate the abstract graphs. A chord diagram consists of
an oriented circle with an even number of points marked along it. These
points are paired with the pairing indicated by arcs or chords connecting
the paired points. See the gure below.
Vassiliev Invariants and Quantum Gravity
This gure illustrates the process of associating a chord diagram to a given
embedded valent graph. Each transversal selfintersection in the embed
ding is matched to a pair of points in the chord diagram.
Vassiliev invariants from the functional integral
In order to examine the Vassiliev invariants associated with the functional
integral, we must rst normalize these invariants to invariants of ambient
isotopy, and then consider the structure of the dierence between posi
tive and negative crossings. The framed dierence formula provides the
necessary information for obtaining the top row.
We have shown that ZK
loc
ZK with ed/D. Hence
PK
wK
ZK
is an ambient isotopy invariant. The equation
ZK
ZK
i/kZT
a
T
a
K
implies that if wK
w, then we have the ambient isotopy dierence
formula
PK
PK
w
i/k
ZT
a
T
a
K
d/DZK
.
We leave the proof of this formula as an exercise for the reader.
This formula tells us that for the Vassiliev invariant associated with P
we have
PK
w
i/k
ZT
a
T
a
K
d/DZK
.
Furthermore, if V
j
K denotes the coecient of i/k
j
in the expansion
of PK in powers of /k, then the ambient dierence formula implies
that /k
j
divides PG when G has j or more nodes. Hence V
j
G if
G has more than j nodes. Therefore V
j
K is a Vassiliev invariant of nite
type. This result was proved by Birman and Lin by dierent methods
and by BarNatan by methods equivalent to ours.
The fascinating thing is that the ambient dierence formula, appropri
ately interpreted, actually tells us how to compute V
k
G when G has k
Louis H. Kauman
nodes. Under these circumstances each node undergoes a Casimir insertion,
and because the Wilson loop is being evaluated abstractly, independent of
the embedding, we insert nothing else into the loop. Thus we take the pair
ing structure associated with the graph the socalled chord diagram and
use it as a prescription for obtaining a trace of a sum of products of Lie al
gebra elements with T
a
and T
a
inserted for each pair or a simple crossover
for the pair multiplied by d/D. This yields the graphical evaluation
implied by the recursion
VG
VT
a
T
a
G
d/DV G
.
At each stage in the process one node of G disappears or it is replaced
by these insertions. After k steps we have a fully inserted sum of abstract
Wilson loops, each of which can be evaluated by taking the indicated trace.
This result is equivalent to BarNatans result, but it is very interesting to
see how it follows from a minimal approach to the Witten integral.
In particular, it follows from BarNatan and Kontsevich that the
condition of topological invariance is translated into the fact that the Lie
bracket is represented as a commutator and that it is closed with respect
to the Lie algebra. Diagrammatically we have
Since
T
a
T
b
T
b
T
a
c
if
abc
T
c
we obtain
Vassiliev Invariants and Quantum Gravity
This relationship on chord diagrams is the seed of all the topology. In
particular, it implies the basic term relation
Proof
.
The presence of this relation on chord diagrams for V
i
G with G i
is the basis for the existence of a corresponding Vassiliev invariant. There
is not room here to go into more detail about this matter, and so we
bring this discussion to a close. Nevertheless, it must be mentioned that
this brings us to the core of the main question about Vassiliev invariants
are there nontrivial Vassiliev invariants of knots and links that cannot be
constructed through combinations of Lie algebraic invariants There are
many other open questions in this arena, all circling this basic problem.
Louis H. Kauman
Quantum gravityloop states
We now discuss the relationship of Wilson loops and quantum gravity that
is forged in the theory of Ashtekar, Rovelli, and Smolin . In this theory
the metric is expressed in terms of a spin connection A, and quantization
involves considering wave functions A. Smolin and Rovelli analyze the
loop transform
K
dAA
KA
where
KA
denotes the Wilson
loop for the knot or singular embedding K. Dierential operators on the
wave function can be referred, via integration by parts, to corresponding
statements about the Wilson loop. It turns out that the condition that
K be a knot invariant without framing dependence is equivalent to the
socalled dieomorphism constraint for these wave functions. In this
way, knots and weaves and their topological invariants become a language
for representing a state of quantum gravity.
The main point that we wish to make in the relationship of the Vassiliev
invariants to these loop states is that the Vassiliev vertex
satisfying
is not simply a transverse intersection of Wilson loops. We have seen that
it follows from the dierence formula that the Vassiliev vertex is much more
complex than thisthat it involves the Casimir insertion at the transverse
intersection up to the rst order of approximation, and that this structure
can be used to compute the top row of the corresponding Vassiliev invariant.
This situation suggests that one should amalgamate the formalism of the
Vassiliev invariants with the structure of the Poisson algebras of loops and
insertions used in the quantum gravity theory , . It also suggests
Vassiliev Invariants and Quantum Gravity
taking the formalism of these invariants in the functional integral form
quite seriously even in the absence of an appropriate measure theory.
One can begin to work backwards, taking the position that invari
ants that do not ostensibly satisfy the dieomorphism constraint owing
to change of value under framing change nevertheless still dene states of
a quantum gravity theory that is a modication of the Ashtekar formula
tion. This theory can be investigated by working the transform methods
backwardsfrom knots and links to dierential operators and dierential
geometry.
All these remarks are the seeds for another paper. We close here, and
ask the reader to stay tuned for further developments.
Acknowledgements
It gives the author pleasure to acknowledge the support of NSF Grant
Number DMS and the Program for Mathematics and Molecular
Biology of the University of California at Berkeley, Berkeley, CA.
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Louis H. Kauman
Geometric Structures and Loop Variables in
dimensional Gravity
Steven Carlip
Department of Physics, University of California,
Davis, California , USA
email carlipdirac.ucdavis.edu
Abstract
In this chapter, I review the relationship between the metric formula
tion of dimensional gravity and the loop observables of Rovelli
and Smolin. I emphasize the possibility of reconstructing the classi
cal geometry, via the theory of geometric structures, from the values
of the loop variables. I close with a brief discussion of implications
for quantization, particularly for covariant canonical approaches to
quantum gravity.
Introduction
Ashtekars connection representation for general relativity , and the
closely related loop variable approach have generated a good deal of
excitement over the past few years. While it is too early to make rm
predictions, there seems to be some real hope that these new variables will
allow the construction of a consistent nonperturbative quantum theory of
gravity. Some important progress has been made a number of observables
have been found, large classes of quantum states have been identied, and
the rst steps have been taken towards establishing a reasonable weakeld
perturbation theory .
Progress has been hampered, however, by the absence of a clear physi
cal interpretation for the observables built out of Ashtekars new variables.
In part, the problem is simply one of unfamiliarityphysicists accustomed
to metrics and their associated connections can nd it dicult to make
the transition to densitized triads and selfdual connections. But there is
a deeper problem as well, inherent in almost any canonical formulation of
general relativity. To dene Ashtekars variables, one must choose a time
slicing, an arbitrary splitting of spacetime into spacelike hypersurfaces.
But real geometry and physics cannot depend on such a choice the true
physical observables must somehow forget any details of the time slicing,
and refer only to the invariant underlying geometry. To a certain extent,
this is already a source of trouble in classical general relativity, where one
must take care to separate physical phenomena from artifacts of coordinate
Steven Carlip
choices. In canonical quantization, however, the problem becomes much
sharperall observables must be dieomorphism invariants, and the need
to reconstruct geometry and physics from such quantities becomes unavoid
able.
In a sense, the Ashtekar program is a victim of its own success. For
the rst time, we can actually write down a large set of dieomorphism
invariant operators, built out of the loop variables of Rovelli and Smolin
in the loop representation of quantum gravity , . But although some
progress has been made in dening area and volume operators in terms
of these variables , the goal of reconstructing spacetime geometry from
such invariant quantities remains out of reach.
The purpose of this chapter is to demonstrate that such a reconstruc
tion is possible in the simple model of dimensional gravity, general
relativity in two spatial dimensions plus time. A reduction in the number of
dimensions greatly simplies general relativity, allowing the use of powerful
techniques not readily available in the realistic dimensional theory.
As a consequence, many of the specic results presented here will not read
ily generalize to higher dimensions. But the success of dimensional
gravity can be viewed as an existence proof for canonical quantum grav
ity, and one may hope that at least some of the technical results have
extensions to our physical spacetime.
From geometry to holonomies
Let us begin with a brief review of dimensional general relativity in
rstorder formalism. As our spacetime we take a manifold M, which we
shall often assume to have a topology R, where is a closed orientable
surface. The fundamental variables are a triad e
a
a section of the bundle
of orthonormal framesand a connection on the same bundle, which can
be specied by a connection form
a
b
.
The EinsteinHilbert action
can be written as
I
grav
M
e
a
d
a
abc
b
c
,.
where e
a
e
a
dx
and
a
abc
bc
dx
. The action is invariant under
local SO, transformations,
e
a
abc
e
b
c
.
a
d
a
abc
b
c
,
Indices , , , . . . are spacetime coordinate indices i, j, k, . . . are spatial coordinate
indices and a, b, c, . . . are Lorentz indices, labelling vectors in an orthonormal basis.
Lorentz indices are raised and lowered with the Minkowski metric
ab
. This notation is
standard in papers in dimensional gravity, but diers from the usual conventions
for Ashtekar variables, so readers should be careful in translation.
Geometric Structures and Loop Variables
as well as local translations,
e
a
d
a
abc
b
c
.
a
.
I
grav
is also invariant under dieomorphisms of M, of course, but this is not
an independent symmetry Witten has shown that when the triad e
a
is invertible, dieomorphisms in the connected component of the identity
are equivalent to transformations of the form ... We therefore
need only worry about equivalence classes of dieomorphisms that are not
isotopic to the identity, that is elements of the mapping class group of M.
The equations of motion coming from the action . are easily derived
de
a
abc
b
e
c
.
and
d
a
abc
b
c
..
These equations have four useful interpretations, which will form the basis
for our analysis
. We can solve . for as a function of e, and rewrite . as an
equation for e. The result is equivalent to the ordinary vacuum
Einstein eld equations,
R
g,.
for the Lorentzian i.e. pseudoRiemannian metric g
e
a
e
b
ab
.
In dimensions, these eld equations are much more powerful
than they are in dimensions the full Riemann curvature tensor
is linearly dependent on the Ricci tensor,
R
g
R
g
R
g
R
g
R
g
g
g
g
R,
.
so . actually implies that the metric g
is at. The space of
solutions of the eld equations can thus be identied with the set of
at Lorentzian metrics on M.
. As a second alternative, note that eqn . depends only on and
not on e. In fact, . is simply the requirement that the curvature
of vanish that is, that be a at SO, connection. Moreover,
eqn . can be interpreted as the statement that e is a cotangent
vector to the space of at SO, connections indeed, if s is a
curve in the space of at connections, the derivative of . gives
d
d
a
ds
abc
b
d
c
ds
,.
Steven Carlip
which can be identied with . with
e
a
d
a
ds
..
To determine the physically inequivalent solutions of the eld equa
tions, we must still factor out the gauge transformations ...
The local Lorentz transformations . act on as ordinary SO,
gauge transformations, and tell us that only gauge equivalence classes
of at SO, connections are relevant. Let us denote the space of
such equivalence classes as
. The transformations of e can once
again be interpreted as statements about the cotangent space if we
consider a curve s of SO, transformations, it is easy to check
that the rst equations of . and . follow from dierentiating
the second equation of ., with
a
d
a
ds
.
and e
a
as in .. A solution of the eld equations is thus determined
by a point in the cotangent bundle T
.
Now, a at connection on the frame bundle of M is determined by
its holonomies, that is by a homomorphism
M SO, , .
and gauge transformations act on by conjugation. We can therefore
write
Hom
M, SO, / ,
if
h
h
, h SO, . .
It remains for us to factor out the dieomorphisms that are not in the
component of the identity, the mapping class group. These transfor
mations act on
through their action as a group of automorphisms
of
M, and in many interesting casesfor example, when M has
the topology Rthis action comprises the entire set of outer au
tomorphisms of
M . If we denote equivalence under this action
by
t
, and let
/
t
, we can express the space of solutions of
the eld equations .. as
T
.
When M has the topology R, this description can be further
rened. In that case, the space
or at least the physically relevant
connected component of
is homeomorphic to the Teichm uller
Strictly speaking, one more subtlety remains. The space of homomorphisms .
is not always connected, and it is often the case that only one connected component cor
responds to physically admissible spacetimes. See , for the mathematical structure
and , , , for physical implications.
Geometric Structures and Loop Variables
space of , and is the corresponding moduli space , . The
set of vacuum spacetimes can thus be identied with the cotangent
bundle of the moduli space of , and many powerful results from
Riemann surface theory become applicable.
. A third approach is available if M has the topology R. For such
a topology, it is useful to split the eld equations into spatial and
temporal components. Let us write
d
d dt
,
e
a
e
a
e
a
dt, .
a
a
a
dt.
This decomposition can be made in a less explicitly coordinate
dependent mannersee, for example, but the nal results are
unchanged. The spatial projections of .. take the same form
as the original equations, with all quantities replaced by their tilded
spatial equivalents. As above, solutions may therefore be labelled by
classes of homomorphisms, now from
to SO, , and the cor
responding cotangent vectors. The temporal components of the eld
equations, on the other hand, now become
e
a
de
a
abc
b
e
c
abc
e
b
c
.
a
d
a
abc
b
c
.
Comparing with .., we see that the time development of
e, is entirely described by a gauge transformation, with
a
a
and
a
e
a
.
This is consistent with our previous results, of course. For a topol
ogy of the form R, the fundamental group is simply that of ,
and an invariant description in terms of holonomies should not be
able to detect a particular choice of spacelike slice. Equation .
shows in detail how this occurs motion in coordinate time is merely
a gauge transformation, and is therefore invisible to the holonomies.
But the central dilemma described in the introduction now stands out
sharply. For despite eqn ., solutions of the dimensional
eld equations are certainly not static as geometriesthey do not, in
general, admit timelike Killing vectors. The real physical dynamics
has somehow been hidden by this analysis, and must be uncovered if
we are to nd a sensible physical interpretation of our solutions. This
puzzle is an example of the notorious problem of time in gravity ,
and exemplies one of the basic issues that must be resolved in order
to construct a sensible quantum theory.
. A nal approach to the eld equations .. was suggested by
Witten , who observed that the triad e and the connection could
Steven Carlip
be combined to form a single connection on an ISO, bundle.
ISO, , the dimensional Poincare group, has a Lie algebra with
generators
a
and T
b
and commutation relations
a
,
b
abc
c
,
a
,T
b
abc
T
c
,
T
a
,T
b
.
.
If we write a single connection form
Ae
a
T
a
a
a
.
and dene a trace, an invariant inner product on the Lie algebra,
by
Tr
a
T
b
ab
, Tr
a
b
Tr
T
a
T
b
,.
then it is easy to verify that the action . is simply the Chern
Simons action for A,
I
CS
M
Tr
A dA
AAA
..
The eld equations now reduce to the requirement that A be a at
ISO, connection, and the gauge transformations .. can
be identied with standard ISO, gauge transformations. Imitat
ing the arguments of our second interpretation, we should therefore
expect solutions of the eld equations to be characterized by gauge
equivalence classes of at ISO, connections, that is by homo
morphisms in the space
/ Hom
M, ISO, / ,
if
h
h
, h ISO, . .
To relate this description to our previous results, observe that
ISO, is itself a cotangent bundle with base space SO, . In
deed, a cotangent vector at the point
SO, can be written in
the form d
, and the multiplication law
,d
,d
,d
,d
d
.
may be recognized as the standard semidirect product composition
law for Poincare transformations. The space of homomorphisms from
M to ISO, inherits this cotangent bundle structure in an
obvious way, leading to the identication
/T
, where
is
the space of homomorphisms .. It remains for us to factor out
the mapping class group. But this group acts in . and . as
the same group of automorphisms of
M writing the quotient as
//
t
/, we thus see that / T
.
Geometric Structures and Loop Variables
Of these four approaches to the dimensional eld equations, only
the rst corresponds directly to our usual picture of spacetime physics.
Trajectories of physical particles, for instance, are geodesics in the at
manifolds of this description. The second approach, on the other hand, is
the one that is closest to the loop variable picture in dimensional
gravity. The loop variables of Rovelli and Smolin , , , may be
expressed as follows. Let
U, x P exp
a
a
.
be the holonomy of the connection form
a
around a closed path t
based at x. Here, P denotes path ordering, and the basepoint x
species the point at which the path ordering begins. We then dene
T
Tr U, x .
and
T
dt Tr
U, xt e
a
t
dx
dt
t
a
..
T
is thus the trace of the SO, holonomy around , while T
is
essentially a cotangent vector to T
indeed, given a curve s in the
space of at SO, connections, we can dierentiate . to obtain
d
ds
T
Tr
U, xt
d
a
ds
t
a
,.
and we have already seen that the derivative d
a
/ds can be identied with
the triad e
a
.
In dimensions, the variables T
and T
depend on particular
loops , and considerable work is still needed to construct dieomorphism
invariant observables that depend only on knot classes. In dimensions,
on the other hand, the loop variables are already invariant, at least under
dieomorphisms isotopic to the identity. The key dierence is that in
dimensions the connection is at, so the holonomy U, x depends only
on the homotopy class of . Some care must be taken in handling the
mapping class group, which acts nontrivially on T
and T
this issue has
been investigated in a slightly dierent context by Nelson and Regge .
Of course, T
and T
are not quite the equivalence classes of holonomies
of our interpretation number above T
is not a holonomy, but only
the trace of a holonomy. But knowledge of T
for a large enough set of
homotopically inequivalent curves may be used to reconstruct a point in
the space
of eqn ., and indeed, the loop variables can serve as local
coordinates on
,,.
Steven Carlip
Geometric structures from holonomies
to geometry
The central problem described in the introduction can now be made ex
plicit. Spacetimes in dimensions can be characterized a la Ashtekar et
al. as points in the cotangent bundle T
, our description number
of the last section. Such a description is fully dieomorphism invariant,
and provides a natural starting point for quantization. But our intuitive
geometric picture of a dimensional spacetime is that of a manifold M
with a at metricdescription number and only in this representation
do we know how to connect the mathematics with ordinary physics. Our
goal is therefore to provide a translation between these two descriptions.
To proceed, let us investigate the space of at spacetimes in a bit more
detail. If M is topologically trivial, the vanishing of the curvature tensor
implies that M, g is simply ordinary Minkowski space V
,
, , or at least
some subset of V
,
, that can be extended to the whole of Minkowski
space. If the spacetime topology is nontrivial, M can still be covered by
contractible coordinate patches U
i
that are each isometric to V
,
, with
the standard Minkowski metric
on each patch. The geometry is then
encoded in the transition functions
ij
on the intersections U
i
U
j
, which
determine how these patches are glued together. Moreover, since the met
rics in U
i
and U
j
are identical, these transition functions must be isometries
of
, that is elements of the Poincare group ISO, .
Such a construction is an example of what Thurston calls a geomet
ric structure , , , , in this case a Lorentzian or ISO, ,V
,
structure. In general, a G, X manifold is one that is locally modelled on
X, just as an ordinary ndimensional manifold is modelled on R
n
. More
precisely, let G be a Lie group that acts analytically on some nmanifold
X, the model space, and let M be another nmanifold. A G, X structure
on M is then a set of coordinate patches U
i
covering M with coordinates
i
U
i
X taking their values in the model space and with transition
functions
ij
i
j
U
i
U
j
in G. While this general formulation may
not be widely known, specic examples are familiar for example, the uni
formization theorem for Riemann surfaces implies that any surface of genus
g gt admits an PSL,R, H
structure.
A fundamental ingredient in the description of a G, X structure is
its holonomy group, which can be viewed as a measure of the failure of a
single coordinate patch to extend around a closed curve. Let M be a G, X
manifold containing a closed path . We can cover with coordinate charts
i
U
i
X, i , . . . , n .
with constant transition functions g
i
G between U
i
and U
i
, i.e.
i
U
i
U
i
g
i
i
U
i
U
i
.
Geometric Structures and Loop Variables
n
U
n
U
g
n
U
n
U
.
Let us now try analytic continuation of the coordinate
from the patch
U
to the whole of . We can begin with a coordinate transformation in U
that replaces
by
t
g
, thus extending
to U
U
. Continuing
this process along the curve, with
t
j
g
...g
j
j
, we will eventually
reach the nal patch U
n
, which again overlaps U
. If the new coordinate
function
t
n
g
...g
n
n
happens to agree with
on U
n
U
, we
will have succeeded in covering with a single patch. Otherwise, the
holonomy H, dened as H g
...g
n
, measures the obstruction to
such a covering.
It may be shown that the holonomy of a curve depends only on its
homotopy class . In fact, the holonomy denes a homomorphism
H
M G. .
Note that if we pass from M to its universal covering space
M, we will
no longer have noncontractible closed paths, and
will be extendable
to all of
M. The resulting map D
M X is called the developing map
of the G, X structure. At least in simple examples, D embodies the
classical geometric picture of development as unrollingfor instance, the
unwrapping of a cylinder into an innite strip.
The homomorphism H is not quite uniquely determined by the geo
metric structure, since we are free to act on the model space X by a xed
element h G, thus changing the transition functions g
i
without altering
the G, X structure of M. It is easy to see that such a transformation
has the eect of conjugating H by h, and it is not hard to prove that H
is in fact unique up to such conjugation . For the case of a Lorentzian
structure, where G ISO, , we are thus led to a space of holonomies
of precisely the form ..
This identication is not a coincidence. Given a G, X structure on a
manifold M, it is straightforward to dene a corresponding at G bundle
. To do so, we simply form the product G U
i
in each patchgiving
the local structure of a G bundleand use the transition functions
ij
of
the geometric structure to glue together the bers on the overlaps. It is
then easy to verify that the at connection on the resulting bundle has a
holonomy group isomorphic to the holonomy group of the geometric struc
ture.
We can now try to reverse this process, and use one of the holonomy
groups of eqn .approach number to the eld equationsto dene
a Lorentzian structure on M, reproducing approach number . In gen
eral, this step may fail the holonomy group of a G, X structure is not
necessarily sucient to determine the full geometry. For spacetimes, it is
easy to see what can go wrong. If we start with a at manifold M and
simply cut out a ball, we can obtain a new at manifold without aecting
Steven Carlip
the holonomy of the geometric structure. This is a rather trivial change,
however, and we would like to show that nothing worse can go wrong.
Mess has investigated this question for the case of spacetimes with
topologies of the form R. He shows that the holonomy group determines
a unique maximal spacetime Mspecically, a spacetime constructed as a
domain of dependence of a spacelike surface . Mess also demonstrates that
the holonomy group H acts properly discontinuously on a region WV
,
of Minkowski space, and that M can be obtained as the quotient space
W/H. This quotient construction can be a powerful tool for obtaining a
description of M in reasonably standard coordinates, for instance in a time
slicing by surfaces of constant mean curvature.
For topologies more complicated than R, I know of very few gen
eral results. But again, a theorem of Mess is relevant if M is a compact
manifold with a at, nondegenerate, timeorientable Lorentzian metric
and a strictly spacelike boundary, then M necessarily has the topology
R, where is a closed surface homeomorphic to one of the bound
ary components of M. This means that for spatially closed dimensional
universes, topology change is classically forbidden, and the full topology is
uniquely xed by that of an initial spacelike slice. Hence, although more
exotic topologies may occur in some approaches to quantum gravity, it is
not physically unreasonable to restrict our attention to spacetimes R.
To summarize, we now have a procedurevalid at least for spacetimes
of the form Rfor obtaining a at geometry from the invariant data
given by AshtekarRovelliSmolin loop variables. First, we use the loop
variables to determine a point in the cotangent bundle T
, establishing
a connection to our second approach to the eld equations. Next, we
associate that point with an ISO, holonomy group H/, as in our
approach number . Finally, we identify the group H with the holonomy
group of a Lorentzian structure on M, thus determining a at spacetime of
approach number . In particular, if we can solve the dicult technical
problem of nding an appropriate fundamental region W V
,
for the
action of H, we can write M as a quotient space W/H.
This procedure has been investigated in detail for the case of a torus
universe, RT
, in and . For a universe containing point particles,
it is implicit in the early descriptions of Deser et al. , and is explored
in some detail in . For the dimensional black hole, the geometric
structure can be read o from and .
And although it is never
stated explicitly, the recent work of t Hooft and Waelbroeck is
really a description of at spacetimes in terms of Lorentzian structures.
For the black hole, a cosmological constant must be added to the eld equations.
Instead of being at, the resulting spacetime has constant negative curvature, and the
geometric structure becomes an SO, , H
,
structure. A related result for the torus
will appear in .
Geometric Structures and Loop Variables
Quantization and geometrical observables
Our discussion so far has been strictly classical. I would like to conclude
by briey describing some of the issues that arise if we attempt to quantize
dimensional gravity.
The canonical quantization of a classical system is by no means uniquely
dened, but most approaches have some basic features in common. A classi
cal system is characterized by its phase space, a Ndimensional symplectic
manifold , with local coordinates consisting of N position variables and
N conjugate momenta. Classical observables are functions of the positions
and momenta, that is maps f, g, . . . from to R. The symplectic form on
determines a set of Poisson brackets f, g among observables, and hence
induces a Lie algebra structure on the space of observables. To quantize
such a system, we are instructed to replace the classical observables with
operators and the Poisson brackets with commutators that is, we are to
look for an irreducible representation of this Lie algebra as an algebra of
operators acting on some Hilbert space.
As stated, this program cannot be carried out Van Hove showed in
that in general, no such irreducible representation of the full Poisson
algebra of classical observables exists . In practice, we must therefore
choose a subalgebra of preferred observables to quantize, one that must be
small enough to permit a consistent representation and yet big enough to
generate a large class of classical observables . Ordinarily, the resulting
quantum theory will depend on this choice of preferred observables, and
we will have to look hard for physical and mathematical justications for
our selection.
In simple classical systems, there is often an obvious set of preferred
observablesthe positions and momenta of point particles, for instance, or
the elds and their canonical momenta in a free eld theory. For gravity,
on the other hand, such a natural choice seems dicult to nd. In
dimensions, where a number of approaches to quantization can be carried
out explicitly, it is known that dierent choices of variables lead to genuinely
dierent quantum theories , , .
In particular, each of the four interpretations of the eld equations
discussed above suggests its own set of fundamental observables. In the
rst interpretationsolutions as at spacetimesthe natural candidates
are the metric and its canonical momentum on some spacelike surface. But
these quantities are not dieomorphism invariant, and it seems that the
best we can do is to dene a quantum theory in some particular, xed time
slicing , , . This is a rather undesirable situation, however, since
the choice of such a slicing is arbitrary, and there is no reason to expect
the quantum theories coming from dierent slicings to be equivalent.
In our second interpretationsolutions as classes of at connections and
their cotangentsthe natural observables are points in the bundle T
.
Steven Carlip
These are dieomorphism invariant, and quantization is relatively straight
forward in particular, the appropriate symplectic structure for quantiza
tion is just the natural symplectic structure of T
as a cotangent bundle.
The procedure for quantizing such a cotangent bundle is well established
, and there seem to be no fundamental diculties in constructing the
quantum theory. But now, just as in the classical theory, the physical
interpretation of the quantum observables is obscure.
It is therefore natural to ask whether we can extend the classical rela
tionships between these approaches to the quantum theories. At least for
simple topologies, the answer is positive. The basic strategy is as follows.
We begin by choosing a set of physically interesting classical observables
of at spacetimes. For example, it is often possible to foliate uniquely a
spacetime by spacelike hypersurfaces of constant mean extrinsic curvature
TrK the intrinsic and extrinsic geometries of such slices are useful ob
servables with clear physical interpretations. Let us denote these variables
generically as QT, where the parameter T labels the time slice on which
the Q are dened for instance, T TrK.
Classically, such observables can be determinedat least in principle
as functions of the geometric structure, and thus of the ISO, holonomies
,
Q Q, T. .
We now adopt these holonomies as our preferred observables for quantiza
tion, obtaining a Hilbert space L
and a set of operators . Finally, we
translate . into an operator equation,
Q
Q , T, .
thus obtaining a set of dieomorphisminvariant but timedependent quan
tum observables to represent the variables Q. Some ambiguity will remain,
since the operator ordering in . is rarely unique, but in the examples
studied so far, the requirement of mapping class group invariance seems to
place major restrictions on the possible orderings .
This program has been investigated in some detail for the simplest non
trivial topology, M RT
, . There, a natural set of geometric
variables are the modulus of a toroidal slice of constant mean curvature
TrK T and its conjugate momentum p
. These can be expressed ex
plicitly in terms of a set of loop variables that characterize the ISO,
holonomies of the spacetime. Following the program outlined above, one
obtains a parameter family of dieomorphisminvariant operators T
and p
T that describe the physical evolution of a spacelike slice. The
Tdependence of these operators can be described by a set of Heisenberg
Geometric Structures and Loop Variables
equations of motion,
d
dT
i
H,
,
dp
dT
i
H, p
,.
with a Hamiltonian
H,p
, T that can again be calculated explicitly.
Let me stress that despite the familiar appearance of ., the pa
rameter T is not a time coordinate in the ordinary sense the operators
T and p
T are fully dieomorphism invariant. We thus have a kind
of time dependence without time dependence, an expression of dynamics
in terms of operators that are individually constants of motion. For more
complicated topologies, such an explicit construction seems quite dicult,
although Unruh and Newbury have taken some interesting steps in that
direction . Ideally, one would like to nd some kind of perturbation
theory for geometrical variables like
Q, but little progress has yet been
made in this direction.
The specic constructions I have described here are unique to di
mensions, of course. But I believe that some of the basic features are
likely to extend to realistic dimensional gravity. The quantization
of holonomies is a form of covariant canonical quantization , , or
quantization of the space of classical solutions. We do not yet understand
the classical solutions of the dimensional eld equations well enough
to duplicate such a strategy, but a similar approach may be useful in mini
superspace models. The invariant but Tdependent operators T and
p
T are examples of Rovellis evolving constants of motion , whose
use has also been suggested in dimensional gravity. Finally, our sim
ple model has strikingly conrmed the power of the RovelliSmolin loop
variables. A full extension to dimensions undoubtedly remains a dis
tant goal, but for the rst time in years, there seems to be some real cause
for optimism.
Acknowledgements
This work was supported in part by US Department of Energy grant DE
FGER.
Bibliography
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. A. Ashtekar, Lectures on Nonperturbative Quantum Gravity World
Scientic, Singapore, .
. C. Rovelli and L. Smolin, Nucl. Phys. B .
. For a recent review, see L. Smolin, in General Relativity and Gravi
tation , ed. R. J. Gleiser, C. N. Kozameh, and O. M. Moreschi
IOP Publishing, Bristol, .
. A. Ashtekar, C. Rovelli, and L. Smolin, Phys. Rev. Lett.
Steven Carlip
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. E. Witten, Nucl. Phys. B .
. W. J. Harvey, in Discrete Groups and Automorphic Functions, ed.
W. J. Harvey Academic Press, New York, .
. W. M. Goldman, Invent. Math. .
. G. Mess, Lorentz Spacetimes of Constant Curvature, Institut des
Hautes Etudes Scientiques preprint IHES/M// .
. S. Carlip, Classical amp Quantum Gravity .
. J. Luoko and D. M. Marolf, Solution Space to Gravity on RT
in Wittens Connection Formulation, Syracuse preprint SUGP/
.
. W. M. Goldman, Adv. Math. .
. L. Smolin, in Knots, Topology, and Quantum Field Theory, ed. L. Lu
sanna World Scientic, Singapore, .
. For an extensive review, see K. Kuchar, in Proc. of the th Canadian
Conf. on General Relativity and Relativistic Astrophysics, ed. G. Kun
statter et al. World Scientic, Singapore, .
. E. Witten, Commun. Math. Phys. .
. A. Ashtekar, V. Husein, C. Rovelli, J. Samuel, and L. Smolin, Classical
amp Quantum Gravity L.
. See also T. Jacobson and L. Smolin, Nucl. Phys. B .
. J. E. Nelson and T. Regge, Commun. Math. Phys.
Phys. Lett. B .
. M. Seppala and T. Sorvali, Ann. Acad. Sci. Fenn. Ser. A. I. Math.
.
. T. Okai, Hiroshima Math. J. .
. J. E. Nelson and T. Regge, Commun. Math. Phys. .
. W.P. Thurston, The Geometry and Topology of ThreeManifolds,
Princeton lecture notes .
. R. D. Canary, D. B. A. Epstein, and P. Green, in Analytical and
Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture
Notes Series , ed. D. B. Epstein Cambridge University Press,
Cambridge, .
. W. M. Goldman, in Geometry of Group Representations, ed. W. M.
Goldman and A. R. Magid American Mathematical Society, Provi
dence, .
. For examples of geometric structures, see D. Sullivan and W.
Thurston, Enseign. Math. .
. S. Carlip, Phys. Rev. D .
. S. Deser, R. Jackiw, and G. t Hooft, Ann. Phys. .
Geometric Structures and Loop Variables
. M. Ba nados et al., Phys. Rev. D .
. D. Cangemi, M. Leblanc, and R. B. Mann, Gauge Formulation of the
Spinning Black Hole in Dimensional Antide Sitter Space, MIT
preprint MITCTP .
. S. Carlip and J. E. Nelson, Equivalent Quantisations of
Dimensional Gravity, Turin preprint DFTT / and Davis preprint
UCD.
. G. t Hooft, Classical amp Quantum Gravity Classical amp
Quantum Gravity .
. H. Waelbroeck, Classical amp Quantum Gravity Phys. Rev.
Lett. Nucl. Phys. B .
. L. Van Hove, Mem. de lAcad. Roy. de Belgique Classe de Sci.
.
. For a useful summary, see C. J. Isham, in Relativity, Groups, and
Topology II, ed. B. S. DeWitt and R. Stora NorthHolland, Amster
dam, .
. S. Carlip, Phys. Rev. D Phys. Rev. D .
. S. Carlip, Six Ways to Quantize Dimensional Gravity, Davis
preprint UCD, to appear in Proc. of the Fifth Canadian Con
ference on General Relativity and Relativistic Astrophysics.
. D. M. Marolf, Loop Representations for Gravity on a Torus,
Syracuse preprint SUGP/ .
. V. Moncrief, J. Math. Phys. .
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Dreibein Formalism, University of British Columbia preprint .
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S. W. Hawking and W. Israel Cambridge University Press, Cam
bridge, .
. C. Rovelli, Phys. Rev. D Phys. Rev. D .
Steven Carlip
From ChernSimons to WZW via Path
Integrals
Dana S. Fine
Department of Mathematics, University of Massachusetts,
Dartmouth, Massachusetts , USA
email dnecis.umassd.edu
Abstract
Direct analysis of the ChernSimons path integral reduces quantum
expectations of unknotted Wilson lines in ChernSimons theory on
S
with group G to npoint functions in the WZW model of maps
fromS
to G. The reduction hinges on the characterization of A/G
n
,
the space of connections modulo those gauge transformations which
are the identity at a point n, as itself a principal ber bundle with
anelinear ber.
Introduction
In this chapter, I will describe how to reduce the path integral for Chern
Simons theory on S
to the path integral for the WessZuminoWitten
WZW model for maps from S
to G, where G is the symmetry group of
the ChernSimons theory. The points I would like to emphasize are, rst,
that the ChernSimons path integral on S
is tractable, and, second, that
it provides an unusually direct link between the ChernSimons theory and
the WZW model. The details I omit appear in .
That the two theories are related has been known at least since Witten
rst described the geometric quantization of the ChernSimons theory in
the axial gauge on manifolds of the form R , where is a Riemann
surface . Although this is frequently described as a path integral ap
proach, it does not use a direct evaluation of the path integral rather, the
formal properties of the path integral provide axioms crucial to extending
the theory from R to compact manifolds. In this approach, the WZW
model appears in a highly rened form, as the source of a certain modular
functor. That is, one must already know how to solve the WZW model or
at least know that its solution denes this modular functor to recognize
its appearance in the ChernSimons theory.
Treatments of the path integral for ChernSimons in the axial gauge on
R tend to make contact with the WZW model at earlier stages. Moore
and Seiberg proceed with the evaluation of the path integral far enough
Dana S. Fine
to obtain a path integral over the phase space LG/G. This links the two
theories as theories of the representations of LG/G. Fr ohlich and King
obtain the KnizhnikZamolodchikov equation of the WZW model from the
same gaugexed ChernSimons path integral.
The approach I will describe is novel in that it treats the path integral
on S
directly, it does not require a choice of gauge, and one need only
know the action of the WZW model, not its solution nor even its current
algebra, to recognize its appearance in the ChernSimons theory. The key
ingredient is a technique for integrating directly on the space of connections
modulo gauge transformations which I developed to evaluate path integrals
for YangMills on Riemann surfaces .
The main result
The ChernSimons path integral to evaluate is
f
CS
Z
,/
n
fAe
iSA
TA,
where A is a connection on the principal bundle P S
G, and SA is
the ChernSimons action
k
S
Tr
A dA
AAA
.
Here / is the space of smooth connections on P,
n
is the space of gauge
transformations which are the identity at the north pole n of S
,/
//
n
is the standard map to the quotient, TA means the standard measure
on /, and
TA is the pushforward to //
n
. Note that even the formal
denition of TA requires a metric. Take this to be the standard metric on
S
. The WZW path integral for
G based maps from S
to G is
f
WZW
Z
G
fXe
iSX
TX,
where the WZW action is
SX
k
S
TrX
X
M
Tr
X
d
X
X
d
X
X
d
X
.
Here M is any manifold with M S
, and
X is any extension of X
from S
to M.
The main result is
f
CS
Z
fXe
i
k
S
TrX
Xi
k
M
Tr
X
dX
TX,
where X
G, and f explicitly determines
f. In particular, if f is
a Wilson line, then
f is the trace of a product of evaluationatapoint
From ChernSimons to WZW
Fig. . The path e
maps. That is, the ChernSimons path integral directly reduces to the
path integral for a WZW model of based maps from S
to G.
The technique is as follows
. Characterize //
n
as a bundle over
G with anelinear bers.
. Integrate along the ber over each point X
G.
. Recognize the resulting measure on
G as a WZW path integral.
//
n
as a bundle over
G
For each connection A, and each point e S
, dene
A
e by holonomy
about paths e of the form pictured in Fig. . That is, e follows a
longitude from the north pole n to the south pole s through a xed point e
of the equator and then returns along the longitude through the variable
point e. Relative to a basepoint in the ber over n,
A
e lies in G. As e
varies throughout the equator identied with S
,
A
e denes, for each
A, a map from S
to G. Since
A
e
, it is a based map, so
A
G.
Moreover, since a gauge transformation aects holonomy about e by
conjugation by the value of the gauge transformation at n,
A
A
for
n
. Thus the assignment A
A
denes a mapping //
n
G
which will serve as our projection.
Turn now to the ber of . Clearly,
A
A
if vanishes along the
longitudes of Fig. . If P
S
,g
S
, g denotes projection to
the longitudinal part, the space of such is ker P
. In fact, the ber over
Dana S. Fine
A
is the set of orbits
A ker P
..
Thus the ber is the ane version of the linear space ker P
, and the
connection A in the above presentation represents a choice of origin in the
ber.
The image of is all of
G. To see this, use a homotopy from X
to the identity to determine lifts to P of the family of paths e in S
which dened . Such a homotopy exists, because
G for any
Lie group G. This determines P
A such that A X. This is a slight
renement of and , which describe a homotopy equivalence between
//
n
for bundles over S
and
G. The reason to redo this is to describe
the topologically trivial ber in an explicit manner.
According to eqn ., the integral over the ber looks like
I
ber
X det
/
D
A
P
D
A
fA e
iSA
T.
The determinant factor relates
TA on the ber to the metriccompatible
measure T. The action along the ber is, after an integration by parts,
SA SA
k
M
Tr D
A
F
A
M
TrA
,
where, for reasons which will soon be clear, the boundary contribution is
retained. To integrate over the ber, choose a specic origin
A to remove
the linear term.
One such choice is determined by
A
X
d
X,
where
XS
G is parallel transport by A along the longitudinal paths
e used in dening . Note that
X is not well dened at n. In fact
X
X, where
is the restriction to a small sphere approaching n.
Thus
A is not continuous at n nor is any other such choice, so to keep
A smooth requires
A
X
dX.
With this choice, the boundary term vanishes, and
S
AS
A
k
S
Tr D
A
.
The integral over the ber becomes
I
ber
X det
/
D
A
P
D
A
e
iS
A
ker P
f
Ae
i
k
TrDA
T,
From ChernSimons to WZW
where
ker P
denotes elements of ker P
with the prescribed discontinuity
at n. Now assume f is independent of and consider
e
i
k
TrDA
T.
But for the discontinuity at n, this would be standard. The result would be
a determinant times the exponential of the etainvariant as in and .
Repeating these arguments, which refer to a nitedimensional analog, with
a minor variation to account for the discontinuity, will evaluate I
ber
. The
metric on
S
, g restricts to ker P
to dene a decomposition of its
complexication ker P
C
, in terms of which
Tr D
A
S
Tr
D
A
S
Tr
S
Tr
D
A
S
TrX
X.
Model this by a vector space V V
V
with a linear mapping T V
V
. Add a small quadratic term, and integrate to nd
e
iTv,v
Tv
Tv
det
/
T
T
.
The determinant analogous to the righthand side is that of P
D
A
D
A
P
acting on elements of ker P
which vanish at n. Now, regularize and account
for negative eigenvalues using the etainvariant that is, take
det
P
D
A
D
A
P
e
i
,
where is the etainvariant, as the analogue for det
/
T
T
. Then
I
ber
X
det
D
A
P
D
A
det
P
D
A
D
A
P
e
i
e
iS
A
e
i
k
TrX
X
.
Performing the integration over the bers in the path integral for f,
assuming f constant along the bers, thus gives
f
Z
f
Ae
iS A
e
i
k
TrX
X
det
D
A
P
D
A
det
P
D
A
D
A
P
e
i
base
,
where
base
is the measure on
G induced by the metric on //
n
. To
interpret this in terms of
G requires four observations
.S
A
k
S
Tr
X
d
X
.
.
base
constant TX.
. Each determinant is constant along the base in //
n
.
. f can be expressed as a function
f of X.
Dana S. Fine
Fig. . The path whose holonomy is Xe
Xe
Absorbing everything independent of X into Z
, then,
f
CS
Z
fXe
i
k
S
TrX
Xi
k
S
Tr
X
dX
TX,
as claimed.
An explicit reduction from ChernSimons to WZW
Let R be a representation of G, and let J
denote the Wilson line corre
sponding to the longitudinal path indicated in Fig. . Then
J
R
CS
Z
Tr
R
Xe
Xe
e
i
k
TrX
Xi
k
S
Tr
X
dX
TX
Tr
R
Xe
Xe
WZW
.
As a description of the relation between ChernSimons theory on S
and the WZW for
G, this work is nearly complete. However, as an eval
uation of the ChernSimons path integral it lacks an essential component
namely, an evaluation of the WZW path integral appearing on the right
hand side of the above equation. For expectations of evaluationatapoint
maps, the WZW path integral is tractable, at least formally, but this is
work in progress. The basic idea is to reinterpret X
G as an element
of PathG with some constraints on its endpoint values. Then, the WZW
From ChernSimons to WZW
path integral becomes a path integral for quantum mechanics on G. At
a formal level, expectations of evaluationatapoint maps are given by the
FeynmanKac formula. Formal calculations lead to some promising pre
liminary observations such as complicated unknot unknot. However,
to come up with an explicit evaluation of even unknot requires going be
yond formal properties to a more detailed understanding of the heat kernel
on G.
Acknowledgements
This material is based upon work supported in part by the National Science
Foundation under Grant DMS.
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. I. M. Singer, Families of Dirac operators with applications to physics,
Societe Mathematique de France Asterisque, hor serie,
. E. Witten, Quantum eld theory and the Jones polynomial, Com
mun. Math. Phys.
Dana S. Fine
Topological Field Theory as the Key to
Quantum Gravity
Louis Crane
Mathematics Department, Kansas State University,
Manhattan, Kansas , USA
email cranemath.ksu.edu
Abstract
Motivated by the similarity between CSW theory and the Chern
Simons state for general relativity in the Ashtekar variables, we ex
plore what a universe would look like if it were in a state correspond
ing to a d TQFT. We end up with a construction of propagating
states for parts of the universe and a Hilbert space corresponding to a
certain approximation. The construction avoids path integrals, using
instead recombination diagrams in a very special tensor category.
Introduction
What I wish to propose is that the quantum theory of gravity can be
constructed from a topological eld theory. Notice, I do not say that it is
a topological eld theory. Physicists will be quick to point out that there
are local excitations in gravity, so it cannot be topological.
Nevertheless, the connections between general relativity in the loop
formulation and the CSW TQFT are very suggestive. Formally, at least,
the CSW action term
e
ik
TrAdA
AAA
.
gives a state for the Ashtekar variable version of quantized general relativ
ity, as mentioned previously in this volume. That is, we can think of this
measure on the space of connections on a manifold either as dening a
theory in dimensions, or as a state in a d theory, which is GR with a
cosmological constant.
Furthermore, states for the Ashtekar or loop variable form of GR are
very scarce, unless one is prepared to consider states of zero volume. Fi
nally, one should note that a number of very talented workers in the eld
have searched in vain for a Hilbert space of states in either representation.
If one attempts to use the loop variables to quantize subsystems of
the universe, one ends up looking at functionals on the set of links in a
manifold with boundary with ends on the boundary. It turns out in
regularizing that the links need to be framed, and one can trace them in
Louis Crane
any representation of SU, so they need to be labelled. The coincidence
of all this with the picture of d TQFT was the initial motivation for this
work.
Another point, which seems important, is that although path integrals
in general do not exist, the CSW path integral can be thought of as a
symbolic shorthand for a d TQFT which can be rigorously dened by
combining combinatorial topology with some very new algebraic structures,
called modular tensor categories. Thus, if we want to explore . as a
state for GR, we have the possibility of switching from the language of
path integrals to a nite, discrete picture, where physics is described by
variables on the edges of a triangulation.
Physics on a lattice is nothing new. What is new here is that the choice
of lattice is unimportant, because the coecients attached to simplices are
algebraically special. Thus we do not have to worry about a continuum
limit, since a nite result is exact.
Motivated by these considerations, I propose to consider the hypothesis
Conjecture The universe is in the CSW state.
This idea is similar to the suggestion of Witten that the CSW theory
is the topological phase of quantum gravity . My suggestion diers from
his in assuming that the universe is still in the topological phase, and that
the concrete geometry we see is not the result of a uctuation of the state of
the universe as a whole, but rather due to the collapse of the wave packet in
the presence of classical observers. As I shall explain below, the machinery
of CSW theory provides spaces of states for parts of the universe, which
I interpret as the relative states an observer might see. This suggestion is
closely related to the manyworlds interpretation of quantum mechanics
the states on surfaces are data in one particular world. The state of the
universe is the sum of all possible worlds, which takes the elegant form of
the CSW invariant of links, or more concretely, of the Jones polynomial or
one of its generalizations.
During the talk I gave at the conference one questioner pointed out,
quite correctly, that it is not necessary to make this conjecture, since other
states exist. Let me emphasize that it is only a hypothesis. I think it is fair
to say that it leads to more beautiful mathematics than any other I know
of. Whether that is a good guide for physics, only time will tell.
Topological quantum eld theory in dimensions , , and is leading
into an extraordinarily rich mathematical eld. In the spirit of the drunk
ard who looks for his keys near a street light, I have been thinking for
several years about an attempt to unite the structures of TQFT with a
suitable reinterpretation of quantum mechanics for the entire universe. I
believe that the two subjects resemble one another strongly enough that
the program I am outlining becomes natural.
Topological Field Theory and Quantum Gravity
Quantum mechanics of the universe. Categorical
physics
I want to propose that a quantum eld theory is the wrong structure to
describe the entire universe. Quantum eld theory, like quantum mechanics
generally, presupposes an external classical observer. There can be no
probability interpretation for the whole universe. In fact, the state of the
universe cannot change, because there is no time, except in the presence
of an external clock. Since the universe as a whole is in a xed state, the
quantum theory of the gravitational eld as a whole is not a physical theory.
It describes all possible universes, while the task of a physical theory is to
explain measurements made in this universe in this state. Thus, what is
needed is a theory which describes a universe in a particular state. This is
similar to what some researchers, such as Penrose, have suggested that the
initial conditions of the universe are determined by the laws. What I am
proposing is that the initial conditions become part of the laws. Naturally,
this requires that the universe be in a very special state.
Philosophically, this is similar to the idea of a wave function of the
universe. One can phrase this proposal as the suggestion that the wave
function of the universe is the CSW functional. There is also a good deal
here philosophically in common with the recent paper of Rovelli and Smolin
. Although they do not assume the universe is in the CSW state, they
couple the gravitational eld to a particular matter eld, which acts as a
clock, so that the gravitational eld is treated as only part of a universe.
What replaces a quantum eld theory is a family of quantum theories
corresponding to parts of the universe. When we divide the universe into
two parts, we obtain a Hilbert space which is associated to the boundary
between them. This is another departure from quantum eld theory it
amounts to abandonment of observation at a distance.
The quantum theories of dierent parts of the universe are not, of
course, independent. In a situation where one observer watches another
there must be maps from the space of states which one observer sees to the
other. Furthermore, these maps must be consistent. If A watches B watch
C watch the rest of the universe, A must see B see C see what A sees C
see.
Since, in the Ashtekar/loop variables the states of quantum gravity
are invariants of embedded graphs, we allow labelled punctures on the
boundaries between parts of the universe, and include embedded graphs in
the parts of the universe we study, i.e. in manifolds with boundary.
The structure we arrive at has a natural categorical avor. Objects are
places where observations take place, i.e. boundaries and morphisms are
d cobordisms, which we think of as A observing B.
Let us formalize this as follows
Denition . An observer is an oriented manifold with boundary
Louis Crane
containing an embedded labelled framed graph which intersects the boundary
in isolated labelled points.
The labelling sets for the edges and vertices of the graph are nite,
and need to be chosen for all of what ensues.
Denition . A skin of observation is a closed oriented surface
with labelled punctures.
Denition . If A and B are skins of observation an inspection, ,
of B by A is an observer whose boundary is identied with A B i.e.
reverse the orientation on A such that the labellings of the components of
the graph which reach the boundary of match the labellings in A B.
This denition requires that the set of labellings possess an involution
corresponding to reversal of orientation on a surface.
To every inspection of B by A there corresponds a dual inspection of
A by B, given by reversing orientation and dualizing on .
Denition . The category of observation is the category whose
objects are skins of observation and whose morphisms are inspections.
Denition . If M
is a closed oriented manifold the category of
observation in M
is the relative i.e. embedded version of the above.
Of course, we can speak of observers, etc., in M
.
Denition . A state for quantum gravity in M
is a functor
from the category of observation in M
to the category of vector spaces.
Nowhere in any of this do we assume that these boundaries are con
nected. In fact, the most important situation to study may be one in
which the universe is crammed full of a gas of classical observers. We
shall discuss this situation below as a key to a physical interpretation of
our theory.
If we examine the mathematical structure necessary to produce a state
for the universe in this sense, we nd that it is identical to a d TQFT.
Thus, the CSW state produces a state in this new sense as well.
Another way to look at this proposal is as follows the CSW state for the
Ashtekar variables is a very special state, in that it factorizes when we cut
the manifold along a surface with punctures, so that a nitedimensional
Hilbert space is attached to each such surface, and the invariant of any link
cut by the surface can be expressed as the inner product of two vectors
corresponding to the two half links. John Baez has pointed out that these
nitedimensional spaces really do possess natural inner products. Thus,
it produces a state also in the sense we have dened above.
I interpret this as saying that the state of the universe is unchanging,
but that because the universe is in a very special state, it can contain a
classical world, i.e. a family of classical observers with consistent mutual
Topological Field Theory and Quantum Gravity
observations. The states on pieces of the universe i.e. links with ends in
manifolds with boundary can be interpreted as changing, once we learn to
interpret vectors in the Hilbert spaces as clocks.
So far, we have a net of nitedimensional Hilbert spaces, rather than
one big one, and no idea how to reintroduce time in the presence of ob
servers. There are natural things to try for both of these problems in the
mathematical context of TQFT.
Before going into a program for solving these problems and thereby
opening the possibility of computing the results of real experiments let us
make a survey of the mathematical toolkit we inherit from TQFT.
Ideas from TQFT
As we have indicated, the notion of a state of quantum gravity we dened
above is equivalent to the notion of a d TQFT which is currently prevalent
in the literature.
The simplest denition of a d TQFT is that it is a machine which
assigns a vector space to a surface, and a linear map to a cobordism between
two surfaces. The empty surface receives a dimensional vector space, so
a closed manifold gets a numerical invariant. Composition of cobordisms
corresponds to composition of the linear maps. A more abstract way to
phrase this is that a TQFT is a functor from the cobordism category to
the category of vector spaces.
The TQFTs which have appeared lately are richer than this. They as
sign vector spaces to surfaces with labelled punctures, and linear maps to
cobordisms containing links or knotted graphs with labelled edges. The
labels correspond to representations. Since it is easy to extend the loop
variables to allow traces in arbitrary representations characters, there is
a great deal of coincidence in the pictures of CSW TQFT and the loop
variables. That constituted much of the initial motivation for this pro
gram. We can describe this as a functor from a richer cobordism category
to V ect. The objects in the richer cobordism category are surfaces with
labelled punctures, and the morphisms are cobordisms containing labelled
links with ends on the punctures.
There are several ways to construct TQFTs in various dimensions. Let
us here discuss the construction of a TQFT from a triangulation. We assign
labels to edges in the triangulation, and some sort of factors combining the
edge labels to dierent dimensional simplices in the triangulation, multiply
the factors together, and sum over labellings.
In order to obtain a topologically invariant theory, we need the combi
nation factors to satisfy some equations. The equations they need to satisfy
are very algebraic in nature as we go through dierent classes of theories in
dierent low dimensions we rst rediscover most of the interesting classes
of associative algebras, then of tensor categories .
Louis Crane
d
d
d
d
d
d
d
d
d
d
d
d
E
Fig. . Move for d TQFT
A simple example in two dimensions may explain why the equations for
a topological theory have a fundamental algebraic avor. For a d TQFT
dened from a triangulation, we need invariance under the move in Fig. .
Now, if we think of the coecients which we use to combine the labels
around a triangle as the structure coecients of an algebra, this is exactly
the associative law. Much of the rest of classical abstract algebra makes
an appearance here too.
As has been described elsewhere , a d TQFT can be constructed
from a modular tensor category. The interesting examples of MTCs can
be realized as quotients of the categories of representations of quantum
groups.
If we try to use the modular tensor categories to construct a TQFT on
a triangulation, we obtain, not the CSW theory, but the weaker TQFT of
Turaev and Viro . Here we label edges, not from a basis for an algebra,
but with irreducible objects from a tensor category. The combinations we
attach to tetrahedra simplices are the quantum j symbols, which come
from the associativity isomorphisms of the category.
We refer to this sort of formula as a state summation.
The full CSW theory can also be reproduced from a modular tensor
category by a slightly subtler construction which uses a Heegaard splitting
which can be easily produced from a triangulation .
The quantum j symbols satisfy some identities, which imply that the
summation formula for a manifold is independent under a change of the
triangulation. The most interesting of these is the ElliotBiedenharn iden
tity. This identity is the consistency relation for the associativity isomor
phism of the MTC. This is another example of the marriage between algebra
and topology which underlies TQFT.
The suggestion that this sort of summation could be related to the
quantum theory of gravity is older than one might think. If we use the
representations of an ordinary Lie group instead of a quantum group, then
the TuraevViro formula becomes the PonzanoRegge formula for the eval
uation of a spin network.
Ponzano and Regge , were able to interpret the evaluation as a sort
of discrete path integral for d Euclidean quantum gravity. The formula
which Regge and Ponzano found for the evaluation of a graph is the analog
for a Lie group of the state sum of Turaev and Viro for a quantum group.
Topological Field Theory and Quantum Gravity
Thus, the evaluation of a tetrahedron for a spin network is a j symbol for
the group SU
labellings
internal edges
dim
q
j
tetrahedra
j. .
The topological invariance of this formula follows from some elementary
properties of representation theory. In ., we have placed our trivalent
labelled graph on the boundary of a manifold with boundary, and then
cut the interior up into tetrahedra, labelling the edges of all the internal
lines with arbitrary spins. The ClebschGordan relations imply that only
nitely many terms in . are nonzero.
Ponzano and Regge then proceeded to interpret . as a discretized
path integral for d quantum GR. The geometric interpretation consisted
in thinking of the Casimirs of the representations as lengths of edges. Rep
resentation theory implied that the summation was dominated by at ge
ometries .
Another way to think of the program in this chapter is as an attempt
to nd the appropriate algebraic structure for extending the spin network
approach to quantum gravity from d to d.
Another idea from TQFT, which seems to have relevance for this phys
ical program, is the ladder of dimensions . TQFTs in adjacent dimen
sions seem to be related algebraically. The spirit of the relationship is like
the relationship of a tensor category to an algebra. Tensor categories look
just like algebras with the operation symbols in circles. The identities of
an algebra correspond to isomorphisms in a tensor category. These isomor
phisms then satisfy higherorder consistency or coherence relations, which
relate to topology up dimension.
The program of construction in TQFT, which I hope will yield the
tools for the quantization of general relativity, is not yet completed, but
is progressing rapidly. There are two outstanding problems, which are
mathematically natural, and which I believe are crucial to the physical
problem as well. They are
Topological problem Find a triangulation version of CSW theory not
merely its absolute square as in .
Topological problem Construct a d TQFT related via the dimensional
ladder to CSW theory.
The solution of these two problems will be in reach, if the program in
succeeds. The program of also suggests that the solutions of these
two topological problems are closely related, both being constructed from
state sums involving the same new algebraic tools.
In this context we should also note that a d TQFT has been con
structed in , out of a modular tensor category. This construction begins
with a triangulation of a manifold, and picks a Heegaard splitting for the
Louis Crane
boundary of each simplex of the triangulation. Thus the theory in
can be thought of as producing a d theory by lling the space with a
network of d subspaces containing observers. This connection between
TQFTs in dimensions and dimensions is the sort of thing I believe
we will need to solve the problem of reintroducing time in a universe in a
TQFT state. What I expect is that the program of will provide richer
examples of such a connection, which will prove to be relevant to quantizing
general relativity.
What we conjecture in is that a new algebraic structure will give
us a new state summation, similar to the one we discussed above, which
will give us CSW in dimensions. If we use the new summation in
dimensions, we should obtain a discrete version of F F theory. This
combination seems very suggestive, since we are supposed to be obtaining
our relative states from CSW theory on a boundary, while F F theory
is a Lagrangian for the topological sector for the quantum theory of GR
. Also, the algebraic structure we need to construct seems to be an
expanded quantum version of the Lorentz group. It is this new, not yet
understood summation, which comes from the representation theory of the
new structure which we call an F algebra in , which I believe should
give us a discretized version of a path integral for quantum gravity.
Hilbert space is deadlong live Hilbert space
or the observer gas approximation
Let us assume that we have found a suitable state summation formula
to solve our two topological problems so that what we suggest will be
predicated on the success of the program in , although one could also
try to use the state sum in together with TuraevViro theory. There
is a natural proposal to make a physical interpretation within it in a
dimensional setting. In the scheme I am suggesting, we can recover a
Hilbert space in a certain sort of classical limit, in which the universe is full
of a gas, of classical observers. The idea is that if we choose a triangulation
for the manifold and a Heegaard splitting for the boundary of each
simplex, then we obtain a family of surfaces which can be thought of as
lling up the manifold. These surfaces should be thought of as skins of
observation for a family of observers who are crowding the spacetime. Now
let us think of a manifold with boundary. In a dimensional boundary
component, we can then combine the vector spaces which we assign to
the surfaces which cut the boundaries of those simplices lying on the
boundary into a larger Hilbert space, which is a quotient of their tensor
product. It is a quotient because the surfaces overlap. Now, if we pass
to a ner triangulation, we will obtain a larger space, with a linear map
to the smaller one. The linear map comes from the fact that we are using
a d TQFT, and the existence of d cobordisms connecting pieces of the
Topological Field Theory and Quantum Gravity
surfaces of the two sets of Heegaard splittings.
We end up with a large vector space for each triangulation of the bound
ary manifold, and a linear map when one triangulation is a renement
of another. This produces a directed graph of vector spaces and linear
maps associated to a manifold. The d state sum can be extended to act
on the vectors in these vector spaces by extending a triangulation for the
boundary manifold to one for the manifold.
Whenever we have a directed set of vector spaces and consistent linear
maps, there is a construction called the inverse limit which can combine
them into a single vector space. The vectors are vectors in any of the spaces,
with images under the maps identied. Since the d state sum is assumed
to be topologically invariant, we can extend the linear map it assigns to
a d cobordism to a map on the inverse limit spaces. I propose that it is
these inverse limit spaces which play the role of the physical Hilbert spaces
of the theory.
The thought here is that physical Hilbert space is the union of the
nitedimensional spaces which a gas of observers can see, in the limit of
an innitely dense gas.
The eect of this suggestion is to reverse the relationship between the
nitedimensional spaces of states which each observer can actually see
and the global Hilbert space. We are regarding the states which can be
observed at one skin of observation as primary. This relational approach
bears some resemblance to the physical ideas of Leibniz, although that is
not an argument for it.
The problem of time
The test of this proposal is whether it can reproduce Einsteins equations
in some classical limit. What I am proposing is that the d state sum
which I hope to construct from the representation theory of the F algebra
can be interpreted as a discretized version of the path integral for general
relativity. The key question is whether the state sum is dominated in
the limit of large spins on edges by assignments of spins whose geometric
interpretation would give a discrete approximation to an Einstein manifold.
This would mean that the theory was a quantum theory whose classical
limit was general relativity.
The analogy with the work of Ponzano and Regge is the rst suggestion
that this program might work. In , they interpreted the summation we
wrote down above as a discretized path integral for d quantum gravity.
The geometric interpretation consisted in thinking of the Casimirs of rep
resentations as lengths. It was somewhat easier to get Einsteins equations
in d , since the solutions are just at metrics.
Should this program work One objection, which was raised in the
discussion, is that the F F model may only be a sector of quantum
Louis Crane
gravity in some weak sense. I am restating this objection somewhat, in
the absence of the questioner. It can be replied that since the theory
we are writing has the right symmetries, the action of the renormalization
group should x the Lagrangian. I do not think that either the objection or
the reply is decisive in the abstract. If the state sum suggested in can
be dened, then we can investigate whether we recover Einsteins equation
or not.
If this does work, one could do the state sum on a manifold with
corners, i.e. x states on surfaces in the dimensional boundaries of a d
cobordism. One could interpret these states as initial conditions for an
experiment, and investigate the results by studying the propagation via
the d state sum.
One could conjecture that this would give a spacetime picture for the
evolution of relative states within the framework of the CSW state of the
universe.
Matter and symmetry
It is natural to ask whether this picture could be expanded to include
matter elds. The obvious thing to try is to pass from SU to a larger
group. It may be noted that Peldan has observed that the CSW state
also satises the YangMills equation. If we really get a picture of gravity
from the state sum associated to SU, it would not be a great leap to try
a larger gauge group.
As I was writing this I became aware of , in which Gambini and
Pullin point out that the CSW state is also a state for GR coupled to
electromagnetism. They also say that the sources for such a theory, thought
of as places where links have ends, are necessarily fermionic. These results
seem to strengthen further the program set forth here.
In general, the constructions which lead up to the F algebras in
can be thought of as expressions of quantum symmetry, i.e. as relatives
of Lie groups in a quantum context. For example, the modular tensor
categories can be described as deformed ClebschGordan coecients. The
F algebras arise as a result of pushing this process further, recategori
fying the categories into categories. These constructions can also be
done from extremely simple starting points, looking at representations of
Dynkin diagrams as quivers. The idea that theories in physics should be
reconstructed from symmetries actually originated with Einstein, who was
thinking of general relativity. It would be tting somehow to reconstruct
truly fundamental theories of physics from fundamental mathematical ex
pressions of symmetries.
Topological Field Theory and Quantum Gravity
Acknowledgements
The author wishes to thank Lee Smolin and Carlo Rovelli for many years
of conversation on this extremely elusive topic. The mathematical ideas
here are largely the result of collaborations with David Yetter and Igor
Frenkel. Louis Kauman has provided many crucial insights. The author
is supported by NSF grant DMS.
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Louis Crane
to appear.
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vector spaces and Zamolodchikovs tetrahedra equation, preprint.
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Strings, Loops, Knots, and Gauge Fields
John C. Baez
Department of Mathematics, University of California,
Riverside, California , USA
email jbaezucrmath.ucr.edu
Abstract
The loop representation of quantum gravity has many formal resem
blances to a backgroundfree string theory. In fact, its origins lie
in attempts to treat the string theory of hadrons as an approxima
tion to QCD, in which the strings represent ux tubes of the gauge
eld. A heuristic pathintegral approach indicates a duality between
backgroundfree string theories and generally covariant gauge theo
ries, with the loop transform relating the two. We review progress
towards making this duality rigorous in three examples d Yang
Mills theory which, while not generally covariant, has symmetry
under all areapreserving transformations, d quantum gravity, and
d quantum gravity. SUN YangMills theory in dimensions has
been given a stringtheoretic interpretation in the largeN limit, but
here we provide an exact stringtheoretic interpretation of the theory
on RS
for nite N. The stringtheoretic interpretation of quan
tum gravity in dimensions gives rise to conjectures about integrals
on the moduli space of at connections, while in dimensions there
may be relationships to the theory of tangles.
Introduction
The notion of a deep relationship between string theories and gauge theo
ries is far from new. String theory rst arose as a model of hadron interac
tions. Unfortunately this theory had a number of undesirable features in
particular, it predicted massless spin particles. It was soon supplanted
by quantum chromodynamics QCD, which models the strong force by an
SU YangMills eld. However, string models continued to be popular as
an approximation of the conning phase of QCD. Two quarks in a meson,
for example, can be thought of as connected by a stringlike ux tube in
which the gauge eld is concentrated, while an excitation of the gauge eld
alone can be thought of as a looped ux tube. This is essentially a modern
reincarnation of Faradays notion of eld lines, but it can be formalized
using the notion of Wilson loops. If A denotes the vector potential, or
connection, in a classical gauge theory, a Wilson loop is simply the trace
of the holonomy of A around a loop in space, typically written in terms
John C. Baez
of a pathordered exponential
Tr Pe
A
.
If instead A denotes the vector potential in a quantized gauge theory, the
Wilson loop may be reinterpreted as an operator on the Hilbert space of
states, and, at least heuristically, applying this operator to the vacuum
state one obtains a state in which the YangMills analog of the electric
eld ows around the loop .
In the late s, Makeenko and Migdal, Nambu, Polyakov, and others
, attempted to derive equations of string dynamics as an approxi
mation to the YangMills equation, using Wilson loops. More recently, D.
Gross and others , , , , have been able to reformulate Yang
Mills theory in dimensional spacetime as a string theory by writing an
asymptotic series for the vacuum expectation values of Wilson loops as a
sum over maps from surfaces the string worldsheet to spacetime. This
development raises the hope that other gauge theories might also be iso
morphic to string theories. For example, recent work by Witten and
Periwal suggests that ChernSimons theory in dimensions is also
equivalent to a string theory.
String theory eventually became popular as a theory of everything be
cause the massless spin particles it predicted could be interpreted as the
gravitons one obtains by quantizing the spacetime metric perturbatively
about a xed background metric. Since string theory appears to avoid the
renormalization problems in perturbative quantum gravity, it is a strong
candidate for a theory unifying gravity with the other forces. However,
while classical general relativity is an elegant geometrical theory relying
on no background structure for its formulation, it has proved dicult to
describe string theory along these lines. Typically one begins with a xed
background structure and writes down a string eld theory in terms of this
only afterwards can one investigate its background independence . The
clarity of a manifestly backgroundfree approach to string theory would be
highly desirable.
On the other hand, attempts to formulate YangMills theory in terms of
Wilson loops eventually led to a fulledged loop representation of gauge
theories, thanks to the work of Gambini and Trias , and others. After
Ashtekar formulated quantum gravity as a sort of gauge theory using
the new variables, Rovelli and Smolin were able to use the loop rep
resentation to study quantum gravity nonperturbatively in a manifestly
backgroundfree formalism. While supercially quite dierent from mod
ern string theory, this approach to quantum gravity has many points of
similarity, thanks to its common origin. In particular, it uses the device
of Wilson loops to construct a space of states consisting of multiloop in
variants, which assign an amplitude to any collection of loops in space.
Strings, Loops, Knots, and Gauge Fields
The resemblance of these states to wavefunctions of a string eld theory is
striking. It is natural, therefore, to ask whether the loop representation of
quantum gravity might be a string theory in disguiseor vice versa.
The present paper does not attempt a denitive answer to this question.
Rather, we begin by describing a general framework relating gauge theories
and string theories, and then consider a variety of examples. Our treatment
of examples is also meant to serve as a review of YangMills theory in
dimensions and quantum gravity in and dimensions.
In Section we describe how the loop representation of a generally
covariant gauge theory is related to a backgroundfree closed string eld
theory. We take a very naive approach to strings, thinking of them simply
as maps from a surface into spacetime, and disregarding any conformal
structure or elds propagating on the surface. We base our treatment
on the path integral formalism, and in order to simplify the presentation
we make a number of overoptimistic assumptions concerning measures on
innitedimensional spaces such as the space // of connections modulo
gauge transformations.
In Section we consider YangMills theory in dimensions as an ex
ample. In fact, this theory is not generally covariant, but it has an innite
dimensional subgroup of the dieomorphism group as symmetries, the
group of all areapreserving transformations. Rather than the path in
tegral approach we use canonical quantization, which is easier to make
rigorous. Gross, Taylor, and others , , , , have already given
dimensional SUN YangMills theory a stringtheoretic interpretation
in the largeN limit. Our treatment is mostly a review of their work, but we
nd it to be little extra eort, and rather enlightening, to give the theory
a precise stringtheoretic interpretation for nite N.
In Section we consider quantum gravity in dimensions. We review
the loop representation of this theory and raise some questions about inte
grals over the moduli space of at connections on a Riemann surface whose
resolution would be desirable for developing a stringtheoretic picture of
the theory. We also briey discuss ChernSimons theory in dimensions.
These examples have nitedimensional reduced conguration spaces,
so there are no analytical diculties with measures on innitedimensional
spaces, at least in canonical quantization. In Section , however, we con
sider quantum gravity in dimensions. Here the classical conguration
space is innite dimensional and issues of analysis become more impor
tant. We review recent work by Ashtekar, Isham, Lewandowski, and the
author , , on dieomorphisminvariant generalized measures on //
and their relation to multiloop invariants and knot theory. We also note
how a stringtheoretic interpretation of the theory leads naturally to the
study of tangles.
John C. Baez
String eld/gauge eld duality
In this section we sketch a relationship between string eld theories and
gauge theories. We begin with a nonperturbative Lagrangian description
of backgroundfree closed string eld theories. From this we derive a Hamil
tonian description, which turns out to be mathematically isomorphic to the
loop representation of a generally covariant gauge theory. We emphasize
that while our discussion here is rigorous, it is schematic, in the sense that
some of our assumptions are not likely to hold precisely as stated in the
most interesting examples. In particular, by measure in this section we
will always mean a positive regular Borel measure, but in fact one should
work with a more general version of this concept. We discuss these analyt
ical issues more carefully in Section .
Consider a theory of strings propagating on a spacetime M that is
dieomorphic to R X, with X a manifold we call space. We do not
assume a canonical identication of M with RX, or any other background
structure metric, etc. on spacetime. We take the classical conguration
space of the string theory to be the space / of multiloops in X
/
n
/
n
with
/
n
MapsnS
, X.
Here nS
denotes the disjoint union of n copies of S
, and we write Maps
to denote the set of maps satisfying some regularity conditions continuity,
smoothness, etc. to be specied. Let T denote a measure on / and
let Fun/ denote some space of squareintegrable functions on /. We
assume that Fun/ and the measure T are invariant both under dieo
morphisms of space and reparametrizations of the strings. That is, both the
identity component of the dieomorphism group of X and the orientation
preserving dieomorphisms of nS
act on /, and we wish Fun/ and
T to be preserved by these actions.
Introduce on FunM the kinematical inner product, which is just the
L
inner product
,
kin
,
T.
We assume for convenience that this really is an inner product, i.e. it is
nondegenerate. Dene the kinematical state space H
kin
to be the Hilbert
space completion of Fun/ in the norm associated to this inner product.
Following ideas from canonical quantum gravity, we do not expect H
kin
to be the true space of physical states. In the space of physical states, any
two states diering by a dieomorphism of spacetime are identied. The
physical state space thus depends on the dynamics of the theory. Taking a
Strings, Loops, Knots, and Gauge Fields
Lagrangian approach, dynamics may be described in terms of path integrals
as follows. Fix a time t gt . Let T denote the set of histories, that is
maps f , t X, where is a compact oriented manifold with
boundary, such that
f , t X f.
Given ,
t
/, we say that f T is a history from to
t
if f
, t X and the boundary of is a disjoint union of circles nS
mS
,
with
fnS
, fmS
t
.
We x a measure, or path integral, on T,
t
. Following tradition, we
write this as e
iSf
Tf, with Sf denoting the action of f, but e
iSf
and
Tf only appear in the combination e
iSf
Tf. Since we are interested in
generally covariant theories, this path integral is assumed to have some
invariance properties, which we note below as they are needed.
Using the standard recipe in topological quantum eld theory, we dene
the physical inner product on H
kin
by
,
phys
,
,
,
t
e
iSf
TfTT
t
assuming optimistically that this integral is well dened. We do not ac
tually assume this is an inner product in the standard sense, for while we
assume , for all H
kin
, we do not assume positive deniteness.
The general covariance of the theory should imply that this inner product
is independent of the choice of time t gt , so we assume this as well.
Dene the space of normzero states I H
kin
by
I,
phys
,
phys
for all H
kin
and dene the physical state space H
phys
to be the Hilbert space com
pletion of H
kin
/I in the norm associated to the physical inner product.
In general I is nonempty, because if g Di
X is a dieomorphism in
the connected component of the identity, we can nd a path of dieomor
phisms g
t
Di
M with g
g and g
t
equal to the identity, and dening
g Di, t X by
gt, x t, g
t
x,
we have
,
phys
,
,
,
t
e
iSf
TfTT
t
,
,
,
gg
t
e
iSf
TfTT
t
John C. Baez
,
,
,
gg
t
e
iS gf
T gfTgT
t
,
,
,
g
t
e
iSf
TfTT
t
g,
phys
for any , . Here we are assuming
e
iS gf
T gf e
iSf
Tf,
which is one of the expected invariance properties of the path integral. It
follows that I includes the space J, the closure of the span of all vectors of
the form g. We can therefore dene the spatially dieomorphism
invariant state space H
diff
by H
diff
H
kin
/J and obtain H
phys
as a
Hilbert space completion of H
diff
/K, where K is the image of I in H
diff
.
To summarize, we obtain the physical state space from the kinematical
state space by taking two quotients
H
kin
H
kin
/J H
diff
H
diff
H
diff
/K H
phys
.
As usual in canonical quantum gravity and topological quantum eld the
ory, there is no Hamiltonian instead, all the information about dynamics
is contained in the physical inner product. The reason, of course, is that
the path integral, which in traditional quantum eld theory described time
evolution, now describes the physical inner product. The quotient map
H
diff
H
phys
, or equivalently its kernel K, plays the role of a Hamil
tonian constraint. The quotient map H
kin
H
diff
, or equivalently its
kernel J, plays the role of the dieomorphism constraint, which is indepen
dent of the dynamics. Strictly speaking, we should call K the dynamical
constraint, as we shall see that it expresses constraints on the initial data
other than those usually called the Hamiltonian constraint, such as the
Mandelstam constraints arising in gauge theory.
It is common in canonical quantum gravity to proceed in a slightly
dierent manner than we have done here, using subspaces at certain points
where we use quotient spaces , . For example, H
diff
may be dened
as the subspace of H
kin
consisting of states invariant under the action
of Di
X, and H
phys
then dened as the kernel of certain operators,
the Hamiltonian constraints. The method of working solely with quotient
spaces has, however, been studied by Ashtekar .
The choice between these dierent approaches will in the end be dic
tated by the desire for convenience and/or rigor. As a heuristic guiding
principle, however, it is worth noting that the subspace and quotient space
approaches are essentially equivalent if we assume that the subspace I is
closed in the norm topology on H
kin
. Relative to the kinematical inner
Strings, Loops, Knots, and Gauge Fields
H
d
d
d
a
d
d
d
b
dd
dd
c
E
E
Fig. . Twostring interaction in dimensional space
product, we can identify H
diff
with the orthogonal complement J
, and
similarly identify H
phys
with I
. From this point of view we have
H
phys
H
diff
H
kin
.
Moreover, H
diff
if and only if is invariant under the action of
Di
X on H
kin
. To see this, rst note that if g for all g Di
X,
then for all H
kin
we have
,gg
,
so J
. Conversely, if J
,
,g
so , , g, and since g acts unitarily on H
kin
the CauchySchwarz
inequality implies g .
The approach using subspaces is the one with the clearest connection to
knot theory. An element H
kin
is a function on the space of multiloops.
If is invariant under the action of Di
X, we call a multiloop in
variant. In particular, denes an ambient isotopy invariant of links in X
when we restrict it to links which are nothing but multiloops that happen
to be embeddings. We see therefore that in this situation the physical
states dene link invariants. As a suggestive example, take X S
, and
take as the Hamiltonian constraint an operator H on H
diff
that has the
property described in Fig. . Here a, b, c C are arbitrary. This Hamil
tonian constraint represents the simplest sort of dieomorphisminvariant
twostring interaction in dimensional space. Dening the physical space
H
phys
to be the kernel of H, it follows that any H
phys
gives a link
invariant that is just a multiple of the HOMFLY invariant . For appro
priate values of the parameters a, b, c, we expect this sort of Hamiltonian
constraint to occur in a generally covariant gauge theory on dimensional
spacetime known as BF theory, with gauge group SUN . A simi
lar construction working with unoriented framed multiloops gives rise to
the Kauman polynomial, which is associated with BF theory with gauge
group SON . We see here in its barest form the path from string
theoretic considerations to link invariants and then to gauge theory.
In what follows, we start from the other end, and consider a generally
covariant gauge theory on M. Thus we x a Lie group G and a principal
Gbundle P M. Fixing an identication M
R X, the classical
conguration space is the space /of connections on P
X
. The physical
John C. Baez
Hilbert space of the quantum theory, it should be emphasized, is supposed
to be independent of this identication M
RX. Given a loop S
X and a connection A /, let T, A be the corresponding Wilson loop,
that is the trace of the holonomy of A around in a xed nitedimensional
representation of G
T, A Tr Pe
A
.
The group of gauge transformations acts on /. Fix a invariant
measure TA on / and let Fun// denote a space of gaugeinvariant
functions on / containing the algebra of functions generated by Wilson
loops. We may alternatively think of Fun// as a space of functions on
// and TA as a measure on //. Assume that TA is invariant under the
action of Di
M on //, and dene the kinematical state space H
kin
to be the Hilbert space completion of Fun// in the norm associated to
the kinematical inner product
,
kin
,/
AA TA.
The relation of this kinematical state space and that described above
for a string eld theory is given by the loop transform. Given any multiloop
,...,
n
/
n
, dene the loop transform
of Fun// by
,...,
n
,/
AT
,AT
n
, A TA.
Take Fun/ to be the space of functions in the range of the loop trans
form. Let us assume, purely for simplicity of exposition, that the loop
transform is onetoone. Then we may identify H
kin
with Fun/ just as
in the string eld theory case.
The process of passing from the kinematical state space to the dieo
morphisminvariant state space and then the physical state space has al
ready been treated for a number of generally covariant gauge theories, most
notably quantum gravity , , . In order to emphasize the resemblance
to the string eld case, we will use a path integral approach.
Fix a time t gt . Given A, A
t
/, let TA, A
t
denote the space
of connections on P
,tX
which restrict to A on X and to A
t
on
t X. We assume the existence of a measure on TA, A
t
which we
write as e
iSa
Ta, using a to denote a connection on P
,tX
. Again, this
generalized measure has some invariance properties corresponding to the
general covariance of the gauge theory. Dene the physical inner product
on H
kin
by
,
phys
,
,
A,A
AA
t
e
iSa
TaTATA
t
Strings, Loops, Knots, and Gauge Fields
again assuming that this integral is well dened and that , for all
. This inner product should be independent of the choice of time t gt .
Letting I H
kin
denote the space of normzero states, the physical state
space H
phys
of the gauge theory is H
kin
/I. As before, we can use the
general covariance of the theory to show that I contains the closed span J
of all vectors of the form g. Letting H
diff
H
kin
/J, and letting K
be the image of I in H
diff
, we again see that the physical state space is
obtained by applying rst the dieomorphism constraint
H
kin
H
kin
/J H
diff
and then the Hamiltonian constraint
H
diff
H
diff
/K H
phys
.
In summary, we see that the Hilbert spaces for generally covariant string
theories and generally covariant gauge theories have a similar form, with
the loop transform relating the gauge theory picture to the string the
ory picture. The key point, again, is that a state in H
kin
can be re
garded either as a wave function on the classical conguration space / for
gauge elds, with A being the amplitude of a specied connection A,
or as a wave function on the classical conguration space / for strings,
with
,...,
n
being the amplitude of a specied ntuple of strings
,...,
n
S
X to be present. The loop transform depends on the
nonlinear duality between connections and loops,
// / C
A,
,...,
n
TA, TA,
n
which is why we speak of string eld/gauge eld duality rather than an
isomorphism between string elds and gauge elds.
At this point it is natural to ask what the dierence is, apart from words,
between the loop representation of a generally covariant gauge theory and
the sort of purely topological string eld theory we have been considering.
From the Hamiltonian viewpoint that is, in terms of the spaces H
kin
,
H
diff
, and H
phys
the dierence is not so great. The Lagrangian for a
gauge theory, on the other hand, is quite a dierent object than that of
a string eld theory. Note that nothing we have done allows the direct
construction of a string eld Lagrangian from a gauge eld Lagrangian
or vice versa. In the following sections we will consider some examples
YangMills theory in dimensions, quantum gravity in dimensions, and
quantum gravity in dimensions. In no case is a string eld action Sf
known that corresponds to the gauge theory in question However, in d
YangMills theory a working substitute for the string eld path integral is
known a discrete sum over certain equivalence classes of maps f M.
This is, in fact, a promising alternative to dealing with measures on the
John C. Baez
space T of string histories. In dimensional quantum gravity, such an
approach might involve a sum over tangles, that is ambient isotopy
classes of embeddings f , t X.
YangMills theory in dimensions
We begin with an example in which most of the details have been worked
out. YangMills theory is not a generally covariant theory since it relies
for its formulation on a xed Riemannian or Lorentzian metric on the
spacetime manifold M. We x a connected compact Lie group G and a
principal Gbundle P M. Classically the gauge elds in question are
connections A on P, and the YangMills action is given by
SA
M
TrF F
where F is the curvature of A and Tr is the trace in a xed faithful unitary
representation of G and hence its Lie algebra g. Extremizing this action
we obtain the classical equations of motion, the YangMills equation
d
A
F,
where d
A
is the exterior covariant derivative.
The action SA is gauge invariant, so it can be regarded as a function
on the space of connections on M modulo gauge transformations. The
group DiM acts on this space, but the action is not dieomorphism
invariant. However, if M is dimensional one may write F f where
is the volume form on M and f is a section of P
Ad
g, and then
SA
M
Trf
.
It follows that the action SA is invariant under the subgroup of dieomor
phisms preserving the volume form . So upon quantization one expects
toand doesobtain something analogous to a topological quantum eld
theory, but in which dieomorphism invariance is replaced by invariance
under this subgroup. Strictly speaking, then, many of the results of the
previous section do not apply. In particular, this theory has an honest
Hamiltonian, rather than a Hamiltonian constraint. Still, it illustrates
some interesting aspects of gauge eld/string eld duality.
The Riemannian case of d YangMills theory has been extensively
investigated. An equation for the vacuum expectation values of Wilson
loops for the theory on Euclidean R
was found by Migdal , and these
expectation values were explicitly calculated by Kazakov . These calcu
lations were made rigorous using stochastic dierential equation techniques
by L. Gross, King and Sengupta , as well as Driver . The classical
YangMills equations on Riemann surfaces were extensively investigated
Strings, Loops, Knots, and Gauge Fields
by Atiyah and Bott , and the quantum theory on Riemann surfaces has
been studied by Rusakov , Fine , and Witten . In particular,
Witten has shown that the quantization of d YangMills theory gives a
mathematical structure very close to that of a topological quantum eld
theory, with a Hilbert space ZS
S
associated to each compact
manifold S
S
, and a vector ZM, ZM for each compact
oriented manifold M with boundary having total area
M
. Similar
results were also obtained by Blau and Thompson .
Here, however, it will be a bit simpler to discuss YangMills theory on
RS
with the Lorentzian metric
dt
dx
,
where t R, x S
, as done by Rajeev . This will simultaneously
serve as a brief introduction to the idea of quantizing gauge theories after
symplectic reduction, which will also be important in d quantum gravity.
This approach is an alternative to the path integral approach of the previous
section, and in some cases is easier to make rigorous.
Any Gbundle P R S
is trivial, so we x a trivialization and
identify a connection on P with a gvalued form on RS
. The classical
conguration space of the theory is the space / of connections on P
S
.
This may be identied with the space of gvalued forms on S
. The
classical phase space of the theory is the cotangent bundle T
/. Note that
a tangent vector v T
A
/ may be identied with a gvalued form on S
.
We may also think of a gvalued form E on S
as a cotangent vector,
using the nondegenerate inner product
E, v
S
TrE v,
We thus regard the phase space T
/as the space of pairs A, E of gvalued
forms on S
.
Given a connection on P solving the YangMills equation we obtain a
point A, E of the phase space T
/ as follows let A be the pullback of the
connection to S
, and let E be its covariant time derivative pulled
back to S
. The pair A, E is called the initial data for the solution,
and in physics A is called the vector potential and E the electric eld. The
YangMills equation implies a constraint on A, E, the Gauss law
d
A
E,
and any pair A, E satisfying this constraint are the initial data for some
solution of the YangMills equation. However, this solution is not unique,
owing to the gauge invariance of the equation. Moreover, the loop group
C
S
, G acts as gauge transformations on /, and this action lifts
John C. Baez
naturally to an action on T
/, given by
g A, E gAg
gdg
, gEg
.
Two points in the phase space T
/ are to be regarded as physically equiv
alent if they dier by a gauge transformation.
In this sort of situation it is natural to try to construct a smaller, more
physically relevant reduced phase space using the process of symplectic re
duction. The phase space T
/ is a symplectic manifold, but the constraint
subspace
A, E d
A
ET
/
is not. However, the constraint d
A
E, integrated against any f C
S
,g
as follows,
S
Trfd
A
E,
gives a function on phase space that generates a Hamiltonian ow coin
ciding with a parameter group of gauge transformations. In fact, all
parameter subgroups of are generated by the constraint in this fash
ion. Consequently, the quotient of the constraint subspace by is again a
symplectic manifold, the reduced phase space.
In the case at hand there is a very concrete description of the reduced
phase space. First, by basic results on moduli spaces of at connections,
the reduced conguration space // may be naturally identied with
Hom
S
, G/AdG, which is just G/AdG. Alternatively, one can see
this quite concretely. We may rst take the quotient of / by only those
gauge transformations that equal the identity at a given point of S
gC
S
,Gg.
This almost reduced conguration space //
is dieomorphic to G itself,
with an explicit dieomorphism taking each equivalence class A to its
holonomy around the circle
A Pe
S
A
.
The remaining gauge transformations form the group /
G, which
acts on the almost reduced conguration space G by conjugation, so //
G/AdG.
Next, writing E edx, the Gauss law says that e C
S
, g is a
at section, and hence determined by its value at the basepoint of S
. It
follows that any point A, E in the constraint subspace is determined by
A / together with e g. The quotient of the constraint subspace
by
, the almost reduced phase space, is thus identied with T
G. It
follows that the quotient of the constraint subspace by all of , the reduced
phase space, is identied with T
G/AdG.
Strings, Loops, Knots, and Gauge Fields
The advantage of the almost reduced conguration space and phase
space is that they are manifolds. Observables of the classical theory can be
identied either with functions on the reduced phase space, or functions on
the almost reduced phase space T
G that are constant on the orbits of the
lift of the adjoint action of G. For example, the YangMills Hamiltonian is
initially a function on T
/
HA, E
E, E
but by the process of symplectic reduction one obtains a corresponding
Hamiltonian on the reduced phase space. One can, however, carry out
only part of the process of symplectic reduction, and obtain a Hamiltonian
function on the almost reduced phase space. This is just the Hamiltonian
for a free particle on G, i.e. for any p T
g
G it is given by
Hg, p
p
with the obvious inner product on T
g
G.
Now let us consider quantizing dimensional YangMills theory. What
should be the Hilbert space for the quantized theory on R S
As de
scribed in the previous section, it is natural to take L
of the reduced
conguration space //. Since the theory is not generally covariant,
the dieomorphism and Hamiltonian constraints do not enter the kine
matical Hilbert space is the physical Hilbert space. However, to dene
L
// requires choosing a measure on // G/AdG. We will choose
the pushforward of normalized Haar measure on G by the quotient map
G G/AdG. This measure has the advantage of mathematical elegance.
While one could also argue for it on physical grounds, we prefer simply to
show ex post facto that it gives an interesting quantum theory consistent
with other approaches to d YangMills theory.
To begin with, note that this measure gives a Hilbert space isomorphism
L
//
L
G
inv
where the right side denotes the subspace of L
G consisting of functions
constant on each conjugacy class of G. Let
denote the character of
an equivalence class of irreducible representations of G. Then by the
Schur orthogonality relations, the set
forms an orthonormal basis of
L
G
inv
. In fact, the Hamiltonian of the quantum theory is diagonalized
by this basis. Since the YangMills Hamiltonian on the almost reduced
phase space T
G is that of a classical free particle on G, we take the
quantum Hamiltonian to be that for a quantum free particle on G
H/
where is the nonnegative Laplacian on G. When we decompose the
John C. Baez
regular representation of G into irreducibles, the function
lies in the
sum of copies of the representation , so
H
c
,.
where c
is the quadratic Casimir of G in the representation . Note that
the vacuum the eigenvector of H with lowest eigenvalue is the function
, which is
for the trivial representation.
In a sense this diagonalization of the Hamiltonian completes the solution
of YangMills theory on R S
. However, extracting the physics from
this solution requires computing expectation values of physically interesting
observables. To take a step in this direction, and to make the connection
to string theory, let us consider the Wilson loop observables. Recall that
given a based loop S
S
, the classical Wilson loop T, A is dened
by
T, A TrPe
A
.
We may think of T T, as a function on the reduced conguration
space //, but it lifts to a function on the almost reduced conguration
space G, and we prefer to think of it as such. In the case at hand these
Wilson loop observables depend only on the homotopy class of the loop,
because all connections on S
are at. In the string eld picture of Section
, we obtain a theory in which all physical states have
,...,
n
,...,
n
when
i
is homotopic to
i
for all i. We will see this again in d quantum
gravity. Letting
n
S
S
be an arbitrary loop of winding number n,
we have
T
n
, g Trg
n
.
Since the classical Wilson loop observables are functions on congura
tion space, we may quantize them by interpreting them as multiplication
operators acting on L
G
inv
T
n
g Trg
n
g.
We can also form elements of L
G
inv
by applying products of these op
erators to the vacuum. Let
n
,...,n
k
T
n
T
n
k
.
The states n
,...,n
k
may also be regarded as states of a string theory in
which k strings are present, with winding numbers n
,...,n
k
, respectively.
For convenience, we dene to be the vacuum state.
The idea of describing the YangMills Hamiltonian in terms of these
string states goes back at least to Jevickis work on lattice gauge the
Strings, Loops, Knots, and Gauge Fields
ory. More recently, string states appear prominently in the work of Gross,
Taylor, Douglas, Minahan, Polychronakos, and others , , , ,
on d YangMills theory as a string theory. All these authors, however,
work primarily with the largeN limit of the theory, for since the work of
t Hooft it has been clear that SUN YangMills theory simplies
as N . In what follows we will use many ideas from these authors,
but give a stringtheoretic formula for the SUN YangMills Hamiltonian
that is exact for arbitrary N, instead of working in the largeN limit.
The resemblance of the string states n
,...,n
k
to states in a bosonic
Fock space should be clear. In particular, the T
n
are analogous to cre
ation operators. However, we do not generally have a representation of the
canonical commutation relations. In fact, the string states do not neces
sarily span L
G
inv
, although they do in some interesting cases. They are
never linearly independent, because the Wilson loops satisfy relations. One
always has T
Tr, for example, and for any particular group G the
Wilson loops will satisfy identities called Mandelstam identities. For exam
ple, for G SU and taking traces in the fundamental representation,
the Mandelstam identity is
T
n
T
m
T
nm
T
nm
.
Note that this implies that
n, m n m n m,
so the total number of strings present in a given state is ambiguous. In
other words, there is no analog of the Fock space number operator on
L
G
inv
.
For the rest of this section we set G SUN and take traces in the
fundamental representation. In this case the string states do span L
G
inv
,
and all the linear dependencies between string states are consequences of
the following Mandelstam identities . Given loops
,...,
k
in S
, let
M
k
,...,
k
k
S
k
sgnTg
j
g
jn
Tg
j
k
g
j
kn
k
where has the cycle structure j
j
n
j
k
j
kn
k
. Then
M
N
,...,
for all loops , and
M
N
,...,
N
for all loops
i
. There are also explicit formulae expressing the string
states in terms of the basis
of characters. These formulae are based
on the classical theory of Young diagrams, which we shall briey review.
The importance of this theory for dimensional physics was stressed by
John C. Baez
Nomura , and plays a primary role in Gross and Taylors work on d
YangMills theory . As we shall see, Young diagrams describe a duality
between the representation theory of SUN and of the symmetric groups
S
n
which can be viewed as a mathematical reection of string eld/gauge
eld duality.
First, note using the Mandelstam identities that the string states
n
,...,n
k
with all the n
i
positive but k possibly equal to zero span L
SUN
inv
.
Thus we will restrict our attention for now to states of this kind, which we
call righthanded. There is a correspondence between righthanded
string states and conjugacy classes of permutations in symmetric groups,
in which the string state n
,...,n
k
corresponds to the conjugacy class
of all permutations with cycles of length n
,...,n
k
. Note that consists
of permutations in S
n
, where n n
n
k
. To take advantage
of this correspondence, we simply dene
n
,...,n
k
.
when is the conjugacy class of permutations with cycle lengths n
,...,n
k
.
We will assume without loss of generality that n
n
k
gt .
The rationale for this description of string states as conjugacy classes of
permutations is in fact quite simple. Suppose we have lengthminimizing
strings in S
with winding numbers n
,...,n
k
. Labelling each strand of
string each time it crosses the point x , for a total of n n
n
k
labels, and following the strands around counterclockwise to x , we
obtain a permutation of the labels, and hence an element of S
n
. How
ever, since the labelling was arbitrary, the string state really only denes a
conjugacy class of elements of S
n
.
In a Young diagram one draws a conjugacy class with cycles of length
n
n
k
gt as a diagram with k rows of boxes, having n
i
boxes in
the ith row. See Fig. . Let Y denote the set of Young diagrams.
On the one hand, there is a map from Young diagrams to equivalence
classes of irreducible representations of SUN. Given Y , we form
an irreducible representation of SUN, which we also call , by taking a
tensor product of n copies of the fundamental representation, one copy for
each box, and then antisymmetrizing over all copies in each column and
symmetrizing over all copies in each row. This gives a correspondence
between Young diagrams with lt N rows and irreducible representations
of SUN. If has N rows it is equivalent to a representation coming
from a Young diagram having lt N rows, and if has gt N rows it is zero
dimensional. We will write
for the character of the representation if
has gt N rows
.
On the other hand, Young diagrams with n boxes are in correspon
Strings, Loops, Knots, and Gauge Fields
Fig. . Young diagram
dence with irreducible representations of S
n
. This allows us to write the
Frobenius relations expressing the string states in terms of characters
and vice versa. Given Y , we write for the corresponding repre
sentation of S
n
. We dene the function
on S
n
to be zero for n , n,
where n is the number of boxes in . Then the Frobenius relations are
Y
,.
and conversely
n
S
n
..
The YangMills Hamiltonian has a fairly simple description in terms
of the basis of characters
. First, recall that eqn . expresses the
Hamiltonian in terms of the Casimir. There is an explicit formula for the
value of the SUN Casimir in the representation
c
Nn N
n
nn
dim
where denotes the conjugacy class of permutations in S
n
with one
cycle of length and the rest of length . It follows that
H
NH
N
H
H
.
John C. Baez
where
H
n
.
and
H
nn
dim
..
To express the operators H
and H
in stringtheoretic terms, it is
convenient to dene string annihilation and creation operators satisfying
the canonical commutation relations. As noted above, there is no natural
way to do this in L
SUN
inv
since the string states are not linearly
independent. The advantage of taking the N limit is that any
nite set of distinct string states becomes linearly independent, in fact
orthogonal, for suciently large N. We will proceed slightly dierently,
simply dening a space in which all the string states are independent. Let
H be a Hilbert space having an orthonormal basis A
Y
indexed by all
Young diagrams. For each Y , dene a vector in H by the Frobenius
relation .. Then a calculation using the Schur orthogonality equations
twice shows that these string states are not only linearly independent,
but orthogonal
t
,
Y
t
A
,A
Y
t
n
where is the number of elements in regarded as a conjugacy class in
S
n
. One can also derive the Frobenius relation . from these denitions
and express the basis A
in terms of the string states
A
n
S
n
.
It follows that the string states form a basis for H.
The YangMills Hilbert space L
SUN
inv
is essentially a quotient
space of the string eld Hilbert space H, with the only densely dened
quotient map
jHL
SUN
inv
being given by
A
.
This quotient map sends the string state in H to the corresponding
Strings, Loops, Knots, and Gauge Fields
string state L
SUN
inv
. It follows that this quotient map is
precisely that which identies any two string states that are related by the
Mandelstam identities. It was noted some time ago by Gliozzi and Virasoro
that Mandelstam identities on string states are strong evidence for a
gauge eld interpretation of a string eld theory. Here in fact we will
show that the Hamiltonian on the YangMills Hilbert space L
SUN
inv
lifts to a Hamiltonian on H with a simple interpretation in terms of string
interactions, so that dimensional SUN YangMills theory is isomorphic
to a quotient of a string theory by the Mandelstam identities. In the
framework of the previous section, the Mandelstam identities would appear
as part of the dynamical constraint K of the string theory.
Following eqns .., we dene a Hamiltonian H on the string eld
Hilbert space H by
H
NH
N
H
H
where
H
A
nA
,H
A
nn
dim
A
.
This clearly has the property that
Hj jH,
so the YangMills dynamics is the quotient of the string eld dynamics.
On H we can introduce creation operators a
j
j gt by
a
j
n
,...,n
k
j, n
,...,n
k
,
and dene the annihilation operator a
j
to be the adjoint of a
j
. These
satisfy the following commutation relations
a
j
,a
k
a
j
,a
k
,a
j
,a
k
j
jk
.
We could eliminate the factor of j and obtain the usual canonical commu
tation relations by a simple rescaling, but it is more convenient not to. We
then claim that
H
jgt
a
j
a
j
and
H
j,kgt
a
jk
a
j
a
k
a
j
a
k
a
jk
.
These follow from calculations which we briey sketch here. The Frobenius
relations and the denition of H
give
H
n, .
John C. Baez
d
d
E
E
H
Fig. . Twostring interaction in dimensional space
and this implies the formula for H
as a sum of harmonic oscillator
Hamiltonians a
j
a
j
. Similarly, the Frobenius relations and the denition
of H
give
H
Y
nn
dim
.
Since there are nn / permutations S
n
lying in the conjugacy
class , we may rewrite this as
H
Y,
dim
.
Since
dim
the Frobenius relations give
H
..
An analysis of the eect of composing with all possible shows that
either one cycle of will be broken into two cycles, or two will be joined to
form one, giving the expression above for H
in terms of annihilation and
creation operators.
We may interpret the Hamiltonian in terms of strings as follows. By eqn
., H
can be regarded as a string tension term, since if we represent a
string state n
,...,n
k
by lengthminimizing loops, it is an eigenvector of
H
with eigenvalue equal to n
n
k
, proportional to the sum of the
lengths of the loops.
By eqn ., H
corresponds to a twostring interaction as in Fig. .
In this gure only the x coordinate is to be taken seriously the other has
been introduced only to keep track of the identities of the strings. Also,
we have switched to treating states as functions on the space of multiloops.
As the gure indicates, this kind of interaction is a dimensional version
of that which gave the HOMFLY invariant of links in dimensional space
in the previous section. Here, however, we have a true Hamiltonian rather
than a Hamiltonian constraint.
Figure can also be regarded as two frames of a movie of a string
worldsheet in dimensional spacetime. Similar movies have been used by
Carter and Saito to describe string worldsheets in dimensional spacetime
Strings, Loops, Knots, and Gauge Fields
. If we draw the string worldsheet corresponding to this movie we
obtain a surface with a branch point. Indeed, in the path integral approach
of Gross and Taylor this kind of term appears in the partition function as
part of a sum over string histories, associated to those histories with branch
points. They also show that the H
term corresponds to string worldsheets
with handles. When considering the /N expansion of the theory, it is
convenient to divide the Hamiltonian H by N, so that it converges to
H
as N . Then the H
term is proportional to /N
. This is in
accord with the observation by t Hooft that in an expansion of the
free energy logarithm of the partition function as a power series in /N,
string worldsheets of genus g give terms proportional to /N
g
.
From the work of Gross and Taylor it is also clear that in addition to
the space H spanned by righthanded string states one should also consider
a space with a basis of lefthanded string states n
,...,n
k
with n
i
lt .
The total Hilbert space of the string theory is then the tensor product
H
H
of righthanded and lefthanded state spaces. This does not
describe any new states in the YangMills theory per se, but it is more
natural from the stringtheoretic point of view. It follows from the work of
Minahan and Polychronakos that there is a Hamiltonian H on H
H
naturally described in terms of string interactions and a densely dened
quotient map j H
H
L
SUN
inv
such that Hj jH.
Quantum gravity in dimensions
Now let us turn to a more sophisticated model, dimensional quantum
gravity. In dimensions, Einsteins equations say simply that the spacetime
metric is at, so there are no local degrees of freedom. The theory is
therefore only interesting on topologically nontrivial spacetimes. Interest
in the mathematics of this theory increased when Witten reformulated
it as a ChernSimons theory. Since then, many approaches to the subject
have been developed, not all equivalent . We will follow Ashtekar,
Husain, Rovelli, Samuel and Smolin , and treat dimensional quantum
gravity using the new variables and the loop transform, and indicate some
possible relations to string theory. It is important to note that there are
some technical problems with the loop transform in Lorentzian quantum
gravity, since the gauge group is then noncompact . These are presently
being addressed by Ashtekar and Loll in the dimensional case, but
for simplicity of presentation we will sidestep them by working with the
Riemannian case, where the gauge group is SO.
It is easiest to describe the various action principles for gravity using the
abstract index notation popular in general relativity, but we will instead
translate them into language that may be more familiar to mathematicians,
since this seems not to have been done yet. In this section we describe the
Witten action, applicable to the dimensional case in the next section
John C. Baez
we describe the Palatini action, which applies to any dimension, and the
Ashtekar action, which applies to dimensions. The relationship between
these action principles has been discussed rather thoroughly by Peldan .
Let the spacetime M be an orientable manifold. Fix a real vector
bundle T over M that is isomorphic tobut not canonically identied
withthe tangent bundle TM, and x a Riemannian metric and an
orientation on T . These dene a volume form on T , that is a nowhere
vanishing section of
T
. The basic elds of the theory are then taken to
be a metricpreserving connection A on T , or SO connection, together
with a T valued form e on M. Using the isomorphism T
T
given
by the metric, the curvature F of A may be identied with a
T valued
form. It follows that the wedge product e F may be dened as a
T
valued form. Pairing this with to obtain an ordinary form and then
integrating over spacetime, we obtain the Witten action
SA, e
M
e F.
The classical equations of motion obtained by extremizing this action
are
F
and
d
A
e.
Note that we can pull back the metric on E by e TM T to obtain a
Riemannian metric on M, which, however, is only nondegenerate when
e is an isomorphism. When e is an isomorphism we can also use it to pull
back the connection to a metricpreserving connection on TM. In this case,
the equations of motion say simply that this connection is the LeviCivita
connection of the metric on M, and that the metric on M is at. The
formalism involving the elds A and e can thus be regarded as a device
for extending the usual Einstein equations in dimensions to the case of
degenerate metrics on M.
Now suppose that M R X, where X is a compact oriented
manifold. The classical conguration and phase spaces and their reduction
by gauge transformations are reminiscent of those for d YangMills theory.
There are, however, a number of subtleties, and we only present the nal
results. The classical conguration space can be taken as the space / of
metricpreserving connections on T X, which we call SO connections
on X. The classical phase space is then the cotangent bundle T
/. Note
that a tangent vector v T
A
/ is a
T valued form on X. We can thus
regard a T valued form
E on X as a cotangent vector by means of the
pairing
Ev
X
E v.
Strings, Loops, Knots, and Gauge Fields
Thus given any solution A, e of the classical equations of motion, we can
pull back A and e to the surface X and get an SO connection and
a T valued form on X, that is a point in the phase space T
/. This is
usually written A,
E, where
E plays a role analogous to the electric eld
in YangMills theory.
The classical equations of motion imply constraints on A,
ET
/
which dene a reduced phase space. These are the Gauss law, which in
this context is
d
A
E,
and the vanishing of the curvature B of the connection A on T X, which
is analogous to the magnetic eld
B.
The latter constraint subsumes both the dieomorphism and Hamiltonian
constraints of the theory. The reduced phase space for the theory turns
out to be T
/
/, where /
is the space of at SO connections on
X, and is the group of gauge transformations . As in d YangMills
theory, it will be attractive to quantize after imposing constraints, taking
the physical state space of the quantized theory to be L
of the reduced
conguration space, if we can nd a tractable description of /
/.
A quite concrete description of /
/ was given by Goldman . The
moduli space T of at SObundles has two connected components, cor
responding to the two isomorphism classes of SO bundles on M. The
component corresponding to the bundle T X is precisely the space /
/,
so we wish to describe this component.
There is a natural identication
T
Hom
X, SO/AdSO,
given by associating to any at bundle the holonomies around homotopy
classes of loops. Suppose that X has genus g. Then the group
X has
a presentation with g generators x
,y
,...,x
g
,y
g
satisfying the relation
Rx
i
,y
i
x
y
x
y
x
g
y
g
x
g
y
g
.
An element of Hom
X, SO may thus be identied with a collection
u
,v
,...,u
g
,v
g
of elements of SO, satisfying
Ru
i
,v
i
,
and a point in T is an equivalence class u
i
,v
i
of such collections.
The two isomorphism classes of SO bundles on M are distinguished
by their second StiefelWhitney number w
Z
. The bundle T X is
trivial so w
T X We can calculate w
for any point u
i
,v
i
T by
the following method. For all the elements u
i
,v
i
SO, choose lifts u
i
,v
i
John C. Baez
to the universal cover
SO
SU. Then
w
Ru
i
,v
i
.
It follows that we may think of points of /
/ as equivalence classes of
gtuples u
i
,v
i
of elements of SO admitting lifts u
i
,v
i
with
Ru
i
,v
i
,
where the equivalence relation is given by the adjoint action of SO.
In fact /
/ is not a manifold, but a singular variety. This has been
investigated by Narasimhan and Seshadri , and shown to be dimension
d g for g , or d for g the case g is trivial and
will be excluded below. As noted, it is natural to take L
/
/ to be
the physical state space, but to dene this one must choose a measure on
/
/. As noted by Goldman , there is a symplectic structure on
/
/ coming from the following form on /
B, C
X
TrB C,
in which we identify the tangent vectors B, C with EndT Xvalued
forms. The dfold wedge product denes a measure on /
/,
the Liouville measure. On the grounds of elegance and dieomorphism
invariance it is customary to use this measure to dene the physical state
space L
/
/.
It would be satisfying if there were a stringtheoretic interpretation of
the inner product in L
/
/ along the lines of Section . Note that we
may dene string states in this space as follows. Given any loop in X,
the Wilson loop observable T is a multiplication operator on L
/
/
that only depends on the homotopy class of . As in the case of d Yang
Mills theory, we can form elements of L
/
/ by applying products of
these operators to the function , so given
,...,
k
X, dene
,...,
k
T
T
k
.
The rst step towards a stringtheoretic interpretation of d quantum grav
ity would be a formula for an inner product of string states
,...,
k
t
,...,
t
k
,
which is just a particular kind of integral over /
/. Note that this sort
of integral makes sense when /
/ is the moduli space of at connections
for a trivial SON bundle over X, for any N. Alternatively, one could
formulate d quantum gravity as a theory of SU connections and then
generalize to SUN.
In fact, it appears that this sort of integral can be computed using
d YangMills theory on a Riemann surface, by introducing a coupling
Strings, Loops, Knots, and Gauge Fields
constant in the action
SA
M
TrF F,
computing the appropriate vacuum expectation value of Wilson loops, and
then taking the limit . This procedure has been discussed by Blau
and Thompson and Witten . One thus expects that this sort of
integral simplies in the N limit just as it does in d YangMills
theory, giving a formula for
,...,
k
t
,...,
t
k
as a sum over ambient isotopy classes of surfaces f , t X having
the loops
i
,
t
i
as boundaries. This, together with a method of treating
the nite N case by imposing Mandelstam identities, would give a string
theoretic interpretation of d quantum gravity. Taylor and the author are
currently trying to work out the details of this program.
Before concluding this section, it is worth noting another generally co
variant gauge theory in dimensions, ChernSimons theory. Here one xes
an arbitrary Lie group G and a Gbundle P M over spacetime, and the
eld of the theory is a connection A on P. The action is given by
SA
k
Tr
A dA
AAA
.
As noted by Witten , d quantum gravity as we have described it is
essentially the same ChernSimons theory with gauge group ISO, the
Euclidean group in dimension, with the SO connection and triad eld
appearing as two parts of an ISO connection. There is a profound con
nection between ChernSimons theory and knot theory, rst demonstrated
by Witten , and then elaborated by many researchers see, for example,
. This theory does not quite t our formalism because in it the space
/
/ of at connections modulo gauge transformations plays the role of a
phase space, with the Goldman symplectic structure, rather than a cong
uration space. Nonetheless, there are a number of clues that ChernSimons
theory admits a reformulation as a generally covariant string eld theory.
In fact, Witten has given such an interpretation using open strings and the
BatalinVilkovisky formalism . Moreover, for the gauge groups SUN
Periwal has expressed the partition function for ChernSimons theory on
S
, in the N limit, in terms of integrals over moduli spaces of Rie
mann surfaces. In the case N there is also, as one would expect, an
expression for the vacuum expectation value of Wilson loops, at least for
the case of a link where it is just the Kauman bracket invariant, in terms
of a sum over surfaces having that link as boundary . It would be very
worth while to reformulate ChernSimons theory as a string theory at the
level of elegance with which one can do so for d YangMills theory, but
John C. Baez
this has not yet been done.
Quantum gravity in dimensions
We begin by describing the Palatini and Ashtekar actions for general rela
tivity. As in the previous section, we will sidestep certain problems with the
loop transformby working with Riemannian rather than Lorentzian gravity.
We shall then discuss some recent work on making the loop representation
rigorous in this case, and indicate some mathematical issues that need to
be explored to arrive at a stringtheoretic interpretation of the theory.
Let the spacetime M be an orientable nmanifold. Fix a bundle T over
M that is isomorphic to TM, and x a Riemannian metric and orientation
on T . These dene a nowherevanishing section of
n
T
. The basic elds
of the theory are then taken to be a metricpreserving connection A on T ,
or SOn connection, and a T valued form e. We require, however, that
e TM T be a bundle isomorphism its inverse is called a frame eld.
The metric denes an isomorphism T
T
and allows us to identify the
curvature F of A with a section of the bundle
T
T
M.
We may also regard e
as a section of T
TM and dene e
e
in
the obvious manner as a section of the bundle
T
TM.
The natural pairing between these bundles gives rise to a function Fe
e
on M. Using the isomorphism e, we can push forward to a volume
form on M. The Palatini action for Riemannian gravity is then
SA, e
M
Fe
e
.
We may use the isomorphism e to transfer the metric and connection
A to a metric and connection on the tangent bundle. Then the classical
equations of motion derived from the Palatini action say precisely that this
connection is the LeviCivita connection of the metric, and that the metric
satises the vacuum Einstein equations i.e. is Ricci at.
In dimensions, the Palatini action reduces to the Witten action, which,
however, is expressed in terms of e rather than e
. In dimensions the
Palatini action can be rewritten in a somewhat similar form. Namely, the
wedge product e e F is a
T valued form, and pairing it with to
obtain an ordinary form we have
SA, e
M
e e F.
The Ashtekar action depends upon the fact that in dimensions the
Strings, Loops, Knots, and Gauge Fields
metric and orientation on T dene a Hodge star operator
T
T
with
not to be confused with the Hodge star operator on dierential
forms. This allows us to write F as a sum F
F
of selfdual and anti
selfdual parts
F
F
.
The remarkable fact is that the action
SA, e
M
eeF
gives the same equations of motion as the Palatini action. Moreover, sup
pose T is trivial, as is automatically the case when M
RX. Then F is
just an sovalued form on M, and its decomposition into selfdual and
antiselfdual parts corresponds to the decomposition so
soso.
Similarly, A is an sovalued form, and may thus be written as a sum
A
A
of selfdual and antiselfdual connections, which are forms
having values in the two copies of so. It is easy to see that F
is the
curvature of A
. This allows us to regard general relativity as the theory
of a selfdual connection A
and a T valued form ethe socalled new
variableswith the Ashtekar action
SA
,e
M
eeF
.
Now suppose that M R X, where X is a compact oriented
manifold. We can take the classical conguration space to be space / of
righthanded connections on T X, or equivalently xing a trivialization of
T , sovalued forms on X. A tangent vector v T
A
/ is thus an so
valued form, and an sovalued form
E denes a cotangent vector by
the pairing
Ev
X
Tr
E v.
A point in the classical phase space T
/ is thus a pair A,
E consisting
of an sovalued form A and an sovalued form
E on X. In the
physics literature it is more common to use the natural isomorphism
T
X
TX
T
X
given by the interior product to regard the gravitational electric eld
E as
an sovalued vector density, that is, a section of so TX
T
X.
A solution A
, e of the classical equations of motion determines a
point A,
ET
/ as follows. The gravitational vector potential A is
simply the pullback of A
to the surface X. Obtaining
E from e is
John C. Baez
a somewhat subtler aair. First, split the bundle T as the direct sum of
a dimensional bundle
T and a line bundle. By restricting to TX and
then projecting down to
T X, the map
e TM T
gives a map
e TX
T X,
called a cotriad eld on X. Since there is a natural isomorphism of the
bers of
T with so, we may also regard this as an sovalued form
on X. Applying the Hodge star operator we obtain the sovalued form
E.
The classical equations of motion imply constraints on A,
ET
/.
These are the Gauss law
d
A
E,
and the dieomorphism and Hamiltonian constraints. The latter two are
most easily expressed if we treat
E as an sovalued vector density. Let
ting B denote the gravitational magnetic eld, or curvature of the con
nection A, the dieomorphism constraint is given by
Tr i
E
B
and the Hamiltonian constraint is given by
Tr i
E
i
E
B.
Here the interior product i
E
B is dened using matrix multiplication
and is an M
R
T
Xvalued form similarly, i
E
i
E
B is an M
R
T
X
T
Xvalued function.
In d YangMills theory and d quantum gravity one can impose enough
constraints before quantizing to obtain a nitedimensional reduced con
guration space, namely the space /
/ of at connections modulo gauge
transformations. In d quantum gravity this is no longer the case, so a
more sophisticated strategy, rst devised by Rovelli and Smolin , is re
quired. Let us rst sketch this without mentioning the formidable technical
problems. The Gauss law constraint generates gauge transformations, so
one forms the reduced phase space T
//. Quantizing, one obtains the
kinematical Hilbert space H
kin
L
//. One then applies the loop
transform and takes H
kin
Fun/ to be a space of functions of multi
loops in X. The dieomorphism constraint generates the action of Di
X
on //, so in the quantum theory one takes H
diff
to be the subspace of
Di
Xinvariant elements of Fun/. One may then either attempt to
represent the Hamiltonian constraint as operators on H
kin
, and dene the
image of their common kernel in H
diff
to be the physical state space H
phys
,
or attempt to represent the Hamiltonian constraint directly as operators
Strings, Loops, Knots, and Gauge Fields
on H
diff
and dene the kernel to be H
phys
. The latter approach is still
under development by Rovelli and Smolin .
Even at this formal level, the full space H
phys
has not yet been de
termined. In their original work, Rovelli and Smolin obtained a large
set of physical states corresponding to ambient isotopy classes of links in
X. More recently, physical states have been constructed from familiar
link invariants such as the Kauman bracket and certain coecients of
the Alexander polynomial to all of /. Some recent developments along
these lines have been reviewed by Pullin . This approach makes use
of the connection between d quantum gravity with cosmological constant
and ChernSimons theory in dimensions. It is this work that suggests a
profound connection between knot theory and quantum gravity.
There are, however, signicant problems with turning all of this work
into rigorous mathematics, so at this point we shall return to where we left
o in Section and discuss some of the diculties. In Section we were
quite naive concerning many details of analysisdeliberately so, to indicate
the basic ideas without becoming immersed in technicalities. In particular,
one does not really expect to have interesting dieomorphisminvariant
measures on the space // of connections modulo gauge transformations
in this case. At best, one expects the existence of generalized measures
sucient for integrating a limited class of functions.
In fact, it is possible to go a certain distance without becoming involved
with these considerations. In particular, the loop transform can be rigor
ously dened without xing a measure or generalized measure on // if
one uses, not the Hilbert space formalism of the previous section, but a
Calgebraic formalism. A Calgebra is an algebra A over the complex
numbers with a norm and an adjoint or operation satisfying
a
a, a
a
,ab
a
b
, ab
b
a
,
ab ab, a
aa
for all a, b in the algebra and C. In the Calgebraic approach to
physics, observables are represented by selfadjoint elements of A, while
states are elements of the dual A
that are positive, a
a , and
normalized, . The number a then represents the expectation
value of the observable a in the state . The relation to the more tradi
tional Hilbert space approach to quantum physics is given by the Gelfand
NaimarkSegal GNS construction. Namely, a state on A denes an
inner product that may, however, be degenerate
a, b a
b.
Let I A denote the subspace of normzero states. Then A/I has an
honest inner product and we let H denote the Hilbert space completion of
A/I in the corresponding norm. It is then easy to check that I is a left
John C. Baez
ideal of A, so that A acts by left multiplication on A/I, and that this action
extends uniquely to a representation of A as bounded linear operators on
H. In particular, observables in A give rise to selfadjoint operators on H.
A Calgebraic approach to the loop transform and generalized mea
sures on // was introduced by Ashtekar and Isham in the context of
SU gauge theory, and subsequently developed by Ashtekar, Lewandow
ski, and the author , . The basic concept is that of the holonomy
Calgebra. Let X be a manifold, space, and let P X be a princi
pal Gbundle over X. Let / denote the space of smooth connections on P,
and the group of smooth gauge transformations. Fix a nitedimensional
representation of G and dene Wilson loop functions T T, on
// taking traces in this representation.
Dene the holonomy algebra to be the algebra of functions on //
generated by the functions T T, . If we assume that G is compact
and is unitary, the functions T are bounded and continuous in the
C
topology on //. Moreover, the pointwise complex conjugate T
equals T
, where
is the orientationreversed loop. We may thus
complete the holonomy algebra in the sup norm topology
f
sup
A,/
fA
and obtain a Calgebra of bounded continuous functions on //, the
holonomy Calgebra, which we denote as Fun// in order to make
clear the relation to the previous section.
While in what follows we will assume that G is compact and is unitary,
it is important to emphasize that for Lorentzian quantum gravity G is not
compact This presents important problems in the loop representation
of both and dimensional quantum gravity. Some progress in solving
these problems has recently been made by Ashtekar, Lewandowski, and
Loll , , .
Recall that in the previous section the loop transform of functions on
// was dened using a measure on //. It turns out to be more natural
to dene the loop transform not on Fun// but on its dual, as this
involves no arbitrary choices. Given Fun//
we dene its loop
transform to be the function on the space / of multiloops given by
,...,
n
T
T
n
.
Let Fun/ denote the range of the loop transform. In favorable cases,
such as G SUN and the fundamental representation, the loop trans
form is onetoone, so
Fun//
Fun/.
This is the real justication for the term string eld/gauge eld duality.
Strings, Loops, Knots, and Gauge Fields
We may take the generalized measures on // to be simply ele
ments Fun//
, thinking of the pairing f as the integral of
f Fun//. If is a state on Fun//, we may construct the kine
matical Hilbert space H
kin
using the GNS construction. Note that the
kinematical inner product
f, g
kin
f
g
then generalizes the L
inner product used in the previous section. Note
that a choice of generalized measure also allows us to dene the loop
transform as a linear map from Fun// to Fun/
f
,...,
n
T
T
n
f
in a manner generalizing that of the previous section. Moreover, there is
a unique inner product on Fun/ such that this map extends to a map
from H
kin
to the Hilbert space completion of Fun/. Note also that
Di
X acts on Fun// and dually on Fun//
. The kinematical
Hilbert space constructed from a Di
Xinvariant state Fun//
thus becomes a unitary representation of Di
X.
It is thus of considerable interest to nd a more concrete description of
Di
Xinvariant states on the holonomy Calgebra Fun//. In fact, it
is not immediately obvious that any exist, in general For technical reasons,
the most progress has been made in the realanalytic case. That is, we take
X to be real analytic, Di
X to consist of the realanalytic dieomor
phisms connected to the identity, and Fun// to be the holonomy C
algebra generated by realanalytic loops. Here Ashtekar and Lewandowski
have constructed a Di
Xinvariant state on Fun// that is closely
analogous to the Haar measure on a compact group . They have also
given a general characterization of such dieomorphisminvariant states.
The latter was also given by the author , using a slightly dierent for
malism, who also constructed many more examples of Di
Xinvariant
states on Fun//. There is thus some real hope that the loop represen
tation of generally covariant gauge theories can be made rigorous in cases
other than the toy models of the previous two sections.
We conclude with some speculative remarks concerning d quantum
gravity and tangles. The correct inner product on the physical Hilbert
space of d quantum gravity has long been quite elusive. A path integral
formula for the inner product has been investigated recently by Rovelli
, but there is as yet no manifestly welldened expression along these
lines. On the other hand, an inner product for relative states of quantum
gravity in the Kauman bracket state has been rigorously constructed by
the author , but there are still many questions about the physics here.
The example of d YangMills theory would suggest an expression for the
John C. Baez
inner product of string states
,...,
n
t
,...,
t
n
as a sum over ambient isotopy classes of surfaces f , t X having
the loops
i
,
t
i
as boundaries. In the case of embeddings, such surfaces are
known as tangles, and have been intensively investigated by Carter and
Saito using the technique of movies.
The relationships between tangles, string theory, and the loop rep
resentation of d quantum gravity are tantalizing but still rather obscure.
For example, just as the Reidemeister moves relate any two pictures of the
same tangle in dimensions, there are a set of movie moves relating any
two movies of the same tangle in dimensions. These moves give a set
of equations whose solutions would give tangle invariants. For example,
the analog of the YangBaxter equation is the Zamolodchikov equation,
rst derived in the context of string theory . These equations can be
understood in terms of category theory, since just as tangles form a braided
tensor category, tangles form a braided tensor category . It is thus
quite signicant that Crane has initiated an approach to generally co
variant eld theory in dimensions using braided tensor categories. This
approach also claries some of the signicance of conformal eld theory
for dimensional physics, since braided tensor categories can be con
structed from certain conformal eld theories. In a related development,
CottaRamusino and Martellini have endeavored to construct tangle
invariants from generally covariant gauge theories, much as tangle invari
ants may be constructed using ChernSimons theory. Clearly it will be
some time before we are able to appraise the signicance of all this work,
and the depth of the relationship between string theory and the loop rep
resentation of quantum gravity.
Acknowledgements
I would like to thank Abhay Ashtekar, Scott Axelrod, Matthias Blau,
Scott Carter, Paolo CottaRamusino, Louis Crane, Jacob Hirbawi, Jerzy
Lewandowski, Renate Loll, Maurizio Martellini, Jorge Pullin, Holger Niel
sen, and Lee Smolin for useful discussions. Wati Taylor deserves special
thanks for explaining his work on YangMills theory to me. Also, I would
like to thank the Center for Gravitational Physics and Geometry as a whole
for inviting me to speak on this subject.
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BF Theories and knots
Paolo CottaRamusino
Dipartimento di Fisica, Universit a di Trento
and INFN, Sezione di Milano,
Via Celoria , Milano, Italy
email cottamilano.infn.it
Maurizio Martellini
Dipartimento di Fisica, Universit a di Milano
and INFN, Sezione di Pavia,
Via Celoria , Milano, Italy
email martellinimilano.infn.it
Abstract
We discuss the relations between topological quantum eld theo
ries in dimensions and the theory of knots embedded spheres
in a manifold. The socalled BF theories allow the construction
of quantum operators whose trace can be considered as the higher
dimensional generalization of Wilson lines for knots in dimensions.
Firstorder perturbative calculations lead to higherdimensional link
ing numbers, and it is possible to establish a heuristic relation be
tween BF theories and Alexander invariants. Functional integration
byparts techniques allow the recovery of an innitesimal version of
the Zamolodchikov tetrahedron equation, in the form considered by
Carter and Saito.
Introduction
In a seminal work, Witten has shown that there is a very deep connection
between invariants of knots and links in a manifold and the quantum
ChernSimons eld theory.
One of the questions that is possible to ask is whether there exists
an analog of Wittens ideas in higher dimensions. After all, knots and
links are dened in any dimension as embeddings of kspheres into a k
dimensional space, and topological quantum eld theories exist in
dimensions , .
The question, unfortunately, is not easy to answer. On one hand, the
theory of higherdimensional knots and links is much less developed the
relevant invariants are, for the time being, more scarce than in the theory
of ordinary knots and links. As far as dimensional topological eld
theories are concerned, only the general BRST structure has been thor
oughly discussed perturbative calculations, at least in the nonAbelian
Paolo CottaRamusino and Maurizio Martellini
case, do not appear to have been carried out. Even the quantum observ
ables have not been clearly dened in the nonAbelian case. Moreover,
ChernSimons theory in dimensions beneted greatly from the results of
dimensional conformal eld theory. No analogous help is available for
dimensional topological eld theories.
In this paper we make some preliminary considerations and proposals
concerning knots and topological eld theories of the BF type. Speci
cally we propose a set of quantum observables associated to surfaces em
bedded in space. These observables can be seen as a generalization of
ordinary Wilson lines in dimensions. Moreover, the framework in which
these observables are dened ts with the picture of knots as movies
of ordinary links or link diagrams considered by Carter and Saito see
references below. It is also possible to nd heuristic arguments that may
relate the expectation values of our observables in the BF theory with the
Alexander invariants of knots.
As far as perturbative calculations are concerned, they are much more
complicated than in the lowerdimensional case. Nevertheless it is pos
sible to recover a relation between rstorder calculations and higher
dimensional linking numbers, which parallels a similar relation for ordinary
links in a dimensional space.
Finally, in dimensions it is also possible to apply functional integration
byparts techniques. In this case, these techniques produce a solution of
an innitesimal Zamolodchikov tetrahedron equation, consistent with the
proposal made by Carter, Saito, and Lawrence.
BF theories and their geometrical signicance
We start by considering a dimensional Riemannian manifold M, a com
pact Lie group G, and a principal bundle PM, G over M. If A denotes a
connection on such a bundle, then its curvature will be denoted by F
A
or
simply by F. The space of all connections will be denoted by the symbol
/ and the group of gauge transformations by the symbol . The group G,
with Lie algebra LieG, will be, in most cases, the group SUn or SOn.
Looking for possible topological actions for a dimensional manifold,
we may consider , the Gibbs measures of the
. Chern action
exp
M
Tr
FF
,.
. BF action
exp
M
Tr
BF
..
In the formulae above and are coupling constants, is a given represen
BF Theories and knots
tation of the group G and of its Lie algebra. The curvature F is a form
on M with values in the associated bundle P
Ad
LieG or, equivalently,
a tensorial form on P with values in LieG.
The eld B in eqn . is assumed classically to be a form of the
same nature as F. We shall see, though, that the quantization procedure
may force B to take values in the universal enveloping algebra of LieG.
It is then convenient to discuss the geometrical aspects of a more general
BF theory, where B is assumed to be a form on M with values in the
associated bundle P
Ad
G where G is the universal enveloping algebra
of LieG. In other words B is a section of the bundle
T
MP
Ad
G.
The space of forms with values in P
Ad
G will be denoted by the sym
bol
M, adP this space naturally contains
M, adP, the space of
forms with values in P
Ad
LieG.
In order to obtain an ordinary form on M from an element of
M, adP, one needs to take the trace with respect to a given represen
tation of LieG. Notice that the wedge product for forms in
M, adP
is obtained by combining the product in G with the exterior product for
ordinary forms.
We now mention some examples of Belds that have a nice geometrical
meaning
. Let LM M be the frame bundle and let be the soldering form.
It is an R
n
valued form, given by the identity map on the tangent
space of M. For any given metric g we consider the reduced SOn
bundle of orthonormal frames O
g
M. A vielbein in physics is dened
as the R
n
valued form on M given by
, for a given section
MO
g
M. We now dene the form B
M, adO
g
M as
B
T
OgM
,
where
T
denotes transposition. In the corresponding BF action, the
curvature F is the curvature of a connection in the orthonormal frame
bundle O
g
M. This action is known to be classically equivalent to
the Einstein action, written in the vielbein formalism.
. The manifold / is an ane space with underlying vector space
M, adP. We denote by the form on / given by the iden
tity map on the tangent space of /, i.e. on
M, adP. From this
we can obtain the form on /
B,.
with values in
M, adP. More explicitly, B can be dened as
Paolo CottaRamusino and Maurizio Martellini
Ba, bX, Y
p,A
aXbY aY bX bXaY bY aX,
.
where a, b T
A
/ X, Y T
p
P.
We now want to show that the BF action considered in the last example
is connected with the ABJ AdlerBellJackiw anomaly in dimensions.
On the principal Gbundle
P/M/.
we can consider the tautological connection dened by the following hori
zontal distribution
Hor
p,A
Hor
A
p
T
A
/.
where Hor
A
p
denotes the horizontal space at p P with respect to the
connection A. The connection form of the tautological connection is given
simply by A, seen as a ,form on P /. Its curvature form is given by
T
p,A
F
A
p
A
..
Let us now use Q
l
to denote an irreducible adinvariant polynomial on
LieG with l entries. Integrating the ChernWeil forms corresponding to
such polynomials over M we obtain for l , ,
M
Q
T, T k
M
Tr
F
A
F
A
.
M
Q
T, T, T k
M
Tr
BF
A
.
M
Q
T, T, T, T k
M
Tr
BB.
where B is dened as in eqns . and ., and k
l
are normalization
factors. In eqn . the wedge product also includes the exterior product
of forms on /.
As Atiyah and Singer pointed out, one can consider together with
BF Theories and knots
. the following principal Gbundle
P/
M
/
..
Any connection on the bundle / // determines a connection on the
bundle . and hence on the bundle .. When we consider such a
connection on . instead of the tautological one, then the lefthand side
of eqn . becomes the term which generates, via the socalled descent
equation, the ABJ anomaly in dimensions , . Hence the BF action
. is a sort of gaugexed version of the generating term for the ABJ
anomaly.
In any BF action one has to integrate forms dened on M /. Hence
we can consider dierent kinds of symmetries ,
. connection invariance this refers to forms on M / which are
closed when they are integrated over cycles of M, they give closed
forms on /.
. gauge invariance this refer to forms which are dened over M /,
but are pullbacks of forms dened over M
/
.
. dieomorphism invariance this can be considered whenever we have
an action of DiM over P and over /. In particular we can consider
dieomorphism invariance when P is a trivial bundle, or when P is
the bundle of linear frames over M. In the latter case
P/
DiM
M/
DiM
.
is a principal bundle
.
. dieomorphism/gauge invariance this refers to the action of AutP,
i.e. the full group of automorphisms of the principal bundle P in
cluding the automorphisms which do not induce the identity map on
the base manifold. This kind of invariance implies gauge invariance
and dieomorphism invariance when the latter is dened.
For instance, the Chern action . is both connection and dieomor
phism/gauge invariant, while the BF action . is gauge invariant and
its connection invariance depends on further specications. In particular
the BF action of example is connection invariant, while the action of
Strictly speaking, in this case, one should consider either the space of irreducible
connections and the gauge group, divided by its center, or the space of all connections
and the group of gauge transformations which give the identity when restricted to a
xed point of M. We always assume that one of the above choices is made.
In fact one should really consider only the group of dieomorphisms of M which
strongly x one point of M. Also, instead of LM A it is possible to consider
O
M
A
metric
, namely the space of all orthonormal frames paired with the corresponding
metric connections see e.g. .
Paolo CottaRamusino and Maurizio Martellini
example is not. More precisely, in example , the integrand of the BF
action denes a closed form on O
g
M /, i.e. we have
,
M
Tr
F
T
OgM
only when the covariant derivative d
A
is zero in other words, the critical
connections are the LeviCivita connections.
When B is given by ., then for any cycle in M
Tr
B
Q
TT
is a closed form on / and so we have a map
Z
MZ
/, .
where Z
Z
denotes cycles cocycles. When we replace the curva
ture of the tautological connection with the curvature of a connection in
the bundle ., then we again obtain a map ., but now this map
descends to a map
H
MH
/
..
The map . is the basis for the construction of the Donaldson polyno
mials .
As a nal remark we would like to mention BF theories in arbitrary
dimensions. When M is a manifold of dimension d, then we can take the
eld B to be a d form, again with values in the bundle P
Ad
G.
The form . then makes perfect sense in any dimension.
A special situation occurs when the group G is SOd. In this case
there is a linear isomorphism
R
d
LieSOd, .
dened by
e
i
e
j
E
i
j
E
j
i
,
where e
i
is the canonical basis of R
d
and E
i
j
is the matrix with entries
E
i
j
m
n
m,i
n,j
. The isomorphism . has the following properties
. It is an isomorphism of inner product spaces, when the inner product
in LieSOd is given by A, B /TrAB.
. The left action of SOd on R
n
corresponds to the adjoint action on
LieSOd.
One can then consider a principal SOdbundle and an associated bundle
E with ber R
d
. In this special case we can consider a dierent kind of
BF theory, where the eld B is assumed to be a d form with values
BF Theories and knots
in the d th exterior power of E. In this case the corresponding BF
action is given by
exp
M
BF
,.
where the wedge product combines the wedge product of forms on M with
the wedge product in
R
d
. The action . is invariant under gauge
transformations. When the SOdbundle is the orthonormal bundle and
the eld B is the d th exterior power of the soldering form, then the
corresponding BF action . gives the classical action for gravity in d
dimensions, in the socalled Palatini rstorder formalism , .
knots and their quantum observables
In WittenChernSimons theory we have a topological action on a
dimensional manifold M
, and the observables correspond to knots or
links in M
. More precisely to each knot we associate the trace of the
holonomy along the knot in a xed representation of the group G, or
Wilson line. In dimensions we have at our disposal the Lagrangians
considered at the beginning of the previous section. It is natural to con
sider as observables, quantities higherdimensional Wilson lines related
to knots.
Let us recall here that while an ordinary knot is a sphere embedded
in S
or R
, a knot is a sphere embedded in S
or in the space
R
. A generalized knot in a dimensional closed manifold M can be
dened as a closed surface embedded in M. Two knots generalized or
not will be called equivalent if they can be mapped into each other by a
dieomorphism connected to the identity of the ambient manifold M.
The theory of knots and links is less developed than the theory of
ordinary knots and links. For instance, it is not known whether one can
have an analogue of the Jones polynomial for knots. On the other hand,
one can dene Alexander invariants for knots see e.g. .
The problem we would like to address here is whether there exists a
connection between dimensional eld theories and invariants of knots.
Namely, we would like to ask whether there exists a generalization to
dimensions of the connection established by Witten between topological
eld theories and knot invariants in dimensions.
Even though we are not able to show rigorously that a consistent set of
nontrivial invariants for knots can be constructed out of dimensional
eld theories, we can show that there exists a connection between BF
theories in dimensions and knots. This connection involves, in dierent
places, the Zamolodchikov tetrahedron equation as well as selflinking and
higherdimensional linking numbers.
In Wittens theory one has to perform functional integration upon an
observable depending on the given ordinary knot in the given manifold
Paolo CottaRamusino and Maurizio Martellini
with respect to a topological Lagrangian. Here we want to do something
similar, namely we want to perform functional integration upon dierent
observables depending on a given embedded surface in M with respect
to a path integral measure given by the BF action.
The rst classical observable we want to consider is
Tr
exp
HolA,
y,x
ByHolA,
z,y
,y..
Here by HolA,
y,x
we mean the holonomy with respect to the connection
A along the path joining two given points y and x belonging to
.
Expression . has the following properties
. It depends on the choice of a map assigning to each y a path
joining x and z and passing through y more precisely, it depends
on the holonomies along such paths. One expects that the functional
integral over the space of all connections will integrate out this de
pendency. More precisely, we recall that the functional measure
over the space of connections is given, up to a Jacobian determinant,
by the measure over the paths over which the holonomy is computed.
See in this regard the analysis of the WessZuminoWitten model
made by Polyakov and Wiegmann .
. It is gauge invariant if we consider only gauge transformations that
are the identity at the points x and z. In the special case when z and
x coincide, then . is gauge invariant without any restriction.
The functional integral of . can be seen as the vacuum expectation
value of the trace in the representation of the quantum surface operator
denoted by O x, z. In other words we set
O x, z exp
HolA,
y,x
ByHolA,
z,y
,y,.
where, in order to avoid a cumbersome notation, we did not write the ex
plicit dependence on the paths , and we did not use dierent symbols for
the holonomy and its quantum counterpart quantum holonomy. More
over, in . a timeordering symbol has to be understood.
Note that we will also allow the operator . to be dened for any
embedded surface closed or with boundary. In the latter case the points
x and z are always supposed to belong to the boundary.
As far as the role of the paths considered above is concerned, we
recall that, in a dimensional theory with knots, a framing is a smooth
assignment of a tangent vector to each point of the knot. In dimensions,
We assume to have chosen once and for all a reference section for the trivial bundle
P
. Hence the holonomy along a path is dened by the comparison of the horizontally
lifted path and the path lifted through the reference section.
BF Theories and knots
instead, we assign a curve to each point of the knot. We will refer to
this assignment as a higherdimensional framing. We will also speak of a
framed knot.
Gauss constraints
In order to study the quantum theory corresponding to ., we rst con
sider the canonical quantization approach. We take the group G to be
SUn and consider its fundamental representation hence we write the
eld B as
B
a
B
a
R
a
. Here
B is a multiple of and R
a
is an or
thonormal basis of
k
G. In the BF action we can disregard the
B
component of the eld B.
We choose a time direction t and write in local coordinates t, x
t, x
,x
,x
dd
t
d
x
,AA
dt A
x
A
dt
A
i
dx
i
i,,,
F
A
i
d
t
A
i
dx
i
d
x
A
dt
i
A
i
,A
dx
i
dt
iltj
F
ij
dx
i
dx
j
B
iltj
B
ij
dx
i
dx
j
B
i
dt dx
i
.
In local coordinates the action is given by
L
M
TrB F
A
.
M
I
i,j,k
TrB
ij
d
t
A
k
B
i
F
jk
dt A
DB
ijk
dx
i
dx
j
dx
k
,
where D denotes the covariant dimensional derivative with respect to the
connection A
x
.
We rst consider the pure BF theory, namely, a BF theory where no
embedded surface is considered. We perform a Legendre transformation for
the Lagrangian L the conjugate momenta to the elds B
i
and A
, namely
L
t
B
i
and
L
t
A
, are zero, so we have the primary constraints
L
B
i
j,k
ijk
F
Ax
jk
,
L
A
i,j,k
ijk
DB
ijk
..
We do not have secondary constraints since the Hamiltonian is zero hence
the time evolution is trivial. At the formal quantum level the constraints
. will be written as
F
Ax
op
physical
, DB
op
physical
,.
Paolo CottaRamusino and Maurizio Martellini
where
op
stands for operator and the vectors
physical
span the physical
Hilbert space of the theory.
We now consider the expectation value of Tr
O x, x see denition
.. We still want to work in the canonical formalism the result will
be that instead of the constraints . and ., we will have a source
represented by the assigned surface . In order to represent the surface
operator . as a source action term to be added to the BF action, we
assume that we have a current singular form J
sing
, concentrated on the
surface , of the form
J
sing
a
J
a
sing
R
a
.
Thus J
sing
can take values in G, but has no component in C G.
We are now in a position to write the operator . as
O z, z exp
M
TrHolA,
y,z
ByHolA,
z,y
J
sing
,.
where for any A, B G, A
a
A
a
R
a
,B
a
B
a
R
a
, we have set
TrAB
a
A
a
B
a
.
From . it follows that the component of J
sing
in LieG does not give
any signicant contribution. We now assume J
sing
G. In this way we
neglect possible higherorder terms in
k
G G, but this is consistent
with our semiclassical treatment.
With the above notation and taking into account the BF action as well,
the observable to be functionally integrated is
Tr
exp
M
Tr
HolA,
y,z
ByHolA,
z,y
J
sing
HolA,
z,y
FyHolA,
y,z
..
At this point the introduction of the singular current allows us to rep
resent formally a theory with sources concentrated on surfaces as a pure
BF theory with a new curvature given by the dierence of the previ
ous curvature and the currents associated to such sources. From eqn
. we conclude, moreover, that J
sing
as a form should be such that
J
sing
yd
y , , y , where d
y is the surface form of .
Now we choose a time direction and proceed to an analysis of the
Gauss constraints as in the pure BF theory. We assume that locally the
manifold M is given by M
I I being a time interval and the inter
In particular
Tr coincides with Tr when A, B LieG.
BF Theories and knots
section of the dimensional manifold M
t with the surface is an
ordinary link L
t
.
In a neighborhood of y the current J
sing
will look like
J
a
sing
xy
x yR
a
dx
dx
,y,.
where is a plane orthogonal to at y with coordinates x
and x
and
n
denotes the ndimensional delta function.
The Gauss constraints imply that in the above neighborhood of y
we have
F
a
x HolA,
z,y
J
a
sing
x yHolA,
y,z
,y,x,.
where is a loop in based at z and passing through y. The righthand
side of . is also equal to
HolA,
z,y,z
A
a
x
y,x.
Notice that the previous analysis implies that the components of the
curvature F
a
and consequently the components of the connection A
a
must take values in a noncommutative algebra, at least when sources are
present. More precisely the components F
a
as well as A
a
should be
proportional to R
a
LieG. Given the fact that B
a
and A
a
are conjugate
elds, we can conclude that the commutation relations of the quantum
theory will require B
a
to be proportional to R
a
as well.
As in the lowerdimensional case ordinary knots or links as sources
in a dimensional theory, the restriction of the connection A to the
plane looks in the approximation for which HolA,
z,x,z
like
A
a
x
R
a
d logx y .
for a given y here d is the exterior derivative.
In order to have some geometrical insight into the above connection, we
consider a linear approximation, where the surface can be approximated
by a collection of dimensional planes in R
, i.e. by a collection of hyper
planes in C
. These hyperplanes, and the hyperplane considered above,
are assumed to be in general position. This means that the intersection of
the hyperplane with any other hyperplane is given by a point.
The quantum connection . on gives a representation of the rst
homotopy group of the manifold of the arrangements of hyperplanes points
in , i.e. more simply, of the pure braid group , . It is then natu
ral to ask whether the same is true in dimensions, namely whether the
dimensional connection, corresponding to the critical curvature .
in the linear approximation dened above, is related to the higher braid
group . We will discuss this point further elsewhere let us mention
only that the dimensional critical connection is related to the existence
Paolo CottaRamusino and Maurizio Martellini
of a higherdimensional version of the KnizhnikZamolodchikov equation
associated with the representation of the higher braid groups, as suggested
by Kohno .
Path integrals and the Alexander invariant of a
knot
In the previous section we worked specically in the canonical approach.
When instead we consider a covariant approach, and compute the expec
tation values, with respect to the BF functional measure only, we can
write
Tr
O x, x
TATB Tr
exp
Tr
M
BxFxHolA,
x,z
J
sing
xHolA,
z,x
.
.
Again, in the approximation in which HolA,
x,z
, we can set
Tr
O x, x
TATB Tr
exp
Tr
M
Bx Fx J
sing
x
.
.
A rough and formal estimate of the previous expectation value . can
be given as follows.
By integration over the B eld we obtain something like
Tr
O x, x dim
TAF J
sing
,.
using a functional Dirac delta. By applying the standard formula for Dirac
deltas evaluated on composite functions, we heuristically obtain
Tr
O x, x dim detD
A
M
,.
where detD
A
M
denotes the regularized determinant of the covariant
derivative operator with respect to a background at connection A
on the
space M.
It is then natural to interpret the expectation value . with the
FaddeevPopov ghosts included as the analytic RaySinger torsion for
the complement of the knot see , . This torsion is related to
the Alexander invariant of the knot .
We omit the ghosts and gaugexing terms , since they are not relevant for a
rough calculation of ..
BF Theories and knots
Firstorder perturbative calculations and
higherdimensional linking numbers
Let us now consider the expectation value of the quantum surface observ
able Tr
O x, x, with respect to the total BF Chern action. The two
elds involved in the quantum surface operator are A
a
the connection
and B
a
,
the Beld with a , . . . , dimG and , , . . . , . The
connection A is present in the quantum surface operator via the holonomy
of the paths
y,x
and
x,y
y as in denitions . and ..
At the rstorder approximation in perturbation theory with a back
ground eld and covariant gauge, we can write the holonomy as
HolA,
y,x
y,x
A
zdz
.
At the same order, the only relevant part of the BF action is the kinetic
part B dA, so we get the following Feynman propagator
A
a
xB
b
y
i
ab
x
y
xy
..
In order to avoid singularities, we perform the usual pointsplitting regu
larization. This is tantamount to lifting the loops
x,y,x
in a neighborhood
of y. The lifting is done in a direction which is normal to the surface .
The complication here is that the loop itself based at x and passing
through y depends on the point y. While the general case seems dicult
to handle, we can make simplifying choices which appear to be legitimate
from the point of view of the general framework of quantum eld theory.
For instance, we can easily assume that when we assign to a point y the
loop
x,y,x
based at x and passing through y, then the same loop will be
assigned to any other point y
t
belonging to .
Moreover, we can consider an arbitrarily ne triangulation T of and
take into consideration only one loop
T
which has nonempty intersection
with each triangle of T. This approximation will break gauge invariance,
which will be, in principle, recovered only in the limit when the size of the
triangles goes to zero. For a related approach see . The advantage of
this particular approximation is that we have only to deal with one loop
T
, which by pointsplitting regularization is then completely lifted from
the surface. We call this lifted path
r
T
, where r stands for regularized.
Finally we get to rstorder approximation in perturbation theory and
Paolo CottaRamusino and Maurizio Martellini
up to the normalization factor
Tr
O x, x
dim
C
i
,,,
dx
dx
r
T
dy
x
y
xy
,
.
where C
is the trace of the quadratic Casimir operator in the represen
tation . When is a sphere, then . is given by
dim
C
i
L
r
T
,
where L
r
T
, is the higherdimensional linking number of with
r
T
.
Wilson channels
Let again represent our knot surface embedded in the manifold
M. Carter and Saito , , inspired by a previous work by Roseman
, describe a knot in fact an embedded sphere in space by the
dimensional analogue of a knot diagram they project down to a
dimensional space, keeping track of the over and undercrossings. One of
the coordinates of this space is interpreted as time. The intersection of
this dimensional diagram of a knot with a plane at a xed time gives
an ordinary link diagram, while the collection of all these link diagrams at
dierent times stills gives a movie representing the knot.
In the Feynman formulation of eld theory, we start by considering the
surface embedded in the manifold M directly, not its projection. At
the local level M will be given by M
I I being a time interval, so
that the intersection of the dimensional manifold M
t with gives
an ordinary link L
t
for all times t in M
. In order to connect quantum
eld theory with the standard theory of knots, we assume more simply
that M is the space R
and that is a sphere. One coordinate axis
in R
is interpreted as time, and the intersection of with space at
xed times is given by ordinary links, with the possible exception of some
critical values of the time parameter.
In other words, we have a space projection
t
R
R
for each time
t, so that
L
t
t
R
is, for noncritical times, an ordinary link. For any time interval t
,t
we
While the normalization factor for a pure dimensional BF theory appears to be
trivial , in a theory with a combined BF Chern action we have a contribution
coming from the Chern action. This contribution has been related to the rst Donaldson
invariant of M .
BF Theories and knots
will also use the notation
t,t
tt,t
t
R
.
We consider a set T of noncritical times t
i
i,,n
. We denote the
components of the corresponding links L
ti
by the symbols K
ji
i
,j
, . . . , si. Here and below we will write ji simply to denote the fact
that the index j ranges over a set depending on i. We choose one base
point x
ji
i
in each knot K
ji
i
and we denote by
ji,ji
i
the surface
contained in whose boundary includes K
ji
i
and K
ji
i
. We then as
sign a framing to
ji,ji
i
as follows for each point p in the interior
of
ji,ji
i
we associate a path with endpoints x
ji
i
and x
ji
i
passing
through p. To the surface
ji,ji
i
with boundary components K
ji
i
and
K
ji
i
, we associate the operator
HolA, K
ji
i
x
ji
i
O
ji,ji
i
x
ji
i
,x
ji
i
HolA, K
ji
i
x
ji
i
,
.
where the subscript to the term HolA, . . . denotes the basepoint.
We refer to the knot equipped with the set T of times as the
temporally sliced knot. For the sliced knot, we dene the Wilson
channels to be the operators of the form
O
,t
HolA, K
x
HolA, K
ji
i
x
ji
i
O
ji,ji
i
x
ji
i
,x
ji
i
HolA, K
ji
i
x
ji
i
O
tn,
x
jn
n
,..
We assume the following requirements for the Wilson channels
. The operators which appear in a Wilson channel are chosen in such
a way that a surface operator is sandwiched between knot operators
only if the relevant knots belong to the boundary of the surface.
. The paths that are included in the surface operators O
ji,ji
i
are not allowed to touch the boundaries of the surface, except at the
initial and nal point.
Finally, to the sliced knot we associate the product of the traces of all
possible Wilson channels. This product can be seen as a possible higher
dimensional generalization of the Wilson operator for ordinary links in
space that was considered by Witten.
We would like now to compare the Wilson channel operators associated
to a sliced knot with the operator
O,.
considered in Section no temporal slicing involved. The main dierence
concerns the prescriptions that are given involving the paths that enter
Paolo CottaRamusino and Maurizio Martellini
the denition of . or, equivalently, the framing of the knot. The
Wilson channels are obtained from . by
. choosing a nite set of times T t
i
, and
. requiring the paths to follow one of the components of the link L
t
,
for each t T .
Finally, recall that in the Wilson channel operators, each of the paths
is forbidden to follow two separate components of the same link L
t
, for
any time t T . This last condition can be understood by noticing that
in order to follow both the jth and the j
t
th components of the link L
ti
,
for some i, ji, j
t
i, one path should follow the jth component, then enter
the surface
ji,ji
i
which is assumed to be equal to
j
i,j
i
i
, and
nally go back to the j
t
th component, thus contradicting a local causality
condition.
Integration by parts
The results of the previous section imply that, given a temporally sliced
knot , we can construct quantum operators by considering all the Wilson
channels relevant to surfaces
ji,ji
i
, i l i l, and links L
ti
for
i lt l i gt l. Let us denote these quantum operators by the terms
W
in
,t
l
,W
out
,t
l
..
They can be seen as a sort of convolution, depending on , of all the
quantum holonomies associated to the links L
ti
for i lt l i gt l.
Ordinary links are the closure of braids, and one may describe the sur
face which generates the dierent links at times t
i
as a sequence movie
of braids . But a link is also obtained by closing a tangle, and the
time evolution of a link represented by a surface can also be described
in terms of tangles, i.e. movies of ordinary tangles. It has been shown
that tangles form a rigid, braided category . The tangle ap
proach with its categorical content may reveal itself as a useful one for
quantum eld theory.
The initial tangle is called the source and the nal tangle is called the
target. In our case, say, the source represents L
t
l
while the target represents
L
t
l
, for l
t
gt l. The quantum counterpart of the source and target is
represented by the quantum holonomies associated to the corresponding
links, while the quantum counterpart of the tangle is represented by the
quantum surface operator relevant to the surface
t
l
,t
l
.
One can think of the quantum surface operators as acting on quantum
holonomies by convolution, but it is dicult to do nite calculations
and write this action explicitly. What we can do instead is to make some
innitesimal calculations using integrationbyparts techniques in Feynman
path integrals.
BF Theories and knots
Fig. . Three strings
Generally speaking, in a nite time evolution t
l
t
l
, the quantum
surface operator dened by the surface with boundaries L
t
l
and L
t
l
will
map the operator W
in
,t
l
into the operator W
in
,t
l
. Let us now con
sider what can be seen as the derivative of this map. Namely, let us see
what happens when we consider the quantum operator corresponding to a
surface bordering two links L
t
l
and L
t
t
l
that dier by an elementary change.
What the functional integrationbyparts rules will show, is that one can in
terchange a variation of the surface operator with a variation of the link
and consequently of its quantum holonomy. So in order to compute the
derivative of the surface operator, we may consider variations of the link
L
t
l
.
In order to describe elementary changes of links, recall that braids or
tangles representing links at a xed time can be seen as collections of
oriented strings, while tangles describe the interaction of those strings
. We consider now two links that dier from each other only in a
small region involving three strings. Let us number these strings with
numbers , , . The string crosses under the string , then the string
crosses under the string , and the string crosses under the string . We
denote by the symbols a and b the part of each string that precedes and,
respectively, follows the rst crossing, as shown in Fig. .
The integrationbyparts rules for the BF theory are as follows. We
consider a knot K
jl
l
, which we denote here by C in order to simplify the
notation, and a point x C. By the symbol
x
we mean the variation of
C obtained by inserting an innitesimally small loop c
x
in C at the point
x. We obtain
x
HolA, C
x
exp
M
TrB F
,,a
d
c
x
HolA, C
x
F
a
R
a
exp
M
TrB F
Paolo CottaRamusino and Maurizio Martellini
,,,,a
d
c
x
exp
M
TrB F
HolA, C
x
R
a
B
a
x
,
.
where d
c
x
denotes the surface form of the surface bounded by the in
nitesimal loop c
x
,R
a
denotes as usual an orthonormal basis of LieSUn,
and the trace is meant to be taken in the fundamental representation. The
second equality has been obtained by functional integration by parts in this
way we produced the functional derivative with respect to the Beld. The
integrationbyparts rules of the BF theory in dimensions completely
parallel the integrationbyparts rules of the ChernSimons theory in
dimensions . When we apply the functional derivative
B
a
x
to a
surface operator O
t
x
t
,x
t
t
for a surface
t
whose boundary includes
C
we obtain
B
a
x
O
t
x
t
,x
t
t
R
a
xx
t
d
t
,
O
t
x
t
,x
t
t
,.
where d
t
is the surface form of
t
.
From eqn . we conclude that only the variations c
x
for which d
t
d
c
x
, matter in the functional integral. But these variations are exactly
the ones for which the small loop c
x
is such that the surface element d
c
x
is transverse to the knot C as in .
Now we come back to the link L
t
l
with the three crossings as in Fig.
. In order to consider the more general situation we assume that the
three strings in Fig. belong to dierent components K
l
,K
l
,K
l
of L
t
l
that will evolve into three dierent components K
l
,K
l
,K
l
of L
t
l
.
By performing three such small variations at each one of the crossings of
the three strings considered in Fig. , we can switch each undercrossing
into an overcrossing. Now as in we compare the expectation value of
the original conguration with the conguration with the three crossings
switched. Such comparison, with the aid of eqns . and . and the
Fierz identity , leads to the conclusion
that when we consider the
action of the surface operators O
j,j
l
x
j
l
,x
j
l
, j , , , on the quantum
holonomies of the knots K
j
l
, its innitesimal variation O can be expressed
as follows
OW
l
W
l
..
Here W
l
is a short notation for the operator
We are considering here only the small variations of
, so we can disregard the
holonomies of the paths in the surface operators the only relevant contribution to be
considered is the one given by the eld B
B
a
R
a
.
Details will be discussed elsewhere.
BF Theories and knots
W
l
HolA, K
l
x
l
O
,
l
x
l
,x
l
HolA, K
l
x
l
HolA, K
l
x
l
O
,
l
x
l
,x
l
HolA, K
l
x
l
HolA, K
l
x
l
O
,
l
x
l
,x
l
HolA, K
l
x
l
the dots in . involve the operators W
in
,t
l
and W
out
,t
l
see .,
the vector space associated to the string j is given by a tensor product
V
j
a
V
j
b
, and nally is a suitable representation of the following operator
/
a
R
a
R
a
a,a
/
a
R
a
R
a
b,a
/
a
R
a
R
a
b,b
.
Now we recall that /
a
R
a
R
a
is the innitesimal approxima
tion of a quantum YangBaxter matrix R. Hence the calculations of this
section suggest that, at the nite level, the solution of the Zamolodchikov
tetrahedron equation which may be more relevant to topological quantum
eld theories is the one considered in , , namely
S
R
a,a
R
b,a
R
b,b
,
where R is a solution of the quantum YangBaxter equation.
Acknowledgements
S. Carter and M. Saito have kindly explained to us many aspects of their
work on knots, both in person and via email. We thank them very much.
We also thank A. Ashtekar, J. Baez, R. Capovilla, L. Kauman, and J.
Stashe very much for discussions and comments. Both of us would also
like to thank J. Baez for having organized a very interesting conference and
for having invited us to participate. P. C.R. would like to thank S. Carter
and R. Capovilla for invitations to the University of South Alabama and
Cinvestav Mexico, respectively.
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actions of surfaces in dimension , Lett. Math. Phys.
.
BF Theories and knots
. J. S. Carter and M. Saito, Reidemeister moves for surface isotopies
and their interpretation as moves to movies, J. Knot Theory Rami
cations, .
. D. Roseman, Reidemeistertype moves for surfaces in four
dimensional space, preprint .
. J. S. Carter and M. Saito, Knotted surfaces, braid movies, and be
yond, this volume.
. J. E. Fischer, Jr, Geometry of categories, PhD Thesis, Yale Uni
versity .
. A. B. Zamolodchikov, Tetrahedra equations and integrable systems
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Paolo CottaRamusino and Maurizio Martellini
Knotted Surfaces, Braid Movies, and Beyond
J. Scott Carter
Department of Mathematics, University of South Alabama,
Mobile, Alabama , USA
email cartermathstat.usouthal.edu
Masahico Saito
Department of Mathematics, University of Texas at Austin,
Austin, Texas , USA
email saitomath.utexas.edu
Abstract
Knotted surfaces embedded in space can be visualized by means
of a variety of diagrammatic methods. We review recent develop
ments in this direction. In particular, generalizations of braids are
explained. We discuss generalizations of the YangBaxter equation
that correspond to diagrams appearing in the study of knotted sur
faces. Our diagrammatic methods of producing new solutions to
these generalizations are reviewed. We observe categorical aspects
of these diagrammatic methods. Finally, we present new observa
tions on algebraic/algebraictopological aspects of these approaches.
In particular, they are related to identities among relations that ap
pear in combinatorial group theory, and cycles in Cayley complexes.
We propose generalizations of these concepts from braid groups to
groups that have certain types of Wirtinger presentations.
Introduction
It has been hoped that ideas from physics can be used to generalize quan
tum invariants to higherdimensional knots and manifolds. Such possibili
ties were discussed in this conference.
In dimension Jonestype invariants were dened algebraically via braid
group representations or diagrammatically via Kauman brackets. The
rst step, then, in the dimensionally higher case is to get algebraic or
diagrammatic ways to describe knotted surfaces embedded in space.
In this paper, we rst review recent developments in this direction.
Reidemeistertype moves were generalized by Roseman and their movie
versions were classied by the authors , . On the other hand, various
braid forms for surfaces were considered by several other authors. In partic
ular, Kamada proved generalized Alexanders and Markovs theorems
for surface braids dened by Viro. We combined the movie and the surface
braid approaches to obtain moves for the braid movie form of knotted sur
J. Scott Carter and Masahico Saito
faces. The braid movie description is rich in algebraic and diagrammatic
avor.
The exposition in Section provides the basic topological tools for nd
ing new invariants along these lines if they exist.
On the other hand, knot theoretical, diagrammatic techniques have
been used to solve algebraic problems related to statistical physics. In par
ticular, we gave solutions to various types of generalizations of the quan
tum YangBaxter equation QYBE. Such generalizations originated in
the work of Zamolodchikov . In Section we review our diagrammatic
methods in solving these equations.
Such generalizations appeared in the context of categories and
higher algebraic structures . In Section we follow Fischer to show
that braid movies form a braided monoidal category in a natural way.
New observations in this paper are in Section , where we discuss close
relations between braid movie moves and various concepts in algebra and
algebraic topology. Some of the moves correspond to cycles in Cayley
complexes. Others are related to Peier equivalences that are studied in
combinatorial group theory and homotopy theory. The charts dened in
Section can be regarded as pictures that are dened for any group
presentations of null homotopy disks in classifying spaces. We propose
generalized chart moves for groups that have Wirtinger presentations.
Movies, surface braids, charts, and isotopies
. Movie moves
The motion picture method for studying knotted surfaces is one of the most
well known. It originated with Fox and has been developed and used
extensively for example , , , , . In , we gave a set of
Reidemeister moves for movies of knotted surfaces such that two movies
described isotopic knottings if and only if one could be obtained from the
other by means of certain local changes in the movies. In order to depict
the movie moves, we rst projected a knot onto a dimensional subspace
of space.
Here a map from a surface to space is called generic if every point
has a neighborhood in which the surface is either embedded, two
planes intersecting along a doublepoint curve, three planes intersecting
at an isolated triple point, or the cone on the gure eight also called
Whitneys umbrella. Such a map is called generic because the collection
of maps of this form is an open dense subset of the space of all the smooth
maps see . Local pictures of these are depicted in Fig. .
Then we x a height function on the space. By cutting space by
level planes, we get immersed circles with nitely many exceptions where
critical points occur. We also can include crossing information by break
ing circles at underpasses. By a movie we mean a sequence of such curves
Knotted Surfaces, Braid Movies, and Beyond
Fig. . Generic intersection points
J. Scott Carter and Masahico Saito
on the planes. Each element in a sequence is called a still. Again without
loss of generality we can assume that successive stills dier by at most a
classical Reidemeister move or a critical point of the surface. The Reide
meister moves are interpreted as critical points of the double points of the
projected surface, and so the movies contain critical information for the
surface and for the projection of the surface. Finally, the moves to these
movies were classied following Roseman by means of an analysis of
foldtype singularities and intersection sets.
The movie moves are depicted in Figs and . In these gures each still
represents a local picture in a larger knot diagram. The dierent stills dier
by either a Reidemeister move or critical point birth/death or saddle on
the surface. These are called the elementary string interactions ESIs.
Figure depicts ESIs. The moves indicate when two dierent sequences
of ESIs depict the same knotting. Any two movie descriptions of the same
knot are related by a sequence of these moves. One rather large class of
moves is not explicitly given in this list namely, distant ESIs commute
that is, when two string interactions occur on widely separated segments,
the order of the interactions can be switched.
. Surface braids
In April , Oleg Viro described to us the braided form of a knotted
surface. Similar notions had been considered by Lee Rudolph and X.
S. Lin . Kamada has studied Viros version of surface braids
we follow Kamadas descriptions.
.. Denitions
Let B
and D
denote disks. A surface braid is a compact oriented surface
F properly embedded in B
D
such that
the projection p B
D
D
to the second factor restricted to F,
p
F
FD
, is a branched covering
The boundary of F is the standard unlink F n points D
B
D
B
D
where n points are xed in the interior of
B
.
Here the positive integer n is called the surface braid index. A surface
braid is called simple if every branch point has branch index . A surface
braid is simple if the covering in condition above is simple. Throughout
the paper we consider simple surface braids only.
Two surface braids are equivalent if they are isotopic under level
preserving isotopy relative to the boundary, where we regard the projection
pB
D
D
as a disk bundle over a disk.
.. Closed surface braids
Let F B
D
be a surface braid with braid index n. Embed B
D
in space standardly. Then F is the unlink in the sphere B
D
Knotted Surfaces, Braid Movies, and Beyond
Fig. . The elementary string interactions
J. Scott Carter and Masahico Saito
Fig. . The Roseman moves
Knotted Surfaces, Braid Movies, and Beyond
Fig. . The remaining movie moves
J. Scott Carter and Masahico Saito
so that we can cap o these circles with the standard embedded disks in
space to obtain a closed, orientable embedded surface in space. The
closed surface,
F called the closure of F, is dened up to ambient isotopy.
A surface obtained this way is called a closed surface braid.
Viro and Kamada have proven an Alexander theorem for surfaces.
Thus, any orientable surface embedded in R
is isotopic to a closed surface
braid. Furthermore, Kamada has recently proven a Markov theorem
for closed surface braids in the PL category. However, in his proof he does
not assume that the branch points are simple. An open problem, then, is
to nd a proof of the Markov theorem for closed surface braids in which
each branch point is simple. Such a theorem would show a clear analogy
between classical and higherdimensional knot theory.
.. Braid movies
Write B
and D
as products of closed unit intervals B
B
B
,
D
D
D
. Factor the projection p B
D
D
into two projections
p
B
B
D
B
D
and p
B
D
D
.
Let F be a surface braid. We may assume without loss of generality
that p
restricted to F is generic. We further regard the D
factor in
B
D
B
D
D
as the time direction. In other words, we
consider families P
t
B
D
ttD
. Except for a nite
number of values of t D
the intersection p
FP
t
is a collection of
properly immersed curves on the disk P
t
.
Exceptions occur when P
t
passes through branch points, or local max
ima/minima of the doublepoint set, or triple points. We may assume that
for any t D
there exists a positive such that at most only one of such
points is contained in
st,t
P
s
. If one of such points is contained,
the dierence between p
FP
s
, s t , is one of the following
smoothing/unsmoothing of a crossing corresponding to a branch point,
type II Reidemeister move corresponding to the maximum/minimum
of the doublepoint set, or type III Reidemeister move corresponding
to a triple point.
Except for such t, p
FP
t
is the projection of a classical braid. Fur
thermore, we can include crossing information by breaking the underarc as
in the classical case. Here the underarc lies below the overarc with respect
to the projection p
. To make this convention more precise we identify the
image of p
with an appropriate subset in B
D
. This idea was used by
Kamada when he constructed a surface with a given chart. See and
also the next section.
By this convention we can describe surface braids by a sequence of
classical braid words. Including another braid group relation
i
j
j
i
,
i j gt we dene a braid movie as follows.
A braid movie is a sequence b
,...,b
k
of words in
i
, i , . . . , n,
such that for any m, b
m
is obtained from b
m
by one of the following
Knotted Surfaces, Braid Movies, and Beyond
changes
insertion/deletion of
i
,
insertion/deletion of a pair
i
i
,
replacement of
i
j
by
j
i
where i j gt ,
replacement of
i
j
i
by
j
i
j
where i j .
These changes are called elementary braid changes or EBCs, for short.
This situation is illustrated in Fig. .
Two braid movies are equivalent if they represent equivalent surface
braids.
. Chart descriptions
A chart is an oriented labelled graph embedded in a disk, the edges are
labelled with generators of the braid group, and the vertices are of one of
three types
A black vertex has valence .
A valent vertex can also be thought of as a crossing point of the
edges of the graph. The labels on the edges at such a crossing must
be braid generators
i
and
j
where i j gt .
A white vertex on the chart has valence . The edges around the vertex
in cyclic order have labels
i
,
j
,
i
,
j
,
i
,
j
where i j . Here,
the orientations are such that the rst three edges in this labelling
are incoming and the last three edges are outgoing.
Consider the double point set of F under the projection p
F
B
D
FB
D
which is a subset of B
D
. Then a chart depicts the
image of this double point set under the projection p
B
D
D
in
the following sense
The birth/death of a braid generator occurs at a saddle point of the
surface at which a doublepoint arc ends at a branch point. This
projects to a black vertex on a chart.
The projection of the braid exchange
i
j
j
i
for i j gt is a
valent vertex or a crossing in the graph.
The valent vertex corresponds to a Reidemeister type III move, and
this in turn is a triple point of the projected surface.
Maximal and minimal points on the graph correspond to the birth or
death of
i
i
, and this is an optimum on the doublepoint set.
The relationships among surface projections, braid movies, and charts
is depicted in Fig. .
.. Reconstructing the surface braid from a chart
A chart gives a complete description of a surface braid in the following way.
Recall that the D
factor of D
is interpreted as a time direction. Thus
J. Scott Carter and Masahico Saito
Fig. . Braid movies, charts, and diagrams
Knotted Surfaces, Braid Movies, and Beyond
Fig. . Chart moves
J. Scott Carter and Masahico Saito
the vertical axis of the disk points in the direction of increasing time. Each
generic intersection of a horizontal line with the graph generic intersections
will miss the vertices and the critical points on the graph gives a sequence
of braid words. The letters in the word are the labels associated to the
edges the exponent on such a generator is positive if the edge is oriented
coherently with the time direction, otherwise the exponent is negative. By
taking a single slice on either side of a vertex or a critical point on the
graph, we can reconstruct a braid movie from the sequence of braid words
that results. Therefore, given a chart and a braid index, we can construct
a surface braid and, indeed, a knotted surface by closing the surface braid.
Theorem . Kamada Every surface braid gives rise to an oriented
labelled chart, and every oriented labelled chart completely describes a sur
face braid.
Furthermore, Kamada provided a set of moves on charts. His list
consists of three moves called CI, II, and III. From the braid movie point of
view, we need to rene his list incorporating movie moves. Such generalized
chart moves, yielding braid movie moves that will be discussed in the next
section, are listed in Fig. .
. Braid movie moves
The following types of changes among braid movies are called braid
movie moves
CIR
i
,
i
j
j
,
j
i
j
i
,
j
j
i
,
j
i
j
,
where i j gt .
CIR
t
i
,
i
i
i
,
i
i
.
CIR
i
j
,
j
i
,
i
j
i
j
,
where i j gt .
CIR
i
k
j
,
k
i
j
,
k
j
i
,
j
k
i
i
k
j
,
i
j
k
,
j
i
k
,
j
k
i
,
where i j gt , j k gt , k i gt .
CIR
k
i
j
i
,
i
k
j
i
,
i
j
k
i
,
i
j
i
k
,
j
i
j
k
k
i
j
i
,
k
j
i
j
,
j
k
i
j
,
j
i
k
j
,
j
i
j
k
,
where i j and i k gt , j k gt .
Knotted Surfaces, Braid Movies, and Beyond
CIM
empty word empty word,
i
i
, empty word.
CIM
i
i
, empty word,
i
i
i
i
.
CIM
i
j
i
,
j
i
j
,
i
j
i
i
j
i
,
where i j .
CIM
i
j
k
i
j
i
,
i
j
i
k
j
i
,
j
i
j
k
j
i
,
j
i
k
j
k
i
,
j
k
i
j
k
i
,
j
k
i
j
i
k
,
j
k
j
i
j
k
,
k
j
k
i
j
k
i
j
k
i
j
i
,
i
j
k
j
i
j
,
i
k
j
k
i
j
,
k
i
j
k
i
j
,
k
i
j
i
k
j
,
k
j
i
j
k
j
,
k
j
i
k
j
k
,
k
j
k
i
j
k
,
where k j i or k j i .
CIM
j
i
,
i
i
j
i
,
i
j
i
j
j
i
,
j
i
j
j
,
i
j
i
j
,
where i j .
CII
j
,
j
i
,
i
j
j
,
i
j
,
where i j gt .
CIII
i
j
,
i
j
i
,
j
i
j
i
j
,
j
i
j
,
where i j .
CIII
j
j
,
j
i
j
,
i
j
i
j
j
, empty word,
i
i
,
i
j
i
,
where i j .
CIV
empty word,
i
i
,
i
empty word,
i
.
Furthermore, we include the variants obtained from the above by the fol
lowing rules
replace
i
with
i
for all or some of i appearing in the sequences if
possible,
if b
,...,b
k
b
t
,...,b
t
k
is one of the above, add b
k
,...,b
b
t
k
,...,b
t
,
J. Scott Carter and Masahico Saito
if b
,...,b
k
b
t
,...,b
t
k
is one of the above, add c
,...,c
k
c
t
,...,c
t
k
where c
j
resp. c
t
j
is the word obtained by reversing the
order of the word b
j
resp. b
t
j
.
combinations of the above.
The braid movie moves are depicted in Figs and . There is an overlap
between the braid movie moves and the ordinary movie moves that overlap
is to be expected.
. Denition locality equivalence
Let b
,...,b
n
and b
t
,...,b
t
n
be two braid movies that satisfy the fol
lowing condition. There exists i, i n, such that
b
j
b
t
j
for all j , i , i, i ,
b
j
resp. b
t
j
, j i , i, i , are written as b
j
u
j
v
j
resp.
b
t
j
u
t
j
v
t
j
where u and v satisfy
au
i
resp. v
t
i
is obtained from u
i
resp. v
t
i
by one of
the elementary braid changes, say , and v
i
v
i
resp.
u
t
i
u
t
i
,
bv
i
resp. u
t
i
is obtained from v
i
resp. u
t
i
by one of
the elementary braid changes, say , and u
i
u
i
resp.
v
t
i
v
t
i
.
Then we say that b
t
,...,b
t
n
is obtained from b
,...,b
n
by a locality
change or vice versa. If two braid movies are related by a sequence of lo
cality changes then they are called equivalent under locality. Loosely speak
ing, equivalence under locality changes means that distant EBCs commute.
Theorem . Two braid movies are equivalent if and only if they
are related by a sequence of braid movie moves and locality changes.
It is interesting to note that in the braid movie point of view, the move
that corresponds to a quadruple point gives a movie move corresponding to
the permutohedral equation rather than to the tetrahedral equation. We
will discuss these equations at some length in Section .
The permutohedral and tetrahedral equations
In , Zamolodchikov presented a system of equations and a solution
to this system that corresponded to the interaction of straight strings in
statistical mechanics. This system of equations is related to the Yang
Baxter equations, and this relationship suggests topological applications.
Baxter showed that the Zamolodchikov solution worked. And in
higherdimensional analogues were dened. In , these systems are
interpreted as consistency conditions for the obstructions to the lower di
mensional equations having solutions. Frenkel and Moore produced
a formulation and a solution of a variant of the Zamolodchikov tetrahe
dral equation. And Kapranov and Voevodsky consider the solution of
Knotted Surfaces, Braid Movies, and Beyond
Fig. . The braid movie moves
J. Scott Carter and Masahico Saito
Fig. . The braid movie moves
Knotted Surfaces, Braid Movies, and Beyond
the Zamolodchikov equation to be a key feature in the construction of a
braided monoidal category. In a parallel development, Ruth Lawrence
described the permutohedral equation and two others as natural equa
tions for which solutions in a algebra could be found.
There are other solutions of these higherorder equations known to us.
In particular, we mention , , , and . In , solutions are
studied in relation to quasicrystals. We expect there are even more solu
tions in the literature and would appreciate being told of these.
Here we review these equations and our methods for solving them.
. The tetrahedral equation
The FrenkelMoore version of the tetrahedral equation is formulated as
S
S
S
S
S
S
S
S
.
where S EndV
, V a module over a xed ring k. Each side of the
equation acts on V
V
V
V
V
, and S
, for example, acts on
V
as the identity. Notice here that eqn . has the nice property that
each index appears three times on either side of the equation.
Compare this equation with the movie move that involves ve frames
on either side. On the lefthand movie, the elementary string interactions
that occur are Reidemeister type III moves. Label the strings to from
left to right in the rst frame of the movie. This index is one more than the
number of strings that cross over the given string. Then the Reidemeister
type III moves occur in order among the strings , , , .
Thus we think of the tensor S as corresponding to a type III Reidemeister
move. The equation corresponds to the given movie move as these type
III moves occur in opposite order on the righthand side of the move. This
correspondence was also pointed out by Towber.
.. A variant
There is a natural variant of this equation
S
S
S
S
S
S
S
S
.
where each side of the equation now acts on V
. In this equation each
index appears twice in each side. This tetrahedral equation was also for
mulated in , , .
We review how to obtain eqn . from eqn . cf. . Dene a
set of pairs of integers among to
C, , , , , ,
with the lexicographical ordering as indicated. There are six elements and
the ordering denes a map C, , . . . , . The rst factor of the
lefthand side of the equation . is S
. Consider the pairs of integers
among this subscript , , in the lexicographical ordering. Under
J. Scott Carter and Masahico Saito
Fig. . Solving the Zamolodchikov equation
Knotted Surfaces, Braid Movies, and Beyond
these are mapped to , , which is the subscript of the rst factor
in eqn .. The other factors are obtained similarly. For example, the
second factor of . has the subscript which yields pairs , ,
which are mapped under to , , giving the second factor of ..
This is how we obtain . from ..
Next we observe that eqn . is solvable appropriate products of QYB
solutions give solutions to .. Let V W W where W is a module
over which there is a QYB solution R R
R
R
R
R
R
. For given
V give a and b as subscripts of W V
i
W
ia
W
ib
for i , . . . , . For
SS
acting on V
V
V
W
a
W
b
W
a
W
b
W
a
W
b
set
S
R
aa
R
ba
R
bb
.
One can compute
S
S
S
S
R
aa
R
ba
R
bb
R
aa
R
ba
R
bb
R
aa
R
ba
R
bb
R
aa
R
ba
R
bb
R
aa
R
aa
R
aa
R
ba
R
ba
R
aa
R
bb
R
ba
R
ba
R
bb
R
bb
R
bb
R
aa
R
aa
R
aa
R
aa
R
ba
R
ba
R
ba
R
ba
R
bb
R
bb
R
bb
R
bb
R
aa
R
ba
R
bb
R
aa
R
ba
R
bb
R
aa
R
ba
R
bb
R
aa
R
ba
R
bb
S
S
S
S
.
Thus . is solvable.
This computation is a bit tedious algebraically. We obtained this result
by considering the preimage of four planes of dimension generically inter
secting in space. The movie version of such planes is depicted in Fig. .
The numbering in the gure matches those described above. Specically,
the intersection points among planes and receive the number since
is the rst in the lexicographical ordering. Each plane is one of the
strings as it evolves on one side of the movie of the corresponding movie
move. The crossing points of the remaining planes give one side of the
Reidemeister type III move on the preimage, and on the other side of the
movie move the other side of the Reidemeister type III move appears. Thus
the solution is a piece of bookkeeping based on this picture.
.. Remarks
In we showed that solutions to the FrenkelMoore equations could
also produce solutions to higherdimensional equations. All of the higher
dimensional solutions are obtained because the equations correspond to the
intersection of hyperplanes in hyperspaces, and the selfintersection sets of
these contain intersections of lowerdimensional strata.
J. Scott Carter and Masahico Saito
Fig. . The permutohedral equation
. The permutohedral equation
This equation was studied in and
S
S
R
R
S
S
R
R
S
S
R
R
S
S
.
The subscripts indicate factors of vector spaces on which the tensor acts
nontrivially, and it acts as the identity on the other factors. The tensors
S and R live in EndV
and EndV
, respectively, and both sides of
the equation live in EndV
.
A pictorial representation is given in Fig. . This gure shows two
sides of a permutohedron truncated octohedron. The gure is the dual to
the CIM move on charts. Each hexagon is dual to a valence vertex,
and each parallelogram is dual to a crossing point on the graph. Thus this
equation is a natural one to consider from the point of view of knotted
surfaces, as it expresses one of the braid movie moves.
The equation expresses the equality among tensors that are associated
to the faces on the sides of the permutohedron. Each hexagon represents
the tensor S and each parallelogram represents R. Parallel edges receive
the same number, and the numbers assigned to the boundary edges of
hexagons and parallelograms determine the subscripts of the tensor.
Finally, observe that by setting R equal to the identity, the equation
reduces to the variant of the tetrahedral equation given above.
Knotted Surfaces, Braid Movies, and Beyond
Fig. . Assigning tensors to the hexagons
.. Solutions
Suppose A, B, M, and R EndV V satisfy the following conditions
AAA AAA, BBB BBB .
MMA AMM, BMM MMB .
MMA AMM, BMM MMB .
AMB BMA .
AMR RMA, RMB BMR .
ARM MRA, MRB BRM .
MAR RAM .
ARB BRA, BRA ARB .
BAR RAB, RAB BAR .
MRM MRM .
where the LHS resp. RHS of every relation has subscripts , , and
resp. , , and and lives in EndV
. The subscripts indicate the
factor in the tensor product on which these operators are acting. For exam
ple, AMM MMA stands for A
M
M
M
M
A
EndV
.
We do not require the equalities with the subscripts of the RHS and LHS
switched. In case those equalities are used, they are listed explicitly as in
eqns . and ..
J. Scott Carter and Masahico Saito
Fig. . The computational technique
Knotted Surfaces, Braid Movies, and Beyond
Proposition . If the conditions stated in eqns .. hold, then
S
A
M
B
is a solution to the permutohedral equation.
The proof is presented in detail in . Here we include the illustrations
that led to the proof. In this way we reinforce the diagrammatic nature of
the subject.
In Fig. , we assign the product ABM to each of the S tensors. Then
in Fig. , we indicate that if A, B, M, and R satisfy YangBaxterlike
relations, then we can get from one side of the permutohedron to the other.
To read this gure start at the upper left, travel left to right, then go
directly down to the second row and read right to left. The zigzag pattern
continues.
Now in the solution given in . set A M R. Then the list of the
conditions reduces to
RRR RRR, BBB BBB .
BRR RRB, RRB BRR .
with the same subscript convention as in eqns ...
To get Rmatrices that satisfy these conditions, we embedded quantum
groups into bigger quantum groups. This idea was inspired by Kapranov
and Voevodskys idea of using projections of quantum groups to obtain
solutions to Zamolodchikovs equation .
There exists a canonical embedding
i
U
qi
sl U
q
sln
corresponding to the ith vertex of the Dynkin diagram, where q
i
q
i
,
i
/, , is the canonical inner product,
i
r
i
is a basis of
simple roots of the root system, and r is the rank p. , . Let
R
t
resp. B be the universal Rmatrix in U
qi
sl U
qi
sl resp.
U
q
sln U
q
sln. Let R
i
R
t
where i is xed.
If we take a representation f U
q
sln EndV for a vector space V
over the complex numbers, both fR and fB satisfy the quantum Yang
Baxter equation RRR RRR, BBB BBB since fR f
i
R
t
is
a representation of U
qi
sl and R
t
is a universal Rmatrix. Furthermore,
in we showed that the remaining relations held between R and B. In
this way, we were able to construct solutions.
. Planar versions
Simplex equations can be represented by planar diagrams when the sub
scripts consist of adjacent sequences of numbers. We can use planar dia
grams to solve such equations.
J. Scott Carter and Masahico Saito
.. Planar tetrahedral equations
One such equation is
S
S
S
S
S
S
S
S
.
We regarded the corresponding planar diagram as the diagrams in the
TemperleyLieb algebra. Elements in the TemperleyLieb algebra are rep
resented by diagrams involving and . Playing with such diagrams, we
found a continuous family of solutions to the above equation expressed
as a linear combination with variables in generators of the Temperley
Lieb algebra. Such generalizations of the Kauman bracket seem useful in
general.
.. The planar permutohedral equation
It was observed by Lawrence that the permutohedral equation becomes
the following equation
r
s
s
r
r
s
s
s
s
r
r
s
s
r
.
after multiplying by permutation maps s P
S, r PR.
We found some new and interesting solutions to this version by using the
Burau representation of the braid group, and these were presented in .
Some of the solutions were noninvertible, and another class of solutions
reected the noninjectivity of the Burau representation.
categories and the movie moves
At the Isle of Thorns Conference in , Turaev described the category
of tangles and laid the foundations for his paper . The objects in the
category are in onetoone correspondence with the nonnegative integers.
And the morphisms are isotopy classes of tangles joining n dots along a
horizontal to m dots along a parallel horizontal. Such a tangle, then,
is a morphism from n to m. A week later in France similar ideas were
being discussed by Joyal . All of the known generalizations of the
Jones polynomial can be understood as representations on this category.
Consequently, there is hope that the yet to be found higherdimensional
generalizations can be found as representations of the category of tangles
that we will describe here.
. Overview of categories
In a category, there are objects. For any two objects in a category there
is a set of morphisms. A category has objects, morphisms, and mor
phisms between morphisms called morphisms. The morphisms can
be composed in essentially two dierent ways either they can be glued
along common morphisms, or they can be glued along common objects.
Knotted Surfaces, Braid Movies, and Beyond
Obviously, they can be glued only if they have either a morphism or an
object in common.
Just as the fundamental problem in an ordinary category is to determine
if two morphisms are the same, the fundamental problem in a category is
to determine if two morphisms are the same. The geometric depiction of a
category consists of dots for objects, arrows between dots for morphisms,
and polygons where poly means two or more for morphisms. Thus,
by composing morphisms we obtain faces of polyhedra, and an equality
among morphisms is a solid bounded by these faces.
. The category of braid movies
This section is an interpretation of Fischers work in the braid movie
scheme. Here we give a description of the category associated to nstring
braids. There is one object, the integer n, and this is identied with n
points arranged along a line. The set of morphisms is the set of nstring
braid diagrams without an equivalence relation of isotopy imposed. Two
diagrams related by a levelpreserving isotopy of a disk which keeps the
crossings are identied, but for example two straight lines and a braid dia
gram represented by
i
i
are distinguished. Equivalently, a morphism is
a word in the letters
,...,
n
. A generating set of morphisms is the
collection of EBCs.
More generally, consider the category in which the collection of objects
consists of the natural numbers , , , . . . each number n is identied
with n dots arranged along a horizontal line. The set of morphisms from
n to m when m , n is empty, and the set of morphisms from n to n is
the set of nstring braid diagrams or equivalently words on the alphabet
,...,
n
. A tensor product is dened by addition of integers, and
acts as an identity for this product. A morphism is a surface braid that
runs between two nstring braid diagrams. Equivalently, a morphism can
be represented by a movie in which stills dier by at most an EBC.
We dene a braiding which is a morphism from n m to m n,
by braiding n strings in front of m strings that is, the braid diagram
represented by the braid word
n
n
nm
n
n
nm
m
.
Assertion With the tensor product, identity, and braiding dened
above, we can dene all the data various morphisms in terms of braid
movies such that the category of braid movies denes a braided monoidal
category.
Sketch of Proof In Figs and a set of movie moves to ribbon
diagrams are depicted. These are illustrations of the KapranovVoevodsky
axioms as interpreted in our setting.
In the middle of gures polytopes are depicted. These are two exam
J. Scott Carter and Masahico Saito
Fig. . A ribbon movie move
Knotted Surfaces, Braid Movies, and Beyond
Fig. . Another ribbon movie move
J. Scott Carter and Masahico Saito
ples of axioms in that are to be satised by morphisms. Letters
represent objects where tensor product notation is abbreviated. Edges in
these polytopes are morphisms. They are in turn represented by braid
words. The ribbon diagrams surrounding the polytopes represent braid
movies. Ribbons represent parallel arcs as in the classical case. Then each
morphism, a braid word, is represented by ribbon diagrams to simplify
movie diagrams. Therefore each ribbon diagram represents a composition
of edges in the polytopes. A morphism is a face in the polytopes, and
changes between two ribbon diagrams. The polytopes are relations satis
ed by compositions of morphisms. In ribbon diagrams, this means that
two ways of deforming ribbon diagrams have to be equal. There are two
ways to deform by braid movie moves one the top ribbon diagram to
the other the bottom by going along the left hand side of the polytope,
or the right hand side. The conditions say that these two ways are equal.
Each frame in a movie can be written as a product of braid generators
while each morphism can be decomposed as a sequence of EBCs. These
EBCs can be written in terms of charts. Then two ribbon movies are
equivalent if and only if the resulting charts are chart equivalent. Thus we
can check the equality by means of chart moves.
Alternatively, we can decompose the polytopes into simpler pieces that
correspond to the polytopes of the braid movie moves.
.. Remark
The existence of a braided monoidal category is not enough to conclude
that one has an invariant of knots. In addition, to the braiding R AB
BA, we need a braiding
R BA AB that also gives a morphism
from
R R to the identity morphism on A B. If R is thought of
as crossing the strings of A over those of B, then
R crosses Bs strings
over As. Since moves CIM and CIM hold, these morphisms are
isomorphisms.
.. Fischers result
Fischers dissertation gives an axiomatization of the category that is as
sociated to regular isotopy classes of knotted surface diagrams. That is,
he denes this category, and he shows that it is the free, rigid, braided,
semistrict monoidal category FRBSMcat generated by a set. Fischers
theorem means that if one has an RBSMcat then one automatically has
constructed an invariant of regular isotopy classes of knotted surface dia
grams.
Algebraic interpretations of braid movie moves
Our motivation in dening braid movies and their moves was to get a
more algebraic description of knotted surfaces. In particular, it is hoped
that this description can be dened and studied in a more general setting
Knotted Surfaces, Braid Movies, and Beyond
for example, for any group. Also, it is desirable if there is an algebraic
topological description of braid movies generalizing the fact that a classical
braid represents an element of the fundamental group of the conguration
space. In this section, we work towards this goal we observe close relations
between braid movie moves and various concepts in algebra and algebraic
topology.
. Identities among relations and Peier elements
In this section we recall denitions of various concepts for group presenta
tions following , where details appear.
.. Identities among relations
Let G be a group. A presentation of G is a triple X R, w such that R
is a set, w R F is a map, where F is the free group on X, and G is
isomorphic to F/NwR, where N is the normal closure.
Let H be the free group on Y R F let H F denote the
homomorphism r, u u
wru. Then H NwR. An identity
among relations is any element in the kernel of .
.. Peier transformations
A Y sequence is a sequence of elements y a
,...,a
n
where a
i
r
i
,u
i
is a generator or its inverse of H for i , , . . . , n. The follow
ing operations are called Peier transformations
A Peier exchange replaces the successive terms r
i
,u
i
,r
i
,u
i
with either
r
i
,u
i
,r
i
,u
i
u
i
wr
i
u
i
or
r
i
,u
i
u
i
wr
i
u
i
,r
i
,u
i
.
A Peier deletion deletes an adjacent pair a, a
in a Y sequence.
A Peier insertion inserts an adjacent pair a, a
into a Y sequence.
A Peier equivalence is a nite sequence of exchanges, deletions, and
insertions to a Y sequence performed in some order.
.. Pictures for presentations
Let X, R, w denote a group presentation. A picture over the presentation
consists of
A disk D with boundary D.
A collection
,...,
k
of disjoint disks in the interior of D that are
labelled by elements of R.
A collection of disjoint oriented arcs labelled by elements of X that
either form simple closed loops or join points on the boundaries of
two or possibly one of the .
J. Scott Carter and Masahico Saito
Fig. . Pictures for group presentations
A base point for each of the and one for the big disk D. The base
point on a is distinct from the endpoints of the connecting arcs.
Finally, starting from the base point and reading around the boundary
of any in a counterclockwise fashion, the labels on the edges that
are encountered should spell out either the relation R or its inverse
that is associated to the small disk. An incoming edge is interpreted
as having a positive exponent, and an outgoing edge has a negative
exponent.
In a spherical picture is dened as a picture in which none of the
edges touch the outer boundary. Figure illustrates a spherical picture.
The dotted arcs indicate how to obtain a Y sequence from a picture. The
Y sequence obtained from a spherical picture is an identity sequence in the
sense that the product a
a
n
in the free group.
A picture determines a map of a disk into BG, the classifying space
of the group with given presentation, that is dened up to homotopy, and
by transversality if such a map is given then there is a picture the little
disks are the inverse images of regular neighborhoods of points at the
center of the disks that correspond to relators. The arcs are the inverse
images of interior points on the edges of the complex.
From a picture a Y sequence can be obtained. Namely, for each
disk a dotted arc is chosen that connects the base point of the small disk
to that of the larger disk. These arcs do not intersect in their interiors.
Furthermore, the dotted arcs are chosen to be transverse to the arcs that
Knotted Surfaces, Braid Movies, and Beyond
interconnect the disks. To each dotted arc we associate a pair r
, u, where
r R and u F. The element u is the word in the free group corresponding
to the intersection sequence of the dotted arc with connecting arcs. The
exponent on a letter is determined by an orientation convention. The
element r is the intersection sequence on the boundary of read in a
counterclockwise fashion. The collection of dotted arcs is ordered in a
counterclockwise fashion at the base point, and this gives the ordering of
the Y sequence.
In , it is noted that an elementary Peier exchange corresponds to
rechoosing a pair of successive arcs from the base points of a pair of to
the base point of D. Thus pictures can be used to study Y sequences and
their Peier equivalences. They provide a diagrammatic tool for the study
of groups from an algebraictopological viewpoint.
Furthermore, they dene deformations of pictures. There are three
types of moves to pictures. The rst type is to insert or delete small circles
labelled by the generating set. This is a direct analogue of the CIM
move on charts. Secondly, surgery between arcs with compatible labels
and orientations is analogous to the CIM move. Third is an insertion
or deletion of a pair of small disks labelled with the same relator, but
oppositely oriented, joined together by arcs corresponding to generators
that appear in the relator. This is an isolated conguration disjoint from
other components of the graph in the picture, but corresponds to CIM
and CIR combined with CIM, CIM.
. Moves on surface braids and identities among relations
In this subsection we observe which moves on surface braids correspond to
Peier transformations dened above, and what the others correspond to.
.. Moves corresponding to Peier transformations
Braid movie moves CIM and CIR are used to perform Peier inser
tions and deletions as noted in the previous section. Peier exchanges
correspond to locality equivalences because ordering the vertices is tanta
mount to choosing a height function.
.. Moves corresponding to cycles
The chart moves CIR, CIR, and CIM are identities among relations
that correspond to nondegenerate cycles in the Cayley complex for the
standard presentation of the braid group. The correspondence is given by
taking duals. The cube, the hexagonal prism, and the permutohedron,
respectively, are the dual cycles in the Cayley complex.
.. Other CI moves
These correspond to rechoosing the dotted arcs in the homotopy classes
relative to the endpoints.
For example, CIR
t
looks like straightening up a cubic curve. Thus if
J. Scott Carter and Masahico Saito
the xaxis in the gure is a part of a dotted arc, then this move corresponds
to rechoosing a dotted arc so that it intersects the edge with two less
intersection points. Other moves can be interpreted similarly.
This rechoosing in turn corresponds to changing the words u
i
in the
terms r
i
,u
i
of the Y sequences, since dotted arcs correspond to these
words.
In summary, we interpret charts without black vertices as null homo
topies of the second homotopy group of the classifying space. Let G be a
braid group B
n
on n strings and BG its classifying space constructed from
the standard presentation. Specically, BG is constructed as follows start
with one vertex, add n oriented edges corresponding to generators, attach
cells corresponding to relators along the skeleton, add cells to kill the
second homotopy group of the skeleton thus constructed, and continue
killing higher homotopies to build BG. Note that pictures are dened for
maps of disks to BG as in the case of maps to the Cayley complexes by
transversality.
Proposition . Let C be a chart of a surface braid S without black
vertices. Note here that S is trivial i.e. S is an unlinked collection of n
spheres. Then there is a map f from a disk to BG, f D
,D
BG, vertex, such that the picture corresponding to f is the same isotopic
as a planar graph as C. Since
BG , this picture is deformed to
the empty picture by a sequence of picture moves. This is the same as a
sequence of CI moves realizing an isotopy of surface braids from S to the
trivial surface braid whose chart is the empty diagram. In other words, CI
moves of isotopy of surface braids correspond to picture moves of
BG
.
Furthermore, this deformation is interpreted as identities among rela
tions. In particular, braid movie moves correspond to Peier transforma
tions, or cells of BG, or rechoosing the dotted arcs in pictures.
. Symmetric equivalence and moves on surface braids
In this section we consider moves on group elements that are conjugates of
generators. This generalizes Kamadas denition of symmetric equivalence
for charts. These moves correspond to CII and CIII moves on charts
which are the moves involving black vertices.
.. Symmetric equivalence and charts
Fix a base point on the boundary of the disk on which the chart of a
surface braid lives. Suppose there are black vertices. Consider a small
disk B centered at a black vertex v. Pick a point d on the boundary of B
which misses the edge coming out of v. Connect d to the base point by
a simple arc which intersects the chart transversely. More precisely, the
arc misses the vertices and crossings of the chart and intersects the edges
transversely.
Knotted Surfaces, Braid Movies, and Beyond
Starting from the base point, travel along the arc and read the labels
as it intersects the edges. Then go around v along B counterclockwise,
and go back to the base point. This yields a word w in braid generators of
the form of a conjugate of a generator w Y
i
Y , where
i
is the label
of the edge coming from v, depends on the orientation of this edge,
and Y is the word we read along the arc.
Then before/after the CII move, the word thus dened changes as
follows
Y
i
YY
j
i
j
Y,
where also denotes and i j gt . Similarly a CIII move causes the
word change
Y
i
YY
j
i
j
i
i
Y,
where i j , , and .
Kamada dened symmetric equivalence between words that are conju
gates of generators in the braid group. The equivalence relation reects the
changes above. He proved that two such words are symmetrically equiv
alent i they represent the same element in the braid group, and then he
used this theorem to prove that C moves suce. Kamadas denition of
symmetric equivalence is dierent from the above. We indicated the word
changes that correspond to CII and CIII moves.
.. Chart moves that involve black vertices
Charts that have black vertices can be interpreted as pictures in the sense
of as follows. Consider a chart with black vertices. Cut the disk open
along the arcs dened above to obtain a picture in which the boundary of
the disk is mapped to the product of conjugates of generators. This gives
a picture associated to a chart that has black vertices.
The chart moves CII and CIII have the following interpretation as
moves on pictures. The words on the boundary of the disk change as
indicated above.
The picture represents a map from the disk into BGwhere Gis the braid
group. Either of these moves corresponds to sliding the disk by homotopy
across the polygon representing the given relation.
.. Generalizations of Wirtinger presentations
Generalizing Kamadas theorem, we consider Wirtinger presentations of
groups. Let G X R be a nite Wirtinger presentation. Thus each of
the relators is of the form r y
V
xV , where x, y X are generators
and V is a word in generators. Let A be the set of all the words in X i.e.,
elements in the free group on X of the form w Y
xY , where x X
and Y is a word in X. Dene symmetric equivalence on A as follows
Two words Y
xY and Z
xZ are symmetrically equivalent if Y and
Z represent the same element in G.
J. Scott Carter and Masahico Saito
Fig. . Generalized chart move
Two words Y
yY and Y
V
xV Y are symmetrically equivalent if
ry
V
xV R where x, y X.
Two words in A are symmetrically equivalent i they are related by a nite
sequence of the above changes.
In terms of pictures, if we allow univalent vertices in pictures, and if
we dene arc systems connecting univalent vertices to a base point on the
boundary of the disk, then we can diagrammatically represent the above
two changes in terms of pictures. They are depicted in Fig. . Such a
generalization of pictures will be discussed next.
We generalize pictures for group presentations to include univalent ver
tices. To make homotopytheoretic sense out of such pictures, consider
small disks centered at each univalent vertex and simple, mutually disjoint
arcs connecting them to a base point. More precisely, our data consist of
a picture on a disk where univalent vertices are allowed,
a small disk centered at each univalent vertex with a specied point
on the boundary of each disk which misses the edge coming from the
univalent vertex,
a system of mutually disjoint simple arcs such that for each univalent
vertex a simple arc from this system connects the specied point on
the boundary of the small disk to a xed base point on the boundary
of the big disk where the picture lives. These arcs miss vertices of
the picture and intersect the edges transversely.
Knotted Surfaces, Braid Movies, and Beyond
Let a
,...,a
n
be the system of arcs dened above. Starting from the
base point, read the labels of the edges intersecting a
when we trace it
towards the univalent vertex. Go around the vertex along the boundary of
the small disk and come back to the base point to get a word w
. Then
we trace a
up to a
n
to get a sequence w
,...,w
n
of words. Each word
is in a conjugate form w
i
Y
i
x
i
Y
i
i , . . . , n, where x
i
is the label
of the edge coming from the ith univalent vertex. Now we require that the
product w
w
n
is in G, the group with which we started. Call this a
generalized picture.
This has the following meaning as explained in the case of braid groups
above. Given a generalized picture, remove the interior of small disks
around each univalent vertex. Then dene a smooth map from the bound
ary big disk cut open along dotted arcs connecting small disks to the
base point, to the skeleton of BG. This edge path represents the product
w
w
n
in G
BG which was assumed to be the identity. Therefore
the map on the boundary of the cutopen disk extends to a map of the
big disk. The transverse preimages of points in cells resp. cells gives
edges resp. vertices of the rest of the picture.
Here we remark that Kamadas theorem on symmetric equivalence
seems to be generalized to other groups. Kamada used Garsides results
, and Garside gave other groups to which his arguments can be applied.
. Concluding remarks
We have indicated relations among braid movie moves, charts and classi
fying spaces of braid groups, and null homotopy disks in such spaces. We
further discussed potential generalizations to Wirtinger presented groups.
We conclude this section with remarks on these aspects.
In the preceding section we discussed null homotopy disks in classifying
spaces in the presence of generalized black vertices. However, we do not
want to allow cancelling black vertices since if we did charts represent just
null homotopies in the case of braid groups and we instead want something
highly nontrivial. Thus we need restrictions on homotopy classes of maps
from cutopen disks to classifying spaces to be able to dene nontrivial ho
motopy classes to represent surface braids. Keeping base point information
is such an example. More natural interpretations are expected in terms of
other standard topological invariants.
By xing a height function on the disk where pictures are dened, we
can dene sequences of words for Wirtinger presented groups generalizing
braid movies. Furthermore we can generalize braid movie moves to such
sequences for Wirtinger presented groups by examining moves on pictures
in the presence of height functions. The moves in this case would consist
of the moves that correspond to Peier equivalences and the moves on
pictures described in , the moves that are dependent on the choice of
height functions, the moves that correspond to cycles in the Cayley
J. Scott Carter and Masahico Saito
complex associated to the presentation, the moves that push black
vertices through Wirtinger relations. The algebraic signicance of such
sequences of words and relations among them deserves further study, since
this is a natural context in which the braid movie moves appear.
Acknowledgements
We thank the following people for valuable conversations and correspon
dence John Baez, Dan Silver, Dan Flath, John Fischer, Seiichi Kamada,
Ruth Lawrence, Dennis McLaughlin, and Kunio Murasugi. Some of the
material here was presented at a special session hosted by YongWu Rong
and at a conference hosted by David Yetter and Louis Crane. Financial
support came from Yetter and Crane and from John Luecke. Cameron
Gordons support nancial and moral is especially appreciated.
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Preface
A fundamental problem with quantum theories of gravity, as opposed to the other forces of
nature, is that in Einsteins theory of gravity, general relativity, there is no background
geometry to work with the geometry of spacetime itself becomes a dynamical variable. This
is loosely summarized by saying that the theory is generally covariant. The standard model
of the strong, weak, and electromagnetic forces is dominated by the presence of a
nitedimensional group, the Poincar group, acting as the e symmetries of a spacetime with a
xed geometrical structure. In general relativity, on the other hand, there is an
innitedimensional group, the group of dieomorphisms of spacetime, which permutes dierent
mathematical descriptions of any given spacetime geometry satisfying Einsteins equations.
This causes two dicult problems. On the one hand, one can no longer apply the usual
criterion for xing the inner product in the Hilbert space of states, namely that it be invariant
under the geometrical symmetry group. This is the socalled inner product problem. On the
other hand, dynamics is no longer encoded in the action of the geometrical symmetry group
on the space of states. This is the socalled problem of time. The prospects for developing a
mathematical framework for quantizing gravity have been considerably changed by a number
of developments in the s that at rst seemed unrelated. On the one hand, Jones discovered a
new polynomial invariant of knots and links, prompting an intensive search for
generalizations and a unifying framework. It soon became clear that the new knot
polynomials were intimately related to the physics of dimensional systems in many ways.
Atiyah, however, raised the challenge of nding a manifestly dimensional denition of these
polynomials. This challenge was met by Witten, who showed that they occur naturally in
ChernSimons theory. ChernSimons theory is a quantum eld theory in dimensions that has
the distinction of being generally covariant and also a gauge theory, that is a theory involving
connections on a bundle over spacetime. Any knot thus determines a quantity called a
Wilson loop, the trace of the holonomy of the connection around the knot. The knot
polynomials are simply the expectation values of Wilson loops in the vacuum state of
ChernSimons theory, and their dieomorphism invariance is a consequence of the general
covariance of the theory. In the explosion of work that followed, it became clear that a useful
framework for understanding this situation is Atiyahs axiomatic description of a topological
quantum eld theory, or TQFT. On the other hand, at about the same time as Jones initial
discovery, Ashtekar discovered a reformulation of general relativity in terms of v
vi
Preface
what are now called the new variables. This made the theory much more closely resemble a
gauge theory. The work of Ashtekar proceeds from the canonical approach to quantization,
meaning that solutions to Einsteins equations are identied with their initial data on a given
spacelike hypersurface. In this approach the action of the dieomorphism group gives rise to
two constraints on initial data the dieomorphism constraint, which generates dieomorphisms
preserving the spacelike surface, and the Hamiltonian constraint, which generates
dieomorphisms that move the surface in a timelike direction. Following the procedure
invented by Dirac, one expects the physical states to be annihilated by certain operators
corresponding to the constraints. In an eort to nd such states, Rovelli and Smolin turned to a
description of quantum general relativity in terms of Wilson loops. In this loop representation,
they saw that at least formally a large space of states annihilated by the constraints can be
described in terms of invariants of knots and links. The fact that link invariants should be
annihilated by the dieomorphism constraint is very natural, since a link invariant is nothing
other than a function from links to the complex numbers that is invariant under
dieomorphisms of space. In this sense, the relation between knot theory and quantum gravity
is a natural one. But the fact that link invariants should also be annihilated by the Hamiltonian
constraint is deeper and more intriguing, since it hints at a relationship between knot theory
and dimensional mathematics. More evidence for such a relationship was found by Ashtekar
and Kodama, who discovered a strong connection between ChernSimons theory in
dimensions and quantum gravity in dimensions. Namely, in terms of the loop representation,
the same link invariant that arises from Chern Simons theorya certain normalization of the
Jones polynomial called the Kauman bracketalso represents a state of quantum gravity with
cosmological constant, essentially a quantization of antideSitter space. The full ramications
of this fact have yet to be explored. The present volume is the proceedings of a workshop on
knots and quantum gravity held on May thth, at the University of California at Riverside,
under the auspices of the Departments of Mathematics and Physics. The goal of the
workshop was to bring together researchers in quantum gravity and knot theory and pursue a
dialog between the two subjects. On Friday the th, Dana Fine began with a talk on
ChernSimons theory and the WessZuminoWitten model. Wittens original work deriving the
Jones invariant of links or, more precisely, the Kauman bracket from ChernSimons theory
used conformal eld theory in dimensions as a key tool. By now the relationship between
conformal eld theory and topological quantum eld theories in dimensions has been explored
from a number of viewpoints. The path integral approach, however, has not yet
and methods for constructing TQFTs in terms of these data. Louis Kauman spoke on
Vassiliev invariants and the loop states in quantum gravity. This concept also plays a major
role in the study of Vassiliev invariants of knots. has been notoriously dicult to make precise.
Introduction . In this talk Fine described work in progress on reducing the ChernSimons path
integral on S to the path integral for the WessZuminoWitten model. She also discussed her
work on applications of the loop representation to lattice gauge theory. going back to the
ReggePonzano model of Euclidean quantum gravity.. The notion of measures on the space
A/G of connections modulo gauge transformations has long been a key concept in gauge
theory. In a sense this is an old idea. His paper with Gambini in this volume is a more
detailed proof of this fact. However. however. Gambini. Kauman explained their relationship
from the path integral viewpoint. largely concerned the paper with Jerzy Lewandowski
appearing in this volume. Abhay Ashtekars talk. cell complexes. this idea was given new life
by Turaev and Viro.Preface vii been worked out in full mathematical rigor. Curiously.
Ashtekar also discussed the implications of this work for the study of quantum gravity. etc.
Recently there has been increasing interest in formulating TQFTs in a manner that relies
upon triangulating spacetime. that the coefcient of the nd term of the AlexanderConway
polynomial represents a state of quantum gravity with zero cosmological constant. an
algebra of observables generated by Wilson loops. Loop transforms. Oleg Viro spoke on
Simplicial topological quantum eld theories. however. and Pullin that is especially of interest
to knot u theorists is that it involves extending the bracket invariant to generalized links
admitting certain kinds of selfintersections. describing his work with Bernd Br gmann and
Rodolfo Gambini u on this subject. Ashtekar and Lewandowski have been able to construct
such a state with remarkable symmetry properties. Pullin spoke on The quantum Einstein
equations and the Jones polynomial. the extensions occurring in the two cases are dierent.
introduced the loop representation and various mathematical issues associated with it.
Sunday morning began with a talk by Gerald Johnson. who rigorously constructed the
ReggePonzano model of d quantum gravity as a TQFT based on the j symbols for the
quantum group SUq . Formalizing the notion of a measure on A/G as a state on this algebra.
Saturday began with a talk by Renate Loll on The loop formulation of gauge theory and
gravity. He also sketched the proof of a new result. which. One aspect of the work of Br
egmann. The goal here is to make the loop representation into rigorous mathematics. On
Saturday afternoon. A key notion developed by Ashtekar and Isham for this purpose is that of
the holonomy Calgebra. in some sense analogous to Haar measure on a compact Lie group.
Viro discussed a variety of approaches of presenting manifolds as simplicial complexes.
Michel Lapidus deserves warm thanks for his help in planning the workshop. Albert Stralka
and Benjamin Shen. Viktor Ginzburg then spoke on Vassiliev invariants of knots. Paolo
CottaRamusino spoke on d quantum gravity and knot theory. includes quantum gravity in the
Ashtekar formulation. and beyond. but the stringtheoretic approach is also related to the
study of knots. The workshop was funded by the Departments of Mathematics and Physics
of U. loops. Lastly. He attempted to clarify the similarity between the loop representation of
quantum gravity and string theory. and Chris Truett. Carlips paper. At a xed time both involve
loops or knots in space. and particularly Susan Spranger. Scott Carter and Masahico Saito.
Knotted surfaces. treats the thorny issue of relating the loop representation of quantum
gravity to the traditional formulation of gravity in terms of a metric. This class. which is an
exactly soluble test case. spin geometry. This talk dealt with his work in progress with
Maurizio Martellini. Riverside. knots and gauge elds. They also discuss the role in
dimensional topology of new algebraic structures such as braided tensor categories. and in
the afternoon. Invaluable help in organizing the workshop and setting things up was provided
by the sta of the Department of Mathematics. and his work with Igor Frenkel on certain
braided tensor categories. As some of the speakers did not submit papers for the
proceedings. In the nal talk of the workshop.viii Preface to the Feynman integral and
Feynmans operational calculus. Just as Chern Simons theory gives a great deal of
information on knots in dimensions. the speakers and other participants are to be
congratulated for making it such a lively and interesting event. The paper by Carter and
Saito. C. his construction with David Yetter of a dimensional TQFT based on the j symbols
for SUq . Geometric structures and loop variables. CottaRamusino and Martellini have
described a way to construct observables associated to knots. contains a review of their work
on knots as well as a number of new results on braids. Arthur Greenspoon kindly volunteered
to help edit the proceedings. and are endeavoring to prove at a perturbative level that they
give knot invariants. This was a review of his work on the relation between ChernSimons
theory and dimensional quantum gravity. He treats quantum gravity in dimensions. The editor
would like to thank many people for making the workshop a success. braid movies. and the
chairs of these departments. as well as Donaldson theory. that is. thanks go to all the
participants in the . Linda Terry. and categorical physics. embedded surfaces in dimensions.
First and foremost. there appears to be a relationship between a certain class of dimensional
theories and socalled knots. Louis Crane spoke on Quantum gravity. called BF theories.
papers were also solicited from Steve Carlip and also from J. John Baez spoke on Strings.
on his work with Michel Lapidus on rigorous path integral methods. were crucial in bringing
this about.
C. Riverside. and especially Jim Gilliam. Javier Muniain.Preface ix Knots and Quantum
Gravity Seminar at U. who helped set things up. . and Mou Roy.
x Preface .
Contents The Loop Formulation of Gauge Theory and Gravity Renate Loll Introduction
YangMills theory and general relativity as dynamics on connection space Introducing loops
Equivalence between the connection and loop formulations Some lattice results on loops
Quantization in the loop approach Acknowledgements Bibliography Representation Theory
of Analytic Holonomy Calgebras Abhay Ashtekar and Jerzy Lewandowski Introduction
Preliminaries The spectrum . U N and SU N connections Acknowledgements Bibliography
The Gauss Linking Number in Quantum Gravity Rodolfo Gambini and Jorge Pullin
Introduction The WheelerDeWitt equation in terms of loops The Gauss selflinking number as
a solution Discussion Acknowledgements xi . Examples of A Integration on A/G . Loop
decomposition . Cylindrical functions . Characterization of A/G . A natural measure
Discussion Appendix A C loops and U connections Appendix B C loops.
Categorical physics Ideas from TQFT Hilbert space is deadlong live Hilbert space or the
observer gas approximation The problem of time . Fine Introduction The main result A/G n as
a bundle over G An explicit reduction from ChernSimons to WZW Acknowledgements
Bibliography Topological Field Theory as the Key to Quantum Gravity Louis Crane
Introduction Quantum mechanics of the universe.xii Bibliography Contents Vassiliev
Invariants and the Loop States in Quantum Gravity Louis H. Kauman Introduction Quantum
mechanics and topology Links and the Wilson loop Graph invariants and Vassiliev invariants
Vassiliev invariants from the functional integral Quantum gravityloop states
Acknowledgements Bibliography Geometric Structures and Loop Variables in dimensional
Gravity Steven Carlip Introduction From geometry to holonomies Geometric structures from
holonomies to geometry Quantization and geometrical observables Acknowledgements
Bibliography From ChernSimons to WZW via Path Integrals Dana S.
Denition locality equivalence The permutohedral and tetrahedral equations . Baez
Introduction String eld/gauge eld duality YangMills theory in dimensions Quantum gravity in
dimensions Quantum gravity in dimensions Acknowledgements Bibliography BF Theories
and knots Paolo CottaRamusino and Maurizio Martellini Introduction BF theories and their
geometrical signicance knots and their quantum observables Gauss constraints Path
integrals and the Alexander invariant of a knot Firstorder perturbative calculations and
higherdimensional linking numbers Wilson channels Integration by parts Acknowledgements
Bibliography Knotted Surfaces. The category of braid movies xiii . Scott Carter and Masahico
Saito Introduction Movies. Braid Movies.Contents Matter and symmetry Acknowledgements
Bibliography Strings. charts. Knots. and Gauge Fields John C. Surface braids . The
tetrahedral equation . Movie moves . surface braids. and isotopies . Chart descriptions .
Planar versions categories and the movie moves . Braid movie moves . and Beyond J.
Loops. The permutohedral equation . Overview of categories .
Concluding remarks Acknowledgements Bibliography . Symmetric equivalence and moves
on surface braids .xiv Contents Algebraic interpretations of braid movie moves . Identities
among relations and Peier elements . Moves on surface braids and identities among
relations .
Pennsylvania . Introduction This chapter summarizes the main ideas and basic
mathematical structures that are necessary to understand the loop formulation of both gravity
and gauge theory.edu Abstract This chapter contains an overview of the loop formulation of
Yang Mills theory and dimensional gravity in the Ashtekar form. It is also meant as a guide to
the literature. YangMills theory and general relativity as dynamics on connection space In
this section I will briey recall the Lagrangian and Hamiltonian formulation of both pure
YangMills gauge theory and gravity. University Park. Since the conguration spaces of these
theories are spaces of gauge potentials. Pennsylvania State University.The Loop
Formulation of Gauge Theory and Gravity Renate Loll Center for Gravitational Physics and
Geometry. and in Section some of the main ingredients of quantum loop representations are
discussed. USA email lollphys. The gravitational theory will be described in the Ashtekar
form. I discuss some mathematical problems that arise in the loop formulation and illustrate
parallels and dierences between the gauge theoretic and gravitational applications.
formulated as theories of connections. in which it most . Section contains a summary of
progress in a corresponding lattice loop approach. their classical and quantum descriptions
may be given in terms of gaugeinvariant Wilson loops. In Section .psu. Section summarizes
the classical and quantum features of gauge theory and gravity. the Wilson loops and their
properties are introduced. I have tried to treat YangMills theory and gravity in parallel. Some
remarks are made on the discretized lattice approach. dened on a dimensional manifold M R
of Lorentzian signature. Many of the mathematical problems arising in the loop approach are
described in a related review article . but also to highlight important dierences. where all the
relevant details may be found. and Section discusses the classical equivalence between the
loop and the connection formulation.
a and b are dimensional spatial indices and i . for gravity. and .. . e M M d x Tr F F M TrF F .
pseudotensorial form gauge potential and E a LieGvalued vector density generalized electric
eld on . and e a vierbein. Note that the action . with the totally antisymmetric tensor. for
YangMills theory and general relativity respectively. The classical actions dening these
theories are up to overall constants SY M A and SGR A. A mathematical characterization of
the action . F denotes in both cases the curvature associated with the dimensional
connection A. In . Assuming the underlying principal bre bundles P . so. Details may be
found in . Ej y i j a x y a . In . for gauge theory.. d x e e e F IJ I J IJKL M eI eJ F KL . we can
identify the relevant phase spaces with cotangent bundles over ane spaces A of LieGvalued
connections on the dimensional manifold assumed compact and orientable. E. is contained
in Baez chapter . C valued connection form. We have G SU N . dening an isomorphism of
vector spaces between the tangent space of M and the xed internal space with the
Minkowski metric IJ . has the unusual feature of being complex. In . and i b b Ai x. I will not
explain here why this still leads to a welldened theory equivalent to the standard formulation
of general relativity. the basic variable A is a LieGvalued connection form on M . The
standard symplectic structure on T A is expressed by the fundamental Poisson bracket
relations i b b Ai x. Renate Loll closely resembles a gauge theory. where A is a LieG valued...
Ej y j a x y a . and . The cotangent bundle T A is coordinatized by canonically conjugate pairs
A. As usual. i The connection A is selfdual in the internal space. G to be trivial. A is a
selfdual. and G SOC for gravity. where G denotes the compact and semisimple Liestructure
group of the YangMills theory typically G SU N . which is a rstorder form of the gravitational
action. say. AMN MN IJ AIJ . for gauge theory. where the factor of i arises because the
connection A is complex valued. Our main interest will be the Hamiltonian formulations
associated with .
and . we have HY M A. is A. The wellknown YangMills transformation law corresponding to
a group element g G. C . However. generates phase space transformations that are
interpreted as corresponding to the time evolution of the spatial slice in M in a spacetime
picture. this manifests itself in the existence of socalled rstclass constraints. is a density. . for
physical congurations and at the same time generate transformations between physically
indistinguishable congurations. the rstclass constraints for the two theories are CY M Da E ai
and b CGR Da E ai . as the Hamiltonian for YangMills theory where for convenience we have
a introduced the generalized magnetic eld Bi abc Fbc i . the socalled scalar or Hamiltonian
constraint C. we see that in the Hamiltonian formulation. The physical interpretation of these
constraints is as follows the so called Gauss law constraints Da E ai . Comparing . because
both theories possess gauge symmetries. These are sets C of functions C on phase space.
pure gravity may be interpreted as a YangMills theory with gauge group G SOC . subject to
four additional constraints in each point of . of constraints in . a b Fab i Ej Ek .The Loop
Formulation of Gauge Theory and Gravity and j internal gauge algebra indices. On the other
hand. At the Hamiltonian level. which are related to invariances of the original Lagrangians
under certain transformations on the fundamental variables. E g Ag g dg. the set of Gvalued
functions on . generates dimensional dieomorphisms of and the last expression. Note the
explicit appearance of a Riemannian background metric g on . E b b d x gab E ai Ei gB ai Bi
g . which are required to vanish. Ca Fab i Ei .. In our case. points in T A do not yet
correspond to physical congurations. However. to contract indices and ensure the integrand
in . g Eg. and at the same time generate local gauge transformations in the internal space. C
ijk . Ca . which are common to both theories. The second set. the dynamics of the two
theories is dierent. The Hamiltonian HGR for gravity is a sum of a . . restrict the allowed
congurations to those whose covariant divergence vanishes. no such additional background
structure is necessary to make the gravitational Hamiltonian well dened.
and therefore vanishes on physical congurations. whose gauge group contains the
dieomorphism group of the underlying manifold. This a means that one rst quantizes on the
unphysical phase space T A. N and N a in . and the presence of gauge symmetries for these
theories. HGR A. After excluding the reducible connections from A. E. this approach has not
yielded enough information about other sectors of the theory. are Lagrange multipliers. the
socalled lapse and shift functions. lead to numerous diculties in their classical and quantum
description. . Similarly. That is. The corresponding physical phase space is then its cotangent
bundle T A/G. one uses a formal operator repre sentation of the canonical variable pairs A. A
further dierence from the gaugetheoretic application is the need for a set of reality conditions
for the gravitational theory. It is nonlinear and has nontrivial topology. The nonlinearities of
both the equations of motion and the physical conguration/phase spaces. as if there were no
constraints. In any case. because the principal bundle A A/G is nontrivial. with Sfree being
just quadratic in A. This latter feature is peculiar to generally covariant theories. subset of the
constraints . a gaugexing term has to be introduced in the sum over all congurations Z A dA
eiSY M . because the connection A is a priori a complex coordinate on phase space. since
the expression .. h . since no good explicit description of T A/G is available. to ensure that
the integration is only taken over one member A of each gauge equivalence class. can be
made meaningful only in a weakeld approximation where one splits SY M Sfree SI . E i
Renate Loll b i d x N a Fab i Ei N ijk a b Fab k Ei Ej . in the canonical quantization. in the path
integral quantization of YangMills theory. a j j a . this quotient can be given the structure of an
innitedimensional manifold for compact and semisimple G . The space of physical
congurations of YangMills theory is the quotient space A/G of gauge potentials modulo local
gauge transformations. Elements of T A/G are called classical observables of the YangMills
theory. where the elds cannot be assumed weak. satisfying the canonical commutation
relations Ai x. one quantizes the theory la Dirac. . However. For example. E b y i i b x y. we
know that a unique and attainable gauge choice does not exist this is the assertion of the
socalled Gribov ambiguity.
If we had such a measure again. DE. Whereas this is not required on the unphysical YM
space F .. one does need such a structure on Fphy in order to make physical statements. i.
Gauss constraint.e. Again. Unfortunately.. The notations F and Fphy indicate that these are
just spaces of complexvalued functions on A and do not yet carry any Hilbert space
structure. Given such a measure. well dened. see . . reads phy . one now needs an inner
product on the space of such states. . Unfortunately. the Hilbert space HGR of states for
quantum gravity would conGR sist of all those elements of Fphy which are squareintegrable
with respect to that measure. an appropriate gaugeinvariant measure dAG in phy . Fphy
consists of all functions that lie in the kernel of the quantized A/G More precisely.e. the
analogue of . YM i. the situation for gravity is not much better. according to YM Fphy A F
DEA .The Loop Formulation of Gauge Theory and Gravity the quantum analogues of the
Poisson brackets . we lack an appropriate scalar product. phy . phy M dA Aphy A. In theup to
nowabsence of explicit measures to make . phy dAG Aphy A. we expect the integral to range
over an extension of the space A/G which includes also distributional elements .. . F A. CA .
where the measure dA is now both gauge and dieomorphism invariant and projects in an
appropriate way to M. for instance about the spectrum of the quantum Hamiltonian H. our
condence in the procedure outlined here stems from the fact that it can be made meaningful
for lowerdimensional model systems such as YangMills theory in and gravity in di . we do
not. the space of physical wave functions would be the Hilbert space YM HY M Fphy of
functions squareintegrable with respect to the inner product . Ca A . For a critical appraisal of
the use of such Dirac YM conditions. phy . Then a subset of YM physical wave functions
Fphy phy A is projected out from the space of all wave functions. Here the space of physical
wave functions is the subspace of F projected out according to GR Fphy A F DEA .
Assuming that the states phy of quantum gravity can be described as the set of
complexvalued functions on some still innitedimensional space M.
we can dene . whereas in gravity the quantum Hamiltonian C is one of the constraints and
therefore a physical state of quantum gravity must lie in its kernel. one searches for solutions
in HY M of the eigenvector equation HY M phy A Ephy A. Note that there is a very important
dierence in the role played by the quantum Hamiltonians in gauge theory and gravity. which
also in this framework are not well understood. where the analogues of the space M are nite
dimensional cf. also . Cphy A . For the case of YangMills theory. However. Many qualitative
and quantitative statements about quantum gauge theory can be derived in a regularized
version of the theory. the most important being the connement property of YangMills theory.
One important ingredient we will be focusing on are the nonlocal loop variables on the space
A of connections. Still there are nonperturbative features. Introducing loops . Similar attempts
to discretize the Hamiltonian theory have so far been not very fruitful in the treatment of
gravity. and a space A of connections. s. because. . Renate Loll mensions. . where the at
background spacetime is approximated by a hypercubic lattice. one then tries to extract for
either weak or strong coupling properties of the continuum theory. in the limit as the lattice
spacing tends to zero. there is no xed background metric with respect to which one could
build a hypercubic lattice. A loop in is a continuous map from the unit interval into . . which
will be the subject of the next section. The set of all such maps will be denoted by . After all
that has been said about the diculties of nding a canonical quantization for gauge theory and
general relativity. An example of this is given by the latticelike structures used in the
construction of dieomorphisminvariant measures on the space A/G . without violating the
dieomorphism invariance of the theory. the loop space of . t. there are more subtle ways in
which discrete structures arise in canonical quantum gravity. Using rened computational
methods. Given such a loop element . the answer is of course in the armative. for exam ple
an approximate spectrum of the Hamiltonian HY M . s a s. In YangMills theory. unlike in
YangMills theory. the reader may wonder whether one can make any statements at all about
their quantum theories.
and N is the dimensionality of this linear representation. It is invariant under
orientationpreserving reparametrizations of the loop . is also known as the holonomy of A
along . T T . there is now one more reason to consider variables that depend on closed
curves in . the socalled Wilson loop. and therefore an observable for YangMills theory. .. It is
independent of the basepoint chosen on . by considering Wilson loops that are constant
along the orbits of the Diaction. . TA FA/G /Di S . etc. The trace is taken in the representation
R of G. The Wilson loop has a number of interesting properties. In contrast. . The
pathordered exponential of the connection A along the loop . since observables in gravity
also have to be invariant under diffeomorphisms of . . Wilson as an indicator of conning
behaviour in the lattice gauge theory . . which is not usually considered in knot theory. The
analogue of . The use of nonlocal holonomy variables in gauge eld theory was pioneered by
Mandelstam in his paper on Quantum electrodynamics . as can easily be seen by expanding
TA for an innitesimal loop . The holonomy measures the change undergone by an internal
vector when parallel transported along . Such a loop variable will just depend on what is
called the generalized knot class K of generalized since may possess selfintersections and
selfoverlaps. It follows from and that the Wilson loop is a function on a crossproduct of the
quotient spaces. However. would then tell us that had to be constant on any connected
component of . since they are not invariant under the full set of gauge transformations of the
theory. Di. x. In gravity. TA TrR P exp N A. This means going one step further in the
construction of such observables. Note that a similar construction for taking care of the
dieomorphism invariance is not very contentful if we start from a local scalar function. say. It
is invariant under the local gauge transformations . the Wilson loops are not observables. U
P exp A. The set of such variables is not big enough to account for all the degrees of
freedom of gravity. It measures the curvature or eld strength in a gaugeinvariant way the
components of the tensor Fab itself are not measurable. number of points of
nondierentiability. The quantity TA was rst introduced by K. loops can carry
dieomorphisminvariant information such as their number of selfintersections.The Loop
Formulation of Gauge Theory and Gravity a complex function on A .
as will be explained in the next section. there is no action principle in terms of loop variables.
Another upsurge in interest in pathdependent formulations of YangMills theory took place at
the end of the s. which are to be thought of as the analogues of the usual local YangMills
equations of motion. this in general will ensure the existence of neither inverse and implicit
function theorems nor theorems on the existence and uniqueness of solutions of dierential
equations. For the YangMills theory itself. . one rst has to introduce a suitable topology. or for
their vacuum expectation values. following the observation of the close formal resemblance
of a particular form of the YangMills equations in terms of holonomy variables with the
equations of motion of the dimensional nonlinear sigma model . and then set up a dierential
calculus on . one expects that not all loop variables will be independent. the holonomy or the
Wilson loop seem like ideal candidates. and has therefore obscured many attempts of
establishing an equivalence between the connection and the loop formulation of gauge
theory. The aim has therefore been to derive suitable dierential equations in loop space for
the holonomy U . the review . Since the space of all loops in is so much larger than the set of
all points in . and are able to dene dierentiation. The hope underlying such approaches has
always been that one may be able to nd a set of basic variables for YangMills theory which
are better suited to its quantization than the local gauge potentials Ax. Renate Loll without
potentials . Still. For example. for example. Even if we give locally the structure of a
topological vector space. Owing to their simple behaviour under local gauge transformations.
This expectation is indeed correct. T . its trace. To make sense of dierential equations on
loop space. perturbation theory in the loop approach always had to fall back on the
perturbative expansion in terms of the connection variable A. Another set of equations for the
Wilson loop was derived by Makeenko and Migdal see. there is a strong physical motivation
to choose . Such issues have only received scant attention in the past. The fact that none of
these attempts have been very successful can be attributed to a number of reasons. and
later generalized to the nonAbelian theory by BialynickiBirula and again Mandelstam . the
ensuing redundancy is hard to control in the continuum theory. which leaves considerable
freedom for deriving loop equations of motion. As another consequence. Equivalence
between the connection and loop formulations In this section I will discuss the motivation for
adopting a pure loop formulation for any theory whose conguration space comes from a
space A/G of connection forms modulo local gauge transformations.
They are typically algebraic. in our case the Wilson loops. For example. /Di S . . Mandelstam
constraints . exhaust the Mandelstam constraints for this particular gauge group and
representation. It turns out that. nonlinear constraints among the T . To obtain a complete set
of such conditions without making reference to the connection space A is a nontrivial
problem that for general G to my knowledge has not been solved. This is the case for
holonomies that take their values in the socalled subgroup of null rotations . at least for
compact G. However. for any structure group G GLN. Therefore all that remains to be shown
is that one can reconstruct the holonomies U from the set of Wilson loops T . this leads only
to an incompleteness of measure zero of the Wilson . In e. although I am not aware of a
formal proof. for G SL. Still. C in the fundamental representation by matrices.e. By
Mandelstam constraints I mean a set of necessary and sucient conditions on generic
complex functions f on /Di S that ensure they are the traces of holonomies of a group G in a
given representation R. The loop in eqn d is the loop with an appendix attached. T T . owing
to the noncompactness of G. it can be shown that not every element of A/G can be
reconstructed from the Wilson loops. . one requires only physical observables O i.The Loop
Formulation of Gauge Theory and Gravity the variables T as a set of basic variables on
physical grounds. is a path starting at a point of and ending at see Fig. with the classical
Poisson relations O. It is a wellknown result that one can reconstruct the gauge potential A
up to gauge transformations from the knowledge of all holonomies U . there is indeed an
equivalence between the two quotient spaces A G T . h There is also a mathematical
justication for reexpressing the classical theory in terms of loop variables. and all conditions
that can be derived from . . the Mandelstam constraints are a b c d e T T T point loop T T T T
T T . It is believed that the set . C. O going over to the quantum commutators i O. The circle
denotes the usual path composition. O . and not the local elds A to be represented by
selfadjoint operators O in the quantum theory. where is any path not necessarily closed
starting at a point of see Fig.
whereas for SU . which now take the form of inequalities . C one derives T T T T T T . T T T
T . and no restrictions for other values of T and T . The loop congurations related by the
Mandelstam constraint T T T T loops on A/G. . one nds the two conditions T . C. . Adding an
appendix to the loop does not change the Wilson loop Fig. . there are further conditions on
the loop variables. . SL. C. Renate Loll Fig. . Unfortunately. If one wants to restrict the group
G to a subgroup of SL. For SU SL.
but real progress in this direction has only been made in the discretized lattice gauge theory
see Section below. . as applied to gauge theory. because in it the gauge elds are taken to be
concentrated on the dimensional links li of the lattice. The existence of such relations makes
it dicult to apply directly mathematical results on the loop space or the space /Di S of
oriented. Finally.e. Note that one may interpret some of the constraints . . One may indeed
hope that at some level. implicitly the nontrivial topological structure of A/G. with the loop
representation being better suited for the description of the nonperturbative structure of the
theory. The Wilson loops are easily formed by multiplying together the holonomies Ul .
although some of the techniques and conclusions may be relevant for gravitational theories
too. the Wilson loops are eectively not functions on /Di S . to the present case. but on some
appropriate quotient with respect to equivalence relations such as . and . There are more
inequalities corresponding to more complicated congurations of more than two intersecting
loops. let me emphasize that the equivalence of the connection and the loop description at
the classical level does not necessarily mean that quantization with the Wilson loops as a set
of basic variables will be equivalent to a quantization in which the connections A are turned
into fundamental quantum operators. Some lattice results on loops This section describes
the idea of taking the Wilson loops as the set of basic variables. Let me nish this section with
some further remarks on the Mandelstam constraints In cases where the Wilson loops form a
complete set of variables. respectively. Another possibility will be mentioned in Section .
which are critically dependent on the geometry of the loop conguration. This is the
overcompleteness I referred to in the previous section. One may try to eliminate as many of
them as possible already in the classical theory. The lattice discretization is particularly
suited to the loop formulation. those two possibilities lead to dierent results. do not exhaust
all inequalities there are for SU and SU . oriented links that form . due to the existence of the
Mandelstam constraints i. they are actually overcomplete. One has to decide how the
Mandelstam constraints are to be carried over to the quantum theory.e. as constraints on the
underlying loop space. . unparametrized loops such as . i.The Loop Formulation of Gauge
Theory and Gravity . Uln of a set of contiguous. .
Again. with a compact. Although the lattice is nite. and hence an innite number of Wilson
loops. and the gauge group G to be SU . using the Mandelstam constraints see. in terms of
which all the others can be expressed. Uln . the number of Mandelstam constraints . Renate
Loll Fig. it is possible to solve this coupled set of equations and obtain an explicit local
description of the nitedimensional physical conguration space Cphy A G lattice T . . is innite.
consisting of no more than six lattice links. in terms of an independent set of Wilson loops.
We take the lattice to be the hypercubic lattice N d . as I showed in . dimensional Riemann
surface g of genus g. One also has a discrete set of loops. . . there are an innite number of
closed contours one can form on it. N . d . . T TrR Ul Ul . a lattice loop Mandelstam
constraints . with periodic boundary conditions N is the number of links in each direction. . At
the same time. . Nevertheless. The nal solution for the independent degrees of freedom is
surprisingly simple it suces to use the Wilson loops of small lattice loops. one is interested in
nding a maximally reduced set of such Wilson loops. At this point it may be noted that a
similar problem arises in gravity in the connection formulation on a manifold g R. again in the
fundamental representation. The only constraints that cause problems are those of type e. .
for example. and can introduce the corresponding Wilson loops. . because of their nonlinear
and highly coupled character. namely. a closed chain on the lattice see Fig. is a loop on the
hypercubic lattice N d . the elements of the homotopy group of g .
and it is clear that all of the aboveposed problems are rather nontrivial. Quantization in the
loop approach I will now return to the discussion of the continuum theory of both gravity and
gauge theory. . even in the discretized lattice theory where the number of degrees of
freedom is eectively nite. and maybe derive a prescription of how to translate local quantities
into their loop space analogues. . What are the consequences of this result for the continuum
theory. where there is no obstruction to shrinking loops down to points Some progress on
points and has been made for small lattices . one also has to show that the dynamics can be
expressed in a manageable form in terms of these variables. Since the space Cphy A/G is
topologically nontrivial. where discrete structures appear too the generalized knot classes
after factoring out by the spatial dieomorphisms. . and review a few concepts that have been
used in their quantization in a loop formulation. What is the explicit form of the Jacobian of
the transformation from the holonomy link variables to the Wilson loops From this one could
derive an eective action/Hamiltonian for gauge theory. once one understands which Wilson
loop congurations contribute most in the path integral. One may be able to apply some of the
techniques used here to dimensional gravity. gaugeinvariant phase space variables T and
not on some gaugecovariant local eld variables. and may lead to new and more ecient ways
of performing calculations in lattice gauge theory. This . It would therefore be desirable to nd
a minimal embedding into a space of loop variables that is topologically trivial together with a
nite number of nonlinear constraints that dene the space Cphy . Lastly. in nding a solution to
the Mandelstam constraints it was crucial to have the concept of a smallest loop size on the
lattice the loops of link length going around a single plaquette. The contributions by Ashtekar
and Lewandowski and Pullin in this volume contain more details about some of the points
raised here. Some important problems are the following. .The Loop Formulation of Gauge
Theory and Gravity Of course it does not suce to establish a set of independent loop
variables on the lattice. say This would be an interesting alternative to the usual perturbation
theory in A or the N expansion. the set of independent lattice Wilson loops cannot provide a
good global chart for Cphy . Is it possible to dene a perturbation theory in loop space. The
main idea is to base the canonical quantization of a theory with conguration space A on the
nonlocal.
For gauge group SU N . but needs an additional geometric ingredient. T a . In any case. T b
. In the derivation of this semidirect product algebra. T a . ut T b . C or any of its subgroups in
the fundamental representation. . You may have noted that so far we have only talked about
conguration space variables. t . the momentum Wilson loop depends not just on a loop. t. but
a vector density. the righthand sides of the Poisson brackets can no longer be written as
linear expressions in T .e. The structure constants are distributional and depend just on the
geometry of loops and intersection points. Also it is strictly speaking not a function on phase
space. ut .e have been used to bring the righthand sides into a form linear in T . The crucial
ingredient in the canonical quantization is the fact that the loop variables T . b b s . T a . a
nondegenerate parameter congruence of curves t s Rs. . This fact was rst established by
Gambini and Trias for the gauge group SU N and later rediscovered by Rovelli and Smolin in
the context of gravity . and it is still gauge invariant under the transformations . T a . The
Poisson bracket of two loop variables is nonvanishing only when an insertion of an electric
eld E in a T a variable coincides in with the holonomy of another loop. x dt t. . but takes
directly into account the nonlinearities of the theory. i.. t T a t . One choice of a generalized
Wilson loop that also depends is on E a TA. . x a t. For a canonical quantization we obviously
need some momentum variables that depend on the generalized electric eld E. This variable
depends now on both a loop and a marked point. This latter fact may be remedied by
integrating T a over a dimensional ribbon or strip. t a . s form a closing algebra with respect
to the canonical symplectic structure on T A. the Wilson loops.E . the Poisson algebra of the
loop variable s is T . the Mandelstam constraints . vs . s T s T s . a . s. s. s Tr U sE a s. T .
Renate Loll is a somewhat unusual procedure in the quantization of a eld theory. For the
special case of G SL. vs T a t . I will use here the unsmeared version since it does not aect
what I am going to say. s T s .
solutions to these quantum constraints have to be found. but there exist a large number of
solutions to the quantum constraint equations . i. Ca T . . These solutions and their relation to
knot invariants are the subject of Pullins chapter . the space of these solutions or an
appropriate subspace has to be given a Hilbert space structure to make a physical
interpretation possible. . The completion of this quantization program is dierent for gauge
theory and gravity. to obtain the spectrum of the theory. Finally. Because of the lack of an
appropriate scalar product in the continuum theory. since they correspond to genuine
observables in the classical theory. has to be reexpressed as a function of Wilson loops. s.
For YangMills theory. T a . Then the YangMills Hamiltonian . as the commutator algebra of a
set of selfadjoint operators T . . For the momentum Wilson loops there are analogous
Mandelstam constraints. s to be represented selfadjointly. one has to solve the eigenvector
equation HY M T . There is one more interesting mathematical structure in the loop ap .The
Loop Formulation of Gauge Theory and Gravity By a quantization of the theory in the loop
representation we will mean a representation of the loop algebra . For the case of general
relativity one does not require the loop variables T . loop functionals that lie in their kernel. T
a . the only progress that has been made along these lines is in the latticeregularized version
of the gauge theory see. which again means that they have to be reexpressed in terms of
loop variables. and nally. Then. cf.e. T a E. .. for example. the Mandelstam constraints must
be incorporated in the quantum theory. hold also for the corresponding quan tum operators T
. s or an appropriately smeared version of the momentum operator T a . . T a . for example
by demanding that relations like . Secondly. one has to nd a Hilbert space of wave
functionals that depend on either the Wilson loops T or directly on the elements of some
appropriate quotient of a loop space. in line with the remarks made in Section above. No
suitable scalar product is known at the moment. and has been extensively used in quantum
gravity. T a CT . The completion of the loop quantization program for gravity requires an
appropriate inclusion of the dimensional dieomorphism constraints Ca and the Hamiltonian
constraint C.. because they do not constitute observables on the classical phase space. The
quantized Wilson loops must be selfadjoint with respect to the inner product on this Hilbert
space. At least one such representation where the generalized Wilson loops are represented
as dierential operators on a space of loop functionals is known.
. etc. that Rovelli and Smolin arrived at the quantum representation of the loop algebra . and
which tries to establish a relation between the quantum connection and the quantum loop
representations. Indeed. a variety of simpler examples such as dimensional gravity. although
in this case the loop transform can be given a rigorous mathematical meaning. is to be
thought of as a nonlinear analogue of the Fourier transform p dx eixp x . which is supposed
to intertwine the two types of representations according to A/G dAG TA phy A . however. ..
Renate Loll proach to which I would like to draw attention. but for this case we know even
less about the welldenedness of expressions like . both classically and
quantummechanically. For the application to gravity. and . This is the socalled loop
transform. and . dx. as explained in Section above. There is. summarizes the relations
between connection and loop dynamics. mentioned above. and similarly for the action of the
T a . rst introduced by Rovelli and Smolin . but does in general not possess an inverse. the
domain space of the integration will be a smaller space M. electrodynamics. for Yang Mills
theory we do not know how to construct a concrete measure dAG that would allow us to
proceed with a loop quantization program. and T A/G dAG TA T phy A . Again. The following
diagram Fig. where the loop transform is a very useful and mathematically welldened tool. on
L R. it was by the heuristic use of . as outlined in Sections and . The relation .
The Loop Formulation of Gauge Theory and Gravity Fig. . .
Fulp.. Ashtekar. A. I. A. SU QCD in the path representation. D . J. I. Arms. Rev.
Representations of the holonomy algebras of gravity and nonAbelian gauge theories.
Bibliography . this volume . Polon. M. Connections on the path bundle of a principal bre
bundle. A. this volume . and Lewandowski. Br gmann. preprint Montevideo. and gauge elds.
J. . Math. loops. R. . O. Gambini. Brylinski. Marsden. Ashtekar. Bull. L. Les Houches lecture
notes.. M. April . Gaugeinvariant variables in the YangMills theory. O. The gauge eld as
chiral eld on the path and its integrability. New loop representations for gravity. Ashtekar. in
preparation . in Proceedings of the Summer Institute on Dierential Geometry. State pure
mathematics report No. The nonintegrable phase factor and gauge theory. Singapore. and
Setaro. A. Lectures on nonperturbative gravity. Mathematical problems of nonperturbative
quantum general relativity. The Kaehler geometry of the space of knots in a smooth
threefold. and Trias. Baez. Symmetry and bifurcations of momentum mappings. World
Scientic. to appear in the Proceedings of the Conference on Quantum Topology. Strings. . R.
V. eds L. J. Ashtekar. J. Gambini. Penn. . J. Yetter. preprint Syracuse SUGP/ . The method
of loops applied to lattice gauge theory. . A. Dieomorphisminvariant generalized measures on
the space of connections modulo gauge transformations. Sci. J. Arefeva. Lett. B. hepth . J.
Baez. Penn. BialynickiBirula. Phys. State University preprint. Representation theory of
analytic Calgebras. Classical amp Quantum Gravity . C. Math. and Loll. Fulp. Ashtekar.
preprint North Carolina State University . Renate Loll Acknowledgements Warm thanks to
John Baez for organizing an inspiring workshop. Ya. Commun. and Moncrief. Math. Gauge
dynamics in the Crepresentation. knots. R. Crane and D. A. Acad. u Phys. Phys. Symposia
in Pure Mathematics Series . PM . E. and Isham. R. R.
Phys. R. Phys. S. B Schper. S. Phys. V. Loop space representation of quantum general
relativity. Geometry of loop spaces. Quantum electrodynamics without potentials. . . . R.. Loll.
Rev. Moncrief. Wilson. A KaluzaKlein type point a of view. R. R. B Loll. A. Nelson. . Phys.
Commun. State University preprint CGPG/. and Regge. . eds M. Connement of quarks.
Feynman rules for electromagnetic and YangMills elds from the gaugeindependent
eldtheoretic formalism. D . L. Rev. and Stornaiolo. I. in Mathematical aspects of classical eld
theory. pp. and Smolin. Gotay. March . K. YangMills theory without Mandelstam constraints.
Math. J. D Goldberg. The Gauss linking number in quantum gravity. J. . Nucl. . Phys.. R.
String representations and hidden symmetries for gauge elds. A. Ann. Penn. preprint
Freiburg THEP /. C. Phys. J. . B Pullin. Migdal. M. Canonical and BRSTquantization of
constrained systems. T. Phys. NY Mandelstam. Degeneracy in loop variables. Phys. A. .
Classical amp Quantum Gravity Mandelstam. Rep. Contemporary Mathematics Series. U. J.
September Loll. Phys. Math. . Nucl. . Lett. R. vol. Loop equations and /N expansion. J.
gravity for genus gt . Loll. this volume Rovelli. Rev. C. Phys. Nucl. Reconstruction of gauge
potentials from Wilson loops. B Giles. Lewandowski. Chromodynamics and gravity as
theories on loop space. . E. hepth . Phys. . N. . Phys. Polyakov. B Loll. Commun. . .The Loop
Formulation of Gauge Theory and Gravity . Nucl. Independent SUloop variables and the
reduced conguration space of SUlattice gauge theory. Marsden. Loop variable inequalities in
gravity and gauge theory.
Renate Loll .
In a typical setup. Gainesville. by the traces of holonomies of connections around closed
loops.psu.pl Abstract Integral calculus on the space A/G of gaugeequivalent connections is
developed. This measure can be used to dene the Hilbert space of quantum states in
theories of connections. Warszawa.edu Jerzy Lewandowski Department of Physics. regular
measures on a suitable completion of A/G are in correspondence with certain functions of
loops and dieomorphism invariant measures correspond to generalized knot and link
invariants. The general setting is provided by the Abelian Calgebra of functions on A/G
generated by Wilson loops i. USA email ashtekarphys. including certain topological theories
and general relativity . Loops. University of Florida. Pennsylvania . The framework is well
suited for quantization of dieomorphisminvariant theories of connections. a faithful. In the
quantum theory.edu. links. dieomorphisminvariant measure is introduced.Representation
Theory of Analytic Holonomy Calgebras Abhay Ashtekar Center for Gravitational Physics
and Geometry. Pennsylvania State University. the Hilbert space of states On leave from
Instytut Fizyki Teoretycznej. Poland . The Wilson loop functionals then serve as the
conguration operators in the quantum theory. Florida . Introduction The space A/G of
connections modulo gauge transformations plays an important role in gauge theories. USA
email lewandfuw. Hoza . University Park.e. and graphs feature prominently in this
description. A/G is the classical conguration space. The representation theory of this algebra
leads to an interesting and powerful duality between gauge equivalence classes of
connections and certain equivalence classes of closed loops. By carrying out a nonlinear
extension of the theory of cylindrical measures on topological vector spaces. then. knots.
Warsaw University ul. In particular.
n will be taken to be and G will be taken to be SU both for concreteness and simplicity. at
the outset. For this. the Wilson loop functions provide a natural set of manifestly
gaugeinvariant conguration observables. Consequently. will be a Cauchy surface in an n
dimensional spacetime in the Lorentzian regime or an ndimensional spacetime in the
Euclidean regime. the Gelfand spectral theory provides a natural setting since any given
Abelian Calgebra with identity is naturally isomorphic with the Calgebra of continuous
functions on a compact. Abhay Ashtekar and Jerzy Lewandowski would consist of all
squareintegrable functions on A/G with respect to some measure. Even if this problem could
be overcome. Since these are conguration observables. The next step is to construct
representations of this algebra. These choices correspond to the simplest nontrivial
applications of the framework to physics. they commute even in the quantum theory. Thus.
the theory of integration over A/G would lie at the heart of the quantization procedure. In a
Hamiltonian approach. As the notation suggests. Fix an nmanifold on which the connections
are to be dened and a compact Lie group G which will be the gauge group of the theory
under consideration. In the main body of this paper. Ashtekar and Isham hereafter referred to
as AI developed an algebraic approach to tackle these problems. The Calgebra is therefore
Abelian. there is a correspondence between regular measures on spC In typical applications.
the wellknown techniques to dene integrals do not go through and. A natural strategy is to
use the Wilson loop functionsthe traces of holonomies of connections around closed loops
on to generate this Calgebra. . since A/G is an innitedimensional nonlinear space. AI pointed
out that. spC can be constructed directly from the given Calgebra its elements are
homomorphisms from the given Calgebra to the algebra of complex numbers. the Gelfand
spectrum spC of that algebra. d for some regular measure d and the Wilson loop operators
act on squareintegrable functions on spC simply by multiplication. manifestly gaugeinvariant
momentum observables is dicult already in the classical theory these observables would
correspond to vector elds on A/G and dierential calculus on this space is not well developed.
Unfortunately. one would still face the issue of constructing selfadjoint operators
corresponding to physically interesting observables. since the elements of the Calgebra are
labelled by loops. The rst step is the construction of a Calgebra of conguration observablesa
suciently large set of gaugeinvariant functions of connections on . every continuous cyclic
representation of the AI Calgebra is of the following type the carrier Hilbert space is L spC .
the problem appears to be rather dicult. The problem of constructing the analogous. Hausdor
space. The AI approach is in the setting of canonical quantization and is based on the ideas
introduced by Gambini and Trias in the context of YangMills theories and by Rovelli and
Smolin in the context of quantum general relativity . Recently.
Holonomy Calgebras
and certain functions on the space of loops. Dieomorphisminvariant measures correspond to
knot and link invariants. Thus, there is a natural interplay between connections and loops, a
point which will play an important role in the present paper. Note that, while the classical
conguration space is A/G, the domain space of quantum states is spC . AI showed that A/G
is naturally embedded in spC and Rendall showed that the embedding is in fact dense.
Elements of spC can be therefore thought of as generalized gaugeequivalent connections.
To emphasize this point and to simplify the notation, let us denote the spectrum of the AI
Calgebra by A/G. The fact that the domain space of quantum states is a completion ofand
hence larger thanthe classical conguration space may seem surprising at rst since in
ordinary i.e. nonrelativistic quantum mechanics both spaces are the same. The enlargement
is, however, characteristic of quantum eld theory, i.e. of systems with an innite number of
degrees of freedom. AI further explored the structure of A/G but did not arrive at a complete
characterization of its elements. They also indicated how one could make certain heuristic
considerations of Rovelli and Smolin precise and associate momentum operators with closed
strips i.e. ndimensional ribbons on . However, without further information on the measure d
on A/G, one cannot decide if these operators can be made selfadjoint, or indeed if they are
even densely dened. AI did introduce certain measures with which the strip operators interact
properly. However, as they pointed out, these measures have support only on
nitedimensional subspaces of A/G and are therefore too simple to be interesting in cases
where the theory has an innite number of degrees of freedom. In broad terms, the purpose of
this paper is to complete the AI program in several directions. More specically, we will
present the following results . We will obtain a complete characterization of elements of the
Gelfand spectrum. This will provide a degree of control on the domain space of quantum
states which is highly desirable since A/G is a nontrivial, innitedimensional space which is not
even locally compact, while A/G is a compact Hausdor space, the completion procedure is
quite nontrivial. Indeed, it is the lack of control on the precise content of the Gelfand
spectrum of physically interesting Calgebras that has prevented the Gelfand theory from
playing a prominent role in quantum eld theory so far. . We will present a faithful,
dieomorphisminvariant measure on A/G. The theory underlying this construction suggests
how one might introduce other dieomorphisminvariant measures. Recently, Baez has
exploited this general strategy to introduce new measures. These developments are
interesting from a strictly mathematical standpoint because the issue of existence of such
measures was a subject of
Abhay Ashtekar and Jerzy Lewandowski
controversy until recently . They are also of interest from a physical viewpoint since one can,
for example, use these measures to regulate physically interesting operators such as the
constraints of general relativity and investigate their properties in a systematic fashion. .
Together, these results enable us to dene a RovelliSmolin loop transform rigorously. This is a
map from the Hilbert space L A/G, d to the space of suitably regular functions of loops on .
Thus, quantum states can be represented by suitable functions of loops. This loop
representation is particularly well suited to dieomorphisminvariant theories of connections,
including general relativity. We have also developed dierential calculus on A/G and shown
that the strip i.e. momentum operators are in fact densely dened, symmetric operators on L
A/G, d i.e. that they interact with our measure correctly. However, since the treatment of the
strip operators requires a number of new ideas from physics as well as certain techniques
from graph theory, these results will be presented in a separate work. The methods we use
can be summarized as follows. First, we make the nonlinear duality between connections
and loops explicit. On the connection side, the appropriate space to consider is the Gelfand
spectrum A/G of the AI Calgebra. On the loop side, the appropriate object is the space of
hoopsi.e. holonomically equivalent loops. More precisely, let us consider based, piecewise
analytic loops and regard two as being equivalent if they give rise to the same holonomy
evaluated at the basepoint for any Gconnection. Call each holonomic equivalence class a
Ghoop. It is straightforward to verify that the set of hoops has the structure of a group. It also
carries a natural topology , which, however, will not play a role in this paper. We will denote it
by HG and call it the hoop group. It turns out that HG and the spectrum A/G can be regarded
as being dual to each other in the following sense . Every element of HG denes a
continuous, complexvalued function on A/G and these functions generate the algebra of all
continuous functions on A/G. . Every element of A/G denes a homomorphism from HG to the
gauge group G which is unique modulo an automorphism of G and every homomorphism
denes an element of A/G.
In the case of topological vector spaces, the duality mapping respects the topology and the
linear structure. In the present case, however, HG has the structure only of a group and A/G
of a topological space. The duality mappings can therefore respect only these structures.
Property above
should add, however, that in most of these debates, it was A/Grather than its completion
A/Gthat was the center of attention.
We
Holonomy Calgebras
provides us with a complete algebraic characterization of the elements of Gelfand spectrum
A/G while property species the topology of the spectrum. The second set of techniques
involves a family of projections from the topological space A/G to certain nitedimensional
manifolds. For each subgroup of the hoop group HG, generated by n independent hoops, we
dene a projection from A/G to the quotient, Gn /Ad, of the nth power of the gauge group by
the adjoint action. An element of Gn /Ad is thus an equivalence class of ntuples, g , . . . , gn ,
with gi G, where g , . . . , gn g g g , . . . , g gn g for any g G. The lifts of funcn tions on G /Ad
are then the nonlinear analogs of the cylindrical functions on topological vector spaces.
Finally, since each Gn /Ad is a compact topological space, we can equip it with measures to
integrate functions in the standard fashion. This in turn enables us to dene cylinder measures
on A/G and integrate cylindrical functions thereon. Using the Haar measure on G, we then
construct a natural, dieomorphisminvariant, faithful measure on A/G. Finally, for
completeness, we will summarize the strategy we have adopted to introduce and analyze the
momentum operators, although, as mentioned above, these results will be presented
elsewhere. The rst observation is that we can dene vector elds on A/G as derivations on the
ring of cylinder functions. Then, to dene the specic vector elds corresponding to the
momentum or strip operators we introduce certain notions from graph theory. These specic
vector elds are divergencefree with respect to our cylindrical measure on A/G, i.e. they leave
the cylindrical measure invariant. Consequently, the momentum operators are densely dened
and symmetric on the Hilbert space of squareintegrable functions on A/G. The paper is
organized as follows. Section is devoted to preliminaries. In Section , we obtain the
characterization of elements of A/G and also present some examples of those elements
which do not belong to A/G, i.e. which represent genuine, generalized gauge equivalence
classes of connections. Using the machinery introduced in this characterization, in Section
we introduce the notion of cylindrical measures on A/G and then dene a natural, faithful,
dieomorphisminvariant, cylindrical measure. Section contains concluding remarks. In all
these considerations, the piecewise analyticity of loops on plays an important role. That is,
the specic constructions and proofs presented here would not go through if the loops were
allowed to be just piecewise smooth. However, it is far from being obvious that the nal results
would not go through in the smooth category. To illustrate this point, in Appendix A we
restrict ourselves to U connections and, by exploiting the Abelian character of this group,
show how one can obtain the main results of this paper using piecewise C loops. Whether
similar constructions are possible in the nonAbelian case is, however, an open question.
Finally, in Appendix
Abhay Ashtekar and Jerzy Lewandowski
B we consider another extension. In the main body of the paper, is a manifold and the gauge
group G is taken to be SU . In Appendix B, we indicate the modications required to
incorporate SU N and U N connections on an nmanifold keeping, however, piecewise
analytic loops.
Preliminaries
In this section, we will introduce the basic notions, x the notation, and recall those results
from the AI framework which we will need in the subsequent discussion. Fix a dimensional,
real, analytic, connected, orientable manifold . Denote by P , SU a principal bre bundle P
over the base space and the structure group SU . Since any SU bundle over a manifold is
trivial we take P to be the product bundle. Therefore, SU connections on P can be identied
with suvalued form elds on . Denote by A the space of smooth say C SU connections,
equipped with one of the standard Sobolev topologies see, e.g., . Every SU valued function g
on denes a gauge transformation in A, A g g Ag g dg. .
Let us denote the quotient of A under the action of this group G of local gauge
transformations by A/G. Note that the ane space structure of A is lost in this projection A/G is
a genuinely nonlinear space with a rather complicated topological structure. The next notion
we need is that of closed loops in . Let us begin by considering continuous, piecewise
analytic C parametrized paths, i.e. maps p , s . . . sn , . which are continuous on the whole
domain and C on the closed intervals sk , sk . Given two paths p , and p , such that p p , we
denote by p p the natural composition p p s p s, for s , p s , for s , . .
The inverse of a path p , is a path p s p s. . A path which starts and ends at the same point is
called a loop. We will be interested in based loops. Let us therefore x, once and for all, a
point x . Denote by Lx the set of piecewise C loops which are based at x , i.e. which start and
end at x . Given a connection A on P , SU , a loop Lx , and a xed point x in the ber over the
point x , we denote the corresponding element of the
A. . where A A/G. We will call it the hoop group . Denote it by HA and call it the holonomy
algebra. A H. the group structure of HG has been systematically exploited and extended by
Di Bartolo. e. Owing to SU trace identities. bounded between and T A Tr H. where. and
where the trace is taken in the fundamental representation of SU . where e is the identity in
SU . on the right. whence the hoop should in fact be called an SU hoop. so that p p is a loop
in Lx . for example.g. we drop this sux. More precisely. A. they belong to the same
hoop.Holonomy Calgebras holonomy group by H. two loops which have the same orientation
and which dier only in their parametrization dene the same hoop. has a natural group
structure . . Thus. since the group is xed to be SU in the main body of this paper. we can now
introduce the AI Calgebra. Lx will be said to be holonomically equivalent. The collection of all
hoops will be denoted by HG. Note that in general the equivalence relation depends on the
choice of the gauge group. Each holonomic equivalence class will be called a hoop. With this
machinery at hand. . if paths p and p are such that p p and p p x . A. any loop in the hoop .
Since P is a product bundle. Since each T is a bounded continuous func The hoop
equivalence relation seems to have been introduced independently by a number of authors
in dierent contexts e. The hoop to which a loop belongs will be denoted by . by Gambini and
collaborators in their investigation of YangMills theory and by Ashtekar in the context of
gravity. it has the structure of a algebra. Recently. if two loops dier just by a retraced
segment. T is called the Wilson loop function in the physics literature. because of the
properties of the holonomy map. for every SU connection A in A. i H. and Griego to develop
a new and potentially powerful framework for gauge theories. products of these functions can
be expressed as sums T T T T . and if is a path starting at the point p p . However. Using the
space of connections A. We begin by assigning to every HG a realvalued function T on A/G.
Gambini. the space of hoops. then p p and p p dene the same hoop. Note that. Therefore.
HG. thus HG Lx / . A is any connection in the gauge equivalence class A. . Similarly. we now
introduce a key equivalence relation on Lx Two loops . we will make the obvious choice x x .
the complex vector space spanned by the T functions is closed under the product. we have
used the composition of hoops in HG.
Therefore. the spectrum spHA. d for some regular measure d on A/G and elements of HA.
each of these representations is uniquely determined via the GelfandNaimark Segal
construction by a positive linear functional on the Calgebra HA ai Ti ai Ti ai Ti . By
construction. it is equipped with the identity element To oK . one can show that the space
A/G of connections modulo gauge transformations is densely embedded in spHA. The
operation. Furthermore. the product. Furthermore. one can make the structure of HA explicit
by constructing it. we can denote the spectrum as A/G. and the norm on HA can be specied
directly on F Lx /K as follows ai i K i i ai i K . zero loop. The completion HA of HA under the
sup norm has therefore the structure of a Calgebra. bounded. Second. from the space of
loops Lx . Thus. HA is an Abelian Calgebra. act simply by pointwise multiplication. Finally. we
can directly apply the Gelfand representation theory. Therefore. where ai is the complex
conjugate of ai . elements of HA can be represented either as ai Ti or as ai i K . cyclic
representation of HA by bounded operators on a Hilbert space has the following form the
representation space is the Hilbert space L A/G. HA is a subalgebra of the Calgebra of all
complexvalued. regarded as functions on A/G. i ai i K bj j K j ij ai i K i AA/G sup ai Ti A. The
functionals naturally dene functions on the space Lx of loopswhich in turn serve as the
generating functionals for the representa . continuous functions on a compact Hausdor
space. As in . . The Calgebra HA is obtained by completing HA in the norm topology. Let us
summarize the main results that follow .e. It is then easy to verify that HA F Lx /K. step by
step. i . Abhay Ashtekar and Jerzy Lewandowski tion on A/G. This is the AI Calgebra. we
know that HA is naturally isomorphic to the Calgebra of complexvalued. Note that the
Kequivalence relation subsumes the hoop equivalence. Denote by F Lx the free vector space
over complexes generated by Lx . every continuous. A A/G. ai bj i j K i j K . where o is the
trivial i. First. continuous functions thereon. Let K be the subspace of F Lx dened as follows i
ai i K i i ai Ti A .
however. and the space of hoops. The measure d is invariant under this induced action on
A/G i is invariant under the induced action on HG. triple and higher products can. more
precisely.j ai aj i j i .. is well dened while the second condition by . it follows that the positive
linear functional extends to the full Calgebra HA. i i.e. conguration operators are labelled by
elements of HG. The elements of the representation space i. obtained by extending by
linearity.Holonomy Calgebras tionvia T . Note. or. one can answer this question armatively A
function on Lx serves as a generating functional for the GNS construction if and only if it
satises the following two conditions ai Ti ai i . a dieomorphism invariant function of hoops is a
function of generalized knotsgeneralized. one cannot get rid of double products. together.
Hence. Conversely. ensures that it satises the positivity property. there is a correspondence
If the gauge group were more general. The question naturally arises can we write down
necessary and sucient conditions for a function on the loop space Lx to qualify as a
generating functional Using the structure of the algebra HA outlined above. they provide a
positive linear functional on the algebra HA. Thus. quantum states are L functions on A/G. its
spectrum A/G. Thus. be reduced to linear combinations . j The rst condition ensures that the
functional on HA. nally.e. i and . that the rst equation in .. intersections. determined by
functions on HG satisfying . HG. a loop function satisfying . We thus have an interesting
interplay between connections and loops. for example. The group of dieomorphisms on has
a natural action on the algebra HA. generalized gaugeequivalent connections elements of
A/G and hoops elements of HG. in turn. Now. we could not have expressed products of the
generators T as sums eqn .. and segments that may be traced more than once. Finally.
ensures that the generating function factors through the hoop equivalence relation . For SU .
every regular measure on A/G provides through vacuum expectation values a loop functional
T satisfying . The generators T of the holonomy algebra i. determines a cyclic representation
of HA and therefore a regular measure on A/G. there is a canonical correspondence between
regular measures on A/G and generating functionals . Regular measures d on A/G are. B
HA. since HA is dense in HA and contains the identity. because our loops are allowed to
have kinks. B B . Thus.
. . the spectrum is the StoneCch compactication e of that space. one does not have as much
control as one would like on the points that lie in its closure see. However. the generating
functional is dened on single and double loops. . . in practice. We shall see in this section
that this is indeed the case. The main point here is that a given set of double products of T
and the T themselves. However.e. i. in the second. and therefore outside the range of
applicability of standard theorems. In the rst. whence the situation is again rather unruly. the
holonomy Calgebra is also rather special it is constructed from natural geometrical
objectsconnections and loopson the manifold . generally we will not explicitly distinguish
between paths and loops and their images in . one might hope that its spectrum can be
characterized through appropriate geometric constructions. the situation seems to be even
more complicated since the algebra HA is generated only by certain functions on A/G. In
particular. of elements of A/G A/G.g. at least at rst sight. . . a good control on the precise
content and structure of the Gelfand spectrum A/G of HA is needed to make further progress.
and in the third we give examples of generalized gaugeequivalent connections. . bounded
functions on a completely regular topological space. In the case of a Calgebra of all
continuous. The spectrum The constructions discussed in Section have several appealing
features. . HA appears to be a proper subalgebra of the Calgebra of all continuous functions
on A/G. . . we introduce the key tools that are needed also in subsequent sections. . and
regular dieomorphisminvariant measures on generalized gaugeequivalent connections. This
section is divided into three parts. . into a nite number of independent hoops. while Rn is
densely embedded in the spectrum. e. Even for simple algebras which arise in nonrelativistic
quantum mechanicssuch as the algebra of almost periodic functions on Rn the spectrum is
rather complicated. whence. Abhay Ashtekar and Jerzy Lewandowski between generalized
knot invariants which in addition satisfy . In this case. n . To keep the discussion simple. . link
invariants can also feature in the denition of the generating function. in addition to knot
invariants. Therefore. k . In the case of the AI algebra HA. we present the main result. Loop
decomposition A key technique that we will use in various constructions is the decomposition
of a nite number of hoops.. The specic techniques we use rest heavily on the fact that the
loops involved are all piecewise analytic. they open up an algebraic approach to the
integration theory on the space of gaugeequivalent connections and bring to the forefront the
duality between loops and connections.
. m . Denote by n the number of edges that result. Our aim then is to show that.e. The edges
and vertices form a piecewise analytically embedded graph. k . as before. The rst condition
makes the sense in which each hoop i can be decom posed in terms of j precise while the
second condition species the sense in which the new hoops . . and has a nonzero parameter
measure along any loop to which it belongs. . . there exists a nite set of loops. k . . . each
edge in the original decomposition of k loops . if the initial set . . . n . . . . such that none of
the i has a piece which is immediately retraced i. Let us call these marked points vertices.
whence. . . . . . . . Next. . then we have HG . . . . . which satises the following properties . Let
us call these oriented paths edges. . with i identity in HG for any i. each of these paths is
piecewise analytic. Let us begin by choosing a loop i Lx in each hoop i . k . k of loops is
extended to include p additional loops. . . none of the loops contains a path of the type . can
be expressed as a composition of a nite number of its n edges. n are independent. where.
this decomposition is minimal in the sense that. . . . If we denote by HG . ignoring
parametrization. Each of the new loops i contains an open segment i. m the subgroup of the
hoop group HG generated by the hoops . . . the holonomies around these loops could be
interrelated. Finally. we will be able to show that any nite set of loops can be decomposed
into a nite number of independent segments and this in turn leads us to the desired
independent hoops. an embedding of an interval which is traced exactly once and which
intersects any of the remaining loops j with i j at most at a nite number of points. . . Now. . . .
divide each loop i into paths which join consecutive vertices. i denotes the hoop to which i
belongs. n Lx . Thus. Ignore the isolated intersection points and mark the end points of all the
overlapping intervals including the end points of selfoverlaps of a loop. k. . .Holonomy
Calgebras of loops need not be holonomically independent every open segment in a given
loop may be shared by other loops in the given set. . . By construction. .e. since the loops are
piecewise analytic. . . However. for any connection A in A. . for all i . kp . k HG . given a nite
number of hoops. . The availability of this decomposition will be used in the next subsection
to obtain a characterization of elements of A/G and in Section to dene cylindrical functions on
A/G. Two distinct paths intersect at a nite set of points. each loop in the list is contained in
the graph and. . any two which overlap do so either on a nite number of nite intervals and/or
intersect in a nite number of points. using the fact that the loops are piecewise analytic. . is
part of at least one of the loops . .
Abhay Ashtekar and Jerzy Lewandowski is expressible as a product of the edges which
feature in the decomposition of the k p loops. . . . k . . . i. . For the moment we only note that
we have achieved our goal the property above ensures that the set . k in our initial set can be
expressed as a nite composition of hoops dened by elements of S and their inverses where a
loop i and its inverse i may occur more than once. . . . where vi are the end points of the
edge ei . if the initial set . and.. . then each hoop i is a nite product of the hoops j . . and that
above implies that the hoop group generated by . n has an interesting consequence which
will be used repeatedly Lemma . k is contained in the hoop group generated by . there exists
a connection A A such that Hi . kp . However. . . Loops i are not unique because of the
freedom in choosing the curves qv. . . that our decomposition satises . such that the paths q
v agree with the paths qv whenever v v . . . S is a nite set and every i S is a nontrivial loop in
the sense that the hoop i it denes is not the identity element of HG. . . . . starting and ending
at the basepoint x . . The next step is to convert this edge decomposition of loops into a
decomposition in terms of elementary loops. .e. n corresponding to . . n. . . We will conclude
this subsection by showing that the independence of hoops . k of loops is extended to
include p additional loops. A gi i . . i qvi ei qvi . . . . and denote by S the set of these n loops.
This is easy to achieve. n of loops is independent in the sense we specied in the point just
below eqn . . . . . . . . piecewise analytic curve qv such that these curves overlap with any
edge ei at most at a nite number of isolated points. . . . . Let us rst note that the
decomposition satises certain conditions which follow immediately from the properties of the
segments noted above . gn SU n . These properties will play an important role throughout the
paper. For every g . . k . . and if S is the set of n loops . . Consider the closed curves i . . kp .
. n . . . . every hoop . we will show that they provide us with the required set of independent
loops associated with the given set . Connect each vertex v to the basepoint x by an
oriented. . . every loop i contains an open interval which is traversed exactly once and no nite
segment of which is shared by any other loop in S. . . .
From now on. w su. consider a connection Ai t g Ai g.. i. Finally. we have the following.
Then. Ai is not the identity element of SU . Similarly. . pick an Abelian connection Ai ai w
where ai is a realvalued form on and w a xed element of the Lie algebra of SU . in particular
gi . it captures the idea that the loops . Hence. .e. w with w a multiple of w . Now. .Holonomy
Calgebras Proof Fix a sequence of elements g . by choosing g and t appropriately. . t R
Then. for each loop i . are independent in the sense of . Then. Let furthermore the support of
ai intersect the ith segment si in a nite interval.. we will say that loops satisfying . . that notion
does not refer to connections at all and is therefore not tied to any specic gauge group. .
since the square root is well dened in SU but not in the sense of . Ai exp tg wg. we can set . .
n . n we constructed above are holonomically independent as far as SU connections are
concerned. The converse is not true. Ai coincide with any element of SU . . g SU . we just
proved can be taken as a statement of the independence of the SU hoops . We conclude this
subsection with two remarks. . The result . Let and be two loops in Lx which do not share any
point other than x and which have no selfintersections or selfoverlaps. and. i . we can choose
connections Ai independently for every i . . if j i. This notion of independence is. a point in SU
n . the connection A A An satises . Then. however. they are necessarily independent in the
sense of . . it is of the form Hi . we can make Hi . Indeed. let us suppose that the connection
is generic in the sense that the holonomy that the holonomy Hi . . . Because of the
independence of the loops i noted above. Ai exp w. More precisely. is independent in the
sense of . We just showed that if loops are independent in the sense of . . we have Hi .
although they are obviously not independent in the sense of . Set and . gn of SU . Next.
intersect the jth segment sj only on a set of measure zero as measured along any loop m
containing that segment. given . Then. it is easy to check that i . weaker than the one
contained in the statement above.
. . to ensure that the measure introduced in Section is well dened. This characterization
continues to hold if the gauge group is replaced by SU N or indeed any group which has the
property that. AI were able to show that Lemma . This second characterization is useful
because it involves only the loops. However. Abhay Ashtekar and Jerzy Lewandowski are
independent and those satisfying . A Tr HA . no mention is made of connections and
holonomies. . However. in particular. Algebraically. preserving homomorphisms from HA to
the algebra of complex numbers. using certain constructions due to Giles. Our aim now is to
establish the converse of this result. we have paraphrased the AI result somewhat because
they did not use the notion of hoops. The availability of the loop decomposition sheds
considerable light on the hoop equivalence one can show that two loops in Lx dene the same
SU hoop if and only if they dier by a combination of reparametrization and immediate
retracings. Weak independence will suce for all considerations in this section. HG. for every
nonnegative integer n. . n . . n to any group G can be extended to every nitely generated
subgroup of HG which contains HG . Although we will not need them explicitly. it is
instructive to spell out some algebraic consequences of these two notions. Thus. . then every
homomorphism from other hand.a They also exhibited the homomorphism explicitly. the
group contains a subgroup which is freely generated by n elements. if HG . Given a
homomorphism H from HG to SU we can simply via A Tr H . Recall rst that elements of the
Gelfand spectrum A/G are in correspondence with multiplicative. Characterization of A/G We
will now use the availability of this hoop decomposition to obtain a complete characterization
of all elements of the spectrum A/G. On the i are independent. Let us make a small
digression to illustrate why the converse cannot be entirely straight forward. . The question is
whether this A can qualify dene A . weak independence of hoops i ensures that i freely
generate a subgroup of HG. are weakly independent. Now. The characterization is based on
results due to Giles . Here. the step involved is completely straightforward. . we will need
hoops which are independent. . . every A A/G acts on T HA and hence on any hoop to
produce a complex number A . Every element A of A/G denes a homomorphism HA from the
hoop group HG to SU such that. This denition extends naturally to hoops. .
Then. the homomorphism A must be continuous. a priori. It turns out. k .b.a. .b Two
homomorphisms H and H dene the same element of the spectrum . . we have H H . and
every nite set of hoops . one might expect that the converse of the AI result would not be
true. . For every homomorphism H from HG to SU . . we have Lemma . From the denition. as
before. We begin our detailed analysis with an immediate application of Lemma . that if the
holonomy algebra is constructed from piecewise analytic loops. .a is in fact equivalent to . .
..Holonomy Calgebras as an element of the spectrum. . . n under H. n . should satisfy Af f . .
We now use this lemma to prove the main result. n . as in Section . . . . . A is the holonomy of
the connection A around any loop in the hoop class . which arise from elements of A/G. . n
generated by . .. Given a homomorphism H from the hoop group HG to SU there exists an
element AH of the spectrum A/G such that AH Tr H . A . the apparently weaker condition .
there exists an SU connection A such that for every i in the set.b appears to be a stronger
requirement than . a set of independent loops corresponding to the given set of hoops. . Use
the construction of Section . . . Since. . n be. . to obtain a connection A such that gi Hi . Proof
Let . Denote by g . On the other hand. . however. Whether it would continue to hold if one
used piecewise smooth loopsas was the case in the AI analysisis not clear see Appendix A.a
for all hoops . . and the converse does hold. That is. . Hi .b for every f ai i K HA. . . . A . . .
Since each of the given i belongs to HG . gn the image of . where. A for all i. . to qualify as an
element of the spectrum A/G. we have. it is clear that A T sup T A AA/G . H i Hi . Recall that
each A A/G is a homomorphism from HA to complex numbers and as before denote by A the
number assigned to T HA by A. . one would expect that there are more homomorphisms H
from the hoop group HG to SU than the HA . It then follows from the denition of the hoop
group that for any hoop in the subgroup HG .e. . Lemma . in particular. i.
we obtain the following characterization of the points of the Gelfand spectrum of the
holonomy Calgebra Theorem . . We will rst show that h passes through the Kequivalence
relation of AI noted in Section and is therefore a welldened complexvalued function on the
holonomy algebra HA F HG/K. Next. . for some independent g in SU . .b provides a
homomorphism from HA to the algebra of complex numbers. given a nite set of hoops. That
h is linear is obvious from its denition. . Tr H for all hoops. Now. Tr H general result given two
homomorphisms H and H from a group G to g Tr H g. g G. Let us begin with the free
complex vector space F HG over the hoop group HG and dene on it a complexvalued
function h as follows h ai i ai Tr H i . we have k i k ai Ti A A A/G h i ai i . Using the hoop group
HG for G.e. it follows that the function h on F HG has a welldened projection to the holonomy
algebra HA. the continuity of this funtion on HA is obvious from . B HA. HG. there exists a
connection A such that k k k h i ai i i ai Ti A sup AA/G i ai Ti A . by combining the results of
the three lemmas. Every point A of A/G gives rise to a homomorphism H from the hoop
group HG to SU and every homomorphism H denes a point of A/G.e. To summarize. there
exists g SU SU such that Tr H such that H g g H g g .b. k . immediately implies that. Note rst
that Lemma . That it respects the operation and is a multiplicative homomorphism follows
from the denitions . suppose H and H give rise to the same element of the spectrum. . denes
a unique element AH via AH B hB. such that A Tr H . Abhay Ashtekar and Jerzy
Lewandowski if and only if H g H g. we obtain the desired uniqueness result. .. Finally.
Therefore. of these operations on HA and the denition of the hoop group HG. i. to show H
that the right side of . Hence h extends uniquely to a homomorphism from the Calgebra HA
to the algebra of complex numbers. then. there is a In particular. Proof The idea is to dene
the required A using . where g is independent of g. i. This is a correspondence modulo the
trivial ambiguity that homomorphisms H and g H g dene Since the left side of this implication
denes the Kequivalence. .
However. however..g. Results obtained in this subsection have a similar avor. It is striking
that Theorem .Holonomy Calgebras the same point of A/G. Therefore. . does not require the
homomorphism H to be continuous. but in a generalized bundle . tells us.. there is a key
dierence our main result shows the existence of a generalized connection modulo gauge.
This is one illustration of the nontriviality of Note. gives rise to the homomorphism HA g x HA
gx . In this case. no reference is made to the topology of the hoop group anywhere. one can
construct a smooth connection which is indistinguish able from AH as far as these hoops are
concerned. Lemma . AH nection.e. A/G is naturally embedded in A/G. there would exist
nontrivial Gprincipal bundles over . This is stronger than the statement that A/G is dense in
A/G. A denes a ho momorphism HA HG SU . that while this proof is tailored to the holonomy
Calgebra HA. in general. Then. This purely algebraic characterization of the elements of A/G
makes it convenient to use it in practice. connections on all possible bundles belong to the
Gelfand spectrum see Appendix B. by Theorem . . A generalized connection can also be
given a geometrical meaning in terms of parallel transport. it can be approximated by smooth
connections in the sense that. as Lemma . . an element of A/G rather than a regular
connection in A/G. indeed. now implies that the embedding is in fact dense. Furthermore.
one can reconstruct a connection modulo gauge such that the given function is the trace of
the holonomy of that connection. i. Nonetheless. A and A dene the same element of the
Gelfand spectrum. Necessary and su cient conditions for H to arise from a smooth
connection were given by Barrett see also . Examples of A Fix a connection A A. There are
several folk theorems in the literature to the eect that given a function on the loop space Lx
satisfying certain conditions. see. even if we begin with a specic bundle in our construction of
the holonomy Calgebra. This provides an alternative proof of the AI and Rendall results
quoted in Section . Rendalls proof is applicable in more general contexts. . the holonomy H.
A g Ag g dg. e. . Had the gauge group been dierent and/or had a dimension greater than . A
gaugeequivalent connection. Note that the homomorphism H determines an element of A/G
and is not. We conclude this subsection with some remarks . given any nite number of
hoops. a smooth gaugeequivalent connot of A/G. For a summary and references.
the homomorphism H from HG to SU it denes sends every hoop to the identity element e of
SU and the multiplicative homomorphism it denes from the Calgebra HA to complex numbers
sends each T to /Tr e. While HA has . we can similarly construct elements which depend . i..
or. if HG x . For each choice of . it does so only a nite number of times. Note rst that. We will
say that A A/G has support on a subset S of if for every hoop which has at least one
representative loop which fails to pass through S. otherwise. we have j e. so that. The
generalized connections we want to dene now will depend only on these tangent directions
at x. is independent of the choice of the loop in the hoop class used in its construction. Let us
denote the incoming and outgoing tan gent directions to the curve at its jth passage through
x by vj . Hence. In this subsection. Furthermore. equivalently. and hence a generalized
connection with support at x. arrives at x and then simply returns by retracing its path in a
neighborhood of x. . It denes a homomorphism from HG to SU which is nontrivial only on HG
x . where HG x is the space of hoops every loop in which passes through x. In particular. is
the most general element of A/G which has support at x and depends only on the tangent
vectors at that point. we will illustrate another facet of this nontriviality we will give examples
of elements of A/G which do not arise from A/G. noted in Theorem . In ii. conditions i and ii
are insensitive to it. we used tangent directions rather than vectors to ensure this. if at the jth
passage. Recall rst that the holonomy of the zero connection around any hoop is identity. we
have i HA e . Let be a not necessarily continuous mapping from the sphere S of directions in
the tangent space at x to SU . if Lx passes through x. . . one can verify that . ii A K is
regarded as a homomorphism of algebras. vj . We are now ready to give the simplest
example of a generalized connec tion A which has support at a single point x .e. the mapping
H is well dened. . the identity element of SU . These are the generalized gaugeequivalent
connections A which have a distributional character in the sense that their support is
restricted to sets of measure zero in . Set j vj vj . / N . Using analyticity of the loops. Now.
dene H HG SU via H e. We now use this information to introduce the notion of the support of
a generalized connection. Abhay Ashtekar and Jerzy Lewandowski the completion procedure
leading from A/G to A/G. A an ambiguity. the identity element of SU . owing to piecewise
analyticity. say N .
We conclude with an example. The key idea in our approach is to replace the standard
linear duality on vector spaces by the duality between connections and loops. which is
nontrivial only on HG M and therefore denes a generalized connection with support on M . g
N g N In this section we shall develop a general strategy to perform integrals on A/G and
introduce a natural. The basic strategy is to carry out an appropriate generalization of the
theory of cylindrical measures . is a welldened homomorphism from HG to SU .M HG SU via
if HG M . The existence of this measure provides considerable condence in the ideas
involving the loop transform and the loop representation introduced in .
dieomorphisminvariant measure on this space. one can check that . In the rst part of This
idea was developed also by Baez using.. Let equal if the orientation j of vj . on topological
vector spaces . where g e. Baez and the authors independently realized that the theory of
cylindrical measures provides the appropriate conceptual setting for this discussion.e. Fix a
dimensional.Holonomy Calgebras on the higherorder behavior of loops at x. Denote by HG M
the subset of HG consisting of hoops. Again. Dene Hg. g e. There is no generalized
connection with support at x which depends only on the zerothorder behavior of the curve.
Subsequently. owing to analyticity. each loop of which intersects M nontangentially. . an
approach which is somewhat dierent from the one presented here. . i. i. the identity element
of SU .M e. only on whether the curve passes through x or not. Denote as before the
incoming and the outgoing tangent directions at the jth intersection by vj and vj respectively.
such that at least one of the incoming or the outgoing tangent direction to the curve is
transverse to M . One can proceed in an analogous manner to produce generalized
connections which have support on . M coincides with the orientation of . Hg. oriented..
Integration on A/G . One can construct more sophisticated examples in which the
homomorphism is sensitive to the higherorder behavior of the loop at the points of
intersection and/or the xed SU element g is replaced by gx M SU . the authors of this paper
rst found the faithful measure introduced in this section and reported this result in the rst draft
of this paper. otherwise.e. faithful.or dimensional submanifolds of . As before. g g . any Lx
can intersect M only a nite number of times. analytic submanifold M of and an element g of
SU . however. and if vj is tangential to M . if the two orientations are opposite.
Chronologically. / .
n . Recall that S is generated by a nite number of linearly independent basis vectors. The
idea is to use the results of Section to nd suitable substitutes for these spaces. Perhaps the
most familiar example is the normalized Gaussian measure. let us introduce the following
equivalence relation on A/G A A i HA g HA g for all S and some hoopindependent g SU . we
proceed as follows. More precisely. S is a nitedimensional vector space and we have a
natural projection map S V S. we can dene an equivalence relation on V v v if and only if v .
Using this S . . . generated by a nite number of independent hoops. if a function f on V is
expressible as a pullback via Sj of functions fj on Sj . For this denition to be meaningful. for
some choice of S .. . using A/G in place of V . the set of measures d that we chose on vector
spaces S must satisfy certain compatibility conditions. . Now suppose we are given n
independent loops. Denote the quotient V / by S. . . we will dene cylindrical functions. we will
present a natural cylindrical measure. Equip each nitedimensional vector space S with a
measure d and set d f V S d f . Clearly. We want to extend these ideas. however. A function f
on V is said to be cylindrical if it is a pullback via S to V of a function f on S. Denote by S the .
s for all v . for j . which is unique up to the freedom HA g HA g for some g SU . Abhay
Ashtekar and Jerzy Lewandowski this section. where v. An immediate problem is that A/G is
a genuinely nonlinear space whence there is no obvious analog of V and S which are central
to the above construction of cylindrical measures. we must have d f S S d f . the role of S .
say. s is the action of s S on v V . we know that each A in A/G is completely characterized by
the homomorphism HA from the hoop group HG to SU . Such compatible measures do exist.
. These are the functions we will be able to integrate. . The appropriate choices turn out to be
the following we will let the hoop group HG play the role of V and its subgroups. From
Theorem . Thus. Given a nitedimensional subspace S of V . Denote by S the subgroup of the
hoop group HG they generate. and in the second part. Let V denote a topological vector
space and V its dual. v V and s S . . s v . Cylindrical functions Let us begin by recalling the
situation in the linear case.
cylindrical measures are generally introduced as follows. Proof By denition of the
equivalence relation .Holonomy Calgebras projection from A/G onto the quotient space A/G/ .
as usual. s of V . However. there is a bijection A/G/ SU n /Ad c Given S as in a. . . SU on the
chosen generators. every A in A/G/ is in correspondence with the restriction to S of an
equivalence class HA of homomorphisms from the full hoop group HG to SU . there is a
correspondence between equivalence classes A and equivalence classes of
homomorphisms from S to SU where. This is similar to the situation in the linear case where
one may begin with a set of linearly independent elements s . . Therefore. . The idea is to
consider functions f on A/G which are pullbacks under S of functions f on A/G/ and dene the
integral of f on A/G to be equal to the integral of f on A/G/ . . Let S be a subgroup of the hoop
group which is generated by a nite number of independent hoops. this is the case Lemma .
Although we began with independent loops . SU /Ad b For each choice of independent
generators . A/G/ should have a simple structure that one can control.. at the end one must
make sure that . Then. In particular. two are equivalent if they dier by the adjoint action of SU
. n . . uses it to dene an isomorphism between S V / and Rn . However. . . a specic basis is
often used in subsequent constructions. in turn. . . a There is a natural bijection A/G/ HomS .
One xes a basis in S . However. Some remarks about this result are in order . the
equivalence relation we introduced makes direct reference only to the subgroup S of the
hoop group they generate. introduces a measure on Rn for each n and uses the
isomorphism to dene measures d on spaces S. . let them generate a subspace n S and then
introduce the equivalence relation on S using S directly. Now. The proof of c is a simple
exercise. Fortunately. n of S . that every homomorphism from S to SU extends to a
homomorphism from the full hoop group to SU . These. . since S admits other bases. dene a
cylindrical measure d on V through . . The statement b follows automatically from a the map
in b is given by the values taken by an element of HomS . the topology induced on A/G/ from
SU n /Ad according to b is independent of the choice of free generators of S . it follows from
Lemma . for this strategy to work.
These are the functions we want to integrate. It follows from Lemma . then they are
equivalent where i is any loop in the hoop class i . Let us note an elementary property of
cylindrical functions which will be used repeatedly. In the remainder of this subsection.. By a
cylindrical function f on A/G. we shall regard S as a projection map from A/G onto BS and
parametrize BS by SU n /Ad. Let S be a subgroup of the hoop group which is generated by a
nite number of independent loops. i. The initial choice of the independent hoops is. Thus.
there is now a natural projection from BS onto BS given by the projection HomS . we can
identify A/G/ with SU n /Ad. for some subgroup S of the hoop group which is generated by a
nite number of independent loops. Finally. SU /Ad Using Lemma . Let S and S be two nitely
generated subgroups of the hoop group and such that S S . it is easy to check that if a
function f on A/G is cylindrical with respect to S . are indistinguishable. . . the projection S
maps the domain space A/G of the continuum quantum theory to the domain space of a
lattice gauge theory where the lattice is formed by the n loops . As we just showed in Lemma
. SU HomS . We can now dene cylindrical functions on A/G. arbitrary and it is this
arbitrariness that is responsible for the richness of the continuum theory. Abhay Ashtekar
and Jerzy Lewandowski the nal measure d on V is welldened. we will follow a similar
strategy. In the case now under consideration. Furthermore. is insensitive to the specic
choice of basis made in its denition. . we can dene a cylindrical measure on A/G by rst
choosing suitable measures on SU n /Ad for each n. Let us set BS HomS . it is also
cylindrical with respect to S . in eect. generated by a nite number of independent loops. .
Then. . . . using the isomorphism to dene measures on A/G/ and showing nally that the nal
integrals are insensitive to the initial choices of generators of S .e. we will analyse the
structure of the space C of cylindrical functions. if A and A are two regular connections
whose pullbacks to all i are the same for i . For the topology on BS we will use the one dened
by its identication with SU n /Ad. says that two generalized connections are equivalent if their
actions on the group S . Therefore. First we have . The equivalence relation of Lemma . SU
and f is the pullback of f via this projection. n. we shall mean the pullback to A/G under S of a
continuous function f on BS . . . given a set of independent loops which generate S . n .
however. that each equivalence class A contains a regular connection A..
. Therefore. and since A/G is embedded densely in A/G. it is clear that the map from F to f is
isometric. .Holonomy Calgebras Lemma . C has the structure of a normed algebra with
respect to the operations of taking linear combinations. Proof It is clear by denition of C that if
f C. Hence we have f C. While the rst n and the last n hoops are independent. . given two
elements. . then any complex multiple f as well as the complex conjugate f also belongs to C.
Consider the set of hoops . . . since Si S . Since the norms of F in HA and f in C are given by
F sup F A AA Theorem . its Gelfand transform is a function f on the spectrum A/G. The
Calgebra C is isomorphic to the Calgebra HA. Finally. k that feature in the sum.. Let Si be
nitely generated subgroups of the hoop group HG such that fi are the pullbacks to A/G of
continuous functions fi on BSi . . fi with i . More precisely. and f sup AA/G f A. Thus C has the
structure of a algebra. it follows that the cylindrical functions are bounded. given by f A ai A i .
and since elements of C are pullbacks to A/G of continuous functions on BS . since BS being
homeomorphic to SU n /Ad is compact for any S . A natural class of functions one would like
to integrate on A/G is given by the Gelfand transforms of the traces of holonomies.
decompose them into independent hoops . We will use the sup norm to complete C and
obtain a Calgebra C. . . Given an element F with F A ai Ti A of HA. since the restriction of f to
A/G coincides with F . nd the set of independent hoops. Then.. . Not only is the answer
armative. Then. . . there may be relations between hoops belonging to the rst set and those
belonging to the second. Finally. complex conjugation and sup norm of cylindrical functions. .
of C. it is clear that the function f is the pullback via S of a continuous function f on BS . we
will show that their sum and their product also belong to C. . multiplication. but in fact the
traces of holonomies generate all of C. Using the technique of Section . . . say. Next. n which
generate the given n n hoops and denote by S the subgroup of the hoop group generated by
. f f and f f are also cylindrical with respect to S . . Consider the sets of n and n independent
hoops generating respectively S and S . The question then is if these functions are contained
in C. and denote by S the subgroup of the hoop group they generate. . since HA is obtained
by a C completion of the space of functions . n as in Section . n . we have Proof Let us begin
by showing that HA is embedded in C. it follows that fi are cylindrical also with respect to S .
We can therefore use the sup norm to endow C with the structure of a normed algebra.
. we could have proceeded as follows. . each A A/G/ contains a regular connection. . Its
projection under the appropriate S is. it is good to keep in mind that this alternate strategy is
available as it may make other issues more transparent. A g H . by the preceeding result. n.
we could have introduced an equivalence relation on A/G not A/G which we do not yet have
as follows A A i H . . suggests that from the very beginning we could have introduced the
Calgebra C in place of HA. . . suppose we are given a cylindrical function f on A/G. From a
mathematical viewpoint. this is unsatisfactory since Wilson loops are the basic observables
of gauge theories. C is contained in HA. Begin with the space A/G. it follows from the
Weierstrass theorem that the Calgebra generated by these projected functions is the entire
Calgebra of continuous functions on BS . More precisely. gn of SU . . We can therefore dene
cylindrical functions. from the knowledge of traces in the fundamental representation of all
elements which belong to the group generated by these gi . i . . We can use it as the starting
point in place of the AI holonomy algebra. Then. Thus. . Now. that the quotient A/G/ is
isomorphic to BS . Next. both stemming from the fact that the Wilson loops are now assigned
a secondary role. Fix n elements g . we can reconstruct g . we have the desired result the
Calgebras HA and C are isomorphic. While this strategy seems natural at rst from an
analysis viewpoint. It therefore follows that the space of functions f on BS considered as n a
copy of SU /Ad which come from the functions F on HA using all the hoops i S suce to
separate points of BS . Nonetheless. but now on A/G rather than A/G as the pullbacks of
continuous functions on BS . It is then again true due to Lemma . as remarked immediately
after Lemma . For physical applications. . we have the result that HA is embedded in the
Calgebra C. A g for all S and some hoop independent g SU . contained in the projection of
some element of HA. . Combining the two results. These functions have a natural Calgebra
structure. gn modulo an adjoint map. . we will show that there is also an inclusion in the
opposite direction. the relation between knot/link invariants and measures on the spectrum of
the algebra would now be obscure. This is indeed the case. Since this space is compact.
Then given a subgroup S of HG generated by a nite number of independent hoops. Abhay
Ashtekar and Jerzy Lewandowski of the form F . it has two drawbacks. Remark Theorem .
This is true inspite of the fact that we are using A/G rather than A/G because. and introduce
the notion of hoops and hoop decomposition in terms of independent hoops. . We rst note a
fact about SU ..
In particular. . Denote by d the normalized Haar measure on SU .. i . We will show that the
measure dS is insensitive to the specic choice of the generators of S used in its construction
and that the resulting family of measures on the various BS satises the appropriate
compatibility conditions. d f A/G BS where f is any cylindrical function compatible with S .
faithful. dieomorphism invariant measure on A/G. the idea is similar to the one used in the
case of topological vector spaces. we have BS dS f BS dS f . . Next. holds. Our main
objective is to introduce a natural. n and use the corresponding map SU n /Ad BS to
transport the measure dn to a measure dS on BS . each of which is generated by a nite
number of independent hoops. consider an arbitrary group S which is generated by n
independent loops. Then. A natural measure In this subsection. Theorem . if fi denote the
projections of f to BSi . as in . . We will then explore the properties of the resulting cylindrical
measure on A/G. Our construction is natural since SU comes equipped with the Haar
measure. we will discuss the issue of integration of cylindrical functions on A/G.Holonomy
Calgebras . Choose a set of independent generators . As mentioned in the remarks following
Lemma . Fix a set of generators in each subgroup and denote by dSi the corresponding
measures on BSi . . set dS f . a Let f be a cylindrical function on A/G with respect to two
subgroups Si . Thus. we have not introduced any additional structure on the underlying
manifold on which the initial connections A are dened. if S S .. we will equip the space BS
with a measure dS and. the integral is independent of the choice of generators used in its
denition. It is clear that for the integral to be welldened. . for each subgroup S of the hoop
group which is generated by a nite number of independent hoops. . hence the resulting
cylindrical measure will be automatically invariant under the action of Di. In particular. . of the
hoop group. the set of measures we choose on different BS should be compatible so that the
analog of . It naturally induces a measure on SU n which is invariant under the adjoint action
of SU and therefore projects down unambiguously to a measure dn on SU n /Ad. We will now
exhibit a natural choice for which the compatibility is automatically satised.
Furthermore. to carry out the decompo sition of these n n hoops in terms of independent
hoops.e. Then gi is expressed in terms of gi as gi . . . . . . . . by showing that each of the two
integrals in that equation equals the analogous integral of f over BS . . is a genuine. we can
ignore the distinction between BSi and SU ni /Ad and between BS and SU n / Ad. Ki Kj .. it
follows that every hoop in the second set can be expressed as a composition of the hoops in
the rst set and their inverses. . Thus. in particular. the construction implies that for each i .
Then. is the pullback of a function f on BS . n . the hoop Kj does not appear at all. . where d
is dened by . Note rst that. . let us denote by g . regular and strictly positive measure on A/G.
with S S . Clearly. and where the cylindrical function f on A/G is the Gelfand transform of the
element F of HA. the hoop Ki or its inverse appears exactly once and if i j . . . the given f is
cylindrical also with respect to S . say . . strictly positive and Diinvariant functional on HA vF
A/G d f . ii in the decomposition of i in terms of unprimed hoops. Si S . n while those . Hence.
. as in the proof of Theorem . and a function f which is cylindrical with respect to both of
them. . .gKi .. We will now establish . with n lt n. It will also be convenient to regard functions
on SU n /Ad as functions on SU n which are invariant under the adjoint SU action. since the
generators have been xed. n . the original n n hoops are contained in the group S generated
by the i . To deal with quantities with suxes and simultaneously. we have nitely generated
subgroups S and S of the hoop group. gn a point of the quotient SU n which is projected to g
.. gn in SU n . and. where . . Thus. . c The cylindrical measure d dened by . The generators
of S are . we can integrate the lifts of these functions to SU ni and SU n respectively. . . d is
Di invariant. there exists Ki . . . . Proof Fix ni independent hoops that generate Si and. For
simplicity. Next. instead of carrying out integrals over BSi and BS of functions on these
spaces. . Abhay Ashtekar and Jerzy Lewandowski b The functional v HA C dened below
extends to a linear. for notational simplicity we will let a prime stand for either the sux or .
since the hoops in each set are independent. . n with the following properties i if i j . and.
Furthermore.. use the construction of Section . From the construction of S Sec of S are n tion
. i. . let us assume that the orientation of the primed hoops is such that it is Ki rather than its
inverse that features in the decomposition of i . . denotes a product of elements of the ..
in the second step we used the invariance property of the Haar measure and simplied
notation for dummy variables being integrated. in turn is the Calgebra of all continuous
functions on the compact Hausdor space A/G. . . v is strictly positive. That it is positive.gK . .
i. Hence. . . .. with n and n . ... . This establishes the required result .Holonomy Calgebras
type gm where m Kj for any j. . which.gK ... Then dg dg f g . satises vF F . .. g dg dg f g g dg f
g .. since the measure is normalized. then vF F gt since the value vF F of v on F F is
obtained by integrating a nonnegative function f f on SU n /Ad with respect to a regular
measure.gKn .. The vacuum expectation value function v is therefore a continuous linear
functional on the algebra of all continuous functions on A/G equipped with the L i. mn dm d
dn f g . gn f . Since the given function f on A/G is a pullback of a function f on SU n as well as
of a function f on SU n . The continuity and linearity of v follow directly from its denition. . . . . .
we simply rearranged the order of integration. . To see that we have a regular measure on
A/G. . Hence. . gn .gKn . . .Kn n dK d dKn f .. and. consider the vacuum expectation value
functional v on HA dened by .e. in the third step. Thus. a stronger result holds if F . . in the rst
step. . . we used the fact that. Next. by the RieszMarkov theorem. where. . . we would for
example have f g . . In fact. Since HA is dense in HA it follows that the functional extends to
HA by continuity. . . recall rst that the Calgebra of cylindrical functions is naturally isomorphic
with the Gelfand transform of the Calgebra HA. ..e... is equally obvious.. where in the second
step we have used the invariance and normalization properties of the Haar measure. sup
norm. the integral of a constant produces just that constant . it follows that f g . gn dm mK.. .
.. it follows that In the simplest case. gn dn f g . g f g g f g . d dn f g . gn f g .
selfadjoint operator on L A/G. unlike the standard invariants. Fix a loop Lx and let i iN N .
Under this action. We will see below that. . This functional on the loop space Lx is
dieomorphism invariant since the measure d is. Finally. giN . We have used the double sux ik
because any one independent loop . the representation of HA can be constructed as follows.
This is the cylindrical function we wish to integrate. .. . the Hilbert space L A/G. Finally. n as
in Section . We will refer to d as the induced Haar measure on A/G. Thus. d and the
elements F of HA act by multiplication so that we have F f for all L A/G. i. This representation
is cyclic and the function dened by A on A/G is the vacuum state. gn Tr gi . . The Hilbert
space is L A/G. . N n. Now. since our construction did not require the introduction of any
extra structure on such as a metric or a volume element. To our knowledge. the vacuum
state is left invariant. only on generalized loops. the projection map of Lemma . Since the
measure d on A/G is invariant under this action. . We now explore some properties of this
measure and of the resulting representation of the holonomy Calgebra HA. This is just our
cylindrical measure. d. it follows from the strictness of the positivity of the measure that the
resulting representation of the Calgebra HA is faithful. this is the rst faithful representation of
the holonomy Calgebra that has been constructed explicitly. Second. . First. restricting the
values of v to the generators T K of HA. where k . . Its action is nontrivial only when the loop
is retraced. Third. . assigns to t the function t on SU n /Ad given by N t g . . Every generator T
of the holonomy Calgebra is represented by a bounded. . be the minimal decomposition of
into independent loops . .e. it follows that the functional v and the measure d are Di invariant.
d carries a unitary representation of Di. where f C is the Gelfand transform of F . n. . d. Abhay
Ashtekar and Jerzy Lewandowski vf df is in fact the integral of the continuous function f on
A/G with respect to a regular measure. . vanishes on all smoothly embedded loops. . keeps
track of the orientation and ik . Let us conclude this section by indicating how one computes
the integral in practice. The loop denes an element T of the holonomy Calgebra HA whose
Gelfand transform t is a function on A/G. . . the dieomorphism group Di of has an induced
action on the Calgebra HA and hence also on its spectrum A/G. It thus dened a generalized
knot invariant. . n can arise more than once in the decomposition. we obtain the generating
functional of this representation dA . .
g. In particular. Now. is naturally isomorphic with the Calgebra of all continuous functions on
the compact Hausdor space A/G and hence. AI had shown that every element of A/G denes
a homomorphism from the hoop group HG to SU . and self . Since C is the Calgebra of all
continuous functions on A/G. Finally. We then carried out a nonlinear extension of the theory
of cylindrical measures to integrate these cylindrical functions on A/G. we completed the AI
program in several directions. This family of projections enables us to dene cylindrical
functions on A/G these are the pullbacks to A/G of the continuous functions on SU n /Ad. the
cylindrical measures in fact turn out to be regular measures on A/G. its generating function
which sends loops based at x to real numbers denes a generalized knot invariant
generalized because we have allowed loops to have kinks. selfintersections. to integrate t on
A/G. say n. The second main result also intertwines the structure of the hoop group with that
of the Gelfand spectrum.Holonomy Calgebras Thus. Discussion In this paper. Hence. Given
a subgroup S of HG generated by a nite number. we need to integrate this t on SU n /Ad.
Since this characterization is rather simple and purely algebraic. of independent hoops.. we
were able to introduce explicitly such a measure on A/G which is natural in the sense that it
arises from the Haar measure on SU . it is useful in practice. if and only if H g H g for some g
SU . e. We analysed the space C of these functions and found that its completion C has the
structure of a Calgebra which. A/G unless every loop i is repeated an even number of times
in the decomposition . These can be now used to evaluate the required integrals on SU n
/Ad. via the inverse Gelfand transform. First. we were able to obtain a complete
characterization of the Gelfand spectrum A/G of the holonomy Calgebra HA. of .e. also to the
holonomy Calgebra HA with which we began. We found that every homomorphism from HG
to SU denes an element of the spectrum and two homomorphisms H and H dene the same
element of A/G only if they dier by an SU automorphism. i. by exploiting the fact that the
loops are piecewise analytic. the representation is faithful and dieomorphism invariant. we
were able to dene a projection S from A/G to the compact space SU n /Ad. . there exist in the
literature useful formulae for integrating polynomials of the matrix element functions over SU
see. In particular. one can readily show the following result d t . The resulting representation
of the holonomy Calgebra has several interesting properties. moreover.
e. in the transform the traces of holonomies play the role of the integral kernel. however. that
these operators also have simple action on the hoop states. denotes the inner product on L
A/G. This condition is awkward to impose in the connection representation. In the connection
representation. we can now make the ideas on the loop transform T of . Having the induced
Haar measure d on A/G at ones disposal. rigorous. The transform T sends it to a function on
the space HG of hoops. In such theories. it is convenient to lift canonically to the space Lx
and regard it as a function on the loop space. a state in the connection representation is a
squareintegrable function on A/G. this action translates to T . one can transform this action to
the hoop states . one can forgo the connection representation and work directly in terms of
hoop states. Elements of the holonomy algebra have a natural action on the connection
states . We will show elsewhere that the transform also interacts well with the momentum
operators which are associated with closed strips. the task is rather simple physical states
depend only on generalized knot and link classes of loops. raises several issues which are .
Di invariant theories. In terms of machinery introduced in this paper. we have T A T A A.
where is the composition of hoops and in the hoop group. On the hoop states. The use of
this representation. These constructions show that the AI program can be realized in detail. It
turns out that the action is surprisingly simple and can be represented directly in terms of
elementary operations on the hoop space. by contrast. and it is dicult to control the structure
of the space of resulting states. thus L A/G. This is the origin of the term loop representation.
d. A/G where t with t A A is the Gelfand transform of the trace of the holonomy function.
Using T . the loop representation has proved to be a powerful tool in quantum general
relativity in and dimensions. and . Consider a generator T of HA. In the loop representation.
We have T with . T . Sometimes. . d t AA t . The transform T sends states in the connection
representation to states in the socalled loop representation. d. associated with the hoop .
Because of this simplication and because the action of the basic operators can be
represented by simple operations on hoops. The hoop states are especially well suited to
deal with dieomorphism i. physical states are required to be invariant under Di. Abhay
Ashtekar and Jerzy Lewandowski overlaps. i. Thus. Thus.e. on A/G.
HG will denote the U hoop group and HA. Now. i. by loops we shall mean continuous.
A.Holonomy Calgebras still open. the U holonomy algebra. d. we would have a nonlinear
generalization of the Plancherel theorem which establishes that the Fourier transform is an
isomorphism between two spaces of L functions. and exploiting. rather than on A/G.
piecewise C loops. unlike SU . We will use the same notation as in the main text but now
those symbols will refer to the U theory. Thus. Nonetheless. Throughout this appendix. our
main results do continue to hold for piecewise C loops. the new arguments make a crucial
use of the Abelian character of U and do not by themselves go over to nonAbelian theories.
Perhaps the most basic of these is that. of elements of L A/G. The third introduces a
dieomorphisminvariant measure on the spectrum and discusses some of its properties. The
appendix is divided into three parts. Neither do we have an expression of the inner product
between hoop states. for example. This restriction was essential in Section . without any
reference to the connection representation. but now on HG. we consider U connections
rather than SU and show that. In this appendix. it is important to nd out if our arguments can
be replaced by more sophisticated ones so that the main results would continue to hold even
if the holonomy Calgebra were constructed from piecewise smooth loops. This may indeed
be possible using again the idea of cylindrical functions.e. we restricted ourselves to
piecewise analytic loops. It would be extremely interesting to develop integration theory also
over the hoop group HG and express the inner product between hoop states directly in terms
of such integrals. this Abelian example is an indication that analyticity may not be
indispensible even in the nonAbelian case. Topological considerations Fix a smooth manifold
. we do not yet have a useful characterization of the hoop states which are images of
connection states. The loop transform would then become an extremely powerful tool. If this
is achieved and if the integrals over A/G and HG can be related. However. The second
proves the analog of the spectrum theorem of Section . for our decomposition of a given set
of a nite number of loops into independent loops which in turn is used in every major result
contained in this paper. Appendix A C loops and U connections In the main body of the
paper. The restriction is not as severe as it might rst seem since every smooth manifold
admits a unique analytic structure. Denote by A the space of all smooth U connections which
belong to an appropriate Sobolev space. as before. the duality between these spaces.
Nonetheless. in this theory. The rst is devoted to certain topological considerations which
arise because manifolds admit nontrivial U bundles. without referring back to the connection
representation.
n . . one could ask that the holonomies . for all i. . . Thus. i . i. A A. . . This lemma has
several useful implications. Let be a globally dened. . there are two equivalence relations on
the space Lx of based loops that one can introduce. A expi A expi. Proof Suppose the
connection A is dened on the bundle P . Denote by A the subspace of A consisting of
connections on the trivial U bundle over . Hi . U dened on a subset V S B which contains all
the loops i . Since the loops i cannot form a dense set in S . since S B is contractible. The
map carries each i into a loop i in S . Given a nite number of loops. . A . As is common in the
physics literature. Now. the connection A denes on the given loops i the same holonomy
elements as A. there exists a smooth section of the Hopf bundle U SU S . or. A Hi . U see
Trautman . We mention two that will be used immediately in what follows. and depends on
both the connection A and the loop . Denition of hoops A priori. Abhay Ashtekar and Jerzy
Lewandowski bundles. One may say that two loops are equivalent if the holonomies of all
connections in A around them are the same. . Hence. for every A A there exists A A such
that Hi . we can now look for the connection A . We will show rst that there exists a local
section s of P which contains the given loops i .e. S B SU . Having obtained this section s. .
we will identify elements A of A with realvalued forms. The question therefore arises as to
whether we should allow arbitrary U connections or restrict ourselves only to the trivial
bundle in the construction of the holonomy Calgebra. We begin with the following result
Lemma A. A Hi . the holonomies dened by any A A will be denoted by H. . Hence. real form
on such that V s A. n. Fortunately. U bundles over manifolds need not be trivial. This is the
main result of this subsection. To construct the section we use a map S which generates the
bundle P . where takes values in . we may remove from the sphere an open ball B which
does not intersect any of i . it turns out that both choices lead to the same Calgebra. there
exists a section s of P .
However. n A . it follows that e. where is any loop in the hoop . Since the holonomies
themselves are complex numbers. Clearly. the Abelian hoop group HG has no torsion. A aj
exp i j A . H. then e. there is only one HA. Then if n e.. for n Z. As we will see below. we will
denote by the hoop to which a loop Lx belongs. A H . is easy to establish every element A of
the spectrum denes . Sup norm Consider functions f on A dened by nite linear combinations
of holonomies on A around hoops in HG. AA A A As a consequence of these implications.
implies that sup f A sup f A . Lemma A. where aj are complex numbers. Hence. Proof Since n
e. in that case. in spite of topological nontrivialities. Let HA denote the complex vector space
of functions f on A of the form n n f A j aj H. Although the proof is more complicated. A. A H .
the trace operation is now redundant. there is only one hoop group HG. the hoop group is
nonAbelian. A. The analog of Lemma . A Hn . A . The result is the required Calgebra HA. A
H. the identity in HG. More precisely. by Lemma A. We conclude by noting another
consequence of Lemma A. A. . Lemma A. Let HG. Equip it with the sup norm and take the
completion. we can use either A or A to construct the holonomy Calgebra. The Gelfand
spectrum of HA We now want to obtain a complete characterization of the spectrum of the U
holonomy Calgebra HA along the lines of Section . As in the main text. it is the Abelian
character of the U hoop group that makes the result of this lemma useful. We can restrict
these functions to A . we have. the analogous result holds also in the SU case treated in the
main text. implies that the two notions are in fact equivalent.Holonomy Calgebras be the
same only for connections in A . To construct this algebra we proceed as follows. A and the
relation given by H. HA has the structure of a algebra with the product law H. we have the
following result Lemma A. for every A A .
On the other hand. . A. . Furthermore. . . . Abhay Ashtekar and Jerzy Lewandowski a
homomorphism HA from the hoop group HG to U now given simply by HA A . . HG . Indeed.
H k U k . .e. i. . mk Zk . which would contradict the assumption that the given set of hoops is
independent. For every homomorphism H from the hoop group HG to U . given such hoops i
. if A. k . we can assume that the hoops i are weakly U independent. . Hence we necessarily
have Vm lt V . i . V Rk . Therefore. This is not surprising. A lt . . . . then ki i.. Now if Vm were
to equal V then we would have m k mk . . The rst map in A. k. Now. without loss of
generality. we must now modify that argument suitably.e i k A U k . . . . . Let m m . it is clear
from the discussion in Section that this lemma continues to hold for piecewise smooth loops
even in the full SU theory.. . which can be expressed as a composition A A A . k A e i A . A.
say . the situation is quite dierent for Lemma . that they satisfy the following condition if k n
kn e. . An appropriate replacement is contained in the following lemma. is satised by i for a
suciently small then it will also be satised by the given j for the given . . the homomorphism H
HG U denes a point H . However. Hence. k . Since HG is Abelian and since it has no
torsion.. and every gt . . is a vector space. . . it follows that Vm lt V. n . . Lemma A. . . .
Denote by Vm the subspace of V orthogonal to m. . . Proof Consider the subgroup HG . mZk
This strict inequality implies that there exists a connection B A such . k is nitely and freely
generated by some elements. Consequently.. . . there also exists a map from A to U k . is
linear and its image. . since a countable union of subsets of measure zero has measure zero.
n HG that is generated by . . . there exists a connection A A such that H i Hi . every nite set
of hoops . because there we made a crucial use of the fact that the loops were continuous
and piecewise analytic. . .
regular measure on the spectrum of HA. in the SU holonomy algebra we have T T T . this
function is strictly positive. Furthermore. otherwise. for the sake of diversity. we will exhibit a
natural measure. into a line which is dense in U k . . . Rather than going through the same
constructions as in the main text. and. extends to a continuous positive linear function on the
holonomy Calgebra HA. to specify a positive linear functional on HA. It is clear that this
functional on Lx is dieomorphism invariant. This is a correspondence. We will present a
strictly positive linear functional on the holonomy Calgebra HA whose properties suce to
ensure the existence of a dieomorphisminvariant. Combining these results. We will now
show that it does have all the properties to be a generating function . Theorem A. it is the
simplest of such functionals on Lx . B is such that every ratio vi /vj of two dierent components
of v is irrational. the line in A dened by B via A t tB is carried by the map A. this strategy fails
because of the SU identities. where o is the identity hoop in HG. This ensures that there
exists an A on this line which has the required property A. we will adopt here a
complementary approach. However. . n B V Vm . . we have the U spectrum theorem
Theorem A. Indeed. For example. We can dene a . We know from the general theory
outlined in Section that. A. mZk In other words. Armed with this substitute of Lemma .. it is
straightforward to show the analog of Lemma . one might imagine using it to dene a positive
linear functional also for the SU holonomy algebra of the main text. A natural measure
Results presented in the previous subsection imply that one can again introduce the notion of
cylindrical functions and measures. Let us simply set . Since this functional is so simple and
natural.. it suces to provide an appropriate generating functional on Lx . Evaluation of the
generating functional A. A. conversely. on this identity would lead to the contradiction .. Every
A in the spectrum of HA denes a homomorphism from the hoop group HG to U . In this
subsection. if o . every homomorphism H H denes an element A of the spectrum such that H
A .Holonomy Calgebras that v B. Thus. faithful.
we have n n n n j aj j j aj j lt sup A A j aj H j . Hence we have A. strictly positive measure on
the Gelfand spectrum of this algebra. .. . given the n complex numbers ai . The inequality A.
this positive linear functional is not strictly positive the measure on A/G it denes is
concentrated just on A . if i j if i j. if T A A A otherwise. Abhay Ashtekar and Jerzy
Lewandowski Proof By its denition. A. the projection is positive denite. However. . the left
side equals aj . f f for all f HA. Furthermore. given the n hoops j . since the norm on this
algebra is the sup norm on A . To see this. However. Finally. faithful representation of HA
and hence a regular. the resulting representation of the SU holonomy Calgebra fails to be
faithful. by denition of . A j aj H j . we will let the manifold have positive linear function on the
SU holonomy Calgebra via .. Finally. implies that the generating functional provides a
continuous. one can nd an A such that the angles j can be made as close as one wishes to a
prespecied ntuple j . note rst that. UN and SUN connections In this appendix. j . Now. implies
that the functional continues to be strictly positive on HA. . where we have regarded A as a
subspace of the space of SU connections. Extend to F HG by linearity. equality holding only
if f . We will now work with analytic manifolds and piecewise analytic loops as in the main
text. such that. Theorem A. one can always nd j such that aj lt aj exp ij aj exp ij . . by Lemma
A. We will rst establish that this extension satises the following property given a nite set of
hoops j . implies that the functional has a welldened projection on the holonomy algebra HA
which is obtained by taking the quotient of F HG by the subspace K consisting of bi i such
that bi H i . it follows from A. n. that the functional is continuous on HA. Consequently. A for
all A A . . A . the loop functional admits a welldened projection on the hoop space HG which
we will denote again by . Let F HG be the free vector space generated by the hoop group. we
will consider another extension of the results presented in the main text. Hence it admits a
unique continuous extension to the Calgebra HA. Appendix B C loops. The right side equals
aj exp ij aj exp ij where j depend on A and the hoop j . on the SU holonomy algebra. Next. It
is not dicult to verify by direct calculations that this is precisely the U analog of the induced
Haar measure discussed in Section . A.
. the products of traces of holonomies can no longer be expressed as sums while the second
arise because the connections in question may be dened on nontrivial principal bundles. we
will refrain from making digressions to the general case. the situation is more involved. with
the relation given by f f . G. G. A. AA and take the completion of HA to obtain a Calgebra.
and HA will be the holonomy Calgebra generated by nite sums of nite products of traces of
holonomies of connections in A around hoops in HG. see eqn . where the trace is taken in
the fun damental representation of U N or SU N . at the end. This was possible because. are
the same as in Section . details will be omitted. for example. The rst arise because. Being
traces of U N and SU N matrices. where the bar stands for complex conjugation. In . For the
groups under consideration. now refer to connections on P . The holonomy algebra HA is.
products of any two T could be expressed as a linear combination of T eqn . the algebra HA
was obtained simply by imposing an equivalence relation K. The key dierence between the
structure of this HA and the one we constructed in Section is the following. T are bounded
functions on A/G. Let us summarize the situation in the slightly more general context of the
group GLN . since most of the arguments are rather similar to those given in the main text.
that they are independent of the initial choice of P . we will see that the main results of the
paper continue to hold in these cases as well. In Section . there are two types of
complications algebraic and topological. and holonomy maps H. and then show. and regard
T as a function on A/G. Denitions of the space Lx of based loops. This is the holonomy
Calgebra. G. the space of gaugeequivalent connections. Ghoop group. Several of the results
also hold when the gauge group is allowed to be any compact Lie group. Nonetheless.
generated by all nite linear combinations with complex coecients of nite products of the T .
We can therefore introduce on HA the sup norm f sup f A. on the free vector space
generated by the loop space Lx . HG will denote the P . Consequently. Thus. However.
obtain the main results. owing to SU Mandelstam identities. B. we will continue to use the
same symbols as in the main text to denote various spaces which. To keep the notation
simple. A.Holonomy Calgebras any dimension and let the gauge group G be either U N or
SU N . Let us begin by xing a principal bre bundle P . however. Also. for brevity of
presentation. We will denote it by HA. by denition. G. A will denote the space of suitably
regular connections on P . Holonomy Calgebra We will set T A N TrH.
the Mandelstam identities follow from the contraction of N matricesH . . the AI result that
there is a natural embedding of A/G into the spectrum A/G of HA goes through as before and
so does the general argument due to Rendall that the image of A/G is dense in A/G. where a.
. B. . if the gauge group is GLN . we have. . The proof requires some minor modications
which consist of using local sections and a suciently large number of generators of the gauge
group. c N . These result from the fact that the determinant of N N matrix M can be
expressed in terms of traces of powers of that matrix. . . for any hoop . using the analog of
Lemma . TN . .. . . Also. TrM . . Consequently. an element of the holonomy algebra HA is
expressible as an equivalence class a. . . . . They enable one to express products of N T
functions as a linear combination of traces of products of . A Gn is onto if the loops i are
independent. A in our casewith the identity j N i . Hn . . it is straightforward to establish the
analog of the . . A. F T . . b . TN F T . . This is true for any group and representation for which
the traces of holonomies separate the points of A/G. for decomposition of a nite number of
loops into independent loops makes no direct reference to the gauge group and therefore
carries over as is. . Finally. namely det M F TrM. . continues to hold. . . Abhay Ashtekar and
Jerzy Lewandowski this case. Finally. we have to begin by considering the free algebra
generated by the hoop group and then impose on it all the suitable identities by extending the
denition of the kernel . . As for the Gelfand spectrum. . . . . . Hence. Loop decomposition and
the spectrum theorem B. . let us note identities that will be useful in what follows. iN j j where
i is the Kronecker delta and the bracket stands for the antisymmetrization. For subgroups of
GLN such as SU N further reductions arise. A. . . In the SU N case a stronger identity holds F
T . . A direct consequence is that the analog of Lemma .a The construction of Section . c are
complex numbers. if the gauge group G is U N . . it is still true that the map A A H . . HN .
TrM N for a certain polynomial F . . Hence.. TN . N T functions. . products involving N and
higher loops are redundant. b. .b B.
. we note from eqn B. . by the identity B.on the other hand requires further analysis based on
Giles results along the lines used by AI in the SU case. . given an ele ment of the spectrum
Ai. . . . If G is SU N then. N for every integer k. Therefore. holds.b. we have the spectrum
theorem every element A of the spectrum A/G denes a homomorphism H from the hoop
group HG to the gauge group G. This suces to conclude that With this result in hand. an A i A
A for all S . The conversethe analog of Lemma . . . we can dene cylindrical functions on A/G
as the pullbacks to A/G of continuous functions on Gn /Ad. as before. For the gauge group
under consideration. . we can now use Giles analysis to conclude that given A we can nd HA
which takes values in U N . the traces of all elements of a nitely generated subgroup in the
fundamental representation suce to characterize the subgroup modulo an overall adjoint
map. for groups under consideration. TrHA N . Two homomorphisms H and H dene the same
element of the spectrum if and only if TrH TrH . First. Thus far. for every i. and.e. . . so that
the analog of Lemma . . n generated by n independent hoops . N of HA . They again . a
continuous homomorphism from HA to the algebra of complex numbersGiles theorem
provides us with a homomorphism HA from the hoop group HG to GLN . n . every
homomorphism H from of the spectrum A/G such the hoop group HG to G denes an element
A that A N TrH for all HG.a that in particular i . the characteristic polynomial of the matrix HA
is expressible directly by the values of A taken on n the hoops . we have det HA . Cylindrical
functions and the induced Haar measure Using the spectrum theorem. . From continuity it
follows that for every hoop . B. we have used only the algebraic properties of A. What we
need to show is that HA can be so chosen that it takes values in the given gauge group G. .
We can establish this by considering the eigenvalues . . . we can again associate with any
subgroup S HG . . After all. Combining these results.Holonomy Calgebras main conclusion of
Lemma . conversely. equivalence relation on A/G A This relation provides us with a family of
projections from A/G onto the compact manifolds Gn /Ad since. . . Substituting for its powers
we conclude that k k N i .
n . and consider a connection A dened on P . A denes also a point in the spectrum of HA . .
be the corresponding projection maps from A/G to Gn /Ad. i. we have n n ai Ti i i ai Ti . Thus.
. Hence. Let us decompose the given hoops . Fix a gauge group G from U N . . Hence.
Abhay Ashtekar and Jerzy Lewandowski form a Calgebra. the construction of Section . P
say. . to GN /Ad are in fact equal. SU N and let P . we know that their I ai Ti i i ai Ti B. we
obtain the same holonomy Calgebra HA. whence it follows that I is an isomorphism of
Calgebras. G and P . as noted at the end of Section . Therefore. we xed a principal bundle P
. G be two principal bundles. Note that the holonomy map of A denes a homomorphism from
the bundleindependent hoop group to G. . it does not matter which bundle we begin with. ..
B. the hoop groups obtained from any two bundles are the same. is an isomorphism from HA
to HA . let i S . First. i . We spectra are naturally isomorphic A/G wish to show that the
algebras are themselves naturally isomorphic. as in Section . Therefore. Using the fact that
traces of products of group elements suce to separate points of Gn /Ad when G U N or G SU
N .e. k into independent hoops .. G A/G . given a manifold M and a gauge group G the
spectrum A/G automatically contains all the connections . . Using the spectrum theorem. Let
the corresponding holonomy Calgebras HomHG. The resulting representation of the
Calgebra HA is again faithful and dieomorphism invariant. Now. . From the denition of the
maps it follows that the projections of the two functions in B. To conclude. . Finally. Let S be
the subgroup of the hoop group they generate and. using the hoop decomposition one can
show that two loops in Lx dene the same hoop if and only if they dier by a combination of
reparametrization and an immediate retracing of segments for gauge groups under
consideration. we will show that the Calgebra HA and hence its spectrum are independent of
this choice. G. Bundle dependence In all the constructions above. x a bundle. it again follows
that the Calgebra of cylindrical functions is isomorphic with HA. that the map I dened by n n
be HA and HA . which led us to the denition of the induced Haar measure goes through step
by step.
. Classical amp Quantum Gravity . .N. gr qc/ . Trias. Nonperturbative canonical quantum
gravity Notes prepared in collaboration with R. DeWittMorette and M. A. Nucl. C. Adv.
Commun. Acknowledgements The authors are grateful to John Baez for several illuminating
comments and to Roger Penrose for suggesting that the piecewise analytic loops may be
better suited for their purpose than the piecewise smooth ones. M. . C. eds L. Analysis. B.
Mitter and C. Singapore . . . Di Bartolo. B. . . DillardBleick. J. Theor. J. New York . Phys.
Quarks. J. . Ashtekar. Isham. Cambridge University Press. Phys. J. Int. R. Giles. Amsterdam
. . Gambini. A. World Scientic. . pp. ChoquetBruhat. Phys. Crane and D. Math. Ashtekar and
C. A. In Proceedings of the Conference on Quantum Topology. Procesi. B. J. Y. and J. Rev.
Cambridge Chapter . A.K. Creutz. by the Polish Committee for Scientic Research KBN
through grant no. C. Bibliography . Phys. hepth/ . D. Rendall. Classical amp Quantum Gravity
. Lewandowski. preprint IFFI/. . Trautman. Kolmogorov. A. Czech. NorthHolland.Holonomy
Calgebras on all the principal Gbundles over M . . . Gambini and A. This work was supported
in part by the National Science Foundation grants PHY and PHY. . Barrett. Phys. Smolin.
Manifolds and Physics. Nucl. Baez. Classical amp Quantum Gravity . Foundations of the
Theory of Probability. Rovelli and L. P. Gluons and Lattices. Math. Viallet. JL also thanks
Cliord Taubes for stimulating discussions. R.W. Yetter to appear. and by the Eberly research
fund of Pennsylvania State University. Chelsea. Tate. C. Phys. R.S. Griego. .
Abhay Ashtekar and Jerzy Lewandowski .
Tristan Narvaja . Uruguay email rgambinising. Montevideo. Using this fact. This does not
appear as reasonable since dierent values of the cosmological constant lead to widely
dierent behaviors in general relativity. Therefore the problem of solving the WheelerDeWitt
equation in loop . Pennsylvania State University.psu. it was later realized that such solutions
are associated with degenerate metrics metrics with zero determinant and this posed
inconsistencies if one wanted to couple the theory . Facultad de Ciencias. University Park.
starting from the connection representation and give details of the proof.The Gauss Linking
Number in Quantum Gravity Rodolfo Gambini Intituto de Fsica. it is straightforward to prove
that the second coecient of the Jones polynomial is a solution of the WheelerDeWitt equation
in loop space with no cosmological constant. For instance.edu Abstract We show that the
exponential of the Gauss selflinking number of a knot is a solution of the WheelerDeWitt
equation in loop space with a cosmological constant. USA email pullinphys.uy Jorge Pullin
Center for Gravitational Physics and Geometry. In the loop representation the Hamiltonian
constraint has nonvanishing action only on functions of intersecting loops. It was rst argued
that by considering wave functions with support on smooth loops one could solve the
constraint straightforwardly . We perform calculations from scratch.edu. Pennsylvania .
Implications for the possibility of generation of other solutions are also discussed.
Introduction The introduction of the loop representation for quantum gravity has made it
possible for the rst time to nd solutions to the WheelerDeWitt equation the quantum
Hamiltonian constraint and therefore to have possible candidates to become physical wave
functions of the gravitational eld. . However. if one considered general relativity with a
cosmological constant it turns out that nonintersecting loop states also solve the
WheelerDeWitt equation for arbitrary values of the cosmological constant.
space is far from solved and has to be tackled by considering the action of the Hamiltonian
constraint in the loop representation on states based on at least triply intersecting loops. as
depicted in Fig. produces a term in Fab that exactly cancels the contribution from the vacuum
Hamiltonian constraint. . . And even in this case the metric is nondegenerate only at the point
of intersection. At least a triple intersection is needed to have a state associated with a
nondegenerate dimensional metric in the loop representation. . Rodolfo Gambini and Jorge
Pullin Fig. Although performing this kind of direct calculations is now possible. is a solution of
the WheelerDeWitt equation in the connection representation with a cosmological term H ijk
Fk Ai Aj ab a b ijk abc Ai a Aj b Ak c . the rst term is just the usual Hamiltonian constraint with
and the second term is just det g where det g is the determinant of the spatial part of the
metric written in terms of triads. since welldened expressions for the Hamiltonian constraint
exist in terms of the loop derivative see also and actually some solutions have been found
with this approach . . To prove this fact one simply needs to notice that for the ChernSimons
state introduced above. another line of reasoning has also proved to be useful. and therefore
the rightmost functional derivative in the cosmological coni stant term of the Hamiltonian .
This other approach is based on the fact that the exponential of the ChernSimons term
based on Ashtekars connection CS A exp d xAi b Ai Ai Aj Ak a c a b c ijk abc . CS A Ai a abc
i Fbc CS A.
should be a solution of the WheelerDeWitt equation in loop space. The Jones polynomial.
since one expects states of quantum gravity to be truly dieomorphisminvariant objects and
framing is always dependent on an external device which should conict with the
dieomorphism invariance of the theory. This last fact can actually be checked in a direct
fashion using the expressions for the Hamiltonian constraint in loop space of . which relates
the Kauman bracket. and we know due to the insight of Witten that this coincides with the
Kauman bracket of the loop. however. Unfortunately it is not clear at present how to settle
this issue since it is tied to the regularization procedures used to dene the constraints. This
raises the issue of up to what extent statements about the Kauman bracket being a state of
gravity are valid.. a are the second and third coecient of the innite expansion of the Jones
polynomial in terms of . where a . the Gauss selflinking number. In order to do this we make
use of the identity Kauman bracket eGauss Jones polynomial .The Gauss Linking Number in
Quantum Gravity This result for the ChernSimons state in the connection representation has
an immediate counterpart in the loop representation. CS dA e SCS W A W . We now expand
both the Jones polynomial and the exponential of the Gauss linking number in terms of and
get the expression Kauman bracket Gauss Gauss a Gauss Gauss a a . and so is the
Kauman bracket. . is framing independent.. with coupling constant . We will return to these
issues in the nal discussion. . can be interpreted as the expectation value of the Wilson loop
in a Chern Simons theory. a is known to coincide with the second coecient of the Conway
polynomial . since the transform of the ChernSimons state into the loop representation CS
dA CS A W A . The Gauss selflinking number is framing dependent. Therefore the state CS
Kauman bracket . and the Jones polynomial.
Therefore we see that by noticing that the Gauss linking number is a state with cosmological
constant. Summarizing. . . D Kauman bracket G . where H is the Hamiltonian constraint
without cosmological constant more recently it has also been shown that H a but we will not
discuss it here. Now. Even if one is reluctant to accept the Kauman bracket as a state
because of its framing dependence. and one would nd that certain conditions have to be
satised if the Kauman bracket is to be a solution. is also a solution of the WheelerDeWitt
equation with cosmological constant. perspective. Rodolfo Gambini and Jorge Pullin One
could now apply the Hamiltonian constraint in the loop representation with a cosmological
constant to the expansion . This means that a should be a solution of the WheelerDeWitt
equation. . Among them it was noticed that H a . Given this fact. . it can be viewed as an
intermediate step of a framingdependent proof that the Jones polynomial which is a
framingindependent invariant solves the WheelerDeWitt equation it should be stressed that
we only have evidence that the rst two nontrivial coecients are solutions. . it is easy to prove
that the second coecient of the innite expansion of the Jones polynomial which coincides
with the second coecient of the Conway polynomial is a solution of the WheelerDeWitt
equation with . . This dierence is of the form D a a Gauss a . in particular for . It can be
viewed as an Abelian limit of the Kauman bracket more on this in the conclusions. and to our
understanding simpler. one can therefore consider their dierence divided by . which is also a
solution of the WheelerDeWitt equation with a cosmological constant. the fact that the
Kauman bracket is a solution of the WheelerDeWitt equation with cosmological constant
seems to have as a direct consequence that the coecients of the Jones polynomial are
solutions of the vacuum WheelerDeWitt equation This partially answers the problem of
framing we pointed out above. . this dierence is a state for all values of . The purpose of this
paper is to present a rederivation of these facts from a dierent. We will show that the
framingdependent knot invariant G eGauss . This conrms the proof given in .
in the case of the Hamiltonian constraint without . In Section we derive the expression of the
Hamiltonian constraint with a cosmological constant in the loop representation for a triply
intersecting loop in terms of the loop derivative.The Gauss Linking Number in Quantum
Gravity The rest of this paper will be devoted to a detailed proof that the exponential of the
Gauss linking number solves the WheelerDeWitt equation with cosmological constant. where
means the adjoint operator with respect to the measure of integration dA. Concretely. Apart
from presenting this new state. On the other hand. To this aim we will derive expressions for
the Hamiltonian constraint with cosmological constant in the loop representation. Applying
the transform OL dAOL W AA . it is clear that OL W A OC W A. this denition should agree
with OL dAW AOC A . Therefore. the operator O in the right member acts on the loop
dependence of the Wilson loop. where OC is the connection representation version of the
operator in question. we just present the explicit proof for a triple intersection since in three
spatial dimensions it represents the most important type of intersection. the more interesting
case for gravity purposes it should be noticed that all the arguments presented above were
independent of the number and order of intersections of the loops. the only eect of taking the
adjoint is to reverse the factor ordering of the operators. Suppose one wants to dene the
action of an operator OL on a wave function in the loop representation . In Section we write
an explicit analytic expression for the Gauss linking number and prove that it is a state of the
theory. . If one assumes that the measure is trivial. The WheelerDeWitt equation in terms of
loops Here we derive the explicit form in the loop representation of the Hamiltonian constraint
with a cosmological constant. we think the calculations exhibited in this paper should help the
reader get into the details of how these calculations are performed and make an intuitive
contact between the expressions in the connection and the loop representation. We will
perform the calculation explicitly for the case of a triply selfintersecting loop. The derivation
proceeds along the following lines. We end in Section with a discussion of the results.
However. It is evident that with this action of the functional derivative the Hamiltonian
operator is not well dened. Here we will only study the operator in the limit in which the
regulator is removed. since it yields a term i i dy a dy b Fab which vanishes due to the
antisymmetry of Fab and the symmea b try of dy dy at points where the loop is smooth. .
continue along to the intersection. . We need to regularize it as HL W A lim f x z ijk k Fab x W
A Ai x Aj z a b . For that we need the expression of the functional derivative of the Wilson
loop with respect to the connection W A Ai x a dy a y xTr Pexp y o dz b Ab i Pexp y o dz b Ab
. One of them is to dene the Hamiltonian inserting pieces of holonomies connecting the
points x and z between the functional derivatives to produce a gaugeinvariant quantity . Ai Aj
a b . Here we compute the contribution at a point of triple selfintersection. There are a
number of ways of xing this situation in the language of loops. By W j k A we really mean
take the holonomy from the basepoint along up to just before the intersection point. A proper
calculation would require their careful study. at intersections there can be nontrivial
contributions. therefore we will not be concerned with these issues. insert a Pauli matrix. ijk k
Fab x Ai x a Aj z b W A ijk k Fab x . since such point splitting breaks the gauge invariance of
the Hamiltonian. ab ab where we have denoted where i are the petals of the loop as
indicated in Fig. It is immediate from the above denitions that the Hamiltonian constraint
vanishes in any regular point of the loop. where o is the basepoint of the loop. ab W j k A W j
k A W j k A. continue along . The Wilson loop is therefore broken at the point of action of the
functional derivative and a Pauli matrix i is inserted. We now need to compute this quantity
explicitly. Rodolfo Gambini and Jorge Pullin cosmological constant HC and therefore HL W A
ijk k Fab ijk Fk Ai Aj ab a b W A. where f x z x z when . Caution should be exercised.
and complete the loop along . so strictly speaking really is a distribution that is nonvanishing
only at a the point of intersection. The result of the application of these identities to the
expression of the Hamiltonian is HW A . . . We will not discuss all its properties here. Its
denition is x o x o x ab ab o . which is a natural generalization to the case of loops with
insertions of the SU Mandelstam identities. and just before the intersection insert another
Pauli matrix. This expression can be further rearranged making use of the loop derivax tive.
This is just shorthand for expressions like dy a x y when the point x is close to the
intersection. which reects the intuitive notion that a holonomy of an innitesimal loop is related
to the eld tensor. . . x o where ab is the element area of the innitesimal loop and by o x we
mean the loop obtained by traversing the path from the basepoint to x. and then the loop .
continue to the intersection. where means the loop opposite to . Therefore we can write
expressions like i Fab W i A . One could pick after the intersection instead of before to
include the Pauli matrices and it would make no dierence since we are concentrating on the
limit in which the regulator is removed. in which the insertions are done at the intersection. is
the dierentiation operator that appears in loop space when one considers two loops to be
close if they x dier by an innitesimal loop appended through a path o going from the
basepoint to a point of the manifold x as shown in Fig. the innitesimal loop . a b a b a b i Fab
W i A W i A W i A . The only one we need is that the loop derivative of a Wilson loop taken
with a path along the loop is given by x ab o W A Tr Fab xPexp dy c Ac . W AW A W A W A.
the path from x to the basepoint. .The Gauss Linking Number in Quantum Gravity . By we
mean the tangent to the petal number just before the intera section where the Pauli matrix
was inserted. The loop derivative ab o . ijk W j k A W i AW A W AW i A. One now uses the
following identity for traces of SU matrices.
we thought that a direct derivation for this particular case would be useful for pedagogical
purposes. The loop dening the loop derivative as ab W A. . . W A Ai Aj Ak a c b abc ijk a b c
W i j k A . .. This latter expression can be rearranged with the following identity between
holonomies with insertions of Pauli matrices ijk W i j k A W A W A W A. We proceed in a
similar fashion. . Rodolfo Gambini and Jorge Pullin Fig. rst computing the action of the
operator in the connection representation det q on a Wilson loop ijk abc ijk abc Ai Aj Ak a c b
. It is therefore immediate to nd the expression of the determinant of the metric in the loop
representation detq a b c abc . . and the nal expression for the Hamiltonian constraint in the
loop representation can therefore be read o as follows H ab a b ab ab ab . . However. We
now have to nd the loop representation form of the operator corresponding to the
determinant of the metric in order to represent the second term in . a b This expression could
be obtained by particularizing that of to the case of a triply selfintersecting loop and
rearranging terms slightly using the Mandelstam identities. .
This is furnished by the wellknown integral expression Gauss dxa dy b abc x yc x y . and the
quantity gab x. y . This formula is most well known when the two loop integrals are computed
along dierent loops. This is in general not well dened without the introduction of a framing .
where x y is the distance between x and y with a ducial metric. gab x. x y . X b y. y is the
propagator of a ChernSimons theory . b and integrate over the manifold for repeated
continuous indices. In the present case we are considering the expression of the linking of a
curve with itself. that is. The Gauss selflinking number as a solution In order to be able to
apply the expressions we derived in the previous section for the constraints to the Gauss
selflinking number we need an expression for it in terms of which it is possible to compute
the loop derivative. We will rewrite the above expression in a more convenient fashion Gauss
d x d yX a x. This notation is also faithful to the fact that the index a behaves as a vector
density index at the point x. where the vector densities X are dened as X a x. where we have
promoted the point dependence in x.The Gauss Linking Number in Quantum Gravity With
these elements we are in a position to perform the main calculation of this paper. to show
that the exponential of the Gauss selflinking number of a loop is a solution of the Hamiltonian
constraint with a cosmological constant. y abc x yc . The only dependence on the loop of the
Gauss selflinking number is through the X. dz a z x . For calculational convenience it is
useful to introduce the notation Gauss X ax X by gax by . y to a continuous index and
assumed a generalized Einstein convention which means sum over repeated indices a. it is
natural to pair a and x together. gab x. so we just need to compute the action of the loop
derivative on one of them to be in a position to perform the calculation straightfor . In that
case the formula gives an integer measuring the extent to which the loops are linked.
. that is we consider the change in the X when one appends an innitesimal loop to the loop
as illustrated in Fig. . the product of the Kronecker and Dirac deltas. v v b uc c x z v c c x z u
a uc v c c x z a o z dy a y z. c c where the notation b x z stands for b x z. . Rearranging one
gets z c ab o X cx a b x z . which we characterize as four segments along the integral curves
of two vector elds ua and v b of associated lengths and . In order to do this we apply the
denition of loop derivative. z ab ab o X ax z o dy a x y ua x z . This is really all we need to
compute the action of the Hamiltonian. and then we continue from there back to the
basepoint along the loop. Now we must integrate by parts. We partition the integral in a
portion going from the basepoint to the point z where we append the innitesimal loop.
Rodolfo Gambini and Jorge Pullin Fig. We therefore now consider the action of the vacuum
part of the Hamiltonian on the exponential of the Gauss linking number. Using the fact that a
X ax and . The last and rst term combine to give back X ax and therefore one can read o the
action of the loop derivative from the other terms. The action of the loop derivative is x ab o
exp X cy X dz gcy dz c a b x yX dz gcy dz exp X ew X f w gew f w . Therefore. The loop
derivative that appears in the denition of the Hamiltonian is evaluated along a path that
follows the loop wardly.
If one is to pointsplit. As a consequence of this. In that case. Discussion We showed that the
exponential of the Gauss selflinking number is a solution of the Hamiltonian constraint of
quantum gravity with a cosmological constant. its form is exactly the same as that of the
exponential of the Gauss linking number if one replaces the propagator of the ChernSimons
theory present in the .The Gauss Linking Number in Quantum Gravity the denition of gcy dz .
We assume such factors coming from all terms to be similar and therefore ignore them. This
concludes the main proof of this paper.. A regularized form of the Hamiltonian constraint in
the loop representation is more complicated than the expression we presented. the
determinant of the metric requires splitting three points whereas the Hamiltonian only needs
two. If one does not take the limit the various terms do not cancel. innitesimal segments of
loop should be used to connect the split points to preserve gauge invariance and a more
careful calculation would be in order. In fact. It is straightforward now to check that applying
the determinant of the metric one the Gauss linking number one gets a contribution exactly
equal and opposite by inspection from expression .. This naturally can be viewed as the
Abelian limit of the solution given by the Kauman bracket. that is. as here. What about the
issue of regularization The proof we presented is only valid in the limit where . there are two
terms in the Hamiltonian that need to be regularized in a dierent way in the case of gravity.
What does all this tell us about the physical relevance of the solutions The situation is
remarkably similar to the one present in the loop representation of the free Maxwell eld . In
the case of the Maxwell eld the vacuum in the loop representation needs to be regularized.
However. the expression for the Hamiltonian constraint we introduced is also only valid when
the regulator is removed. it is not surprising that the wave functions that solve the constraint
have some regularization dependence. The expression is only formal since in order to do this
a divergent factor should be kept in front. when the regulator is removed. cy dz abc X X X
exp X X gcy dz . where we again have replaced the distributional tangents at the point of
intersection by an expression only involving the tangents. cx cx cx Therefore the action of the
Hamiltonian constraint on the Gauss linking number is HeGauss abc eGauss abc . we get x
ab o exp X cy X dz gcy dz .
Lett. B. Phys. Phys. Nucl. Pullin. How could these regularization ambiguities be cured In the
Maxwell case they are solved by considering an extended loop representation in which one
allows the quantities X ax to become smooth vector densities on the manifold without
reference to any particular loop . R G wishes to thank Karel Kucha and r Richard Price for
hospitality at the University of Utah where part of this work was accomplished. although it is
more complicated. Phys. Br gmann. Smolin. . J. Rovelli. Br gmann. . NSFPHY. . In the
gravitational case such construction is being actively pursued . B. u . Smolin. If one allows
the X to become smooth functions the framing problem disappears and one is left with a
solution that is a function of vector elds and only reduces to the Gauss linking number in a
very special singular limit. J P wishes to thank John Baez for inviting him to participate in the
conference and hospitality in Riverside. It would certainly provide new insights into how to
construct nonperturbative quantum states of the gravitational eld. L. R. Phys. . Rodolfo
Gambini and Jorge Pullin latter by the propagator of the Maxwell eld. L. Support from
PEDECIBA Uruguay is also acknowledged. and by research funds of the University of Utah
and Pennsylvania State University. . Nucl. B. . It would be interesting to study if such a limit
could be pursued in a systematic way order by order. In the extended representation. J. This
similarity is remarkable. This work was supported by grants NSFPHY. It has been proved
that the extensions of the Kauman bracket and Jones polynomials to the case of smooth
density elds are solutions of the extended constraints. the wave functions inherit
regularization dependence since the regulator does not appear as an overall factor of the
wave equation. B. It is in this context that the present solutions really make sense. . u . T. B.
The Abelian limit of the Kauman bracket the Gauss linking number appears as the restriction
of the extended Kauman bracket to the case in which higherorder multivector densities
vanish. . Rev. Gambini. Lett. Rev. . . C. The problem is therefore the same. Bibliography .
Phys. Pullin. Acknowledgements This paper is based on a talk given by J P at the Riverside
conference. Jacobson. A similar proof goes through for the extended Gauss linking number.
R. Phys. B. Nucl. there are additional multivector densities needed in the representation. .
Gambini. Lett.
Nucl. Lett. Trias. Guadagnini. Pullin. A. . . Leal. . The extended loop representation a rst
approach. Cim. . . . . Classical loop actions of gauge theories. M. ArmandUgon. Witten. . in
preparation. Gambini. Phys. Gambini. preprint hepth . R. Br gmann. J. . Math. L. . Gambini.
Nori. . Phys. R. preprint . E. F. C. R. Griego. L. General Relativity Gravitation u . Griego.
Trias. . Rev. E. Commun. J. Gambini. Gambini. Pullin. D. B. Mintchev. D. J.The Gauss
Linking Number in Quantum Gravity . Griego. J. . Setaro. R. Martellini. J. B. M. . Di Bartolo.
C. A. . Phys. Nuovo. R. Di Bartolo.
Rodolfo Gambini and Jorge Pullin .
Vassiliev Invariants and the Loop States in Quantum Gravity Louis H. This gives a neat point
of view on the results of BarNatan. Statistics. These methods are strong enough to give the
top row evaluations of Vassiliev invariants for the classical Lie algebras. The formula is then
used to evaluate the top row of the corresponding Vassiliev invariants and it is shown to
match the results of BarNatan obtained via perturbation expansion of the integral. They give
an insight into the structure of these invariants without using the full perturbation expansion
of the integral. Chicago. The same level of handling the functional integral is commonly used
in the loop transform for quantum gravity. One reason for examining the invariants in this
light is the possible applications to the loop variables approach to quantum gravity.edu
Abstract This paper derives at the physical level of rigor a general switching formula for the
link invariants dened via the Witten functional integral.uic. The paper is organized as follows.
and Computer Science. Section is a brief discussion of the nature of the functional integral.
We see that this vertex is not just a transversal intersection of Wilson loops. Section details
the formalism of the functional integral that we shall use. Section denes the Vassiliev
invariants and shows how to formulate them in terms of the functional integral. Kauman
Department of Mathematics. These are applied to the case of an SU N gauge group. In
particular we derive the specic expressions for the top rows of Vassiliev invariants
corresponding to the fundamental representation of SU N . Introduction The purpose of this
paper is to expose properties of Vassiliev invariants by using the simplest of the approaches
to the functional integral denition of link invariants. but rather has the structure . and also
gives a picture of the structure of the graphical vertex associated with the Vassiliev invariant.
and works out a dierence formula for changing crossings in the link invariant and a formula
for the change of framing. Illinois . University of Illinois at Chicago. USA email Uuicvm. It is
explained how this formalism for Vassiliev invariants can be used in the context of the loop
transform for quantum gravity.
The invariants associated with this integral have been given rigorous combinatorial
descriptions see . The gauge eld is a form on M with values in a representation of a Lie
algebra. support the existence of a large class of topological invariants of manifolds and
associated invariants of knots and links in these manifolds. The group corresponding to this
Lie algebra is said to be the gauge group for this particular eld. the integral ZM integrates
over all gauge elds modulo gauge equivalence see for a discussion of the denition and
meaning of gauge equivalence. The formalism of the integral. We need the Wilson loop. .
This marks the beginning of a study that will be carried out in detail elsewhere. In this integral
the action SM. A . but questions and conjectures arising from the integral formulation are still
outstanding see . Here M denotes a manifold without boundary and A is a gauge eld also
called a gauge potential or gauge connection dened on M . Louis H. The analogue of all
paths from point a to point b is all elds from eld A to eld B. Section is a quick remark about
the loop formalism for quantum gravity and its relationships with the invariants studied in the
previous sections. The question posed in this domain is to nd the value of an amplitude for
starting with one eld conguration and ending with another. in its form. Links and the Wilson
loop We now look at the formalism of the Witten integral and see how it implicates invariants
of knots and links corresponding to each classical Lie algebra. a typical integral in quantum
eld theory. This generalization from paths to elds is characteristic of quantum eld theory. .
Wittens integral ZM is. The Wilson loop is an exponentiated . In quantum electrodynamics
the classical entity is the electromagnetic eld. . This claries an issue raised by John Baez in .
In its content ZM is highly unusual. Instead of integrating over paths. Quantum mechanics
and topology In Edward Witten proposed a formulation of a class of manifold invariants as
generalized Feynman integrals taking the form ZM where ZM dA exp ik/SM. Quantum eld
theory was designed in order to accomplish the quantization of electromagnetism. The
product is the wedge product of dierential forms. Kauman of Casimir insertion up to rst order
of approximation coming from the dierence formula in the functional integral. . A is taken to
be the integral over M of the trace of the ChernSimons threeform CS AdA/AAA. and its
internal logic.
With the help of the Wilson loop functional on knots and links. We can write Ax Aa xTa dxk
where the index a ranges from to m with the Lie k algebra basis T . This analysis depends
upon key facts relating the curvature of the gauge eld to both the Wilson loop and the
ChernSimons Lagrangian. K dA exp ik/SM. The reader may compare with the exposition in .
It is the . For each choice of a and k. There is summation over repeated indices. as in the
previous discussion. where x is a point in space. Witten writes down a functional integral for
link invariants in a manifold M ZM. It is usually indicated by the symbolism Tr P exp KA . .
The curvature tensor is written Frs xTa dxr dy s . . the manifold M will be the dimensional
sphere S . Tm .Vassiliev Invariants and Quantum Gravity version of integrating the gauge eld
along a loop K in space that we take to be an embedding knot or a curve with transversal
selfintersections. y xy yx. Tb ifabc Tc where x. A as S. . The Lie algebra generators Ta are
actually matrices corresponding to a given representation of an abstract Lie algebra. let us
recall the local coordinate structure of the gauge eld Ax. T . and fabc the matrix of structure
constants is totally antisymmetric. To this end. The rst fact is the relation of Wilson loops and
curvature for small loops Fact The result of evaluating a Wilson loop about a very small
planar circle around a point x is proportional to the area enclosed by this circle times the
corresponding value of the curvature tensor of the gauge eld a evaluated at x. Ak x is a
smooth function dened on space. The index k goes from to . We assume some properties of
these matrices as follows . An analysis of the formalism of this functional integral reveals
quite a bit about its role in knot theory. the Wilson loop will be denoted by the notation KA to
denote the dependence on the loop K and the eld A. TrTa Tb ab / where ab is the Kronecker
delta ab if a b and zero otherwise. We abbreviate SM. T . Unless otherwise mentioned. We
also assume some facts about curvature. Thus KA Tr P exp KA . Ta . But note the dierence
in conventions on the use of i in the Wilson loops and curvature denitions. A is the
ChernSimons action. For the purpose of this discussion. . A Tr P exp dA exp ik/S KA . KA
Here SM. . Here the P denotes pathordered integration. The symbol Tr denotes the trace of
the resulting matrix. In Ax a we sum over the values of repeated indices.
but inserting the matrix Ta into the loop at the point x. The ordering is forced by the
dimensional nature of the loop. Thus one can write symbolically. . See the gure below. over
partitions of the loop K. If T is a given matrix then it is understood that T KA denotes the
insertion of T into some point of the loop. Insertion of a given matrix into this product at a
point on the loop is then a welldened concept. KA xK Ax xK Aa xTa dxk . Let KA denote the
change in the value of the loop. This is the rstorder approximation to the change in the
Wilson loop. KA Ta KA Remark on insertion The Wilson loop is the limit. Approximate the
change according to Fact . and that we are summing over repeated indices. Kauman local
coordinate expression of dA AA. of products of the matrices Ax where x runs over the
partition. it is understood from the context of the formula a dsr dxs Frs xTa KA that the
insertion is to be performed at the point x indicated in the argument of the curvature.
Application of Fact Consider a given Wilson loop KA . KA is given by the formula a KA dxr
dxs Frs xTa KA . In the case above. Ask how its value will change if it is deformed
innitesimally in the neighborhood of a point x on the loop. In this formula it is understood that
the Lie algebra matrices Ta are to be inserted into the Wilson loop at the point x. and regard
the point x as the place of curvature evaluation. This means that each Ta KA is a new Wilson
loop obtained from the original loop KA by leaving the form of the loop unchanged. Louis H. k
It is understood that a product of matrices around a closed loop connotes the trace of the
product.
Let ZK ZK. . k Aa yTa dy k k Fact The variation of the ChernSimons action S with respect to
the gauge potential at a given point in space is related to the values of the curvature tensor
at that point by the following formula where abc is the epsilon symbol for three indices a Frs x
rst S . The volume element rst dxr dy s dz t is taken with regard to the innitesimal directions
of the loop deformation from this point on the original loop. Aa x t With these facts at hand we
are prepared to determine how the Witten integral behaves under a small deformation of the
loop K. S and let ZK denote the change of ZK under an innitesimal change in the loop K. to
the case where x is a double point of transversal selfintersection of a loop .Vassiliev
Invariants and Quantum Gravity Remark The previous remark implies the following formula
for the variation of the Wilson loop with respect to the gauge eld KA dxk Ta KA . and the
insertion is taken of the matrix products Ta Ta at the chosen point x on the loop K that is
regarded as the center of the deformation. The sum is taken over repeated indices. .
Proposition Compare . All statements of equality in this proposition are up to order /k . with a
dierent interpretation. Then ZK i/k dA exp ik/S r s t rst dx dy dz Ta Ta KA . The same formula
applies. Aa x k Varying the Wilson loop with respect to the gauge eld results in the insertion
of an innitesimal Lie algebra element into the loop. Proof KA Aa x k yltx Aa x k yK Aa yTa dy
k k Ta dxk ygtx Aa yTa dy k k dx Ta KA .
This completes the proof of the proposition. with two directions coming from the plane of
movement of one arc. Louis H. Applying Proposition As the formula of Proposition shows.
one Ta is inserted into each of the transversal crossing segments so that Ta Ta KA denotes
a Wilson loop with a selfintersection at x and insertions of Ta at x and x . Kauman K. where
and denote small displacements along the two arcs of K that intersect at x. the change of
interpretation occurs at the point in the argument when the Wilson loop is dierentiated. the
other term is the one that is described in part . This completes the formalism of the proof.
One of these derivatives gives rise to a term with volume form equal to zero. In this case.
and the deformation consists in shifting one of the crossing segments perpendicularly to the
plane of intersection so that the selfintersection point disappears. In this case. the in . In the
case of part . the volume form is nonzero. Proof ZK dA exp ik/S KA a dA exp ik/S dxr dy s
Frs xTa KA Fact Fact dA exp ik/S dxr dy s dA exp ik/S dA rst S Ta KA Aa x t r s rst dx dy Ta
S Aa x t KA KA i/k i/k exp ik/S Aa x t r s rst dx dy Ta dA exp ik/S r s rst dx dy Ta KA Aa x t KA
integration by parts i/k dA exp ik/S r s t rst dx dy dz Ta Ta dierentiating the Wilson loop.
Dierentiating a selfintersecting Wilson loop at a point of selfintersection is equivalent to
dierentiating the corresponding product of matrices at a variable that occurs at two points in
the product corresponding to the two places where the loop passes through the point. and
the perpendicular direction is the direction of the other arc.
Vassiliev Invariants and Quantum Gravity
tegral ZK is unchanged if the movement of the loop does not involve three independent
space directions since rst dxr dy s dz t computes a volume. This means that ZK ZS , K is
invariant under moves that slide the knot along a plane. In particular, this means that if the
knot K is given in the nearly planar representation of a knot diagram, then ZK is invariant
under regular isotopy of this diagram. That is, it is invariant under the Reidemeister moves II
and III. We expect more complicated behavior under move I since this deformation does
involve three spatial directions. This will be discussed momentarily. We rst determine the
dierence between ZK and ZK where K and K denote the knots that dier only by switching a
single crossing. We take the given crossing in K to be the positive type, and the crossing in K
to be of negative type.
The strategy for computing this dierence is to use K as an intermediate, where K is the link
with a transversal selfcrossing replacing the given crossing in K or K . Thus we must
consider ZK ZK and ZK ZK . The second part of Proposition applies to each of these
dierences and gives i/k dA exp ik/S
r s t rst dx dy dz Ta Ta
KA
where, by the description in Proposition , this evaluation is taken along the loop K with the
singularity and the Ta Ta insertion occurs along the two transversal arcs at the singular point.
The sign of the volume element will be opposite for and consequently we have that . The
volume element rst dxr dy s dz t must be given a conventional value in our calculations.
There is no reason to assign it dierent absolute values for the cases of and since they are
symmetric except for the sign. Therefore ZK ZK ZK ZK . Hence ZK / ZK ZK .
Louis H. Kauman
This result is central to our further calculations. It tells us that the evaluation of a singular
Wilson loop can be replaced with the average of the results of resolving the singularity in the
two possible ways. Now we are interested in the dierence ZK ZK ZK ZK i/k dA exp ik/S
r s t rst dx dy dz Ta Ta
KA.
Volume convention It is useful to make a specic convention about the volume form. We take
Thus ZK ZK i/k dA exp ik/S Ta Ta K A .
r s t rst dx dy dz
for and for .
Integral notation Let ZTa Ta K denote the integral ZTa Ta K dA exp ik/S Ta Ta K A .
Dierence formula Write the dierence formula in the abbreviated form ZK ZK i/kZTaTa K . This
formula is the key to unwrapping many properties of the knot invariants. For diagrammatic
work it is convenient to rewrite the dierence equation in the form shown below. The crossings
denote small parts of otherwise identical larger diagrams, and the Casimir insertion Ta Ta K
is indicated with crossed lines entering a disk labelled C.
Vassiliev Invariants and Quantum Gravity
The Casimir The element a Ta Ta of the universal enveloping algebra is called the Casimir.
Its key property is that it is in the center of the algebra. Note that by our conventions Tr a Ta
Ta a aa / d/ where d is the dimension of the Lie algebra. This implies that an unknotted loop
with one singularity and a Casimir insertion will have Zvalue d/.
In fact, for the classical semisimple Lie algebras one can choose a basis so that the Casimir
is a diagonal matrix with identical values d/D on its diagonal. D is the dimension of the
representation. We then have the loc loc general formula ZTa Ta K d/DZK for any knot K.
Here K denotes the singular knot obtained by placing a local selfcrossing loop in K as shown
below
loc Note that ZK ZK. Let the at loop shrink to nothing. The Wilson loop is still dened on a loop
with an isolated cusp and it is equal to the Wilson loop obtained by smoothing that cusp.
Louis H. Kauman
loc Let K denote the result of adding a positive local curl to the knot loc K, and K the result of
adding a negative local curl to K.
Then by the above discussion and the dierence formula, we have
loc ZK
loc loc ZK i/kZTa Ta K
Thus,
ZK i/kd/DZK.
loc ZK i/kd/D ZK.
Similarly,
loc ZK i/kd/D ZK.
These calculations show how the dierence equation, the Casimir, and properties of Wilson
loops determine the framing factors for the knot invariants. In some cases we can use
special properties of the Casimir to obtain skein relations for the knot invariant. For example,
in the fundamental representation of the Lie algebra for SU N the Casimir obeys the following
equation see , Ta ij Ta kl
a
il jk
ij kl . N
Hence ZTa Ta K ZK ZK N
where K denotes the result of smoothing a crossing as shown below
commutativeringvalued invariant of knots. This last equation shows that P K is a
specialization of the Homy polynomial for arbitrary N . Therefore. so the framing factor is i/k
N N e N /N . then it can be naturally extended to an invariant of rigid vertex graphs by dening
the invariant of graphs in terms of the . and that for N SU it is a specialization of the Jones
polynomial. or. Here d N and D N . we obtain ZK ZK i/k ZK / Hence i/N kZK i/N kZK i/kZK or e
ZK e ZK e e ZK N N N ZK ZK . Graph invariants and Vassiliev invariants We now apply this
integral formalism to the structure of rigid vertex graph invariants that arise naturally in the
context of knot polynomials.Vassiliev Invariants and Quantum Gravity Using ZK ZK ZK and
the dierence identity. then e/N P K e/N P K e e P K . Hence eN P K eN P K e e P K . if P K
wK ZK denotes the normalized invariant of ambient isotopy associated with ZK with wK the
sum of the crossing signs of K. more generally. where ex expi/kx taken up to O/k . If V K is a
Laurentpolynomialvalued.
constitute the important special case of these graph invariants where a . It is not hard to see
that if V K is an ambient isotopy invariant of knots. See the gure below. Here K indicates an
embedding with a transversal valent vertex . We use the symbol to distinguish this choice of
vertex designation from the previous one involving a selfcrossing Wilson loop. this means
that we dene V G for an embedded valent graph G by taking the sum over ai S bi S ci S V S
for all knots S obtained from G by replacing a node of G either with a crossing of positive or
negative type. If V is the nite type k. Thus V G is a Vassiliev invariant if V K V K V K . then
this extension is a rigid vertex isotopy invariant of graphs. so that the vertex behaves like a
rigid disk with exible strings attached to it at specic points. For purposes of enumeration it is
convenient to use chord diagrams to enumerate and indicate the abstract graphs. and c . b .
Kauman knot invariant via an unfolding of the vertex as indicated below V K aV K bV K cV K
. Formally. . Louis H. then V G is independent of the embedding type of the graph G when G
has exactly k nodes. The values of V G on all the graphs of k nodes is called the top row of
the invariant V . or with a smoothing denoted . The Vassiliev invariants . . These points are
paired with the pairing indicated by arcs or chords connecting the paired points. V G is said
to be the nite type k if V G whenever G gt k where G denotes the number of valent nodes in
the graph G. In rigid vertex isotopy the cyclic order at the vertex is preserved. This follows at
once from the denition of nite type. There is a rich class of graph invariants that can be
studied in this manner. A chord diagram consists of an oriented circle with an even number
of points marked along it.
Therefore Vj K is a Vassiliev invariant of nite type. loc We have shown that ZK ZK with ed/D.
The fascinating thing is that the ambient dierence formula. appropriately interpreted. Hence
Vj G if G has more than j nodes.Vassiliev Invariants and Quantum Gravity This gure
illustrates the process of associating a chord diagram to a given embedded valent graph.
Hence P K wK ZK is an ambient isotopy invariant. Each transversal selfintersection in the
embedding is matched to a pair of points in the chord diagram. This formula tells us that for
the Vassiliev invariant associated with P we have P K w i/k ZTa Ta K d/DZK . actually tells us
how to compute Vk G when G has k . then we have the ambient isotopy dierence formula P
K P K w i/k ZTa Ta K d/DZK . The framed dierence formula provides the necessary
information for obtaining the top row. Furthermore. and then consider the structure of the
dierence between positive and negative crossings. then the ambient dierence formula implies
that /kj divides P G when G has j or more nodes. if Vj K denotes the coecient of i/kj in the
expansion of P K in powers of /k. Vassiliev invariants from the functional integral In order to
examine the Vassiliev invariants associated with the functional integral. This result was
proved by Birman and Lin by dierent methods and by BarNatan by methods equivalent to
ours. we must rst normalize these invariants to invariants of ambient isotopy. We leave the
proof of this formula as an exercise for the reader. The equation ZK ZK i/kZTa Ta K implies
that if wK w .
it follows from BarNatan and Kontsevich that the condition of topological invariance is
translated into the fact that the Lie bracket is represented as a commutator and that it is
closed with respect to the Lie algebra. each of which can be evaluated by taking the
indicated trace. we insert nothing else into the loop. Louis H. but it is very interesting to see
how it follows from a minimal approach to the Witten integral. This yields the graphical
evaluation implied by the recursion V G V Ta Ta G d/DV G . This result is equivalent to
BarNatans result. After k steps we have a fully inserted sum of abstract Wilson loops.
Kauman nodes. independent of the embedding. In particular. and because the Wilson loop is
being evaluated abstractly. Under these circumstances each node undergoes a Casimir
insertion. At each stage in the process one node of G disappears or it is replaced by these
insertions. Thus we take the pairing structure associated with the graph the socalled chord
diagram and use it as a prescription for obtaining a trace of a sum of products of Lie algebra
elements with Ta and Ta inserted for each pair or a simple crossover for the pair multiplied
by d/D. Diagrammatically we have Since Ta Tb Tb Ta we obtain ifabc Tc c .
and so we bring this discussion to a close. all circling this basic problem. . In particular. it
must be mentioned that this brings us to the core of the main question about Vassiliev
invariants are there nontrivial Vassiliev invariants of knots and links that cannot be
constructed through combinations of Lie algebraic invariants There are many other open
questions in this arena. Nevertheless. There is not room here to go into more detail about
this matter. it implies the basic term relation Proof The presence of this relation on chord
diagrams for Vi G with G i is the basis for the existence of a corresponding Vassiliev
invariant.Vassiliev Invariants and Quantum Gravity This relationship on chord diagrams is the
seed of all the topology.
This situation suggests that one should amalgamate the formalism of the Vassiliev invariants
with the structure of the Poisson algebras of loops and insertions used in the quantum gravity
theory . The main point that we wish to make in the relationship of the Vassiliev invariants to
these loop states is that the Vassiliev vertex satisfying is not simply a transverse intersection
of Wilson loops. Louis H. to corresponding statements about the Wilson loop. Rovelli. It also
suggests . via integration by parts. In this way. We have seen that it follows from the
dierence formula that the Vassiliev vertex is much more complex than thisthat it involves the
Casimir insertion at the transverse intersection up to the rst order of approximation. In this
theory the metric is expressed in terms of a spin connection A. . and that this structure can
be used to compute the top row of the corresponding Vassiliev invariant. Kauman Quantum
gravityloop states We now discuss the relationship of Wilson loops and quantum gravity that
is forged in the theory of Ashtekar. and quantization involves considering wave functions A.
Smolin and Rovelli analyze the loop transform K dAA KA where KA denotes the Wilson loop
for the knot or singular embedding K. knots and weaves and their topological invariants
become a language for representing a state of quantum gravity. It turns out that the condition
that K be a knot invariant without framing dependence is equivalent to the socalled
dieomorphism constraint for these wave functions. Dierential operators on the wave function
can be referred. and Smolin .
Atiyah. Math. Lin. One can begin to work backwards. Baez. Ashtekar. R. Quantum
Mechanics and Path Integrals. M. Hibbs. Knot invariants as nondeu generate quantum
geometries. Berkeley. Link invariants of nite type and perturbation theory. The Geometry and
Physics of Knots. R. and L. spin geometry and quantum gravity. . Crane. On the Vassiliev
knot invariants. C. M. This theory can be investigated by working the transform methods
backwardsfrom knots and links to dierential operators and dierential geometry. . . . Crane
and D. . A categorical construction of D topological quantum eld theories. . L. Pisa . and J.
Math. Preprint . L. Yetter. Bibliography . . . Preprint . Phys. Geometry of YangMills Fields. Br
gmann. Lett. Lett. A. Principles of Quantum Mechanics. Preprint . Commun. S. Phys. Math.
taking the position that invariants that do not ostensibly satisfy the dieomorphism constraint
owing to change of value under framing change nevertheless still dene states of a quantum
gravity theory that is a modication of the Ashtekar formulation. R. F. Phys. P. . . Gambini. . A.
Weaving a classical geometry with quantum threads. All these remarks are the seeds for
another paper. B. F. Phys. Knot polynomials and Vassilievs invariants. . J. Freed and R.
Smolin. Invent. Pullin. CA. Rovelli. . Atiyah. McGrawHill . Acknowledgements It gives the
author pleasure to acknowledge the support of NSF Grant Number DMS and the Program for
Mathematics and Molecular Biology of the University of California at Berkeley. Rev. J.
Cambridge University Press . BarNatan. . D. Computer calculations of Wittens manifold
invariants. D. Feynman and A. and ask the reader to stay tuned for further developments. M.
B . We close here. Dirac. Accademia Nazionale dei Lincei Scuola Superiore Lezioni
Fermiare. to appear. Conformal eld theory. Birman and X. Oxford University Press . Gompf.
.Vassiliev Invariants and Quantum Gravity taking the formalism of these invariants in the
functional integral form quite seriously even in the absence of an appropriate measure
theory. Lett.
F. Oxford . . Lett. . Jones. Kauman. . . . J. Preprint . Am. . L. H. . Gravitation. L. Louis H. L.
Thesis. Math. On the manifold invariants of ReshetikhinTuraev for sl. H. . B. V. V. Princeton
University Press. Knot theory and quantum gravity a primer. Lett. World Scientic Pub. Hecke
algebra representations of braid groups and link polynomials. Preprint . K. H. Kauman and P.
Melvin. manifolds and the Temperley Lieb algebra. R. . . Soc. J. Jerey. Kontsevich. Kauman.
Kauman. V. Ooguri. Jones. Math. W. . . B . C. H. L. Am. . . . homotopical algebra and low
dimensional topology. and J. Perry. L. . Temperley Lieb Recoupling Theory and Invariants of
Manifolds to appear as Annals monograph. J. Kauman . H. B. A polynomial invariant for links
via von Neumann algebras. R. Math. V. H. Ramications. Preprint . . Jones. Not. Graphs. H.
Index for subfactors. R. Lins. . . Discrete and continuum approaches to threedimensional
quantum gravity. L. Applications of TQFT to invariants in low dimensional topology. R. Soc.
Invent. Invent. Topology . AMS Contemp. Kauman. W. A . Garoufalidis. Kirby and P. Bull.
Hasslacher and M. Statistical mechanics and the Jones polynomial. . Kauman and S. Ooguri.
. Math. On knot invariants related to some statistical mechanics models. . R. Knots and
Physics. Misner. Thorne. R. . . S. R. Math. V. Phys. Pullin. A. . Vogel. Preprint . Princeton
University Press . Ann. . Math. On Some Aspects of ChernSimons Gauge Theory. Lickorish.
Pacic J. Freeman . Math. F. F. F. Mod. Ann. Topological lattice models in four dimensions. H.
Annals of Mathematics Studies Number . Spin networks are simplicial quantum gravity. C. .
H. W. Ser. . A new knot polynomial and von Neumann algebras. C. Jones. L. F. On Knots. S.
Math. Jones. . Phys. Knot Theor. M. Wheeler. State models and the Jones polynomial. Link
polynomials and a graphical calculus. .
ed. Quantum invariants of manifolds associated with classical simple Lie algebras. . . Wenzl.
. Quantum gravity in the selfdual representation. G. Commun. Large k asymptotics of Wittens
invariant of Seifert manifolds. Phys. On Wittens manifold invariants. Quantum eld theory and
the Jones polynomial. Math. . dimensional gravity as an exactly soluble system. . . The
TuraevViro state sum model and dimensional quantum gravity. G. Nucl. Turaev and H.
Finitetype invariants of knots. Contemp. Viro. . Math. . G. Preprint. G. B . . . . Phys.
Reshetikhin and V. . V. Invariants of three manifolds via link polynomials and quantum
groups. Preprint . Quantum invariants of manifolds and a glimpse of shadow topology. . links
and graphs. N. L. T. V. . Stanford. Turaev and O. Witten. Preprint . . Turaev. Invent. Preprint .
I. Turaev. Preprint . Cohomology of knot spaces. . V. V. Arnold. N. Phys. pp. Reshetikhin and
V. Turaev. .. Commun. Quantum invariants of links and valent graphs in manifolds. V. Witten.
Preprint . Soc. L. State sum invariants of manifolds and quantum j symbols. Walker. Preprint
.Vassiliev Invariants and Quantum Gravity . Topology . R. Am. Math. . Williams and F. . G. K.
Ribbon graphs and their invariants derived from quantum groups. Turaev. E. In Theory of
Singularities and its Applications V. Y. Archer. . Smolin. Rozansky. E. Turaev. . Math.
Topology of shadows. Math. Preprint . V. . . Y. Vassiliev.
Louis H. Kauman .
from the values of the loop variables. an arbitrary splitting of spacetime into spacelike
hypersurfaces. particularly for covariant canonical approaches to quantum gravity.ucdavis.
this is already a source of trouble in classical general relativity. To a certain extent.
Introduction Ashtekars connection representation for general relativity . I emphasize the
possibility of reconstructing the classical geometry. I close with a brief discussion of
implications for quantization. and refer only to the invariant underlying geometry. however.
USA email carlipdirac. and the rst steps have been taken towards establishing a reasonable
weakeld perturbation theory . where one must take care to separate physical phenomena
from artifacts of coordinate . But real geometry and physics cannot depend on such a choice.
University of California. California . Some important progress has been made a number of
observables have been found. the true physical observables must somehow forget any
details of the time slicing.edu Abstract In this chapter. inherent in almost any canonical
formulation of general relativity. by the absence of a clear physical interpretation for the
observables built out of Ashtekars new variables.Geometric Structures and Loop Variables in
dimensional Gravity Steven Carlip Department of Physics. But there is a deeper problem as
well. Davis. Progress has been hampered. and the closely related loop variable approach
have generated a good deal of excitement over the past few years. via the theory of
geometric structures. While it is too early to make rm predictions. In part. the problem is
simply one of unfamiliarityphysicists accustomed to metrics and their associated connections
can nd it dicult to make the transition to densitized triads and selfdual connections. large
classes of quantum states have been identied. To dene Ashtekars variables. one must
choose a time slicing. I review the relationship between the metric formulation of dimensional
gravity and the loop observables of Rovelli and Smolin. there seems to be some real hope
that these new variables will allow the construction of a consistent nonperturbative quantum
theory of gravity.
This notation is standard in papers in dimensional gravity. are spacetime coordinate indices.
The fundamental variables are a triad e a a section of the bundle of orthonormal framesand a
connection on the same bundle. c. are spatial coordinate indices. . which we shall often
assume to have a topology R. and a. . however. where ea e a dx and a local SO. From
geometry to holonomies Let us begin with a brief review of dimensional general relativity in
rstorder formalism. As our spacetime we take a manifold M . so readers should be careful in
translation. and one may hope that at least some of the technical results have extensions to
our physical spacetime. general relativity in two spatial dimensions plus time.
transformations. . . allowing the use of powerful techniques not readily available in the
realistic dimensional theory. But the success of dimensional gravity can be viewed as an
existence proof for canonical quantum gravity. j. A reduction in the number of dimensions
greatly simplies general relativity. But although some progress has been made in dening
area and volume operators in terms of these variables . the goal of reconstructing spacetime
geometry from such invariant quantities remains out of reach. . . Steven Carlip choices. .
where is a closed orientable surface. and the need to reconstruct geometry and physics from
such quantities becomes unavoidable. b. . which can be specied by a connection form a b . .
we can actually write down a large set of dieomorphisminvariant operators. In canonical
quantization. many of the specic results presented here will not readily generalize to higher
dimensions. The action is invariant under eb c abc ea a abc a . As a consequence. i. b c . d
Indices . built out of the loop variables of Rovelli and Smolin in the loop representation of
quantum gravity . k. bc dx . The EinsteinHilbert action can be written as Igrav M ea da abc
abc b c . . The purpose of this chapter is to demonstrate that such a reconstruction is
possible in the simple model of dimensional gravity. are Lorentz indices. but diers from the
usual conventions for Ashtekar variables. labelling vectors in an orthonormal basis. the
problem becomes much sharperall observables must be dieomorphism invariants. For the rst
time. Lorentz indices are raised and lowered with the Minkowski metric ab . In a sense. . the
Ashtekar program is a victim of its own success. . . .
. can be interpreted as the statement that e is a cotangent vector to the space of at SO.
actually implies that the metric g is at. indeed. for as a function of e. As a second alternative.
connections. In fact. but this is not an independent symmetry Witten has shown that when
the triad e a is invertible. is simply the requirement that the curvature of vanish. In
dimensions. which will form the basis for our analysis . We therefore need only worry about
equivalence classes of dieomorphisms that are not isotopic to the identity. abc These
equations have four useful interpretations. . . if s is a curve in the space of at connections.
these eld equations are much more powerful than they are in dimensions the full Riemann
curvature tensor is linearly dependent on the Ricci tensor. the derivative of . that be a at SO..
that is elements of the mapping class group of M . so . dieomorphisms in the connected
component of the identity are equivalent to transformations of the form . R g R g R g R g R g
g g g R. and rewrite . . are easily derived dea and d a abc b ec b c . Moreover. that is. as an
equation for e. . ea da . connection.Geometric Structures and Loop Variables as well as local
translations. gives d d a ds abc b dc ds . note that eqn .e. We can solve . The space of
solutions of the eld equations can thus be identied with the set of at Lorentzian metrics on M
. . for the Lorentzian i. Igrav is also invariant under dieomorphisms of M . . depends only on
and not on e. a abc b c . of course. The equations of motion coming from the action . R g .
The result is equivalent to the ordinary vacuum Einstein eld equations.. eqn . .
pseudoRiemannian metric g e a e b ab .
and it is often the case that only one connected component corresponds to physically
admissible spacetimes. If we denote equivalence under this action by . if h h .. SO. for
physical implications. with a d a ds .. it is easy to check that the rst equations of . Now.. is not
always connected. These transfor mations act on N through their action as a group of
automorphisms of M . we must still factor out the gauge transformations . transformations.
this description can be further rened. The transformations of e can once again be interpreted
as statements about the cotangent space if we consider a curve s of SO. the mapping class
group. and let N / N . A solution of the eld equations is thus determined by a point in the
cotangent bundle T N . follow from dierentiating the second equation of . . the space N or at
least the physically relevant is homeomorphic to the Teichm ller connected component of N u
Strictly speaking. . we can express the space of solutions of the eld equations . and tell us
that only gauge equivalence classes of at SO. as T N . . To determine the physically
inequivalent solutions of the eld equations. act on as ordinary SO. In that case. Steven Carlip
which can be identied with . See . h SO. and . We can therefore write N Hom M . ds . a at
connection on the frame bundle of M is determined by its holonomies. The space of
homomorphisms . / . for the mathematical structure and . Let us denote the space of such
equivalence classes as N . one more subtlety remains. connections are relevant. .. when M
has the topology Rthis action comprises the entire set of outer automorphisms of M . and in
many interesting casesfor example.. When M has the topology R. It remains for us to factor
out the dieomorphisms that are not in the component of the identity. and ea as in . . with ea d
a . that is by a homomorphism M SO. . and gauge transformations act on by conjugation.
gauge transformations. The local Lorentz transformations . .
This is consistent with our previous results. abc Comparing with . now from to SO. solutions
of the dimensional eld equations are certainly not static as geometriesthey do not. For a
topology of the form R . For despite eqn . it is useful to split the eld equations into spatial and
temporal components. and N is the corresponding moduli space . who observed that the triad
e and the connection could . and is therefore invisible to the holonomies. solutions may
therefore be labelled by classes of homomorphisms.Geometric Structures and Loop
Variables space of . But the central dilemma described in the introduction now stands out
sharply. The temporal components of the eld equations. and must be uncovered if we are to
nd a sensible physical interpretation of our solutions. on the other hand. A third approach is
available if M has the topology R. The set of vacuum spacetimes can thus be identied with
the cotangent bundle of the moduli space of . with a a e and a e a .. is entirely described by a
gauge transformation. ea ea e a dt. with all quantities replaced by their tilded spatial
equivalents.. of course. . Let us write d d dt . This puzzle is an example of the notorious
problem of time in gravity . the fundamental group is simply that of . The spatial projections of
. shows in detail how this occurs motion in coordinate time is merely a gauge transformation.
. dt... This decomposition can be made in a less explicitly coordinatedependent mannersee.
Equation .. and exemplies one of the basic issues that must be resolved in order to construct
a sensible quantum theory. b ec b c . for example. was suggested by Witten . in general. A
nal approach to the eld equations . For such a topology. we see that the time development of
. admit timelike Killing vectors. now become ea de a a d a abc a a a . take the same form as
the original equations. and an invariant description in terms of holonomies should not be able
to detect a particular choice of spacelike slice. The real physical dynamics has somehow
been hidden by this analysis. and many powerful results from Riemann surface theory
become applicable. abc eb c . As above. and the corresponding cotangent vectors. . but the
nal results are unchanged. .
may be recognized as the standard semidirect product composition law for Poincar
transformations. . if h h . as the same group of automorphisms of M . . ISO. that is by
homomorphisms in the space M Hom M . can be identied with standard ISO. inherits this
cotangent bundle structure in an obvious way. Imitating the arguments of our second
interpretation. . . and the multiplication law . . . d .. has a Lie algebra with e generators J a
and P b and commutation relations J a. is simply the Chern Simons action for A. To relate
this description to our previous results. an invariant inner product on the Lie algebra. But this
group acts in . d . and the gauge transformations . we thus see that M T N .. ICS Tr A dA A A
A . . leading to the identication M T N . and . can be written in the form d . observe that ISO. .
Steven Carlip be combined to form a single connection on an ISO. connection. is itself a
cotangent bundle with base space SO. P b abc Pc . where N is the space of homomorphisms
. h ISO. d . The space of homomorphisms from e M to ISO. a cotangent vector at the point
SO. gauge transformations. we should therefore expect solutions of the eld equations to be
characterized by gauge equivalence classes of at ISO. Indeed. the dimensional Poincar
group. J a. P a . writing the quotient as M/ M. P b . . connections. then it is easy to verify that
the action . Tr J a J b Tr P a P b . / . d d . It remains for us to factor out the mapping class
group. by Tr J a P b ab . J b abc Jc . bundle. If we write a single connection form A ea Pa a
Ja and dene a trace. ISO. M The eld equations now reduce to the requirement that A be a at
ISO.
and considerable work is still needed to construct dieomorphisminvariant observables that
depend only on knot classes. The loop variables of Rovelli and Smolin . T and T are not quite
the equivalence classes of holonomies of our interpretation number above T is not a
holonomy. the loop variables can serve as local coordinates on N . ds . on the other hand. In
dimensions. is the one that is closest to the loop variable picture in dimensional gravity. Here.
given a curve s in the space of at SO.. holonomy around . and we have already seen that the
derivative d a /ds can be identied with the triad ea . and indeed. to obtain d T ds Tr U . while
T is essentially a cotangent vector to T indeed. . In dimensions. . The key dierence is that in
dimensions the connection is at. at least under dieomorphisms isotopic to the identity. the
variables T and T depend on particular loops . may be expressed as follows. this issue has
been investigated in a slightly dierent context by Nelson and Regge . only the rst
corresponds directly to our usual picture of spacetime physics. and the basepoint x species
the point at which the path ordering begins.Geometric Structures and Loop Variables Of
these four approaches to the dimensional eld equations. on the other hand. But knowledge of
T for a large enough set of homotopically inequivalent curves may be used to reconstruct a
point in the space N of eqn . are geodesics in the at manifolds of this description. . Of course.
We then dene T Tr U . T is thus the trace of the SO. connections. dx tJa . . x and T dt Tr U .
xt d a tJa . Let U . x P exp a Ja . xt e a t . P denotes path ordering. we can dierentiate . so the
holonomy U . Some care must be taken in handling the mapping class group. which acts
nontrivially on T and T . The second approach. be the holonomy of the connection form a
around a closed path t based at x. . Trajectories of physical particles. for instance. the loop
variables are already invariant. but only the trace of a holonomy. x depends only on the
homotopy class of . dt .
specic examples are familiar for example. i Ui Ui gi i Ui Ui . the vanishing of the curvature
tensor implies that M. i. A fundamental ingredient in the description of a G. Steven Carlip
Geometric structures from holonomies to geometry The central problem described in the
introduction can now be made explicit. .R. . . M can still be covered by contractible
coordinate patches Ui that are each isometric to V . just as an ordinary ndimensional
manifold is modelled on Rn . Moreover. Our goal is therefore to provide a translation
between these two descriptions. Such a description is fully dieomorphism invariant.e. which
determine how these patches are glued together. We can cover with coordinate charts i Ui X.
A G. or at least some subset of V . e Such a construction is an example of what Thurston
calls a geometric structure . these transition functions must be isometries of . . If the
spacetime topology is nontrivial. Spacetimes in dimensions can be characterized a la
Ashtekar et al. as points in the cotangent bundle T N . a G. X manifold containing a closed
path . structure. H structure. . that is elements of the Poincar group ISO. with constant
transition functions gi G between Ui and Ui .V . If M is topologically trivial. . X structure on M
is then a set of coordinate patches Ui covering M with coordinates i Ui X taking their values in
the model space and with transition functions ij i j Ui Uj in G. X structure is its holonomy
group. While this general formulation may not be widely known. To proceed. let us
investigate the space of at spacetimes in a bit more detail. with the standard Minkowski
metric on each patch. . our description number of the last section. n . More precisely. which
can be viewed as a measure of the failure of a single coordinate patch to extend around a
closed curve. since the metrics in Ui and Uj are identical. and provides a natural starting
point for quantization. in this case a Lorentzian or ISO. the uniformization theorem for
Riemann surfaces implies that any surface of genus g gt admits an P SL. But our intuitive
geometric picture of a dimensional spacetime is that of a manifold M with a at
metricdescription number and only in this representation do we know how to connect the
mathematics with ordinary physics. g is simply ordinary Minkowski space V . . Let M be a G.
the model space. . . . let G be a Lie group that acts analytically on some nmanifold X. and let
M be another nmanifold. . . X manifold is one that is locally modelled on X. The geometry is
then encoded in the transition functions ij on the intersections Ui Uj . . i . In general. that can
be extended to the whole of Minkowski space.
which again overlaps U .. For spacetimes. . . To do so. . . We can begin with a coordinate
transformation in U that replaces by g . . . Otherwise. . where G ISO. the holonomy H. In fact.
X structure of M . gn . . dened as H g . gj j . we simply form the product G Ui in each
patchgiving the local structure of a G bundleand use the transition functions ij of the
geometric structure to glue together the bers on the overlaps. The homomorphism H is not
quite uniquely determined by the geometric structure. It is then easy to verify that the at
connection on the resulting bundle has a holonomy group isomorphic to the holonomy group
of the geometric structure. reproducing approach number . If we start with a at manifold M
and simply cut out a ball. The resulting map D M X is called the developing map of the G. we
can obtain a new at manifold without aecting . Note that if we pass from M to its universal
covering space M . In general. and use one of the holonomy groups of eqn . the holonomy
denes a homomorphism H M G. we will eventually reach the nal patch Un . with j g . If the
new coordinate function n g . measures the obstruction to such a covering. we are thus led to
a space of holonomies of precisely the form . At least in simple examples. it is easy to see
what can go wrong.approach number to the eld equationsto dene a Lorentzian structure on
M . we will have succeeded in covering with a single patch. Continuing this process along the
curve. D embodies the classical geometric picture of development as unrollingfor instance.
the unwrapping of a cylinder into an innite strip. thus changing the transition functions gi
without altering the G. We can now try to reverse this process. It may be shown that the
holonomy of a curve depends only on its homotopy class . gn n happens to agree with on Un
U . and will be extendable to all of M . It is easy to see that such a transformation has the
eect of conjugating H by h. X structure is not necessarily sucient to determine the full
geometry. X structure. X structure on a manifold M . it is straightforward to dene a
corresponding at G bundle . and it is not hard to prove that H is in fact unique up to such
conjugation . For the case of a Lorentzian structure. thus extending to U U . since we are free
to act on the model space X by a xed element h G. This identication is not a coincidence.
Given a G. we will no longer have noncontractible closed paths. Let us now try analytic
continuation of the coordinate from the patch U to the whole of .Geometric Structures and
Loop Variables n Un U gn Un U . this step may fail the holonomy group of a G.
. holonomy group H M. for instance in a time slicing by surfaces of constant mean curvature.
This is a rather trivial change. if we can solve the dicult technical problem of nding an
appropriate fundamental region W V . Hence. For the black hole. For the dimensional black
hole. although more exotic topologies may occur in some approaches to quantum gravity. as
in our approach number . the geometric structure can be read o from and . a spacetime
constructed as a domain of dependence of a spacelike surface . in and . First. He shows that
the holonomy group determines a unique maximal spacetime M specically. This means that
for spatially closed dimensional universes. This quotient construction can be a powerful tool
for obtaining a description of M in reasonably standard coordinates. and the full topology is
uniquely xed by that of an initial spacelike slice. however. we can write M as a quotient space
W/H. and is explored in some detail in . and that M can be obtained as the quotient space
W/H. Finally. Steven Carlip the holonomy of the geometric structure. Next. Mess has
investigated this question for the case of spacetimes with topologies of the form R. For
topologies more complicated than R. topology change is classically forbidden. a
cosmological constant must be added to the eld equations. structure. To summarize. Instead
of being at. a theorem of Mess is relevant if M is a compact manifold with a at. then M
necessarily has the topology R . and we would like to show that nothing worse can go wrong.
of Minkowski space. RT . the resulting spacetime has constant negative curvature.
establishing a connection to our second approach to the eld equations. This procedure has
been investigated in detail for the case of a torus universe. we identify the group H with the
holonomy group of a Lorentzian structure on M . A related result for the torus will appear in .
it is not physically unreasonable to restrict our attention to spacetimes R. thus determining a
at spacetime of approach number . Mess also demonstrates that the holonomy group H acts
properly discontinuously on a region W V . . we now have a procedurevalid at least for
spacetimes of the form Rfor obtaining a at geometry from the invariant data given by
AshtekarRovelliSmolin loop variables. For a universe containing point particles. where is a
closed surface homeomorphic to one of the boundary components of M . In particular. But
again. timeorientable Lorentzian metric and a strictly spacelike boundary. nondegenerate.
the recent work of t Hooft and Waelbroeck is really a description of at spacetimes in terms of
Lorentzian structures. we associate that point with an ISO. H. it is implicit in the early
descriptions of Deser et al. I know of very few general results. and the geometric structure
becomes an SO. for the action of H. we use the loop variables to determine a point in the
cotangent bundle T N . And although it is never stated explicitly. .
each of the four interpretations of the eld equations discussed above suggests its own set of
fundamental observables. or the elds and their canonical momenta in a free eld theory. In our
second interpretationsolutions as classes of at connections and their cotangentsthe natural
observables are points in the bundle T N . from to R. for instance.Geometric Structures and
Loop Variables Quantization and geometrical observables Our discussion so far has been
strictly classical. In dimensions. Classical observables are functions of the positions and
momenta. and it seems that the best we can do is to dene a quantum theory in some
particular. . I would like to conclude by briey describing some of the issues that arise if we
attempt to quantize dimensional gravity. . In practice. and there is no reason to expect the
quantum theories coming from dierent slicings to be equivalent. one that must be small
enough to permit a consistent representation and yet big enough to generate a large class of
classical observables . no such irreducible representation of the full Poisson algebra of
classical observables exists . where a number of approaches to quantization can be carried
out explicitly. The canonical quantization of a classical system is by no means uniquely
dened. there is often an obvious set of preferred observablesthe positions and momenta of
point particles. that is maps f. on the other hand. The symplectic form on determines a set of
Poisson brackets f. . g. we are to look for an irreducible representation of this Lie algebra as
an algebra of operators acting on some Hilbert space. and we will have to look hard for
physical and mathematical justications for our selection. As stated. we are instructed to
replace the classical observables with operators and the Poisson brackets with commutators.
In particular. with local coordinates consisting of N position variables and N conjugate
momenta. A classical system is characterized by its phase space. that is. . Ordinarily. In the
rst interpretationsolutions as at spacetimesthe natural candidates are the metric and its
canonical momentum on some spacelike surface. For gravity. and hence induces a Lie
algebra structure on the space of observables. such a natural choice seems dicult to nd. g
among observables. the resulting quantum theory will depend on this choice of preferred
observables. it is known that dierent choices of variables lead to genuinely dierent quantum
theories . but most approaches have some basic features in common. . But these quantities
are not dieomorphism invariant. xed time slicing . we must therefore choose a subalgebra of
preferred observables to quantize. In simple classical systems. this program cannot be
carried out Van Hove showed in that in general. . since the choice of such a slicing is
arbitrary. . however. This is a rather undesirable situation. . a N dimensional symplectic
manifold . To quantize such a system.
Q Q. one obtains a parameter family of dieomorphisminvariant operators T and p T that
describe the physical evolution of a spacelike slice. Classically. . We begin by choosing a set
of physically interesting classical observables of at spacetimes. a natural set of geometric
variables are the modulus of a toroidal slice of constant mean curvature TrK T and its
conjugate momentum p . the appropriate symplectic structure for quantization is just the
natural symplectic structure of T N as a cotangent bundle. and there seem to be no
fundamental diculties in constructing the quantum theory. There. The procedure for
quantizing such a cotangent bundle is well established . such observables can be
determinedat least in principle as functions of the geometric structure. Steven Carlip These
are dieomorphism invariant. It is therefore natural to ask whether we can extend the classical
relationships between these approaches to the quantum theories. Q Q. into an operator
equation. Some ambiguity will remain. This program has been investigated in some detail for
the simplest nontrivial topology. the answer is positive. M RT . . just as in the classical theory.
holonomies of the spacetime. We now adopt these holonomies as our preferred observables
for quantization. These can be expressed explicitly in terms of a set of loop variables that
characterize the ISO. Finally. the requirement of mapping class group invariance seems to
place major restrictions on the possible orderings . T . since the operator ordering in . where
the parameter T labels the time slice on which the Q are dened for instance. Let us denote
these variables generically as QT . in particular. and thus of the ISO. holonomies . Following
the program outlined above. But now. but in the examples studied so far. thus obtaining a set
of dieomorphisminvariant but timedependent quantum observables to represent the variables
Q. we translate . is rarely unique. At least for simple topologies. The basic strategy is as
follows. the physical interpretation of the quantum observables is obscure. obtaining a Hilbert
space L N and a set of operators . The T dependence of these operators can be described
by a set of Heisenberg . . T TrK. For example. it is often possible to foliate uniquely a
spacetime by spacelike hypersurfaces of constant mean extrinsic curvature TrK. T . and
quantization is relatively straightforward. the intrinsic and extrinsic geometries of such slices
are useful observables with clear physical interpretations.
But I believe that some of the basic features are likely to extend to realistic dimensional
gravity. C. Phys. in General Relativity and Gravitation . Rovelli. R. Rovelli and L. A. the
parameter T is not a time coordinate in the ordinary sense. ed. of course. Ashtekar. . Smolin.
. Smolin. our simple model has strikingly conrmed the power of the RovelliSmolin loop
variables. We do not yet understand the classical solutions of the dimensional eld equations
well enough to duplicate such a strategy. Moreschi IOP Publishing. one would like to nd
some kind of perturbation theory for geometrical variables like Q. but for the rst time in years.
The invariant but T dependent operators T and p T are examples of Rovellis evolving
constants of motion . see L. Rev. but little progress has yet been made in this direction. C.
but a similar approach may be useful in minisuperspace models. . Lett. the operators T and p
T are fully dieomorphism invariant. Acknowledgements This work was supported in part by
US Department of Energy grant DEFGER. . Ashtekar.. For more complicated topologies.
such an explicit construction seems quite dicult. . Ashtekar. For a recent review. A. Gleiser. .
or quantization of the space of classical solutions. there seems to be some real cause for
optimism. Ideally. with a Hamiltonian H. . T that can again be calculated explicitly. dT d p i H.
Smolin. A full extension to dimensions undoubtedly remains a distant goal. and O. d i H. an
expression of dynamics in terms of operators that are individually constants of motion. The
quantization of holonomies is a form of covariant canonical quantization . Phys. N. D . Phys.
J. The specic constructions I have described here are unique to dimensions. dT . p . . We
thus have a kind of time dependence without time dependence. Kozameh. p . Singapore. .
although Unruh and Newbury have taken some interesting steps in that direction . B . C.
Nucl. Bristol. Bibliography . whose use has also been suggested in dimensional gravity. Rev.
Lectures on Nonperturbative Quantum Gravity World Scientic. Finally. A. Let me stress that
despite the familiar appearance of . M. and L.Geometric Structures and Loop Variables
equations of motion.
Singapore. . . B. ed. . Smolin. Carlip. . For examples of geometric structures. Ser. in
Geometry of Group Representations. Institut des Hautes Etudes Scientiques preprint
IHES/M// . L. Goldman and A. Harvey Academic Press. Solution Space to Gravity on RT in
Wittens Connection Formulation. Regge. aa . New York. . . S. Invent. on General Relativity
and Relativistic Astrophysics. Sci. Phys. B . Harvey. Hiroshima Math. and P. C. Marolf.
Husein. of the th Canadian r Conf. . Regge. E. in Knots. . . Lusanna World Scientic.
Princeton lecture notes . B . World Scientic. in Analytical and Geometric Aspects of
Hyperbolic Space. Witten. Adv. Ann. Ashtekar. Rev. . Lett. Classical amp Quantum Gravity L.
Phys. J. Phys. ed. Phys. Math. W. Math. M. Math. D. Kucha. A. I. J. M. ed. Thurston. . M.
Goldman. Math. S. . and G. G. Math. Epstein Cambridge University Press. Enseign.
Goldman. . Singapore. E. Phys. J. . Jackiw. The Geometry and Topology of ThreeManifolds.
Fenn. Acad. J. Providence. E. Samuel. S. Sorvali. R. M. Syracuse preprint SUGP/ . W.
Sullivan and W. Topology. Phys. Witten. Math. see D. D . Nelson and T. Phys. D. R. M. A. .
Ann. see K. W. Canary. Luoko and D. . B . . W. Soc. Commun. and Quantum Field Theory.
B. . Nucl. . . Magid American Mathematical Society. Commun. and L. J. . T. Classical amp
Quantum Gravity . G. Smolin. . . . Thurston. . A. . in Proc. ed. Commun. Seppl and T. . ed. D.
Okai. . . Lorentz Spacetimes of Constant Curvature. M. W. . Cambridge. R. . Goldman. .
Steven Carlip . Jacobson and L. Math. Nelson and T. J. W. V. in Discrete Groups and
Automorphic Functions. . Smolin. For an extensive review. Epstein. Nucl.P. L. Mess. Carlip. .
Green. Rovelli. W. . Lecture Notes Series . t Hooft. E. Kunstatter et al. J. . Deser. Phys. See
also T. London Math.
M. Witten. . Loop Representations for Gravity on a Torus. W. of the Fifth Canadian
Conference on General Relativity and Relativistic Astrophysics. Ba ados et al. G. Hosoya
and K. . Roy. . Phys. . Van Hove. Lett. Classical amp Quantum Gravity . Groups. . D . Rev. J.
de lAcad. H. University of British Columbia preprint . Solution to Gravity in the Dreibein
Formalism. t Hooft. D . Isham. Magnon. Carlip. Carlip and J. Rev. Cangemi. . and R. Phys.
Six Ways to Quantize Dimensional Gravity. MIT preprint MITCTP . DeWitt and R. A. . B.
Turin preprint DFTT / and Davis preprint UCD. For a useful summary. D. Rovelli. Phys. in
Relativity. London A . . n . D . D. . . . Gauge Formulation of the Spinning Black Hole in
Dimensional Antide Sitter Space. Cambridge. Phys. Rev. Equivalent Quantisations of
Dimensional Gravity. Israel Cambridge University Press.Geometric Structures and Loop
Variables . Hawking and W. C. M. J. S. and Topology II. . Nelson. Davis preprint UCD.
Amsterdam. Rev. Phys. G. . . de Belgique Classe de Sci. D . see C. R. B. to appear in Proc. .
E. Stora NorthHolland. M. Mann. D . Nakao. . Classical amp Quantum Gravity . Proc.
Leblanc. S. W. Syracuse preprint SUGP/ . Rev. . Mem. B . Phys. Marolf. Crnkovic and E. L.
ed. C. ed. Waelbroeck. Math. Classical amp Quantum Gravity . Ashtekar and A. Unruh and
P. Soc. . Nucl. Phys. A. Rev. in Three Hundred Years of Gravity. Moncrief. . S. Carlip.
Newbury. S. . Classical amp Quantum Gravity . V. S.. Phys.
Steven Carlip .
Fine Department of Mathematics. where is a Riemann surface . That is. second. Introduction
In this chapter. Treatments of the path integral for ChernSimons in the axial gauge on R tend
to make contact with the WZW model at earlier stages. one must already know how to solve
the WZW model or at least know that its solution denes this modular functor to recognize its
appearance in the ChernSimons theory. Massachusetts . The details I omit appear in . That
the two theories are related has been known at least since Witten rst described the
geometric quantization of the ChernSimons theory in the axial gauge on manifolds of the
form R . In this approach. rst.edu Abstract Direct analysis of the ChernSimons path integral
reduces quantum expectations of unknotted Wilson lines in ChernSimons theory on S with
group G to npoint functions in the WZW model of maps from S to G. the formal properties of
the path integral provide axioms crucial to extending the theory from R to compact manifolds.
USA email dnecis. the WZW model appears in a highly rened form. it does not use a direct
evaluation of the path integral.From ChernSimons to WZW via Path Integrals Dana S. as the
source of a certain modular functor. that the ChernSimons path integral on S is tractable. as
itself a principal ber bundle with anelinear ber. that it provides an unusually direct link
between the ChernSimons theory and the WZW model. The points I would like to emphasize
are. where G is the symmetry group of the ChernSimons theory. Although this is frequently
described as a path integral approach. I will describe how to reduce the path integral for
Chern Simons theory on S to the path integral for the WessZuminoWitten WZW model for
maps from S to G. Dartmouth. the space of connections modulo those gauge transformations
which are the identity at a point n. The reduction hinges on the characterization of A/G n .
University of Massachusetts. rather.umassd. and. Moore and Seiberg proceed with the
evaluation of the path integral far enough .
DA means the standard measure on A. to recognize its appearance in the ChernSimons
theory. Take this to be the standard metric on S . The approach I will describe is novel in that
it treats the path integral on S directly. The WZW path integral for G based maps from S to G
is f WZW Z G f XeiSX DX. and f explicitly determines f . In particular. and X is any extension
of X from S to M . if f is is the trace of a product of evaluationatapoint a Wilson line. Note that
even the formal denition of DA requires a metric. The main result f Z f AeiSA DA. A/G n The
ChernSimons path integral to evaluate is CS where A is a connection on the principal bundle
P S G. where X G. S Here A is the space of smooth connections on P . and DA is the
pushforward to A/G n . not its solution nor even its current algebra. it does not require a
choice of gauge. then f . and SA is the ChernSimons action k Tr A dA A A A . The main result
is f CS Z i f Xe k S k TrXXi M Tr X dX DX. and one need only know the action of the WZW
model. Dana S. Fine to obtain a path integral over the phase space LG/G. A A/G n is the
standard map to the quotient. This links the two theories as theories of the representations of
LG/G. Frhlich and King o obtain the KnizhnikZamolodchikov equation of the WZW model
from the same gaugexed ChernSimons path integral. Here M is any manifold with M S . The
key ingredient is a technique for integrating directly on the space of connections modulo
gauge transformations which I developed to evaluate path integrals for YangMills on
Riemann surfaces . where the WZW action is SX k S TrX X M TrX dX X dX X dX . Gn is the
space of gauge transformations which are the identity at the north pole n of S .
A A for Gn . Moreover. the ChernSimons path integral directly reduces to the path integral
for a WZW model of based maps from S to G. That is. so A G. Relative to a basepoint in the
ber over n. As e varies throughout the equator identied with S . since a gauge transformation
aects holonomy about e by conjugation by the value of the gauge transformation at n. The
path e maps. the space of such is ker P . and each point e S . That is. In fact. If P S .
Integrate along the ber over each point X G. . g S . g denotes projection to the longitudinal
part. e follows a longitude from the north pole n to the south pole s through a xed point e of
the equator and then returns along the longitude through the variable point e. Recognize the
resulting measure on G as a WZW path integral. Since A e . for each A. the ber over A/G n
as a bundle over G . dene A e by holonomy about paths e of the form pictured in Fig. . A e
denes. The technique is as follows . For each connection A. Turn now to the ber of . Thus the
assignment A A denes a mapping A/G n G which will serve as our projection. A e lies in G. A
A if vanishes along the longitudes of Fig. Characterize A/G n as a bundle over G with
anelinear bers. it is a based map. . Clearly.From ChernSimons to WZW Fig. a map from S to
G. . .
where is the restriction to a small sphere approaching n. The action along the ber is. . One
such choice is determined by A X dX. choose a specic origin A to remove the linear term..
the boundary term vanishes. To see this. In fact X X. k With this choice. use a homotopy
from X to the identity to determine lifts to P of the family of paths e in S which dened . The
reason to redo this is to describe the topologically trivial ber in an explicit manner. after an
integration by parts. Such a homotopy exists. which describe a homotopy equivalence
between A/G n for bundles over S and G. This determines P A such that A X. According to
eqn . To integrate over the ber. The determinant factor relates DA on the ber to the
metriccompatible measure D . and the connection A in the above presentation represents a
choice of origin in the ber. because G for any Lie group G. Fine A ker P . the boundary
contribution is retained. so to keep A smooth requires A X dX. the integral over the ber looks
like Iber X det/ DA P DA f A eiSA D. for reasons which will soon be clear. The integral over
the ber becomes Iber X det/ DA P DA eiSA f A ei ker P k Tr DA D. where. and SA SA S Tr
DA . SA SA k M Tr DA FA M TrA . Note that X is not well dened at n. The image of is all of G.
This is a slight renement of and . where X S G is parallel transport by A along the longitudinal
paths e used in dening . Thus the ber is the ane version of the linear space ker P . Thus A is
not continuous at n nor is any other such choice. A is the set of orbits Dana S. .
with a minor variation to account for the discontinuity. that is. Now. The determinant
analogous to the righthand side is that of P DA DA P acting on elements of ker P which
vanish at n. where is the etainvariant. Each determinant is constant along the base in A/G n .
Repeating these arguments. . f can be expressed as a function f of X. which refer to a
nitedimensional analog. where base is the measure on G induced by the metric on A/G n . S
A k S Tr X dX . . in terms of which Tr DA S S k Tr DA Tr DA S Tr TrX X. assuming f constant
along the bers. Add a small quadratic term. will evaluate Iber . Performing the integration
over the bers in the path integral for f . this would be standard. take det P DA DA P ei . The
result would be a determinant times the exponential of the etainvariant as in and . The metric
on S . as the analogue for det/ T T .v Dv Dv det/ T T . base constant DX. Then det DA P DA
Iber X det P DA DA P ei eiSA ei k TrXX . . Now assume f is independent of and consider ei
Tr DA D . To interpret this in terms of G requires four observations . But for the discontinuity
at n. S Model this by a vector space V V V with a linear mapping T V V . g restricts to ker P to
dene a decomposition of its C complexication ker P . and integrate to nd ei T v . thus gives f
Z f AeiSA ei k TrXX det DA P DA det P DA DA P ei base .From ChernSimons to WZW where
ker P denotes elements of ker P with the prescribed discontinuity at n. . regularize and
account for negative eigenvalues using the etainvariant.
f CS Z i f Xe k S k TrXXi S Tr X dX DX. For expectations of evaluationatapoint maps. at least
formally. Dana S. and let WR denote the Wilson line corresponding to the longitudinal path
indicated in Fig. The basic idea is to reinterpret X G as an element of PathG with some
constraints on its endpoint values. but this is work in progress. an evaluation of the WZW
path integral appearing on the righthand side of the above equation. this work is nearly
complete. the WZW path integral is tractable. However. Fine Fig. then. The path whose
holonomy is Xe Xe Absorbing everything independent of X into Z . An explicit reduction from
ChernSimons to WZW Let R be a representation of G. As a description of the relation
between ChernSimons theory on S and the WZW for G. . Then. . Then WR Z TrR Xe Xe e k i
CS S k TrXXi Tr X dX DX TrR Xe Xe WZW . namely. the WZW . as claimed. as an evaluation
of the ChernSimons path integral it lacks an essential component.
At a formal level. . . Topological aspects of YangMills theory.From ChernSimons to WZW
path integral becomes a path integral for quantum mechanics on G. Some remarks on the
Gribov ambiguity. Phys. YangMills on a Riemann surface. King. B I. Atiyah and J. . Phys.
Phys. M. Singer. Witten. Math. D. hor srie. Fine. expectations of evaluationatapoint maps are
given by the FeynmanKac formula. Socit Mathmatique de France Astrisque. Phys. D. Frhlich
and C. . to come up with an explicit evaluation of even unknot requires going beyond formal
properties to a more detailed understanding of the heat kernel on G. M. Commun. D. Taming
the conformal zoo. Commun. Acknowledgements This material is based upon work
supported in part by the National Science Foundation under Grant DMS. Math. . . Math. . ee
e e e E. Fine. M. Singer. However. Jones. Fine. I. S. Bibliography . F. preprint J. Moore and
N. ChernSimons on S via path integrals. Families of Dirac operators with applications to
physics. Math. Commun. G. Formal calculations lead to some promising preliminary
observations such as complicated unknot unknot . YangMills on the twosphere. Quantum eld
theory and the Jones polynomial. Commun. Seiberg. Phys. Phys. Phys. . . Commun. The
ChernSimons theory and knot polyo nomials. Math. . D. Lett. Commun. Math.
Fine . Dana S.
It turns out in regularizing that the links need to be framed. which is GR with a cosmological
constant. one should note that a number of very talented workers in the eld have searched in
vain for a Hilbert space of states in either representation. the connections between general
relativity in the loop formulation and the CSW TQFT are very suggestive. Notice. I do not say
that it is a topological eld theory. If one attempts to use the loop variables to quantize
subsystems of the universe. gives a state for the Ashtekar variable version of quantized
general relativity. at least. Kansas State University. Formally. so it cannot be topological.
unless one is prepared to consider states of zero volume. we can think of this measure on
the space of connections on a manifold either as dening a theory in dimensions. USA email
cranemath. using instead recombination diagrams in a very special tensor category. the
CSW action term e ik TrAdA AAA . and one can trace them in .edu Abstract Motivated by the
similarity between CSW theory and the Chern Simons state for general relativity in the
Ashtekar variables. or as a state in a d theory. That is. Nevertheless. states for the Ashtekar
or loop variable form of GR are very scarce. The construction avoids path integrals.
Introduction What I wish to propose is that the quantum theory of gravity can be constructed
from a topological eld theory. We end up with a construction of propagating states for parts of
the universe and a Hilbert space corresponding to a certain approximation. Finally. we
explore what a universe would look like if it were in a state corresponding to a d TQFT.
Kansas . one ends up looking at functionals on the set of links in a manifold with boundary
with ends on the boundary.Topological Field Theory as the Key to Quantum Gravity Louis
Crane Mathematics Department. Manhattan.ksu. Furthermore. as mentioned previously in
this volume. Physicists will be quick to point out that there are local excitations in gravity.
I think it is fair to say that it leads to more beautiful mathematics than any other I know of.
Topological quantum eld theory in dimensions . which seems important. Whether that is a
good guide for physics. This idea is similar to the suggestion of Witten that the CSW theory is
the topological phase of quantum gravity . Another point. During the talk I gave at the
conference one questioner pointed out. Thus we do not have to worry about a continuum
limit. called modular tensor categories. as a state for GR. As I shall explain below. This
suggestion is closely related to the manyworlds interpretation of quantum mechanics the
states on surfaces are data in one particular world. but rather due to the collapse of the wave
packet in the presence of classical observers. and is leading into an extraordinarily rich
mathematical eld. Let me emphasize that it is only a hypothesis. because the coecients
attached to simplices are algebraically special. which takes the elegant form of the CSW
invariant of links. Motivated by these considerations. we have the possibility of switching from
the language of path integrals to a nite. of the Jones polynomial or one of its generalizations.
and that the concrete geometry we see is not the result of a uctuation of the state of the
universe as a whole. if we want to explore . I have been thinking for several years about an
attempt to unite the structures of TQFT with a suitable reinterpretation of quantum mechanics
for the entire universe. The coincidence of all this with the picture of d TQFT was the initial
motivation for this work. . so they need to be labelled. In the spirit of the drunkard who looks
for his keys near a street light. where physics is described by variables on the edges of a
triangulation. Physics on a lattice is nothing new. the CSW path integral can be thought of as
a symbolic shorthand for a d TQFT which can be rigorously dened by combining
combinatorial topology with some very new algebraic structures. only time will tell. quite
correctly. since a nite result is exact. My suggestion diers from his in assuming that the
universe is still in the topological phase. since other states exist. that it is not necessary to
make this conjecture. I propose to consider the hypothesis Conjecture The universe is in the
CSW state. or more concretely. Louis Crane any representation of SU . . is that although
path integrals in general do not exist. What is new here is that the choice of lattice is
unimportant. the machinery of CSW theory provides spaces of states for parts of the
universe. The state of the universe is the sum of all possible worlds. discrete picture. Thus.
which I interpret as the relative states an observer might see. I believe that the two subjects
resemble one another strongly enough that the program I am outlining becomes natural.
except in the presence of an external clock. because there is no time. What I am proposing
is that the initial conditions become part of the laws. which acts as a clock. A must see B see
C see what A sees C see. The quantum theories of dierent parts of the universe are not.
Although they do not assume the universe is in the CSW state. Thus. One can phrase this
proposal as the suggestion that the wave function of the universe is the CSW functional.
while the task of a physical theory is to explain measurements made in this universe in this
state. these maps must be consistent. we allow labelled punctures on the boundaries
between parts of the universe. so that the gravitational eld is treated as only part of a
universe. What replaces a quantum eld theory is a family of quantum theories corresponding
to parts of the universe.Topological Field Theory and Quantum Gravity Quantum mechanics
of the universe. have suggested that the initial conditions of the universe are determined by
the laws. they couple the gravitational eld to a particular matter eld. which we think of as A
observing B. we obtain a Hilbert space which is associated to the boundary between them.
such as Penrose. It describes all possible universes. i. of course. and include embedded
graphs in the parts of the universe we study. When we divide the universe into two parts. in
the Ashtekar/loop variables the states of quantum gravity are invariants of embedded graphs.
Since. it amounts to abandonment of observation at a distance.e. like quantum mechanics
generally. and morphisms are d cobordisms. Naturally. Let us formalize this as follows
Denition . Objects are places where observations take place.e. There can be no probability
interpretation for the whole universe. Philosophically. An observer is an oriented manifold
with boundary . this is similar to the idea of a wave function of the universe. The structure we
arrive at has a natural categorical avor. Furthermore. independent. presupposes an external
classical observer. the state of the universe cannot change. i. Quantum eld theory.
boundaries. what is needed is a theory which describes a universe in a particular state. In
fact. This is another departure from quantum eld theory. in manifolds with boundary. There is
also a good deal here philosophically in common with the recent paper of Rovelli and Smolin
. Since the universe as a whole is in a xed state. physics Categorical I want to propose that a
quantum eld theory is the wrong structure to describe the entire universe. this requires that
the universe be in a very special state. the quantum theory of the gravitational eld as a whole
is not a physical theory. If A watches B watch C watch the rest of the universe. In a situation
where one observer watches another there must be maps from the space of states which
one observer sees to the other. This is similar to what some researchers.
a family of classical observers with consistent mutual . Louis Crane containing an embedded
labelled framed graph which intersects the boundary in isolated labelled points. Thus. and
the invariant of any link cut by the surface can be expressed as the inner product of two
vectors corresponding to the two half links.e. If M is a closed oriented manifold the category
of observation in M is the relative i. Denition . but that because the universe is in a very
special state. To every inspection of B by A there corresponds a dual inspection of A by B.
reverse the orientation on A such that the labellings of the components of the graph which
reach the boundary of match the labellings in A B. etc. If we examine the mathematical
structure necessary to produce a state for the universe in this sense. John Baez has pointed
out that these nitedimensional spaces really do possess natural inner products..e. A state for
quantum gravity in M is a functor from the category of observation in M to the category of
vector spaces. we can speak of observers. in that it factorizes when we cut the manifold
along a surface with punctures. it can contain a classical world. it produces a state also in the
sense we have dened above. Denition . The category of observation is the category whose
objects are skins of observation and whose morphisms are inspections. of B by A is an
observer whose boundary is identied with A B i. This denition requires that the set of
labellings possess an involution corresponding to reversal of orientation on a surface. A skin
of observation is a closed oriented surface with labelled punctures. If A and B are skins of
observation an inspection. Denition . the CSW state produces a state in this new sense as
well. given by reversing orientation and dualizing on . In fact.e. Another way to look at this
proposal is as follows the CSW state for the Ashtekar variables is a very special state.
Denition . Thus. i. We shall discuss this situation below as a key to a physical interpretation
of our theory. Nowhere in any of this do we assume that these boundaries are connected. we
nd that it is identical to a d TQFT. The labelling sets for the edges and vertices of the graph
are nite. embedded version of the above. so that a nitedimensional Hilbert space is attached
to each such surface. Denition . in M . and need to be chosen for all of what ensues. Of
course. . the most important situation to study may be one in which the universe is crammed
full of a gas of classical observers. I interpret this as saying that the state of the universe is
unchanging.
and no idea how to reintroduce time in the presence of observers. The objects in the richer
cobordism category are surfaces with labelled punctures. and linear maps to cobordisms
containing links or knotted graphs with labelled edges. There are several ways to construct
TQFTs in various dimensions. The TQFTs which have appeared lately are richer than this.
there is a great deal of coincidence in the pictures of CSW TQFT and the loop variables.
once we learn to interpret vectors in the Hilbert spaces as clocks. and the morphisms are
cobordisms containing labelled links with ends on the punctures. . then of tensor categories .
so a closed manifold gets a numerical invariant. A more abstract way to phrase this is that a
TQFT is a functor from the cobordism category to the category of vector spaces. we need the
combination factors to satisfy some equations. The labels correspond to representations. we
have a net of nitedimensional Hilbert spaces. The simplest denition of a d TQFT is that it is a
machine which assigns a vector space to a surface. and a linear map to a cobordism
between two surfaces. links with ends in manifolds with boundary can be interpreted as
changing. So far. There are natural things to try for both of these problems in the
mathematical context of TQFT. as we go through dierent classes of theories in dierent low
dimensions we rst rediscover most of the interesting classes of associative algebras. multiply
the factors together. We assign labels to edges in the triangulation. Since it is easy to extend
the loop variables to allow traces in arbitrary representations characters. Before going into a
program for solving these problems and thereby opening the possibility of computing the
results of real experiments let us make a survey of the mathematical toolkit we inherit from
TQFT. Let us here discuss the construction of a TQFT from a triangulation.Topological Field
Theory and Quantum Gravity observations. The equations they need to satisfy are very
algebraic in nature. The states on pieces of the universe i. We can describe this as a functor
from a richer cobordism category to V ect. In order to obtain a topologically invariant theory.
Ideas from TQFT As we have indicated. and some sort of factors combining the edge labels
to dierent dimensional simplices in the triangulation. The empty surface receives a
dimensional vector space. rather than one big one. the notion of a state of quantum gravity
we dened above is equivalent to the notion of a d TQFT which is currently prevalent in the
literature. Composition of cobordisms corresponds to composition of the linear maps. That
constituted much of the initial motivation for this program. They assign vector spaces to
surfaces with labelled punctures. and sum over labellings.e.
As has been described elsewhere . Much of the rest of classical abstract algebra makes an
appearance here too. The most interesting of these is the ElliotBiedenharn identity. . . a d
TQFT can be constructed from a modular tensor category. then the TuraevViro formula
becomes the PonzanoRegge formula for the evaluation of a spin network. The combinations
we attach to tetrahedra simplices are the quantum j symbols. Move for d TQFT A simple
example in two dimensions may explain why the equations for a topological theory have a
fundamental algebraic avor. This identity is the consistency relation for the associativity
isomorphism of the MTC. . For a d TQFT dened from a triangulation. but the weaker TQFT of
Turaev and Viro . not the CSW theory. The interesting examples of MTCs can be realized as
quotients of the categories of representations of quantum groups. Ponzano and Regge . The
full CSW theory can also be reproduced from a modular tensor category by a slightly subtler
construction which uses a Heegaard splitting which can be easily produced from a
triangulation . If we use the representations of an ordinary Lie group instead of a quantum
group. but with irreducible objects from a tensor category. Now. The suggestion that this sort
of summation could be related to the quantum theory of gravity is older than one might think.
The formula which Regge and Ponzano found for the evaluation of a graph is the analog for
a Lie group of the state sum of Turaev and Viro for a quantum group. We refer to this sort of
formula as a state summation. we need invariance under the move in Fig. not from a basis
for an algebra. The quantum j symbols satisfy some identities. which imply that the
summation formula for a manifold is independent under a change of the triangulation. were
able to interpret the evaluation as a sort of discrete path integral for d Euclidean quantum
gravity. This is another example of the marriage between algebra and topology which
underlies TQFT. if we think of the coecients which we use to combine the labels around a
triangle as the structure coecients of an algebra. this is exactly the associative law. If we try
to use the modular tensor categories to construct a TQFT on a triangulation. which come
from the associativity isomorphisms of the category. we obtain. d d d d d d Louis Crane d d d
d d d E Fig. Here we label edges.
These isomorphisms then satisfy higherorder consistency or coherence relations. out of a
modular tensor category. The spirit of the relationship is like the relationship of a tensor
category to an algebra. TQFTs in adjacent dimensions seem to be related algebraically. is
the ladder of dimensions . The topological invariance of this formula follows from some
elementary properties of representation theory. This construction begins with a triangulation
of a manifold. The geometric interpretation consisted in thinking of the Casimirs of the
representations as lengths of edges. is not yet completed. Representation theory implied that
the summation was dominated by at geometries . if the program in succeeds. The
ClebschGordan relations imply that only nitely many terms in . labelling the edges of all the
internal lines with arbitrary spins. The program of also suggests that the solutions of these
two topological problems are closely related. the evaluation of a tetrahedron for a spin
network is a j symbol for the group SU dimq j labellings internal edges tetrahedra j. as a
discretized path integral for d quantum GR. Ponzano and Regge then proceeded to interpret
.Topological Field Theory and Quantum Gravity Thus. The identities of an algebra
correspond to isomorphisms in a tensor category. The program of construction in TQFT.. .
and which I believe are crucial to the physical problem as well. Topological problem
Construct a d TQFT related via the dimensional ladder to CSW theory. Another idea from
TQFT. They are Topological problem Find a triangulation version of CSW theory not merely
its absolute square as in . but is progressing rapidly. In . and picks a Heegaard splitting for
the . which I hope will yield the tools for the quantization of general relativity. Another way to
think of the program in this chapter is as an attempt to nd the appropriate algebraic structure
for extending the spin network approach to quantum gravity from d to d. Tensor categories
look just like algebras with the operation symbols in circles. The solution of these two
problems will be in reach. both being constructed from state sums involving the same new
algebraic tools. and then cut the interior up into tetrahedra. There are two outstanding
problems. which are mathematically natural. we have placed our trivalent labelled graph on
the boundary of a manifold with boundary. In this context we should also note that a d TQFT
has been constructed in . which seems to have relevance for this physical program. which
relate to topology up dimension. are nonzero.
which I believe should give us a discretized version of a path integral for quantum gravity.
The idea is that if we choose a triangulation for the manifold and a Heegaard splitting for the
boundary of each simplex. not yet understood summation. Thus the theory in can be thought
of as producing a d theory by lling the space with a network of d subspaces containing
observers. of classical observers. and the existence of d cobordisms connecting pieces of
the . in which the universe is full of a gas. since we are supposed to be obtaining our relative
states from CSW theory on a boundary. Now. if we pass to a ner triangulation. we can then
combine the vector spaces which we assign to the surfaces which cut the boundaries of
those simplices lying on the boundary into a larger Hilbert space. which will prove to be
relevant to quantizing general relativity. There is a natural proposal to make a physical
interpretation within it in a dimensional setting. we should obtain a discrete version of F F
theory. This combination seems very suggestive. It is this new. If we use the new summation
in dimensions. In the scheme I am suggesting. we will obtain a larger space. This connection
between TQFTs in dimensions and dimensions is the sort of thing I believe we will need to
solve the problem of reintroducing time in a universe in a TQFT state. The linear map comes
from the fact that we are using a d TQFT. These surfaces should be thought of as skins of
observation for a family of observers who are crowding the spacetime. What I expect is that
the program of will provide richer examples of such a connection. the algebraic structure we
need to construct seems to be an expanded quantum version of the Lorentz group. In a
dimensional boundary component. Also. which comes from the representation theory of the
new structure which we call an F algebra in . although one could also try to use the state sum
in together with TuraevViro theory. What we conjecture in is that a new algebraic structure
will give us a new state summation. we can recover a Hilbert space in a certain sort of
classical limit. Hilbert space is deadlong live Hilbert space or the observer gas approximation
Let us assume that we have found a suitable state summation formula to solve our two
topological problems so that what we suggest will be predicated on the success of the
program in . with a linear map to the smaller one. then we obtain a family of surfaces which
can be thought of as lling up the manifold. similar to the one we discussed above. which will
give us CSW in dimensions. which is a quotient of their tensor product. Louis Crane
boundary of each simplex of the triangulation. It is a quotient because the surfaces overlap.
Now let us think of a manifold with boundary. while F F theory is a Lagrangian for the
topological sector for the quantum theory of GR .
The vectors are vectors in any of the spaces. they interpreted the summation we wrote down
above as a discretized path integral for d quantum gravity. we can extend the linear map it
assigns to a d cobordism to a map on the inverse limit spaces. What I am proposing is that
the d state sum which I hope to construct from the representation theory of the F algebra can
be interpreted as a discretized version of the path integral for general relativity. The
geometric interpretation consisted in thinking of the Casimirs of representations as lengths.
The eect of this suggestion is to reverse the relationship between the nitedimensional spaces
of states which each observer can actually see and the global Hilbert space. Since the d
state sum is assumed to be topologically invariant. This would mean that the theory was a
quantum theory whose classical limit was general relativity. although that is not an argument
for it. Should this program work One objection. The analogy with the work of Ponzano and
Regge is the rst suggestion that this program might work. It was somewhat easier to get
Einsteins equations in d . This produces a directed graph of vector spaces and linear maps
associated to a manifold. there is a construction called the inverse limit which can combine
them into a single vector space.Topological Field Theory and Quantum Gravity surfaces of
the two sets of Heegaard splittings. is that the F F model may only be a sector of quantum .
In . I propose that it is these inverse limit spaces which play the role of the physical Hilbert
spaces of the theory. in the limit of an innitely dense gas. This relational approach bears
some resemblance to the physical ideas of Leibniz. The d state sum can be extended to act
on the vectors in these vector spaces by extending a triangulation for the boundary manifold
to one for the manifold. The key question is whether the state sum is dominated in the limit of
large spins on edges by assignments of spins whose geometric interpretation would give a
discrete approximation to an Einstein manifold. Whenever we have a directed set of vector
spaces and consistent linear maps. The thought here is that physical Hilbert space is the
union of the nitedimensional spaces which a gas of observers can see. which was raised in
the discussion. and a linear map when one triangulation is a renement of another. The
problem of time The test of this proposal is whether it can reproduce Einsteins equations in
some classical limit. We end up with a large vector space for each triangulation of the
boundary manifold. with images under the maps identied. since the solutions are just at
metrics. We are regarding the states which can be observed at one skin of observation as
primary.
.e. Louis Crane gravity in some weak sense. It can be replied that since the theory we are
writing has the right symmetries. They also say that the sources for such a theory. Matter
and symmetry It is natural to ask whether this picture could be expanded to include matter
elds. in which Gambini and Pullin point out that the CSW state is also a state for GR coupled
to electromagnetism. thought of as places where links have ends. If this does work.e. then
we can investigate whether we recover Einsteins equation or not. It would be tting somehow
to reconstruct truly fundamental theories of physics from fundamental mathematical
expressions of symmetries. in the absence of the questioner. The F algebras arise as a
result of pushing this process further. If the state sum suggested in can be dened. x states on
surfaces in the dimensional boundaries of a d cobordism. In general. are necessarily
fermionic. I do not think that either the objection or the reply is decisive in the abstract. who
was thinking of general relativity. i. I am restating this objection somewhat. looking at
representations of Dynkin diagrams as quivers. One could interpret these states as initial
conditions for an experiment. one could do the state sum on a manifold with corners. The
idea that theories in physics should be reconstructed from symmetries actually originated
with Einstein. For example. as relatives of Lie groups in a quantum context. the action of the
renormalization group should x the Lagrangian. i. and investigate the results by studying the
propagation via the d state sum. These constructions can also be done from extremely
simple starting points. It may be noted that Peldan has observed that the CSW state also
satises the YangMills equation. the constructions which lead up to the F algebras in can be
thought of as expressions of quantum symmetry. If we really get a picture of gravity from the
state sum associated to SU . These results seem to strengthen further the program set forth
here. As I was writing this I became aware of . recategorifying the categories into categories.
the modular tensor categories can be described as deformed ClebschGordan coecients. The
obvious thing to try is to pass from SU to a larger group. One could conjecture that this would
give a spacetime picture for the evolution of relative states within the framework of the CSW
state of the universe. it would not be a great leap to try a larger gauge group.
P. G. L. . . . Smolin. and J. World Scientic. . Quantum eld theory and the Jones polynomial.
and quantum gravity. K. . Phys. ed. . Moore and N. . Commun. Thesis. in Proceedings of the
XXth International Conference on Dierential Geometric Methods in Theoretical Physics. . . G.
Crane and I. . personal communication. link invariants and quantum geometry. Commun. in
Quantum Theory and Beyond. . Phys. eds S. L. . Bloch. On Wittens manifold invariants.
Ponzano and T. . Regge. in Spectroscopic and Group Theoretical Methods in Physics. B .
Math. Peldan. Math. . ed. D . R. Phys. F. unpublished. . Nucl. Classical and quantum
conformal eld theory. Crane. Smolin. . Crane. Rocha. . Pullin. Penrose. . The author is
supported by NSF grant DMS. Catto and A. T. personal communication. Cambridge
University Press. . d physics and d topology. Frenkel. spin geometry. RR. B . Quantum
symmetry. J. Rev. . L. What is an observable in classical and quantum gravity Classical amp
Quantum Gravity . Oxford University . R. Louis Kauman has provided many crucial insights.
C. Nucl. Jones polynomials for inu tersecting knots as physical states for quantum gravity.
Bibliography . Math. an approach to combinatorial space time. Loop representation for
quantum general relativity. . Conformal eld theory. Semiclassical limits of Racah coecients.
Seiberg. Singapore. . Lett.Topological Field Theory and Quantum Gravity
Acknowledgements The author wishes to thank Lee Smolin and Carlo Rovelli for many years
of conversation on this extremely elusive topic. The mathematical ideas here are largely the
result of collaborations with David Yetter and Igor Frenkel. B. Phys. Ashtekars variables for
arbitrary gauge group. L. Crane. Commun. Quantum Models of Space Time Based on
Recoupling Theory. Gambini. Phys. . . . Bastin. L. Moncrief. Phys. Moussouris. . C. Walker.
Rovelli. New York. B . V. . . Witten. Angular momentum. Phys. Hopf categories and their
representations. Br gmann. E. . Wiley. Rovelli and L.
to appear in Proceedings of AMS conference. L. Viro. . A categorical construction of d
TQFTs. N. . A representation theoretic approach to dimensional topological quantum eld
theory. . to appear in Proceedings of AMS conference. Rovelli and L. Braided monoidal
categories. . Kapranov and V. Topology . The physical hamiltonian in nonperturbative
quantum gravity. State sum invariants of manifolds and quantum j symbols. V. Mt Holyoke.
Voevodsky. preprint. Lett. Smolin. Quantum EinsteinMaxwell elds a unied viewpoint from the
loop representation. preprint. . Soo. preprint. L. Rovelli and L. Louis Crane to appear. Pullin.
June . C. BRST cohomology and invariants of D gravity in Ashtekar variables. Rev. . Crane
and D. Crane and I. . Turaev and O. . Smolin. M. vector spaces and Zamolodchikovs
tetrahedra equation. Gambini and J. L. . preprint. . . Ohio. October . Massachusetts. C.
Chang and C. . Phys. Frenkel. Yetter. R. Knot theory and quantum gravity. Dayton.
We review progress towards making this duality rigorous in three examples d Yang Mills
theory which. In fact. in a classical gauge theory. Baez Department of Mathematics. while an
excitation of the gauge eld alone can be thought of as a looped ux tube. a Wilson loop is
simply the trace of the holonomy of A around a loop in space. California . Two quarks in a
meson. A heuristic pathintegral approach indicates a duality between backgroundfree string
theories and generally covariant gauge theories. If A denotes the vector potential.edu
Abstract The loop representation of quantum gravity has many formal resemblances to a
backgroundfree string theory. but here we provide an exact stringtheoretic interpretation of
the theory on R S for nite N . typically written in terms . Unfortunately this theory had a
number of undesirable features. can be thought of as connected by a stringlike ux tube in
which the gauge eld is concentrated. d quantum gravity. with the loop transform relating the
two. However. while in dimensions there may be relationships to the theory of tangles. has
symmetry under all areapreserving transformations. and Gauge Fields John C. Knots. it
predicted massless spin particles. University of California.Strings. which models the strong
force by an SU YangMills eld. or connection. Loops. while not generally covariant. USA email
jbaezucrmath. and d quantum gravity. its origins lie in attempts to treat the string theory of
hadrons as an approximation to QCD. but it can be formalized using the notion of Wilson
loops. SU N YangMills theory in dimensions has been given a stringtheoretic interpretation in
the largeN limit. The stringtheoretic interpretation of quantum gravity in dimensions gives rise
to conjectures about integrals on the moduli space of at connections. String theory rst arose
as a model of hadron interactions. in particular. in which the strings represent ux tubes of the
gauge eld. Introduction The notion of a deep relationship between string theories and gauge
theories is far from new. This is essentially a modern reincarnation of Faradays notion of eld
lines. string models continued to be popular as an approximation of the conning phase of
QCD.ucr. for example. It was soon supplanted by quantum chromodynamics QCD.
Riverside.
and. In the late s. applying this operator to the vacuum state one obtains a state in which the
YangMills analog of the electric eld ows around the loop . Baez Tr P e A . . More recently. On
the other hand. However. recent work by Witten and Periwal suggests that ChernSimons
theory in dimensions is also equivalent to a string theory. After Ashtekar formulated quantum
gravity as a sort of gauge theory using the new variables. Polyakov. In particular. have been
able to reformulate Yang Mills theory in dimensional spacetime as a string theory by writing
an asymptotic series for the vacuum expectation values of Wilson loops as a sum over maps
from surfaces the string worldsheet to spacetime. and others . thanks to its common origin.
Typically one begins with a xed background structure and writes down a string eld theory in
terms of this. If instead A denotes the vector potential in a quantized gauge theory. it is a
strong candidate for a theory unifying gravity with the other forces. Rovelli and Smolin were
able to use the loop representation to study quantum gravity nonperturbatively in a manifestly
backgroundfree formalism. it uses the device of Wilson loops to construct a space of states
consisting of multiloop invariants. while classical general relativity is an elegant geometrical
theory relying on no background structure for its formulation. String theory eventually
became popular as a theory of everything because the massless spin particles it predicted
could be interpreted as the gravitons one obtains by quantizing the spacetime metric
perturbatively about a xed background metric. of a pathordered exponential John C. D. The
clarity of a manifestly backgroundfree approach to string theory would be highly desirable.
While supercially quite dierent from modern string theory. this approach to quantum gravity
has many points of similarity. only afterwards can one investigate its background
independence . attempts to formulate YangMills theory in terms of Wilson loops eventually
led to a fulledged loop representation of gauge theories. attempted to derive equations of
string dynamics as an approximation to the YangMills equation. and others. it has proved
dicult to describe string theory along these lines. For example. Nambu. . Gross and others .
Makeenko and Migdal. . at least heuristically. . This development raises the hope that other
gauge theories might also be isomorphic to string theories. using Wilson loops. Since string
theory appears to avoid the renormalization problems in perturbative quantum gravity. which
assign an amplitude to any collection of loops in space. thanks to the work of Gambini and
Trias . the Wilson loop may be reinterpreted as an operator on the Hilbert space of states.
but we nd it to be little extra eort. we begin by describing a general framework relating gauge
theories and string theories. We review recent work by Ashtekar. have already given
dimensional SU N YangMills theory a stringtheoretic interpretation in the largeN limit. . the
group of all areapreserving transformations. we consider quantum gravity in dimensions. In
Section we describe how the loop representation of a generally covariant gauge theory is
related to a backgroundfree closed string eld theory. but it has an innitedimensional
subgroup of the dieomorphism group as symmetries. In Section we consider quantum gravity
in dimensions. thinking of them simply as maps from a surface into spacetime. to ask
whether the loop representation of quantum gravity might be a string theory in disguiseor
vice versa. These examples have nitedimensional reduced conguration spaces. therefore. .
In fact. Taylor. at least in canonical quantization. We also note how a stringtheoretic
interpretation of the theory leads naturally to the study of tangles. We take a very naive
approach to strings. to give the theory a precise stringtheoretic interpretation for nite N . We
base our treatment on the path integral formalism. Loops. which is easier to make rigorous.
Rather. and rather enlightening. We also briey discuss ChernSimons theory in dimensions. .
and disregarding any conformal structure or elds propagating on the surface. We review the
loop representation of this theory and raise some questions about integrals over the moduli
space of at connections on a Riemann surface whose resolution would be desirable for
developing a stringtheoretic picture of the theory. In Section we consider YangMills theory in
dimensions as an example. and Gauge Fields The resemblance of these states to
wavefunctions of a string eld theory is striking. Our treatment of examples is also meant to
serve as a review of YangMills theory in dimensions and quantum gravity in and dimensions.
and the author . and then consider a variety of examples. Rather than the path integral
approach we use canonical quantization. Knots. so there are no analytical diculties with
measures on innitedimensional spaces. In Section . this theory is not generally covariant. on
dieomorphisminvariant generalized measures on A/G and their relation to multiloop
invariants and knot theory. and others . Our treatment is mostly a review of their work. The
present paper does not attempt a denitive answer to this question. however. Here the
classical conguration space is innite dimensional and issues of analysis become more
important. Isham.Strings. . It is natural. and in order to simplify the presentation we make a
number of overoptimistic assumptions concerning measures on innitedimensional spaces
such as the space A/G of connections modulo gauge transformations. . Gross. Lewandowski.
e. X. to be specied. Here nS denotes the disjoint union of n copies of S . We assume that
FunM and the measure D are invariant both under dieomorphisms of space and
reparametrizations of the strings. Baez String eld/gauge eld duality In this section we sketch
a relationship between string eld theories and gauge theories. by measure in this section we
will always mean a positive regular Borel measure. etc. and we write Maps to denote the set
of maps satisfying some regularity conditions continuity. We discuss these analytical issues
more carefully in Section . and we wish FunM and D to be preserved by these actions. That
is. in the sense that some of our assumptions are not likely to hold precisely as stated in the
most interesting examples. We assume for convenience that this really is an inner product.
which is just the L inner product . We take the classical conguration space of the string
theory to be the space M of multiloops in X M with Mn MapsnS . we do not expect Hkin to be
the true space of physical states. on spacetime. both the identity component of the
dieomorphism group of X and the orientationpreserving dieomorphisms of nS act on M. kin n
Mn M D. with X a manifold we call space. but in fact one should work with a more general
version of this concept. it is nondegenerate. etc. We emphasize that while our discussion
here is rigorous. We do not assume a canonical identication of M with RX. In particular.
Taking a . John C. The physical state space thus depends on the dynamics of the theory. it is
schematic. Let D denote a measure on M and let FunM denote some space of
squareintegrable functions on M. i. or any other background structure metric. In the space of
physical states. Consider a theory of strings propagating on a spacetime M that is
dieomorphic to R X. From this we derive a Hamiltonian description. Introduce on FunM the
kinematical inner product. We begin with a nonperturbative Lagrangian description of
backgroundfree closed string eld theories. which turns out to be mathematically isomorphic
to the loop representation of a generally covariant gauge theory. smoothness. Following
ideas from canonical quantum gravity. Dene the kinematical state space Hkin to be the
Hilbert space completion of FunM in the norm associated to this inner product. any two
states diering by a dieomorphism of spacetime are identied.
Strings, Loops, Knots, and Gauge Fields
Lagrangian approach, dynamics may be described in terms of path integrals as follows. Fix a
time t gt . Let P denote the set of histories, that is maps f , t X, where is a compact oriented
manifold with boundary, such that f , t X f . Given , M, we say that f P is a history from to if f , t
X and the boundary of is a disjoint union of circles nS mS , with f nS , f mS . We x a
measure, or path integral, on P, . Following tradition, we write this as eiSf Df , with Sf
denoting the action of f , but eiSf and Df only appear in the combination eiSf Df . Since we
are interested in generally covariant theories, this path integral is assumed to have some
invariance properties, which we note below as they are needed. Using the standard recipe in
topological quantum eld theory, we dene the physical inner product on Hkin by ,
phys
M M P,
eiSf Df DD
assuming optimistically that this integral is well dened. We do not actually assume this is an
inner product in the standard sense, for while we assume , for all Hkin , we do not assume
positive deniteness. The general covariance of the theory should imply that this inner product
is independent of the choice of time t gt , so we assume this as well. Dene the space of
normzero states I Hkin by I , ,
phys phys
for all Hkin
and dene the physical state space Hphys to be the Hilbert space completion of Hkin /I in the
norm associated to the physical inner product. In general I is nonempty, because if g Di X is
a dieomorphism in the connected component of the identity, we can nd a path of
dieomorphisms gt Di M with g g and gt equal to the identity, and dening g Di, t X by gt, x t, gt
x, we have ,
phys
M M P,
eiSf Df DD gg eiSf Df DD
M M P,
MM
John C. Baez
g gg eiSf Dgf DgD
P,
M M P,
g eiSf Df DD
g,
phys
for any , . Here we are assuming
g eiSf Dgf eiSf Df,
which is one of the expected invariance properties of the path integral. It follows that I
includes the space J, the closure of the span of all vectors of the form g. We can therefore
dene the spatially dieomorphisminvariant state space Hdif f by Hdif f Hkin /J and obtain
Hphys as a Hilbert space completion of Hdif f /K, where K is the image of I in Hdif f . To
summarize, we obtain the physical state space from the kinematical state space by taking
two quotients Hdif f Hdif f /K Hphys . As usual in canonical quantum gravity and topological
quantum eld theory, there is no Hamiltonian instead, all the information about dynamics is
contained in the physical inner product. The reason, of course, is that the path integral, which
in traditional quantum eld theory described time evolution, now describes the physical inner
product. The quotient map Hdif f Hphys , or equivalently its kernel K, plays the role of a
Hamiltonian constraint. The quotient map Hkin Hdif f , or equivalently its kernel J, plays the
role of the dieomorphism constraint, which is independent of the dynamics. Strictly speaking,
we should call K the dynamical constraint, as we shall see that it expresses constraints on
the initial data other than those usually called the Hamiltonian constraint, such as the
Mandelstam constraints arising in gauge theory. It is common in canonical quantum gravity
to proceed in a slightly dierent manner than we have done here, using subspaces at certain
points where we use quotient spaces , . For example, Hdif f may be dened as the subspace
of Hkin consisting of states invariant under the action of Di X, and Hphys then dened as the
kernel of certain operators, the Hamiltonian constraints. The method of working solely with
quotient spaces has, however, been studied by Ashtekar . The choice between these dierent
approaches will in the end be dictated by the desire for convenience and/or rigor. As a
heuristic guiding principle, however, it is worth noting that the subspace and quotient space
approaches are essentially equivalent if we assume that the subspace I is closed in the norm
topology on Hkin . Relative to the kinematical inner Hkin Hkin /J Hdif f
Strings, Loops, Knots, and Gauge Fields d H d d d d a d b c d d
EE
Fig. . Twostring interaction in dimensional space product, we can identify Hdif f with the
orthogonal complement J , and similarly identify Hphys with I . From this point of view we
have Hphys Hdif f Hkin . Moreover, Hdif f if and only if is invariant under the action of Di X on
Hkin . To see this, rst note that if g for all g Di X, then for all Hkin we have , g g , so J .
Conversely, if J , , g so , , g , and since g acts unitarily on Hkin the CauchySchwarz inequality
implies g . The approach using subspaces is the one with the clearest connection to knot
theory. An element Hkin is a function on the space of multiloops. If is invariant under the
action of Di X, we call a multiloop invariant. In particular, denes an ambient isotopy invariant
of links in X when we restrict it to links which are nothing but multiloops that happen to be
embeddings. We see therefore that in this situation the physical states dene link invariants.
As a suggestive example, take X S , and take as the Hamiltonian constraint an operator H on
Hdif f that has the property described in Fig. . Here a, b, c C are arbitrary. This Hamiltonian
constraint represents the simplest sort of dieomorphisminvariant twostring interaction in
dimensional space. Dening the physical space Hphys to be the kernel of H, it follows that any
Hphys gives a link invariant that is just a multiple of the HOMFLY invariant . For appropriate
values of the parameters a, b, c, we expect this sort of Hamiltonian constraint to occur in a
generally covariant gauge theory on dimensional spacetime known as BF theory, with gauge
group SU N . A similar construction working with unoriented framed multiloops gives rise to
the Kauman polynomial, which is associated with BF theory with gauge group SON . We see
here in its barest form the path from stringtheoretic considerations to link invariants and then
to gauge theory. In what follows, we start from the other end, and consider a generally
covariant gauge theory on M . Thus we x a Lie group G and a principal Gbundle P M . Fixing
an identication M R X, the classical conguration space is the space A of connections on P X .
The physical
John C. Baez
Hilbert space of the quantum theory, it should be emphasized, is supposed to be
independent of this identication M R X. Given a loop S X and a connection A A, let T , A be
the corresponding Wilson loop, that is the trace of the holonomy of A around in a xed
nitedimensional representation of G T , A Tr P e
A
.
The group G of gauge transformations acts on A. Fix a Ginvariant measure DA on A and let
FunA/G denote a space of gaugeinvariant functions on A containing the algebra of functions
generated by Wilson loops. We may alternatively think of FunA/G as a space of functions on
A/G and DA as a measure on A/G. Assume that DA is invariant under the action of Di M on
A/G, and dene the kinematical state space Hkin to be the Hilbert space completion of
FunA/G in the norm associated to the kinematical inner product ,
kin
A/G
AA DA.
The relation of this kinematical state space and that described above for a string eld theory is
given by the loop transform. Given any multiloop , . . . , n Mn , dene the loop transform of
FunA/G by , . . . , n
A/G
AT , A T n , A DA.
Take FunM to be the space of functions in the range of the loop transform. Let us assume,
purely for simplicity of exposition, that the loop transform is onetoone. Then we may identify
Hkin with FunM just as in the string eld theory case. The process of passing from the
kinematical state space to the dieomorphisminvariant state space and then the physical state
space has already been treated for a number of generally covariant gauge theories, most
notably quantum gravity , , . In order to emphasize the resemblance to the string eld case,
we will use a path integral approach. Fix a time t gt . Given A, A A, let PA, A denote the
space of connections on P ,tX which restrict to A on X and to A on t X. We assume the
existence of a measure on PA, A which we write as eiSa Da, using a to denote a connection
on P ,tX . Again, this generalized measure has some invariance properties corresponding to
the general covariance of the gauge theory. Dene the physical inner product on Hkin by ,
phys
A A PA,A
AA eiSa DaDADA
Knots. quantum gravity in dimensions. . In the following sections we will consider some
examples YangMills theory in dimensions. The Lagrangian for a gauge theory. Letting Hdif f
Hkin /J. Loops. A/G M C A. . with . or as a wave function on the classical conguration space
M for strings. At this point it is natural to ask what the dierence is. In summary. The loop
transform depends on the nonlinear duality between connections and loops. apart from
words. . between the loop representation of a generally covariant gauge theory and the sort
of purely topological string eld theory we have been considering. Hdif f . with the loop
transform relating the gauge theory picture to the string theory picture. with A being the
amplitude of a specied connection A. and quantum gravity in dimensions. for all . . . As
before. In no case is a string eld action Sf known that corresponds to the gauge theory in
question However. and letting K be the image of I in Hdif f . is quite a dierent object than that
of a string eld theory. This is. .Strings. n T A. n S X to be present. is that a state in Hkin can
be regarded either as a wave function on the classical conguration space A for gauge elds. n
being the amplitude of a specied ntuple of strings . a promising alternative to dealing with
measures on the . . on the other hand. we can use the general covariance of the theory to
show that I contains the closed span J of all vectors of the form g. Note that nothing we have
done allows the direct construction of a string eld Lagrangian from a gauge eld Lagrangian or
vice versa. the physical state space Hphys of the gauge theory is Hkin /I. in fact. . From the
Hamiltonian viewpoint that is. we see that the Hilbert spaces for generally covariant string
theories and generally covariant gauge theories have a similar form. The key point. and
Gauge Fields again assuming that this integral is well dened and that . we again see that the
physical state space is obtained by applying rst the dieomorphism constraint Hkin Hkin /J
Hdif f and then the Hamiltonian constraint Hdif f Hdif f /K Hphys . in terms of the spaces Hkin
. n which is why we speak of string eld/gauge eld duality rather than an isomorphism
between string elds and gauge elds. . again. in d YangMills theory a working substitute for
the string eld path integral is known a discrete sum over certain equivalence classes of maps
f M . T A. Letting I Hkin denote the space of normzero states. and Hphys the dierence is not
so great. . This inner product should be independent of the choice of time t gt . . . .
that is ambient isotopy classes of embeddings f . so it can be regarded as a function on the
space of connections on M modulo gauge transformations. as well as Driver . then. and the
YangMills action is given by SA TrF F M where F is the curvature of A and Tr is the trace in a
xed faithful unitary representation of G and hence its Lie algebra g. The Riemannian case of
d YangMills theory has been extensively investigated. An equation for the vacuum
expectation values of Wilson loops for the theory on Euclidean R was found by Migdal . We x
a connected compact Lie group G and a principal Gbundle P M . Baez space P of string
histories. the YangMills equation dA F . rather than a Hamiltonian constraint. t X. The group
DiM acts on this space. YangMills theory in dimensions We begin with an example in which
most of the details have been worked out. Gross. where dA is the exterior covariant
derivative. M It follows that the action SA is invariant under the subgroup of dieomorphisms
preserving the volume form . and these expectation values were explicitly calculated by
Kazakov . but in which dieomorphism invariance is replaced by invariance under this
subgroup. this theory has an honest Hamiltonian. Still. Strictly speaking. King and Sengupta .
John C. Extremizing this action we obtain the classical equations of motion. So upon
quantization one expects toand doesobtain something analogous to a topological quantum
eld theory. In particular. These calculations were made rigorous using stochastic dierential
equation techniques by L. but the action is not dieomorphism invariant. it illustrates some
interesting aspects of gauge eld/string eld duality. if M is dimensional one may write F f
where is the volume form on M and f is a section of P Ad g. However. In dimensional
quantum gravity. YangMills theory is not a generally covariant theory since it relies for its
formulation on a xed Riemannian or Lorentzian metric on the spacetime manifold M . such an
approach might involve a sum over tangles. Classically the gauge elds in question are
connections A on P . The classical YangMills equations on Riemann surfaces were
extensively investigated . and then SA Trf . many of the results of the previous section do not
apply. The action SA is gauge invariant.
v TrE v. Here. The classical phase space of the theory is the cotangent bundle T A. Witten
has shown that the quantization of d YangMills theory gives a mathematical structure very
close to that of a topological quantum eld theory. the Gauss law dA E . using the
nondegenerate inner product E. Note that a tangent vector v TA A may be identied with a
gvalued form on S . where t R. so we x a trivialization and identify a connection on P with a
gvalued form on R S . and let E be its covariant time derivative pulled back to S . E satisfying
this constraint are the initial data for some solution of the YangMills equation. Similar results
were also obtained by Blau and Thompson . owing to the gauge invariance of the equation.
and in physics A is called the vector potential and E the electric eld. This approach is an
alternative to the path integral approach of the previous section. x S . ZM for each compact
oriented manifold M with boundary having total area M . This will simultaneously serve as a
brief introduction to the idea of quantizing gauge theories after symplectic reduction.
however. E of gvalued forms on S . Given a connection on P solving the YangMills equation
we obtain a point A. Loops. as done by Rajeev . the loop group G C S . S We thus regard the
phase space T A as the space of pairs A. Moreover. and in some cases is easier to make
rigorous. Knots. and Witten . G acts as gauge transformations on A. this solution is not
unique. E. We may also think of a gvalued form E on S as a cotangent vector. Fine . In
particular. it will be a bit simpler to discuss YangMills theory on R S with the Lorentzian
metric dt dx . and the quantum theory on Riemann surfaces has been studied by Rusakov .
The pair A. E of the phase space T A as follows let A be the pullback of the connection to S .
and a vector ZM. This may be identied with the space of gvalued forms on S . The classical
conguration space of the theory is the space A of connections on P S . The YangMills
equation implies a constraint on A. Any Gbundle P R S is trivial. which will also be important
in d quantum gravity. with a Hilbert space ZS S associated to each compact manifold S S .
and this action lifts . and any pair A. and Gauge Fields by Atiyah and Bott .Strings. However.
E is called the initial data for the solution.
the quotient of the constraint subspace by G is again a symplectic manifold. However.
writing E edx. Two points in the phase space T A are to be regarded as physically equivalent
if they dier by a gauge transformation. so A/G G/AdG. which acts on the almost reduced
conguration space G by conjugation. gives a function on phase space that generates a
Hamiltonian ow coinciding with a parameter group of gauge transformations. The quotient of
the constraint subspace by G . In this sort of situation it is natural to try to construct a smaller.
the Gauss law says that e C S . integrated against any f C S . one can see this quite
concretely. by basic results on moduli spaces of at connections. . is identied with T G/AdG.
and hence determined by its value at the basepoint of S . Baez naturally to an action on T A.
g is a at section. G/AdG. which is just G/AdG. gEg . It follows that any point A. Consequently.
Alternatively. all parameter subgroups of G are generated by the constraint in this fashion.
We may rst take the quotient of A by only those gauge transformations that equal the identity
at a given point of S G g C S . the constraint dA E. E dA E T A is not. the almost reduced
phase space. The phase space T A is a symplectic manifold. Next. In the case at hand there
is a very concrete description of the reduced phase space. the reduced phase space. Trf dA
S E. more physically relevant reduced phase space using the process of symplectic
reduction. is thus identied with T G. given by g A. the reduced phase space. John C. g as
follows. but the constraint subspace A. E gAg gdg . with an explicit dieomorphism taking
each equivalence class A to its holonomy around the circle A P e S A . The remaining gauge
transformations form the group G/G G. the reduced conguration space A/G may be naturally
identied with Hom S . E in the constraint subspace is determined by A A together with e g. In
fact. It follows that the quotient of the constraint subspace by all of G. G g . First. This almost
reduced conguration space A/G is dieomorphic to G itself.
e. Since the YangMills Hamiltonian on the almost reduced phase space T G is that of a
classical free particle on G. In fact. However.Strings. the Hamiltonian of the quantum theory
is diagonalized by this basis. we take the quantum Hamiltonian to be that for a quantum free
particle on G H / where is the nonnegative Laplacian on G. carry out only part of the process
of symplectic reduction. This measure has the advantage of mathematical elegance. What
should be the Hilbert space for the quantized theory on R S As described in the previous
section. i. p p The advantage of the almost reduced conguration space and phase space is
that they are manifolds. note that this measure gives a Hilbert space isomorphism L A/G L
Ginv where the right side denotes the subspace of L G consisting of functions constant on
each conjugacy class of G. and obtain a Hamiltonian function on the almost reduced phase
space. Let denote the character of an equivalence class of irreducible representations of G.
and Gauge Fields E. Loops. This is just the Hamiltonian for a free particle on G. we prefer
simply to show ex post facto that it gives an interesting quantum theory consistent with other
approaches to d YangMills theory. Since the theory is not generally covariant. Observables
of the classical theory can be identied either with functions on the reduced phase space. E
Hg. however. For example. We will choose the pushforward of normalized Haar measure on
G by the quotient map G G/AdG. Now let us consider quantizing dimensional YangMills
theory. Knots. E but by the process of symplectic reduction one obtains a corresponding
Hamiltonian on the reduced phase space. or functions on the almost reduced phase space T
G that are constant on the orbits of the lift of the adjoint action of G. for any p Tg G it is given
by HA. Then by the Schur orthogonality relations. To begin with. the set forms an
orthonormal basis of L Ginv . it is natural to take L of the reduced conguration space A/G. the
kinematical Hilbert space is the physical Hilbert space. When we decompose the . the
dieomorphism and Hamiltonian constraints do not enter. While one could also argue for it on
physical grounds. One can. to dene L A/G requires choosing a measure on A/G G/AdG. the
YangMills Hamiltonian is initially a function on T A with the obvious inner product on Tg G.
. For convenience. . In the string eld picture of Section . nk T n T nk . Note that the vacuum
the eigenvector of H with lowest eigenvalue is the function . . we obtain a theory in which all
physical states have . . we may quantize them by interpreting them as multiplication
operators acting on L Ginv T n g Trg n g. because all connections on S are at. To take a step
in this direction. so H c . . we dene to be the vacuum state. . In the case at hand these
Wilson loop observables depend only on the homotopy class of the loop. let us consider the
Wilson loop observables. John C. We may think of T T . We will see this again in d quantum
gravity. n when i is homotopic to i for all i. . n . In a sense this diagonalization of the
Hamiltonian completes the solution of YangMills theory on R S . we have T n . We can also
form elements of L Ginv by applying products of these operators to the vacuum. The states n
. respectively. . g Trg n . . and we prefer to think of it as such. . . . However. Since the
classical Wilson loop observables are functions on conguration space. and to make the
connection to string theory. Letting n S S be an arbitrary loop of winding number n. . nk . but
it lifts to a function on the almost reduced conguration space G. . The idea of describing the
YangMills Hamiltonian in terms of these string states goes back at least to Jevickis work on
lattice gauge the . . A is dened by T . with winding numbers n . as a function on the reduced
conguration space A/G. . . where c is the quadratic Casimir of G in the representation . the
function lies in the sum of copies of the representation . Recall that given a based loop S S .
nk may also be regarded as states of a string theory in which k strings are present. Baez
regular representation of G into irreducibles. Let n . extracting the physics from this solution
requires computing expectation values of physically interesting observables. . A TrPe A . the
classical Wilson loop T . . which is for the trivial representation. . .
They are never linearly independent. . . . . However. In what follows we will use many ideas
from these authors. More recently. . Taylor. All these authors. because the Wilson loops
satisfy relations. . . work primarily with the largeN limit of the theory. . For the rest of this
section we set G SU N and take traces in the fundamental representation. . In fact.Strings.
Note that this implies that n. . . . although they do in some interesting cases. Knots. so the
total number of strings present in a given state is ambiguous. . for example. there is no
analog of the Fock space number operator on L Ginv . . which we shall briey review. the T n
are analogous to creation operators. and others . Then MN . instead of working in the largeN
limit. . One always has T Tr. In other words. Douglas. Polychronakos. let Mk . The
resemblance of the string states n . on d YangMills theory as a string theory. nk to states in a
bosonic Fock space should be clear. the Mandelstam identity is T n T m T nm T nm . N for all
loops i . but give a stringtheoretic formula for the SU N YangMills Hamiltonian that is exact
for arbitrary N . for since the work of t Hooft it has been clear that SU N YangMills theory
simplies as N . however. . . the string states do not necessarily span L Ginv . and all the
linear dependencies between string states are consequences of the following Mandelstam
identities . . we do not generally have a representation of the canonical commutation
relations. Minahan. The importance of this theory for dimensional physics was stressed by .
and Gauge Fields ory. Loops. These formulae are based on the classical theory of Young
diagrams. There are also explicit formulae expressing the string states in terms of the basis
of characters. k k sgnT gj gjn T gjk gjknk Sk where has the cycle structure j jn jk jknk . and
MN . . and for any particular group G the Wilson loops will satisfy identities called
Mandelstam identities. m n m n m . In particular. In this case the string states do span L Ginv
. for G SU and taking traces in the fundamental representation. . . k in S . for all loops . Given
loops . For example. . . string states appear prominently in the work of Gross.
. . when is the conjugacy class of permutations with cycle lengths n . Young diagrams
describe a duality between the representation theory of SU N and of the symmetric groups
Sn which can be viewed as a mathematical reection of string eld/gauge eld duality. nk with all
the ni positive but k possibly equal to zero span L SU N inv . . Young diagrams with n boxes
are in correspon . . and following the strands around counterclockwise to x . To take
advantage of this correspondence. . where n n nk . since the labelling was arbitrary. . In a
Young diagram one draws a conjugacy class with cycles of length n nk gt as a diagram with
k rows of boxes. we form an irreducible representation of SU N . Suppose we have
lengthminimizing strings in S with winding numbers n . . We will write for the character of the
representation . which we also call . . in which the string state n . one copy for each box. .
The rationale for this description of string states as conjugacy classes of permutations is in
fact quite simple. if has gt N rows . . Thus we will restrict our attention for now to states of this
kind. and then antisymmetrizing over all copies in each column and symmetrizing over all
copies in each row. there is a map from Young diagrams to equivalence classes of
irreducible representations of SU N . which we call righthanded. nk corresponds to the
conjugacy class of all permutations with cycles of length n . . nk . Note that consists of
permutations in Sn . nk . the string state really only denes a conjugacy class of elements of
Sn . First. . However. . . . for a total of n n nk labels. note using the Mandelstam identities that
the string states n . we obtain a permutation of the labels. and plays a primary role in Gross
and Taylors work on d YangMills theory . John C. As we shall see. Baez Nomura . Let Y
denote the set of Young diagrams. There is a correspondence between righthanded string
states and conjugacy classes of permutations in symmetric groups. If has N rows it is
equivalent to a representation coming from a Young diagram having lt N rows. . by taking a
tensor product of n copies of the fundamental representation. . We will assume without loss
of generality that n nk gt . Labelling each strand of string each time it crosses the point x . nk
. See Fig. . Given Y . we simply dene n . . On the other hand. . . nk . . On the one hand. and
hence an element of Sn . . and if has gt N rows it is zero dimensional. . having ni boxes in the
ith row. This gives a correspondence between Young diagrams with lt N rows and irreducible
representations of SU N . .
recall that eqn . Y . Knots. . First. Given Y . and Gauge Fields Fig. . Loops. Sn The
YangMills Hamiltonian has a fairly simple description in terms of the basis of characters .
where n is the number of boxes in . This allows us to write the Frobenius relations expressing
the string states in terms of characters and vice versa. . Then the Frobenius relations are and
conversely n . We dene the function on Sn to be zero for n n. we write for the corresponding
repre sentation of Sn .Strings. . It follows that H N H N H H . Young diagram dence with
irreducible representations of Sn . There is an explicit formula for the value of the SU N
Casimir in the representation c N n N n nn dim where denotes the conjugacy class of
permutations in Sn with one cycle of length and the rest of length . expresses the
Hamiltonian in terms of the Casimir.
As noted above. To express the operators H and H in stringtheoretic terms. from these
denitions and express the basis X in terms of the string states X n . Sn It follows that the
string states form a basis for H. for suciently large N . X Y n where is the number of elements
in regarded as a conjugacy class in Sn .. Y X . but orthogonal . Baez H n and H nn . Then a
calculation using the Schur orthogonality equations twice shows that these string states are
not only linearly independent. For each Y . This quotient map sends the string state in H to
the corresponding . dim . with the only densely dened quotient map j H L SU N inv being
given by X . Let H be a Hilbert space having an orthonormal basis X Y indexed by all Young
diagrams. in fact orthogonal. One can also derive the Frobenius relation . simply dening a
space in which all the string states are independent. where John C. We will proceed slightly
dierently. . there is no natural way to do this in L SU N inv since the string states are not
linearly independent. The YangMills Hilbert space L SU N inv is essentially a quotient space
of the string eld Hilbert space H. The advantage of taking the N limit is that any nite set of
distinct string states becomes linearly independent. it is convenient to dene string annihilation
and creation operators satisfying the canonical commutation relations. dene a vector in H by
the Frobenius relation .
. . Knots. Here in fact we will show that the Hamiltonian on the YangMills Hilbert space L SU
N inv lifts to a Hamiltonian on H with a simple interpretation in terms of string interactions. . In
the framework of the previous section. dim This clearly has the property that Hj jH. k We
could eliminate the factor of j and obtain the usual canonical commutation relations by a
simple rescaling. so that dimensional SU N YangMills theory is isomorphic to a quotient of a
string theory by the Mandelstam identities. j k aj . On H we can introduce creation operators
a j gt by j a n .Strings. It follows that this quotient map is precisely that which identies any two
string states that are related by the Mandelstam identities. we dene a Hamiltonian H on the
string eld Hilbert space H by H where H X nX . . . . H X N H N H H nn X . so the YangMills
dynamics is the quotient of the string eld dynamics.. . n . nk j. We then claim that H a aj j jgt
and H j. but it is more convenient not to. and Gauge Fields string state L SU N inv .kgt a aj ak
a a ajk . a jjk . . a . . Loops. the Mandelstam identities would appear as part of the dynamical
constraint K of the string theory.. j k jk These follow from calculations which we briey sketch
here. It was noted some time ago by Gliozzi and Virasoro that Mandelstam identities on
string states are strong evidence for a gauge eld interpretation of a string eld theory. ak a .
The Frobenius relations and the denition of H give H n . j and dene the annihilation operator
aj to be the adjoint of a . . Following eqns . These j satisfy the following commutation
relations aj . nk .
. An analysis of the eect of composing with all possible shows that either one cycle of will be
broken into two cycles. however. Here.. Similarly. By eqn . H can be regarded as a string
tension term. we have switched to treating states as functions on the space of multiloops.
since if we represent a string state n . . H dim Y Since there are nn / permutations Sn lying in
the conjugacy class . Figure can also be regarded as two frames of a movie of a string
worldsheet in dimensional spacetime. We may interpret the Hamiltonian in terms of strings
as follows. . Similar movies have been used by Carter and Saito to describe string
worldsheets in dimensional spacetime . it is an eigenvector of H with eigenvalue equal to n
nk . giving the expression above for H in terms of annihilation and creation operators. the
Frobenius relations and the denition j of H give nn . Twostring interaction in dimensional
space and this implies the formula for H as a sum of harmonic oscillator Hamiltonians a aj . .
. we may rewrite this as H Since Y. or two will be joined to form one.. this kind of interaction
is a dimensional version of that which gave the HOMFLY invariant of links in dimensional
space in the previous section. H corresponds to a twostring interaction as in Fig. Baez H d d
E E Fig. Also. John C. dim dim the Frobenius relations give H . the other has been
introduced only to keep track of the identities of the strings. nk by lengthminimizing loops. . .
In this gure only the x coordinate is to be taken seriously. As the gure indicates. we have a
true Hamiltonian rather than a Hamiltonian constraint. proportional to the sum of the lengths
of the loops. . By eqn .
where the gauge group is SO. Knots. applicable to the dimensional case. so that it
converges to H as N . since the gauge group is then noncompact . When considering the /N
expansion of the theory. it is convenient to divide the Hamiltonian H by N . Loops. since this
seems not to have been done yet. so there are no local degrees of freedom. and Gauge
Fields . The theory is therefore only interesting on topologically nontrivial spacetimes. This
does not describe any new states in the YangMills theory per se. .Strings. . nk with ni lt . .
Rovelli. Since then. Samuel and Smolin . We will follow Ashtekar. Quantum gravity in
dimensions Now let us turn to a more sophisticated model. It follows from the work of
Minahan and Polychronakos that there is a Hamiltonian H on H H naturally described in
terms of string interactions and a densely dened quotient map j H H L SU N inv such that Hj
jH. but for simplicity of presentation we will sidestep them by working with the Riemannian
case. From the work of Gross and Taylor it is also clear that in addition to the space H
spanned by righthanded string states one should also consider a space with a basis of
lefthanded string states n . and treat dimensional quantum gravity using the new variables
and the loop transform. It is easiest to describe the various action principles for gravity using
the abstract index notation popular in general relativity. many approaches to the subject have
been developed. and indicate some possible relations to string theory. Interest in the
mathematics of this theory increased when Witten reformulated it as a ChernSimons theory.
Then the H term is proportional to /N . string worldsheets of genus g give terms proportional
to /N g . . The total Hilbert space of the string theory is then the tensor product H H of
righthanded and lefthanded state spaces. in the next section . Husain. dimensional quantum
gravity. Indeed. It is important to note that there are some technical problems with the loop
transform in Lorentzian quantum gravity. but it is more natural from the stringtheoretic point
of view. If we draw the string worldsheet corresponding to this movie we obtain a surface
with a branch point. They also show that the H term corresponds to string worldsheets with
handles. Einsteins equations say simply that the spacetime metric is at. This is in accord with
the observation by t Hooft that in an expansion of the free energy logarithm of the partition
function as a power series in /N . associated to those histories with branch points. not all
equivalent . In this section we describe the Witten action. in the path integral approach of
Gross and Taylor this kind of term appears in the partition function as part of a sum over
string histories. In dimensions. These are presently being addressed by Ashtekar and Loll in
the dimensional case. but we will instead translate them into language that may be more
familiar to mathematicians.
Note that we can pull back the metric on E by e T M T to obtain a Riemannian metric on M .
which we call SO connections on X. and that the metric on M is at. M The classical equations
of motion obtained by extremizing this action are F and dA e . The basic elds of the theory
are then taken to be a metricpreserving connection A on T . and we only present the nal
results. or SO connection. Using the isomorphism T T given by the metric. which applies to
dimensions. X . Baez we describe the Palatini action. the curvature F of A may be identied
with a T valued form. is only nondegenerate when e is an isomorphism. a number of
subtleties. we obtain the Witten action SA. however. Note that a tangent vector v TA A is a T
valued form on X. These dene a volume form on T . Fix a real vector bundle T over M that is
isomorphic tobut not canonically identied withthe tangent bundle T M . where X is a compact
oriented manifold. the equations of motion say simply that this connection is the LeviCivita
connection of the metric on M . The relationship between these action principles has been
discussed rather thoroughly by Peldan . together with a T valued form e on M . that is a
nowherevanishing section of T . It follows that the wedge product e F may be dened as a T
valued form. and x a Riemannian metric and an orientation on T . John C. In this case. There
are. which. Now suppose that M R X. We can thus regard a T valued form E on X as a
cotangent vector by means of the pairing Ev E v. The formalism involving the elds A and e
can thus be regarded as a device for extending the usual Einstein equations in dimensions to
the case of degenerate metrics on M . The classical conguration and phase spaces and their
reduction by gauge transformations are reminiscent of those for d YangMills theory. When e
is an isomorphism we can also use it to pull back the connection to a metricpreserving
connection on T M . e e F . The classical phase space is then the cotangent bundle T A. The
classical conguration space can be taken as the space A of metricpreserving connections on
T X. which applies to any dimension. Pairing this with to obtain an ordinary form and then
integrating over spacetime. however. and the Ashtekar action. Let the spacetime M be an
orientable manifold.
. and Gauge Fields Thus given any solution A. vi SO. The two isomorphism classes of SO
bundles on M are distinguished by their second StiefelWhitney number w Z . Loops. xg . The
bundle T X is trivial so w T X We can calculate w for any point ui . where E plays a role
analogous to the electric eld in YangMills theory. which in this context is dA E . e of the
classical equations of motion. ug . . so we wish to describe this component. and the
vanishing of the curvature B of the connection A on T X. taking the physical state space of
the quantized theory to be L of the reduced conguration space. and a point in F is an
equivalence class ui .Strings. v . yi x y x y xg yg x yg . g An element of Hom X. A quite
concrete description of A /G was given by Goldman . There is a natural identication F Hom X.
Knots. The latter constraint subsumes both the dieomorphism and Hamiltonian constraints of
the theory. that is a point in the phase space T A. SO may thus be identied with a collection u
. which is analogous to the magnetic eld B . choose lifts ui . For all the elements ui .
SO/AdSO. vi . given by associating to any at bundle the holonomies around homotopy
classes of loops. if we can nd a tractable description of A /G. corresponding to the two
isomorphism classes of SO bundles on M . vi . Then the group X has a presentation with g
generators x . satisfying Rui . y . . vi F by the following method. The classical equations of
motion imply constraints on A. yg satisfying the relation Rxi . we can pull back A and e to the
surface X and get an SO connection and a T valued form on X. where A is the space of at
SO connections on X. . This is usually written A. . As in d YangMills theory. The moduli
space F of at SObundles has two connected components. vg of elements of SO. E. . E T A
which dene a reduced phase space. it will be attractive to quantize after imposing
constraints. . . vi of such collections. The reduced phase space for the theory turns out to be
T A /G. The component corresponding to the bundle T X is precisely the space A /G.
Suppose that X has genus g. and G is the group of gauge transformations . These are the
Gauss law.
but to dene this one must choose a measure on A /G. u It follows that we may think of points
of A /G as equivalence classes of gtuples ui . k T T k . . The rst step towards a stringtheoretic
interpretation of d quantum gravity would be a formula for an inner product of string states . It
would be satisfying if there were a stringtheoretic interpretation of the inner product in L A /G
along the lines of Section . vi . and shown to be dimension d g for g . . so given . vi . we can
form elements of L A /G by applying products of these operators to the function . the Liouville
measure. . . which is just a particular kind of integral over A /G. . In fact A /G is not a
manifold. k X. dene . The dfold wedge product denes a measure on A /G. . but a singular
variety. Alternatively. In fact. by introducing a coupling . . or d for g the case g is trivial and
will be excluded below. Then w Ri . there is a symplectic structure on A /G coming from the
following form on A B. k . u where the equivalence relation is given by the adjoint action of
SO. Note that we may dene string states in this space as follows. John C. This has been
investigated by Narasimhan and Seshadri . As in the case of d Yang Mills theory. . Given any
loop in X. C with EndT Xvalued forms. . for any N . On the grounds of elegance and
dieomorphism invariance it is customary to use this measure to dene the physical state
space L A /G. . one could formulate d quantum gravity as a theory of SU connections and
then generalize to SU N . Note that this sort of integral makes sense when A /G is the moduli
space of at connections for a trivial SON bundle over X. k . it is natural to take L A /G to be
the physical state space. . As noted. . vi of elements of SO admitting lifts ui . . As noted by
Goldman . in which we identify the tangent vectors B. vi with Ri . . Baez to the universal
cover SO SU . . it appears that this sort of integral can be computed using d YangMills theory
on a Riemann surface. . the Wilson loop observable T is a multiplication operator on L A /G
that only depends on the homotopy class of . C X TrB C.
Strings. in terms of a sum over surfaces having that link as boundary . . Here one xes an
arbitrary Lie group G and a Gbundle P M over spacetime. with the Goldman symplectic
structure. and Gauge Fields constant in the action SA M TrF F . k as a sum over ambient
isotopy classes of surfaces f . and the eld of the theory is a connection A on P . In fact. . . t X
having the loops i . k . One thus expects that this sort of integral simplies in the N limit just as
it does in d YangMills theory. but . Witten has given such an interpretation using open strings
and the BatalinVilkovisky formalism . Nonetheless. as one would expect. . computing the
appropriate vacuum expectation value of Wilson loops. and then taking the limit . As noted by
Witten . . . rst demonstrated by Witten . This theory does not quite t our formalism because in
it the space A /G of at connections modulo gauge transformations plays the role of a phase
space. The action is given by SA k Tr A dA A A A . Before concluding this section. This
procedure has been discussed by Blau and Thompson and Witten . for example. . It would
be very worth while to reformulate ChernSimons theory as a string theory at the level of
elegance with which one can do so for d YangMills theory. . it is worth noting another
generally covariant gauge theory in dimensions. the Euclidean group in dimension. Taylor
and the author are currently trying to work out the details of this program. Moreover.
ChernSimons theory. This. together with a method of treating the nite N case by imposing
Mandelstam identities. in terms of integrals over moduli spaces of Riemann surfaces. d
quantum gravity as we have described it is essentially the same ChernSimons theory with
gauge group ISO. Loops. Knots. . i as boundaries. giving a formula for . an expression for the
vacuum expectation value of Wilson loops. there are a number of clues that ChernSimons
theory admits a reformulation as a generally covariant string eld theory. in the N limit. at least
for the case of a link where it is just the Kauman bracket invariant. for the gauge groups SU
N Periwal has expressed the partition function for ChernSimons theory on S . In the case N
there is also. There is a profound connection between ChernSimons theory and knot theory.
rather than a conguration space. with the SO connection and triad eld appearing as two parts
of an ISO connection. and then elaborated by many researchers see. would give a
stringtheoretic interpretation of d quantum gravity.
In dimensions the Palatini action can be rewritten in a somewhat similar form. As in the
previous section. Then the classical equations of motion derived from the Palatini action say
precisely that this connection is the LeviCivita connection of the metric. John C. the wedge
product e e F is a T valued form. we will sidestep certain problems with the loop transform by
working with Riemannian rather than Lorentzian gravity. or SOn connection. We shall then
discuss some recent work on making the loop representation rigorous in this case. which.
and x a Riemannian metric and orientation on T . The metric denes an isomorphism T T and
allows us to identify the curvature F of A with a section of the bundle T T M. is expressed in
terms of e rather than e . the Palatini action reduces to the Witten action. We may also
regard e as a section of T T M and dene e e in the obvious manner as a section of the
bundle T T M. however. however. e e e F .e. that e T M T be a bundle isomorphism. Namely.
and a T valued form e. Fix a bundle T over M that is isomorphic to T M . The basic elds of the
theory are then taken to be a metricpreserving connection A on T . M The Ashtekar action
depends upon the fact that in dimensions the . and indicate some mathematical issues that
need to be explored to arrive at a stringtheoretic interpretation of the theory. and that the
metric satises the vacuum Einstein equations i. In dimensions. Baez Quantum gravity in
dimensions We begin by describing the Palatini and Ashtekar actions for general relativity.
The natural pairing between these bundles gives rise to a function F e e on M . this has not
yet been done. Using the isomorphism e. The Palatini action for Riemannian gravity is then
SA. is Ricci at. and pairing it with to obtain an ordinary form we have SA. M We may use the
isomorphism e to transfer the metric and connection A to a metric and connection on the
tangent bundle. We require. its inverse is called a frame eld. we can push forward to a
volume form on M . e F e e . These dene a nowherevanishing section of n T . Let the
spacetime M be an orientable nmanifold.
that is. a section of so T X T X. e M e e F . Then F is just an sovalued form on M . We can
take the classical conguration space to be space A of righthanded connections on T X.
Similarly. as is automatically the case when M R X. or equivalently xing a trivialization of T .
A tangent vector v TA A is thus an so valued form. and its decomposition into selfdual and
antiselfdual parts corresponds to the decomposition so soso. Loops. The gravitational vector
potential A is simply the pullback of A to the surface X. This allows us to write F as a sum F F
of selfdual and antiselfdual parts F F . E consisting on X. The remarkable fact is that the
action SA. This allows us to regard general relativity as the theory of a selfdual connection A
and a T valued form ethe socalled new variableswith the Ashtekar action SA . suppose T is
trivial. and an sovalued form E denes a cotangent vector by the pairing Ev TrE v. Moreover.
Knots. and Gauge Fields metric and orientation on T dene a Hodge star operator T T with not
to be confused with the Hodge star operator on dierential forms. A solution A . X A point in
the classical phase space T A is thus a pair A. It is easy to see that F is the curvature of A . A
is an sovalued form. Obtaining E from e is . E T A as follows. In the of an sovalued form A
and an sovalued form E physics literature it is more common to use the natural isomorphism
T X T X T X given by the interior product to regard the gravitational electric eld E as an
sovalued vector density.Strings. which are forms having values in the two copies of so.
where X is a compact oriented manifold. e M e e F gives the same equations of motion as
the Palatini action. and may thus be written as a sum A A of selfdual and antiselfdual
connections. e of the classical equations of motion determines a point A. sovalued forms on
X. Now suppose that M R X.
The dieomorphism constraint generates the action of Di X on A/G. we may also regard this
as an sovalued form on X. so one forms the reduced phase space T A/G. or curvature of the
connection A. the map e T M T gives a map e T X T X. the dieomorphism constraint is given
by Tr iE B and the Hamiltonian constraint is given by Tr iE iE B . so in the quantum theory
one takes Hdif f to be the subspace of Di Xinvariant elements of FunM. so a more
sophisticated strategy. and dene the image of their common kernel in Hdif f to be the
physical state space Hphys . E T A. These are the Gauss law dA E . called a cotriad eld on
X. or attempt to represent the Hamiltonian constraint directly as operators . By restricting to T
X and then projecting down to T X. Baez a somewhat subtler aair. rst devised by Rovelli and
Smolin . Quantizing. Letting B denote the gravitational magnetic eld. one obtains the
kinematical Hilbert space Hkin L A/G. First. In d quantum gravity this is no longer the case.
Let us rst sketch this without mentioning the formidable technical problems. split the bundle T
as the direct sum of a dimensional bundle T and a line bundle. One may then either attempt
to represent the Hamiltonian constraint as operators on Hkin . One then applies the loop
transform and takes Hkin FunM to be a space of functions of multiloops in X. The latter two
are most easily expressed if we treat E as an sovalued vector density. similarly. is required.
Since there is a natural isomorphism of the bers of T with so. Applying the Hodge star
operator we obtain the sovalued form E. and the dieomorphism and Hamiltonian constraints.
In d YangMills theory and d quantum gravity one can impose enough constraints before
quantizing to obtain a nitedimensional reduced conguration space. The Gauss law constraint
generates gauge transformations. The classical equations of motion imply constraints on A.
iE iE B is an M R T X T Xvalued function. namely the space A /G of at connections modulo
gauge transformations. John C. Here the interior product iE B is dened using matrix
multiplication and is an M R T Xvalued form.
b in the algebra and C. Then A/I has an honest inner product and we let H denote the Hilbert
space completion of A/I in the corresponding norm. In particular. be degenerate a. More
recently. There are. Knots. observables are represented by selfadjoint elements of A. the full
space Hphys has not yet been determined. it is possible to go a certain distance without
becoming involved with these considerations. The latter approach is still under development
by Rovelli and Smolin . In fact. one does not really expect to have interesting
dieomorphisminvariant measures on the space A/G of connections modulo gauge
transformations in this case. It is then easy to check that I is a left .Strings. a a . the loop
transform can be rigorously dened without xing a measure or generalized measure on A/G if
one uses. Even at this formal level. In the Calgebraic approach to physics. however. b a b. a
a . b . for all a. In particular. a state on A denes an inner product that may. signicant
problems with turning all of this work into rigorous mathematics. and normalized. Rovelli and
Smolin obtained a large set of physical states corresponding to ambient isotopy classes of
links in X. It is this work that suggests a profound connection between knot theory and
quantum gravity. while states are elements of the dual A that are positive. so at this point we
shall return to where we left o in Section and discuss some of the diculties. not the Hilbert
space formalism of the previous section. a a a ab b a . Let I A denote the subspace of
normzero states. In their original work. but a Calgebraic formalism. ab a a b a b . physical
states have been constructed from familiar link invariants such as the Kauman bracket and
certain coecients of the Alexander polynomial to all of M. and Gauge Fields on Hdif f and
dene the kernel to be Hphys . Some recent developments along these lines have been
reviewed by Pullin . The number a then represents the expectation value of the observable a
in the state . Namely. This approach makes use of the connection between d quantum
gravity with cosmological constant and ChernSimons theory in dimensions. to indicate the
basic ideas without becoming immersed in technicalities. Loops. A Calgebra is an algebra A
over the complex numbers with a norm and an adjoint or operation satisfying a a. one
expects the existence of generalized measures sucient for integrating a limited class of
functions. . The relation to the more traditional Hilbert space approach to quantum physics is
given by the Gelfand NaimarkSegal GNS construction. At best. In Section we were quite
naive concerning many details of analysisdeliberately so. however.
the loop transform is onetoone. Some progress in solving these problems has recently been
made by Ashtekar. and let P X be a principal Gbundle over X. . the pointwise complex
conjugate T equals T . Baez ideal of A. and the author . . It turns out to be more natural to
dene the loop transform not on FunA/G but on its dual. observables in A give rise to
selfadjoint operators on H.and dimensional quantum gravity. Given FunA/G we dene its loop
transform to be the function on the space M of multiloops given by . . While in what follows
we will assume that G is compact and is unitary. A Calgebraic approach to the loop transform
and generalized measures on A/G was introduced by Ashtekar and Isham in the context of
SU gauge theory. The basic concept is that of the holonomy Calgebra. which we denote as
FunA/G in order to make clear the relation to the previous section. n T T n . John C. such as
G SU N and the fundamental representation. so that A acts by left multiplication on A/I. as
this involves no arbitrary choices. and that this action extends uniquely to a representation of
A as bounded linear operators on H. In particular. . so FunA/G FunM. the functions T are
bounded and continuous in the C topology on A/G. the holonomy Calgebra. . it is important to
emphasize that for Lorentzian quantum gravity G is not compact This presents important
problems in the loop representation of both . and G the group of smooth gauge
transformations. where is the orientationreversed loop. In favorable cases. space. and
subsequently developed by Ashtekar. Lewandowski. . on A/G taking traces in this
representation. Dene the holonomy algebra to be the algebra of functions on A/G generated
by the functions T T . We may thus complete the holonomy algebra in the sup norm topology
f sup f A AA/G and obtain a Calgebra of bounded continuous functions on A/G. . Fix a
nitedimensional representation of G and dene Wilson loop functions T T . Recall that in the
previous section the loop transform of functions on A/G was dened using a measure on A/G.
Lewandowski. Moreover. and Loll . Let FunM denote the range of the loop transform. This is
the real justication for the term string eld/gauge eld duality. . Let A denote the space of
smooth connections on P . If we assume that G is compact and is unitary. . Let X be a
manifold.
but there is as yet no manifestly welldened expression along these lines. using a slightly
dierent formalism. who also constructed many more examples of Di Xinvariant states on
FunA/G. g kin f g then generalizes the L inner product used in the previous section. the most
progress has been made in the realanalytic case. an inner product for relative states of
quantum gravity in the Kauman bracket state has been rigorously constructed by the author .
We conclude with some speculative remarks concerning d quantum gravity and tangles.
there is a unique inner product on FunM such that this map extends to a map from Hkin to
the Hilbert space completion of FunM. Di X to consist of the realanalytic dieomorphisms
connected to the identity. If is a state on FunA/G. thinking of the pairing f as the integral of f
FunA/G. and Gauge Fields We may take the generalized measures on A/G to be simply
elements FunA/G . Knots. in general For technical reasons. Note that the kinematical inner
product f . A path integral formula for the inner product has been investigated recently by
Rovelli . . we may construct the kinematical Hilbert space Hkin using the GNS construction.
The kinematical Hilbert space constructed from a Di Xinvariant state FunA/G thus becomes a
unitary representation of Di X. n T T n f in a manner generalizing that of the previous section.
. They have also given a general characterization of such dieomorphisminvariant states.
Loops. Here Ashtekar and Lewandowski have constructed a Di Xinvariant state on FunA/G
that is closely analogous to the Haar measure on a compact group . it is not immediately
obvious that any exist. In fact. and FunA/G to be the holonomy Calgebra generated by
realanalytic loops. That is. Note that a choice of generalized measure also allows us to dene
the loop transform as a linear map from FunA/G to FunM f . .Strings. On the other hand. we
take X to be real analytic. It is thus of considerable interest to nd a more concrete description
of Di Xinvariant states on the holonomy Calgebra FunA/G. There is thus some real hope that
the loop representation of generally covariant gauge theories can be made rigorous in cases
other than the toy models of the previous two sections. The example of d YangMills theory
would suggest an expression for the . Note also that Di X acts on FunA/G and dually on
FunA/G . but there are still many questions about the physics here. . The latter was also
given by the author . The correct inner product on the physical Hilbert space of d quantum
gravity has long been quite elusive. Moreover.
These equations can be understood in terms of category theory. Maurizio Martellini. and Lee
Smolin for useful discussions. . Scott Axelrod. I would like to thank the Center for
Gravitational Physics and Geometry as a whole for inviting me to speak on this subject.
Jacob Hirbawi. Also. Clearly it will be some time before we are able to appraise the
signicance of all this work. D . . Holger Nielsen. n as a sum over ambient isotopy classes of
surfaces f . New variables for classical and quantum gravity. A. string theory. Bibliography .
For example. i as boundaries. just as the Reidemeister moves relate any two pictures of the
same tangle in dimensions. since just as tangles form a braided tensor category. Rev. . such
surfaces are known as tangles. Jerzy Lewandowski. since braided tensor categories can be
constructed from certain conformal eld theories. t X having the loops i . Renate Loll. and the
depth of the relationship between string theory and the loop representation of quantum
gravity. . Ashtekar. It is thus quite signicant that Crane has initiated an approach to generally
covariant eld theory in dimensions using braided tensor categories. . Acknowledgements I
would like to thank Abhay Ashtekar. . Baez . tangles form a braided tensor category . rst
derived in the context of string theory . Phys. This approach also claries some of the
signicance of conformal eld theory for dimensional physics. Phys. the analog of the
YangBaxter equation is the Zamolodchikov equation. n . and the loop representation of d
quantum gravity are tantalizing but still rather obscure. Matthias Blau. For example.
CottaRamusino and Martellini have endeavored to construct tangle invariants from generally
covariant gauge theories. inner product of string states John C. In a related development.
These moves give a set of equations whose solutions would give tangle invariants. and have
been intensively investigated by Carter and Saito using the technique of movies. Paolo
CottaRamusino. much as tangle invariants may be constructed using ChernSimons theory.
In the case of embeddings. . Louis Crane. Wati Taylor deserves special thanks for explaining
his work on YangMills theory to me. there are a set of movie moves relating any two movies
of the same tangle in dimensions. The relationships between tangles. . . Lett. Jorge Pullin.
Rev. New Hamiltonian formulation of general relativity. Scott Carter. .
Strings, Loops, Knots, and Gauge Fields
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Lectures on Nonperturbative Canonical Quantum Gravity, Singapore, World Scientic, . A.
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Universit di Milano a and INFN. only the general BRST structure has been thoroughly
discussed. Introduction In a seminal work. at least in the nonAbelian . Italy email
martellinimilano. Sezione di Pavia. unfortunately. Via Celoria . The question. Functional
integrationbyparts techniques allow the recovery of an innitesimal version of the
Zamolodchikov tetrahedron equation. One of the questions that is possible to ask is whether
there exists an analog of Wittens ideas in higher dimensions.it Maurizio Martellini
Dipartimento di Fisica. Sezione di Milano. perturbative calculations. is not easy to
answer.infn. Via Celoria . The socalled BF theories allow the construction of quantum
operators whose trace can be considered as the higherdimensional generalization of Wilson
lines for knots in dimensions.BF Theories and knots Paolo CottaRamusino Dipartimento di
Fisica. and it is possible to establish a heuristic relation between BF theories and Alexander
invariants. On one hand. in the form considered by Carter and Saito. the theory of
higherdimensional knots and links is much less developed. Witten has shown that there is a
very deep connection between invariants of knots and links in a manifold and the quantum
ChernSimons eld theory. Firstorder perturbative calculations lead to higherdimensional
linking numbers. Universit di Trento a and INFN. Milano. .it Abstract We discuss the relations
between topological quantum eld theories in dimensions and the theory of knots embedded
spheres in a manifold. After all. Italy email cottamilano. for the time being. Milano. the
relevant invariants are. and topological quantum eld theories exist in dimensions . knots and
links are dened in any dimension as embeddings of kspheres into a k dimensional space. As
far as dimensional topological eld theories are concerned. more scarce than in the theory of
ordinary knots and links.infn.
Chern action exp . . Looking for possible topological actions for a dimensional manifold. is a
given represen . These observables can be seen as a generalization of ordinary Wilson lines
in dimensions. Finally. . and a principal bundle P M. The group G. No analogous help is
available for dimensional topological eld theories. consistent with the proposal made by
Carter. In this case. do not appear to have been carried out. Saito. BF action exp M M Tr F F
. in dimensions it is also possible to apply functional integrationbyparts techniques. Even the
quantum observables have not been clearly dened in the nonAbelian case. these techniques
produce a solution of an innitesimal Zamolodchikov tetrahedron equation. It is also possible
to nd heuristic arguments that may relate the expectation values of our observables in the BF
theory with the Alexander invariants of knots. with Lie algebra LieG. and Lawrence. will be. a
compact Lie group G. Specically we propose a set of quantum observables associated to
surfaces embedded in space. As far as perturbative calculations are concerned. they are
much more complicated than in the lowerdimensional case. BF theories and their
geometrical signicance We start by considering a dimensional Riemannian manifold M .
Moreover. the group SU n or SOn. G over M . the Gibbs measures of the . In this paper we
make some preliminary considerations and proposals concerning knots and topological eld
theories of the BF type. ChernSimons theory in dimensions beneted greatly from the results
of dimensional conformal eld theory. then its curvature will be denoted by FA or simply by F .
Nevertheless it is possible to recover a relation between rstorder calculations and
higherdimensional linking numbers. Tr B F . in most cases. which parallels a similar relation
for ordinary links in a dimensional space. we may consider . The space of all connections will
be denoted by the symbol A and the group of gauge transformations by the symbol G. the
framework in which these observables are dened ts with the picture of knots as movies of
ordinary links or link diagrams considered by Carter and Saito see references below. In the
formulae above and are coupling constants. If A denotes a connection on such a bundle.
Paolo CottaRamusino and Maurizio Martellini case. Moreover.
It is an Rn valued form. the space of forms with values in P Ad LieG. for a given section M
Og M. It is then convenient to discuss the geometrical aspects of a more general BF theory.
We now dene the form B M. In the corresponding BF action. where T denotes transposition.
The eld B in eqn .e. given by the identity map on the tangent space of M . adP . UadP . the
curvature F is the curvature of a connection in the orthonormal frame bundle Og M . UadP .
written in the vielbein formalism. though. with values in M. adP . on M. We denote by the
form on A given by the identity map on the tangent space of A. We now mention some
examples of Belds that have a nice geometrical meaning . In order to obtain an ordinary form
on M from an element of M. . B can be dened as . UadP is obtained by combining the
product in UG with the exterior product for ordinary forms. Let LM M be the frame bundle and
let be the soldering form.BF Theories and knots tation of the group G and of its Lie algebra.
UadP this space naturally contains M. Notice that the wedge product for forms in M. We shall
see. More explicitly. For any given metric g we consider the reduced SOnbundle of
orthonormal frames Og M . where B is assumed to be a form on M with values in the
associated bundle P Ad UG where UG is the universal enveloping algebra of LieG. a
tensorial form on P with values in LieG. From this we can obtain the form on A B . that the
quantization procedure may force B to take values in the universal enveloping algebra of
LieG. equivalently. adP . The manifold A is an ane space with underlying vector space M.
one needs to take the trace with respect to a given representation of LieG. This action is
known to be classically equivalent to the Einstein action. A vielbein in physics is dened as the
Rn valued form on M given by . adOg M as B T Og M . The space of forms with values in P
Ad UG will be denoted by the symbol M. is assumed classically to be a form of the same
nature as F . In other words B is a section of the bundle T M P Ad UG. The curvature F is a
form on M with values in the associated bundle P Ad LieG or. i. .
Paolo CottaRamusino and Maurizio Martellini Ba. the wedge product also includes the
exterior product of forms on A. F k M Tr B FA . and . M Q F . M M M Q F . F k M Tr B B . F .
As Atiyah and Singer pointed out. b TA A. On the principal Gbundle P A M A . F k Tr FA FA .
we can consider the tautological connection dened by the following horizontal distribution
Horp.A HorA TA A p . Q F . Integrating the ChernWeil forms corresponding to such
polynomials over M we obtain for l . where HorA denotes the horizontal space at p P with
respect to the p connection A. . F . one can consider together with . where a. where B is
dened as in eqns .A FA p A . bX. X. .A aXbY aY bX bXaY bY aX. . Its curvature form is given
by Fp.. seen as a . Let us now use Ql to denote an irreducible adinvariant polynomial on
LieG with l entries. Y Tp P. Y p.form on P A. F . and kl are normalization factors. We now
want to show that the BF action considered in the last example is connected with the ABJ
AdlerBellJackiw anomaly in dimensions. In eqn . The connection form of the tautological
connection is given simply by A.
. We always assume that one of the above choices is made. When we consider such a
connection on . via the socalled descent equation. In fact one should really consider only the
group of dieomorphisms of M which strongly x one point of M . . then the lefthand side of eqn
. i. instead of LM A it is possible to consider OMAmetric . they give closed forms on A.
dieomorphism/gauge invariance this refers to the action of AutP . the full group of
automorphisms of the principal bundle P including the automorphisms which do not induce
the identity map on the base manifold. G dieomorphism invariance this can be considered
whenever we have an action of DiM over P and over A.g. gauge invariance this refer to forms
which are dened over M A. . the Chern action . and hence on the bundle . is gauge invariant
and its connection invariance depends on further specications. . becomes the term which
generates.BF Theories and knots . in this case. namely the space of all orthonormal frames
paired with the corresponding metric connections see e. is a principal bundle . Hence we can
consider dierent kinds of symmetries . or when P is the bundle of linear frames over M ..
Also. Hence the BF action . . while the BF action . the following principal Gbundle P A A M .
divided by its center. . . Any connection on the bundle A A/G determines a connection on the
bundle . the ABJ anomaly in dimensions . connection invariance this refers to forms on M A
which are closed when they are integrated over cycles of M . one should consider either the
space of irreducible connections and the gauge group. is a sort of gaugexed version of the
generating term for the ABJ anomaly. This kind of invariance implies gauge invariance and
dieomorphism invariance when the latter is dened. In any BF action one has to integrate
forms dened on M A. In particular the BF action of example is connection invariant. In
particular we can consider dieomorphism invariance when P is a trivial bundle. A but are
pullbacks of forms dened over M . G G . In the latter case P A M A DiM DiM . while the action
of Strictly speaking. is both connection and dieomorphism/gauge invariant. For instance. or
the space of all connections and the group of gauge transformations which give the identity
when restricted to a xed point of M .e. instead of the tautological one.
. where Z Z denotes cycles cocycles. has the following properties . .j . B /TrAB. The form . It
is an isomorphism of inner product spaces. again with values in the bundle P Ad UG. i. then
makes perfect sense in any dimension. When M is a manifold of dimension d. in example . .
we have A M Tr F T Og M only when the covariant derivative dA is zero. One can then
consider a principal SOdbundle and an associated bundle E with ber Rd . G The map . In this
case there is a linear isomorphism Rd LieSOd. then we can take the eld B to be a d form.
Paolo CottaRamusino and Maurizio Martellini example is not.i n. As a nal remark we would
like to mention BF theories in arbitrary dimensions. The left action of SOd on Rn corresponds
to the adjoint action on LieSOd. A special situation occurs when the group G is SOd. is the
basis for the construction of the Donaldson polynomials . but now this map descends to a
map A H M H . then for any cycle in M Tr B Q F F is a closed form on A and so we have a
map Z M Z A. When we replace the curvature of the tautological connection with the
curvature of a connection in the bundle . dened by j i ei ej Ej Ei . More precisely.. When B is
given by .. when the inner product in LieSOd is given by A. then we again obtain a map . the
critical connections are the LeviCivita connections. in other words. i where ei is the canonical
basis of Rd and Ej is the matrix with entries i m Ej n m. The isomorphism . In this special
case we can consider a dierent kind of BF theory.. the integrand of the BF action denes a
closed form on Og M A.e. where the eld B is assumed to be a d form with values . .
On the other hand. A generalized knot in a dimensional closed manifold M can be dened as
a closed surface embedded in M . it is not known whether one can have an analogue of the
Jones polynomial for knots. In this case the corresponding BF action is given by exp M BF . .
knots and their quantum observables In WittenChernSimons theory we have a topological
action on a dimensional manifold M . we can show that there exists a connection between BF
theories in dimensions and knots. The theory of knots and links is less developed than the
theory of ordinary knots and links. we would like to ask whether there exists a generalization
to dimensions of the connection established by Witten between topological eld theories and
knot invariants in dimensions. Even though we are not able to show rigorously that a
consistent set of nontrivial invariants for knots can be constructed out of dimensional eld
theories.BF Theories and knots in the d th exterior power of E. Two knots generalized or not
will be called equivalent if they can be mapped into each other by a dieomorphism connected
to the identity of the ambient manifold M . or Wilson line. where the wedge product combines
the wedge product of forms on M with the wedge product in Rd . one can dene Alexander
invariants for knots see e. The problem we would like to address here is whether there exists
a connection between dimensional eld theories and invariants of knots.g. gives the classical
action for gravity in d dimensions. This connection involves. quantities higherdimensional
Wilson lines related to knots. a knot is a sphere embedded in S or in the space R . then the
corresponding BF action . in the socalled Palatini rstorder formalism . in dierent places. .
More precisely to each knot we associate the trace of the holonomy along the knot in a xed
representation of the group G. The action . It is natural to consider as observables. Let us
recall here that while an ordinary knot is a sphere embedded in S or R . In Wittens theory
one has to perform functional integration upon an observable depending on the given
ordinary knot in the given manifold . the Zamolodchikov tetrahedron equation as well as
selflinking and higherdimensional linking numbers. . When the SOdbundle is the orthonormal
bundle and the eld B is the d th exterior power of the soldering form. is invariant under gauge
transformations. Namely. For instance. and the observables correspond to knots or links in M
. In dimensions we have at our disposal the Lagrangians considered at the beginning of the
previous section.
One expects that the functional integral over the space of all connections will integrate out
this dependency. to be dened for any embedded surface closed or with boundary. and we
did not use dierent symbols for the holonomy and its quantum counterpart quantum
holonomy. Note that we will also allow the operator . x. y. In dimensions. It is gauge invariant
if we consider only gauge transformations that are the identity at the points x and z. . we
recall that the functional measure over the space of connections is given. z. we recall that.
Expression . z. in order to avoid a cumbersome notation. x. See in this regard the analysis of
the WessZuminoWitten model made by Polyakov and Wiegmann . z. by the measure over
the paths over which the holonomy is computed. a framing is a smooth assignment of a
tangent vector to each point of the knot.x we mean the holonomy with respect to the
connection A along the path joining two given points y and x belonging to . is gauge invariant
without any restriction. we did not write the explicit dependence on the paths . In the special
case when z and x coincide. . Moreover. Hence the holonomy along a path is dened by the
comparison of the horizontally lifted path and the path lifted through the reference section.
namely we want to perform functional integration upon dierent observables depending on a
given embedded surface in M with respect to a path integral measure given by the BF action.
y. has the following properties . Paolo CottaRamusino and Maurizio Martellini with respect to
a topological Lagrangian. it depends on the holonomies along such paths.y . can be seen as
the vacuum expectation value of the trace in the representation of the quantum surface
operator denoted by O. We assume to have chosen once and for all a reference section for
the trivial bundle P . In the latter case the points x and z are always supposed to belong to
the boundary. As far as the role of the paths considered above is concerned.x ByHolA. y.
then . The functional integral of . where. a timeordering symbol has to be understood. In
other words we set O.x ByHolA. . Here we want to do something similar. . in a dimensional
theory with knots. up to a Jacobian determinant. Here by HolA. more precisely. y . It depends
on the choice of a map assigning to each y a path joining x and z and passing through y. in .
z exp HolA. y .y . More precisely. The rst classical observable we want to consider is Tr exp
HolA.
We will also speak of a framed knot. In local coordinates the action is given by L TrB FA . M
M I i.k .BF Theories and knots instead. namely L L and . Here B is a multiple of and Ra is an
or thonormal basis of k UG. FA i A A dt Ax A dt Ai dxi i . . where D denotes the covariant
dimensional derivative with respect to the connection Ax . In the BF action we can disregard
the B component of the eld B.j.k TrBij dt Ak Bi Fjk dt A DBijk dxi dxj dxk . x d dt dx . are zero.
A dxi dt iltj Bij dxi dxj Bi dt dxi . We do not have secondary constraints since the Hamiltonian
is zero. will be written as FAx op physical . . We take the group G to be SU n and consider its
fundamental representation. . L A ijk DBijk i. . we rst consider the canonical quantization
approach. We rst consider the pure BF theory. a BF theory where no embedded surface is
considered.k . x t. x . namely. We choose a time direction t and write in local coordinates t..
Fij dxi dxj dt Ai dxi dx A dt B iltj i Ai . We perform a Legendre transformation for the
Lagrangian L. .j. hence we write the eld B as B a B a Ra . hence the time evolution is trivial.
the conjugate momenta to the elds Bi and A . At the formal quantum level the constraints . so
we have the primary constraints t Bi t A L Bi ijk FAx jk j. DBop physical . x . We will refer to
this assignment as a higherdimensional framing. Gauss constraints In order to study the
quantum theory corresponding to . we assign a curve to each point of the knot.
Paolo CottaRamusino and Maurizio Martellini where op stands for operator and the vectors
physical span the physical Hilbert space of the theory. y.. moreover. We assume that locally
the manifold M is given by M I I being a time interval and the inter In particular Tr coincides
with Tr when A. and . x. Jsing HolA. it follows that the component of Jsing in LieG does not
give any signicant contribution. y. y. z. x see denition . a . B UG. the observable to be
functionally integrated is Tr exp M Tr HolA. where d y is the surface form of . but has no
component in C UG. B Aa B a . . We now consider the expectation value of Tr O.y F yHolA.
z. B LieG. a Thus Jsing can take values in UG. . In order to represent the surface operator .
we assume that we have a current singular form Jsing . With the above notation and taking
into account the BF action as well. we have set TrAB From .z At this point the introduction of
the singular current allows us to represent formally a theory with sources concentrated on
surfaces as a pure BF theory with a new curvature given by the dierence of the previous
curvature and the currents associated to such sources. In this way we neglect possible
higherorder terms in U k G UG. we conclude. y .y Jsing . We are now in a position to write
the operator . z. z.z ByHolA. as a source action term to be added to the BF action. We still
want to work in the canonical formalism. that Jsing as a form should be such that Jsing y d y
. of the form a Jsing Jsing Ra . Now we choose a time direction and proceed to an analysis
of the Gauss constraints as in the pure BF theory.z ByHolA. but this is consistent with our
semiclassical treatment. a a B a Ra . We now assume Jsing U G. concentrated on the
surface . z exp M TrHolA. the result will be that instead of the constraints . as O..y . where for
any A. we will have a source represented by the assigned surface . A Aa Ra . From eqn .
is related to the higher braid group . the restriction of the connection A to the plane looks in
the approximation for which HolA.x. z. corresponding to the critical curvature . Aa x Notice
that the previous analysis implies that the components of the curvature F a and consequently
the components of the connection Aa must take values in a noncommutative algebra. y. let
us mention only that the dimensional critical connection is related to the existence . It is then
natural to ask whether the same is true in dimensions. In order to have some geometrical
insight into the above connection. y .BF Theories and knots section of the dimensional
manifold M t with the surface is an ordinary link Lt . in the linear approximation dened above.
The righthand side of . x . i. where the surface can be approximated by a collection of
dimensional planes in R . The quantum connection . y . y . y x yRa dx dx . for a given y here
d is the exterior derivative. and the hyperplane considered above. Given the fact that B a and
Aa are conjugate elds. x where is a plane orthogonal to at y with coordinates and and n
denotes the ndimensional delta function. where is a loop in based at z and passing through
y. by a collection of hyperplanes in C . More precisely the components F a as well as Aa
should be proportional to Ra LieG. z. x . We will discuss this point further elsewhere. is also
equal to HolA. at least when sources are present. .e. are assumed to be in general position.
we consider a linear approximation. yHolA. This means that the intersection of the
hyperplane with any other hyperplane is given by a point. z. of the pure braid group . on
gives a representation of the rst homotopy group of the manifold of the arrangements of
hyperplanes points in . These hyperplanes.z like Aa x Ra d logx y .z . As in the
lowerdimensional case ordinary knots or links as sources in a dimensional theory.y Jsing x.y.
x . In a neighborhood of y the current Jsing will look like a Jsing x. we can conclude that the
commutation relations of the quantum theory will require B a to be proportional to Ra as
well.z .e. The Gauss constraints imply that in the above neighborhood of y we have a F a x
HolA. namely whether the dimensional connection. i. more simply. .
x. where detDA M denotes the regularized determinant of the covariant derivative operator
with respect to a background at connection A on the space M . x. . since they are not
relevant for a rough calculation of . We omit the ghosts and gaugexing terms . x DADB Tr
exp Tr BxF xHolA. Paolo CottaRamusino and Maurizio Martellini of a higherdimensional
version of the KnizhnikZamolodchikov equation associated with the representation of the
higher braid groups. we can set Tr O. x dim detDA M .x . x dim DAF Jsing . Path integrals
and the Alexander invariant of a knot In the previous section we worked specically in the
canonical approach. x DADB Tr exp Tr Bx F x Jsing x . M . can be given as follows.z . . z.
with respect to the BF functional measure only. By applying the standard formula for Dirac
deltas evaluated on composite functions. with the FaddeevPopov ghosts included as the
analytic RaySinger torsion for the complement of the knot see . x. A rough and formal
estimate of the previous expectation value . we can write Tr O. as suggested by Kohno . This
torsion is related to the Alexander invariant of the knot .z Jsing xHolA. Again. x. By
integration over the B eld we obtain something like Tr O. . . . and compute the expectation
values. x. we heuristically obtain Tr O. When instead we consider a covariant approach.. in
the approximation in which HolA. x. M using a functional Dirac delta. It is then natural to
interpret the expectation value .
For instance. with respect to the total BF Chern action. . we can consider an arbitrarily ne
triangulation T of and take into consideration only one loop T which has nonempty
intersection with each triangle of T . so we get the following Feynman propagator b Aa xB y i
ab x y . The connection A is present in the quantum surface operator via the holonomy of the
paths y. x. . dimG and . . . We call this lifted path T . While the general case seems dicult to
handle. . in principle. The complication here is that the loop itself based at x and passing
through y depends on the point y. For a related approach see . the Beld with a . we perform
the usual pointsplitting regularization.y.x in a neighborhood of y. In order to avoid
singularities. . Moreover. where r stands for regularized. the only relevant part of the BF
action is the kinetic part B dA. This is tantamount to lifting the loops x. This approximation will
break gauge invariance. The lifting is done in a direction which is normal to the surface . y.
which will be.BF Theories and knots Firstorder perturbative calculations and
higherdimensional linking numbers Let us now consider the expectation value of the
quantum surface observable Tr O. .y y as in denitions . which by pointsplitting regularization
is then completely lifted from r the surface.y.x A zdz . we can write the holonomy as HolA.
Finally we get to rstorder approximation in perturbation theory and . The advantage of this
particular approximation is that we have only to deal with one loop T .x and x. y. we can
make simplifying choices which appear to be legitimate from the point of view of the general
framework of quantum eld theory. . At the rstorder approximation in perturbation theory with
a background eld and covariant gauge. recovered only in the limit when the size of the
triangles goes to zero. x y . we can easily assume that when we assign to a point y the loop
x. and . x.x based at x and passing through y. The two elds involved in the quantum surface
operator are Aa the connection a and B. .x At the same order. then the same loop will be
assigned to any other point y belonging to . ..
x. we assume more simply that M is the space R and that is a sphere. where C is the trace
of the quadratic Casimir operator in the representation . Carter and Saito . x y . One
coordinate axis in R is interpreted as time. t we While the normalization factor for a pure
dimensional BF theory appears to be trivial . keeping track of the over. Paolo CottaRamusino
and Maurizio Martellini up to the normalization factor Tr O. is given by dim C i r LT . x y r r
where LT . In other words. inspired by a previous work by Roseman . for noncritical times.
then . dx dx r T dy . One of the coordinates of this space is interpreted as time. is the
higherdimensional linking number of with T . so that Lt t R is. .and undercrossings. we have a
space projection t R R for each time t. x dim C i . in a theory with a combined BF Chern
action we have a contribution coming from the Chern action. so that the intersection of the
dimensional manifold M t with gives an ordinary link Lt for all times t in M . For any time
interval t .. At the local level M will be given by M I I being a time interval. . When is a sphere.
not its projection. we start by considering the surface embedded in the manifold M directly.
The intersection of this dimensional diagram of a knot with a plane at a xed time gives an
ordinary link diagram. describe a knot in fact an embedded sphere in space by the
dimensional analogue of a knot diagram they project down to a dimensional space.. while the
collection of all these link diagrams at dierent times stills gives a movie representing the knot.
This contribution has been related to the rst Donaldson invariant of M . In the Feynman
formulation of eld theory. with the possible exception of some critical values of the time
parameter. Wilson channels Let again represent our knot surface embedded in the manifold
M . In order to connect quantum eld theory with the standard theory of knots. an ordinary link.
and the intersection of with space at xed times is given by ordinary links.
ji point xi in each knot Ki and we denote by i the surface ji ji contained in whose boundary
includes Ki and Ki . si.t t R . The operators which appear in a Wilson channel are chosen in
such a way that a surface operator is sandwiched between knot operators only if the relevant
knots belong to the boundary of the surface. ji.n . . Ki ji xji i jn Otn .BF Theories and knots will
also use the notation t .ji ji ji of i we associate a path with endpoints xi and xi passing through
p. i . Ki ji ji. denotes the basepoint.t HolA. except at the initial and nal point. . For the sliced
knot. . xi ji . We choose one baseji ji ji. . xi ji ji and xji Oi i ji.ji . xi ji HolA. . We refer to the knot
equipped with the set T of times as the temporally sliced knot.ji .ji ji. Here and below we will
write ji simply to denote the fact that the index j ranges over a set depending on i.. Finally.t tt
. we associate the operator HolA. The paths that are included in the surface operators Oi are
not allowed to touch the boundaries of the surface. K x HolA. . To the surface i ji Ki . where
the subscript to the term HolA. . to the sliced knot we associate the product of the traces of
all possible Wilson channels. We denote the ji components of the corresponding links Lti by
the symbols Ki . K i ji xji . xn . . Ki ji xji Oi i ji. . . This product can be seen as a possible
higherdimensional generalization of the Wilson operator for ordinary links in space that was
considered by Witten. The main dierence concerns the prescriptions that are given involving
the paths that enter . xi ji HolA. We would like now to compare the Wilson channel operators
associated to a sliced knot with the operator O. considered in Section no temporal slicing
involved. . We consider a set T of noncritical times ti i. j . we dene the Wilson channels to be
the operators of the form O. We then as sign a framing to i as follows for each point p in the
interior ji.ji . .ji with boundary components Ki . We assume the following requirements for the
Wilson channels .
for some i. The tangle approach with its categorical content may reveal itself as a useful one
for quantum eld theory. recall that in the Wilson channel operators. One can think of the
quantum surface operators as acting on quantum holonomies by convolution. we can
construct quantum operators by considering all the Wilson ji. Finally. The initial tangle is
called the source and the nal tangle is called the target. for each t T . choosing a nite set of
times T ti . movies of ordinary tangles.e. They can be seen as a sort of convolution. say.
each of the paths is forbidden to follow two separate components of the same link Lt . tl .
then enter ji.tl . i. while the quantum counterpart of the tangle is represented by the quantum
surface operator relevant to the surface tl . What we can do instead is to make some
innitesimal calculations using integrationbyparts techniques in Feynman path integrals. and
one may describe the surface which generates the dierent links at times ti as a sequence
movie of braids . or. and the time evolution of a link represented by a surface can also be
described in terms of tangles. The quantum counterpart of the source and target is
represented by the quantum holonomies associated to the corresponding links. i l i l.
Integration by parts The results of the previous section imply that. braided category . for l gt
l.j i the surface i which is assumed to be equal to i . and requiring the paths to follow one of
the components of the link Lt . but it is dicult to do nite calculations and write this action
explicitly. the framing of the knot. of all the quantum holonomies associated to the links Lti for
i lt l i gt l.ji j i. and links Lti for i lt l i gt l. Ordinary links are the closure of braids. j i. and nally
go back to the j th component. Let us denote these quantum operators by the terms W in .
given a temporally sliced knot . Paolo CottaRamusino and Maurizio Martellini the denition of .
equivalently. by . one path should follow the jth component. It has been shown that tangles
form a rigid. . This last condition can be understood by noticing that in order to follow both the
jth and the j th components of the link Lti . for any time t T . the source represents Ltl while
the target represents Ltl . depending on . . In our case. tl . But a link is also obtained by
closing a tangle. . The Wilson channels are obtained from . ji. W out .ji channels relevant to
surfaces i . thus contradicting a local causality condition.
We denote by the symbols a and b the part of each string that precedes and. and the string
crosses under the string . So in order to compute the derivative of the surface operator. Let
us number these strings with numbers . . Three strings Generally speaking. The
integrationbyparts rules for the BF theory are as follows. we may consider variations of the
link Ltl . respectively. recall that braids or tangles representing links at a xed time can be
seen as collections of oriented strings. We consider now two links that dier from each other
only in a small region involving three strings. let us see what happens when we consider the
quantum operator corresponding to a surface bordering two links Ltl and Ltl that dier by an
elementary change. Cx F Ra exp x M . We obtain x HolA. as shown in Fig. is that one can
interchange a variation of the surface operator with a variation of the link and consequently of
its quantum holonomy. . in a nite time evolution tl tl . then the string crosses under the string .
. the quantum surface operator dened by the surface with boundaries Ltl and Ltl will map the
operator W in .. By the symbol x we mean the variation of C obtained by inserting an
innitesimally small loop cx in C at the point x.BF Theories and knots Fig. tl . tl into the
operator W in . What the functional integrationbyparts rules will show. In order to describe
elementary changes of links. and a point x C. Let us now consider what can be seen as the
derivative of this map.a M TrB F TrB F a d c HolA. Namely. follows the rst crossing. The
string crosses under the string . Cx exp . . which we denote here by C in order to simplify the
notation. We jl consider a knot Kl . while tangles describe the interaction of those strings .
x . Cx Ra a B x . . Kl of Ltl that will evolve into three dierent components Kl . But these
variations are exactly the ones for which the small loop cx is such that the surface element d
cx is transverse to the knot C as in . . In order to consider the more general situation we
assume that the three strings in Fig. . . j . so we can disregard the holonomies of the paths in
the surface operators. with the aid of eqns . .. x for a surface whose boundary includes C we
obtain O . O . Here Wl is a short notation for the operator We are considering here only the
small variations of .a Paolo CottaRamusino and Maurizio Martellini d c exp x TrB F HolA.
where d is the surface form of . we can switch each undercrossing into an overcrossing.
Such comparison. Details will be discussed elsewhere.j . its innitesimal variation O can be
expressed as follows O Wl R Wl . x . By performing three such small variations at each one
of the crossings of the three strings considered in Fig. When we apply the functional
derivative to a a B x surface operator O . . x . and the Fierz identity . M . we conclude that
only the variations cx for which d d cx matter in the functional integral. in this way we
produced the functional derivative with respect to the Beld. The integrationbyparts rules of
the BF theory in dimensions completely parallel the integrationbyparts rules of the
ChernSimons theory in dimensions . xj . Kl . x Ra x x d a B x . Kl of Ltl .. Ra denotes as usual
an orthonormal basis of LieSU n. xj . where d cx denotes the surface form of the surface
bounded by the innitesimal loop cx . belong to dierent components Kl . Now we come back to
the link Ltl with the three crossings as in Fig. leads to the conclusion that when we consider
the action of the surface operators Oj. . Now as in we compare the expectation value of the
original conguration with the conguration with the three crossings switched.. Kl . . on the
quantum l l l holonomies of the knots Klj . The second equality has been obtained by
functional integration by parts. and the trace is meant to be taken in the fundamental
representation. and . the only relevant contribution to be considered is the one given by the
eld B B a Ra . From eqn . x .
Now we recall that / a Ra Ra is the innitesimal approximation of a quantum YangBaxter
matrix R. Phys. C. Capovilla for invitations to the University of South Alabama and Cinvestav
Mexico. F. Ashtekar. involve the operators W in . E. where R is a solution of the quantum
YangBaxter equation. R. and nally R is a suitable representation of the following operator / a
Ra Ra a. namely S Ra. x . Phys. Kl x O. . Carter and R. M. M. Saito have kindly explained to
us many aspects of their work on knots. Kl x O.a Rb.R. Math. . . Kl x l l l l l HolA. Baez for
having organized a very interesting conference and for having invited us to participate.
Quantum eld theory and the Jones polynomial. Carter and M. Dirac operators coupled to
vector . both in person and via email. Atiyah and I. Kl x O. Kauman. Rakowski. Phys. x .
Hence the calculations of this section suggest that. D.a / / a a Ra Ra b. respectively. Baez. x
HolA.BF Theories and knots Wl HolA. Rep. L.b . Math. Commun.b . the vector space
associated to the string j is given by a tensor product j Va Vbj . and G. J. Bibliography . x
HolA. Kl x l l l l l the dots in . Birmingham. Singer. Blau. M. Witten. at the nite level. E. M. We
also thank A. Commun. Kl x . would like to thank S. P. We thank them very much. l l l l l
HolA. Both of us would also like to thank J. Topological quantum eld theory. Topological eld
theory. x .a Ra Ra b. Acknowledgements S. . . .. . . Stashe very much for discussions and
comments. Thompson. and J. . . Witten.a Rb. x HolA. Capovilla. tl see . the solution of the
Zamolodchikov tetrahedron equation which may be more relevant to topological quantum eld
theories is the one considered in . tl and W out .
Manin and V. Math. Blau and G. . Phys. Turaev. Schwarz. Paolo CottaRamusino and
Maurizio Martellini potentials. . Ann. B. . Suppl. I. Phys. Schechtman. Phys. Rev. Singer.
Publish or Perish. Phys. Teor. Fiz. K. CottaRamusino. M. and L. Wiegmann. sigmamodels
and strings. B . Math. Saito. Proc. Reidemeister torsion in knot theory. Polyakov and P.
Bonora. M. Commun. M. Phys. T. Phys. S. A. Braids and quantum group symmetry in
ChernSimons theory. Guadagnini. Classical amp Quantum Gravity LL. Thompson.
Kronheimer. M. Arrangements of hyperplanes. . and J. . Ashtekar. Russ. Syzygies among
elementary string interactions of surfaces in dimension . Ya. Fourier . Donaldson and P. .
Weaving a classical metric with quantum threads. K. . A quantum eld theoretic description of
linking numbers and their generalization. Lett. Lett. . . Kohno. Horowitz and M. B . Lett. D.
and J. Srednicki. Surv. V. Arefeva. . Smolin. Wilmington. Studies Pure Math. . V. T. Rinaldi.
Stashe. and M. The partition function of degenerate quadratic functionals and RaySinger
invariants. Delaware . Rolfsen. Martellini. M. Kohno. Non Abelian Stokes formula. Capovilla..
Knots and Links. T. . The evaluation map in eld theory. . S. higher braid groups and higher
Bruhat orders. preprint. P. Jacobson. Phys. Natl. B . E. . Integrable connections related to
Manin and Schechtmans higher braid groups. Adv. Topological YangMills symmetry. and .
Nucl. . . A. Dell. . Inst. R. G. Goldstone elds in two dimensions with multivalued actions. B .
Mat. Inc. L. C. The Geometry of FourManifolds. Mintchev. S. . . Math. L. Nucl. J. Yu. Phys.
Rovelli. Acad. Baulieu and I. . . A new class of topological eld theories and the RaySinger
torsion. . T. G. Carter and M. . . Oxford University Press . Lett. Math. A. Monodromy
representation of braid groups and Yang Baxter equations. . Lett. Math. Proc. Phys. .
Commun. Sci. Gravitational instantons as SU gauge elds.
Saito. Carter and M. . R. Roseman. . . JETP . Knot Theory Ramications. Saito. J.
Zamolodchikov. . D. CottaRamusino. B. S. Saito. Jr. On formulations and solutions of
simplex equations. preprint . J. Mintchev. J.BF Theories and knots . . E. B . PhD Thesis.
Algebras and triangle relations. Carter and M. Phys. Quantum eld theory and link invariants.
and beyond. Sov. S. Guadagnini. Phys. Knotted surfaces. Fischer. Martellini. preprint . this
volume. Carter and M. . . S. and M. preprint . J. Tetrahedra equations and integrable systems
in dimensional space. M. J. Reidemeistertype moves for surfaces in fourdimensional space.
Yale University . braid movies. P. A. J. . Lawrence. Geometry of categories. . Nucl. E.
Reidemeister moves for surface isotopies and their interpretation as moves to movies.
Paolo CottaRamusino and Maurizio Martellini .
in the dimensionally higher case is to get algebraic or diagrammatic ways to describe
knotted surfaces embedded in space. In this paper. Kamada proved generalized Alexanders
and Markovs theorems for surface braids dened by Viro. we rst review recent developments
in this direction.Knotted Surfaces. On the other hand. We propose generalizations of these
concepts from braid groups to groups that have certain types of Wirtinger presentations. Our
diagrammatic methods of producing new solutions to these generalizations are reviewed.
Braid Movies. We observe categorical aspects of these diagrammatic methods.
Austin.usouthal. Scott Carter Department of Mathematics. In particular. The rst step. In
particular. . USA email cartermathstat. USA email saitomath. University of South Alabama.
various braid forms for surfaces were considered by several other authors. then. University of
Texas at Austin. Alabama . Texas . Such possibilities were discussed in this conference. We
review recent developments in this direction. We combined the movie and the surface braid
approaches to obtain moves for the braid movie form of knotted sur . Introduction It has been
hoped that ideas from physics can be used to generalize quantum invariants to
higherdimensional knots and manifolds. Reidemeistertype moves were generalized by
Roseman and their movie versions were classied by the authors . Finally. we present new
observations on algebraic/algebraictopological aspects of these approaches.utexas. We
discuss generalizations of the YangBaxter equation that correspond to diagrams appearing
in the study of knotted surfaces. Mobile. and Beyond J. and cycles in Cayley complexes.
generalizations of braids are explained. they are related to identities among relations that
appear in combinatorial group theory. In particular. In dimension Jonestype invariants were
dened algebraically via braid group representations or diagrammatically via Kauman
brackets.edu Abstract Knotted surfaces embedded in space can be visualized by means of a
variety of diagrammatic methods.edu Masahico Saito Department of Mathematics.
charts. diagrammatic techniques have been used to solve algebraic problems related to
statistical physics. . The braid movie description is rich in algebraic and diagrammatic avor.
The exposition in Section provides the basic topological tools for nding new invariants along
these lines if they exist. In Section we review our diagrammatic methods in solving these
equations. . we gave a set of Reidemeister moves for movies of knotted surfaces such that
two movies described isotopic knottings if and only if one could be obtained from the other by
means of certain local changes in the movies. In order to depict the movie moves. Here a
map from a surface to space is called generic if every point has a neighborhood in which the
surface is either embedded. Local pictures of these are depicted in Fig. we gave solutions to
various types of generalizations of the quantum YangBaxter equation QYBE. three planes
intersecting at an isolated triple point. or the cone on the gure eight also called Whitneys
umbrella. In Section we follow Fischer to show that braid movies form a braided monoidal
category in a natural way. We also can include crossing information by breaking circles at
underpasses. The charts dened in Section can be regarded as pictures that are dened for
any group presentations of null homotopy disks in classifying spaces. In . Such
generalizations originated in the work of Zamolodchikov . J. New observations in this paper
are in Section . knot theoretical. Movies. we rst projected a knot onto a dimensional
subspace of space. Others are related to Peier equivalences that are studied in
combinatorial group theory and homotopy theory. we get immersed circles with nitely many
exceptions where critical points occur. . By cutting space by level planes. Then we x a height
function on the space. two planes intersecting along a doublepoint curve. Some of the moves
correspond to cycles in Cayley complexes. By a movie we mean a sequence of such curves .
Such generalizations appeared in the context of categories and higher algebraic structures .
Such a map is called generic because the collection of maps of this form is an open dense
subset of the space of all the smooth maps. . Movie moves The motion picture method for
studying knotted surfaces is one of the most well known. On the other hand. and isotopies .
where we discuss close relations between braid movie moves and various concepts in
algebra and algebraic topology. Scott Carter and Masahico Saito faces. We propose
generalized chart moves for groups that have Wirtinger presentations. see . . surface braids.
It originated with Fox and has been developed and used extensively for example . In
particular.
Generic intersection points . . and Beyond Fig. Braid Movies.Knotted Surfaces.
Similar notions had been considered by Lee Rudolph and X. Throughout the paper we
consider simple surface braids only. the order of the interactions can be switched. is a
branched covering. The dierent stills dier by either a Reidemeister move or critical point
birth/death or saddle on the surface. One rather large class of moves is not explicitly given in
this list namely. In these gures each still represents a local picture in a larger knot diagram.
Lin . Closed surface braids Let F B D be a surface braid with braid index n. Oleg Viro
described to us the braided form of a knotted surface. where we regard the projection p B D
D as a disk bundle over a disk. These are called the elementary string interactions ESIs.
Here the positive integer n is called the surface braid index. Each element in a sequence is
called a still. Kamada has studied Viros version of surface braids. Surface braids In April . A
surface braid is a compact oriented surface F properly embedded in B D such that the
projection p B D D to the second factor restricted to F . A surface braid is called simple if
every branch point has branch index . . Figure depicts ESIs. . . that is. The Reidemeister
moves are interpreted as critical points of the double points of the projected surface. S.
Finally. when two string interactions occur on widely separated segments. The movie moves
are depicted in Figs and . Embed B D in space standardly. Then F is the unlink in the sphere
B D . distant ESIs commute. Any two movie descriptions of the same knot are related by a
sequence of these moves. A surface braid is simple if the covering in condition above is
simple.. we follow Kamadas descriptions. and so the movies contain critical information for
the surface and for the projection of the surface. pF F D . Scott Carter and Masahico Saito on
the planes.. The boundary of F is the standard unlink F n points D B D B D where n points
are xed in the interior of B. The moves indicate when two dierent sequences of ESIs depict
the same knotting. the moves to these movies were classied following Roseman by means of
an analysis of foldtype singularities and intersection sets. Denitions Let B and D denote
disks. Two surface braids are equivalent if they are isotopic under levelpreserving isotopy
relative to the boundary. J. Again without loss of generality we can assume that successive
stills dier by at most a classical Reidemeister move or a critical point of the surface.
Braid Movies. The elementary string interactions . and Beyond Fig.Knotted Surfaces. .
. J. Scott Carter and Masahico Saito Fig. The Roseman moves .
and Beyond Fig. Braid Movies. . The remaining movie moves .Knotted Surfaces.
. See and also the next section. is to nd a proof of the Markov theorem for closed surface
braids in which each branch point is simple. A surface obtained this way is called a closed
surface braid. . Scott Carter and Masahico Saito so that we can cap o these circles with the
standard embedded disks in space to obtain a closed. Kamada has recently proven a
Markov theorem for closed surface braids in the PL category. i j gt we dene a braid movie as
follows. Except for such t. Here the underarc lies below the overarc with respect to the
projection p . is one of the following smoothing/unsmoothing of a crossing corresponding to a
branch point. the dierence between p F Ps . bk of words in i . However. . Braid movies Write
B and D as products of closed unit intervals B B B . Factor the projection p B D D into two
projections p B B D B D and p B D D . If one of such points is contained. F called the closure
of F . We may assume that for any t D there exists a positive such that at most only one of
such points is contained in st . or local maxima/minima of the doublepoint set. we can include
crossing information by breaking the underarc as in the classical case. To make this
convention more precise we identify the image of p with an appropriate subset in B D . .
Furthermore. . type II Reidemeister move corresponding to the maximum/minimum of the
doublepoint set. Viro and Kamada have proven an Alexander theorem for surfaces. p F Pt is
the projection of a classical braid. any orientable surface embedded in R is isotopic to a
closed surface braid. in his proof he does not assume that the branch points are simple. In
other words. Except for a nite number of values of t D the intersection p F Pt is a collection of
properly immersed curves on the disk Pt . Furthermore. Let F be a surface braid. This idea
was used by Kamada when he constructed a surface with a given chart. We further regard
the D factor in B D B D D as the time direction. . Such a theorem would show a clear analogy
between classical and higherdimensional knot theory. A braid movie is a sequence b . s t . n.
or triple points. An open problem. J. or type III Reidemeister move corresponding to a triple
point. then. By this convention we can describe surface braids by a sequence of classical
braid words. is dened up to ambient isotopy. bm is obtained from bm by one of the following .
Exceptions occur when Pt passes through branch points. . D D D . orientable embedded
surface in space. i . The closed surface. such that for any m. we consider families Pt B D t t
D . Thus. Including another braid group relation i j j i . .t Ps .. . We may assume without loss
of generality that p restricted to F is generic.
Consider the double point set of F under the projection p F B D F B D which is a subset of B
D . A white vertex on the chart has valence . and this is an optimum on the doublepoint
set.Knotted Surfaces. . . This situation is illustrated in Fig. The projection of the braid
exchange i j j i for i j gt is a valent vertex or a crossing in the graph. A valent vertex can also
be thought of as a crossing point of the edges of the graph. j where i j . and charts is depicted
in Fig. and Beyond changes insertion/deletion of i . insertion/deletion of a pair i i . j . . for
short. the orientations are such that the rst three edges in this labelling are incoming and the
last three edges are outgoing. and this in turn is a triple point of the projected surface. Thus .
. Reconstructing the surface braid from a chart A chart gives a complete description of a
surface braid in the following way. j . Braid Movies. Then a chart depicts the image of this
double point set under the projection p B D D in the following sense The birth/death of a
braid generator occurs at a saddle point of the surface at which a doublepoint arc ends at a
branch point. Here. Chart descriptions A chart is an oriented labelled graph embedded in a
disk. Maximal and minimal points on the graph correspond to the birth or death of i i . The
relationships among surface projections. replacement of i j i by j i j where i j . Recall that the
D factor of D is interpreted as a time direction. The labels on the edges at such a crossing
must be braid generators i and j where i j gt . i . Two braid movies are equivalent if they
represent equivalent surface braids. braid movies. the edges are labelled with generators of
the braid group. This projects to a black vertex on a chart. These changes are called
elementary braid changes or EBCs. replacement of i j by j i where i j gt . The valent vertex
corresponds to a Reidemeister type III move. and the vertices are of one of three types A
black vertex has valence . The edges around the vertex in cyclic order have labels i . i ..
Braid movies. . charts. J. and diagrams . Scott Carter and Masahico Saito Fig.
Chart moves . .Knotted Surfaces. Braid Movies. and Beyond Fig.
j k i i k j . CIR i k j . j k i j . j k gt . j i j k k i j i . Kamada Every surface braid gives rise to an
oriented labelled chart. Kamada provided a set of moves on charts. j k gt . and III. k i gt . .
From the braid movie point of view. indeed. k j i j . yielding braid movie moves that will be
discussed in the next section. i j i k . i j . Theorem . j i k j . we can construct a surface braid
and. Furthermore. otherwise the exponent is negative. CIR k i j i . Such generalized chart
moves. where i j gt . By taking a single slice on either side of a vertex or a critical point on the
graph. where i j and i k gt . Each generic intersection of a horizontal line with the graph
generic intersections will miss the vertices and the critical points on the graph gives a
sequence of braid words. Scott Carter and Masahico Saito the vertical axis of the disk points
in the direction of increasing time. j j i . and every oriented labelled chart completely
describes a surface braid. i j k i . Therefore. . The letters in the word are the labels
associated to the edges. CIR CIR i . are listed in Fig. given a chart and a braid index. the
exponent on such a generator is positive if the edge is oriented coherently with the time
direction. we need to rene his list incorporating movie moves. i i i . i j k . j i . j i j k . i k j i . j i j i
. where i j gt . k i j . II. Braid movie moves The following types of changes among braid
movies are called braid movie moves CIR i . His list consists of three moves called CI. . i j i j .
j k i . j i k . i j j . a knotted surface by closing the surface braid. j i j . i i . J. we can reconstruct
a braid movie from the sequence of braid words that results. k j i . where i j gt .
Furthermore. k i j i k j . j i j . . k j i k j k . i i . b bk . . where i j . k i j k i j . . i j i . i j i k j i . j i j .
where i j . empty word. empty word. we include the variants obtained from the above by the
following rules replace i with i for all or some of i appearing in the sequences if possible. i j i j
j . i k j k i j . j i k j k i . j k i j i k . b . k j i j k j . i j i j . CII j . . j k j i j k . add bk . . i j i . . i j j . k j k i j
k i j k i j i . CIII j j . CIM i j k i j i .Knotted Surfaces. j i j k j i . . Braid Movies. where i j . i . if b . i j
i . . i i . . j i j i j . . and Beyond CIM empty word empty word. . . where k j i or k j i . bk b . j i j . i j
i j j i . . where i j gt . . CIM j i . i empty word. CIV empty word. i i j i . . where i j . j i j j . CIM CIM
i i . empty word. i j k j i j . i j . j i . i i i i . i i . k j k i j k . i j i i j i . bk is one of the above. . j k i j k i .
CIII i j . .
are written as bj uj vj resp. . . i . bj resp. Then we say that b . Scott Carter and Masahico
Saito if b . . and ui ui resp. b vi resp. . . bj uj vj where u and v satisfy a ui resp. . . and vi vi
resp. . . . There is an overlap between the braid movie moves and the ordinary movie moves.
i. . . bk is one of the above. Zamolodchikov presented a system of equations and a solution
to this system that corresponded to the interaction of straight strings in statistical mechanics.
There exists i. Denition locality equivalence Let b . . It is interesting to note that in the braid
movie point of view. ui by one of the elementary braid changes. i . . . . J. and this relationship
suggests topological applications. ck c . ui ui . i n. bn and b . say . bj . . bk b . Theorem . In .
add c . . The permutohedral and tetrahedral equations In . vi vi . Frenkel and Moore
produced a formulation and a solution of a variant of the Zamolodchikov tetrahedral equation.
these systems are interpreted as consistency conditions for the obstructions to the lower
dimensional equations having solutions. the move that corresponds to a quadruple point
gives a movie move corresponding to the permutohedral equation rather than to the
tetrahedral equation. bj . Baxter showed that the Zamolodchikov solution worked. . And
Kapranov and Voevodsky consider the solution of . i. that overlap is to be expected. And in
higherdimensional analogues were dened. . vi is obtained from ui resp. vi by one of the
elementary braid changes. bn is obtained from b . . bn by a locality change or vice versa. . ck
where cj resp. . Two braid movies are equivalent if and only if they are related by a sequence
of braid movie moves and locality changes. . . combinations of the above. say . The braid
movie moves are depicted in Figs and . bn be two braid movies that satisfy the following
condition. . This system of equations is related to the Yang Baxter equations. such that bj bj
for all j i . . . cj is the word obtained by reversing the order of the word bj resp. If two braid
movies are related by a sequence of locality changes then they are called equivalent under
locality. . j i . . . We will discuss these equations at some length in Section . . equivalence
under locality changes means that distant EBCs commute. Loosely speaking. ui is obtained
from vi resp. .
Braid Movies.Knotted Surfaces. and Beyond Fig. . The braid movie moves .
. Scott Carter and Masahico Saito Fig. The braid movie moves . J.
is S . acts on V as the identity. the elementary string interactions that occur are
Reidemeister type III moves. for example. Braid Movies.. The rst factor of the lefthand side of
the equation . . We expect there are even more solutions in the literature and would
appreciate being told of these. Dene a set of pairs of integers among to C. where S EndV .
The equation corresponds to the given movie move as these type III moves occur in opposite
order on the righthand side of the move. . A variant There is a natural variant of this equation
S S S S S S S S . We review how to obtain eqn . . This tetrahedral equation was also
formulated in .Knotted Surfaces. we mention . Under . Ruth Lawrence described the
permutohedral equation and two others as natural equations for which solutions in a algebra
could be found. Here we review these equations and our methods for solving them. Compare
this equation with the movie move that involves ve frames on either side. . . Then the
Reidemeister type III moves occur in order among the strings . On the lefthand movie. In . . .
. and . The tetrahedral equation The FrenkelMoore version of the tetrahedral equation is
formulated as S S S S S S S S . Label the strings to from left to right in the rst frame of the
movie. There are other solutions of these higherorder equations known to us. and Beyond
the Zamolodchikov equation to be a key feature in the construction of a braided monoidal
category. Each side of the equation acts on V V V V V . . There are six elements and the
ordering denes a map C. solutions are studied in relation to quasicrystals. In a parallel
development. . . This index is one more than the number of strings that cross over the given
string. This correspondence was also pointed out by Towber. with the lexicographical
ordering as indicated. . . . from eqn . . Thus we think of the tensor S as corresponding to a
type III Reidemeister move. where each side of the equation now acts on V . V a module
over a xed ring k. . In particular. . . . . In this equation each index appears twice in each side.
Notice here that eqn . cf. . has the nice property that each index appears three times on
either side of the equation. . in the lexicographical ordering. and S . Consider the pairs of
integers among this subscript .
J. Solving the Zamolodchikov equation . Scott Carter and Masahico Saito Fig. .
. . which is the subscript of the rst factor in eqn . has the subscript which yields pairs . This is
how we obtain . Braid Movies. . and Beyond these are mapped to . and the selfintersection
sets of these contain intersections of lowerdimensional strata. . Specically. . the intersection
points among planes and receive the number since is the rst in the lexicographical ordering.
For S S acting on V V V Wa Wb Wa Wb Wa Wb set S Raa Rba Rbb . the second factor of .
Thus .. is solvable.. The numbering in the gure matches those described above.Knotted
Surfaces. . One can compute S S S S Raa Rba Rbb Raa Rba Rbb Raa Rba Rbb Raa Rba
Rbb Raa Raa Raa Rba Rba Raa Rbb Rba Rba Rbb Rbb Rbb Raa Raa Raa Raa Rba Rba
Rba Rba Rbb Rbb Rbb Rbb Raa Rba Rbb Raa Rba Rbb Raa Rba Rbb Raa Rba Rbb S S S
S . . and on the other side of the movie move the other side of the Reidemeister type III move
appears. . . which are mapped under to . We obtained this result by considering the
preimage of four planes of dimension generically intersecting in space. from . giving the
second factor of . For example. The other factors are obtained similarly. The crossing points
of the remaining planes give one side of the Reidemeister type III move on the preimage.
Each plane is one of the strings as it evolves on one side of the movie of the corresponding
movie move. Thus the solution is a piece of bookkeeping based on this picture. For given V
give a and b as subscripts of W Vi Wia Wib for i . . This computation is a bit tedious
algebraically. Let V W W where W is a module over which there is a QYB solution R R R R R
R R . . All of the higherdimensional solutions are obtained because the equations correspond
to the intersection of hyperplanes in hyperspaces.. Remarks In we showed that solutions to
the FrenkelMoore equations could also produce solutions to higherdimensional equations.. is
solvable appropriate products of QYB solutions give solutions to . The movie version of such
planes is depicted in Fig. Next we observe that eqn . .
and both sides of the equation live in EndV . The tensors S and R live in EndV and EndV .
Thus this equation is a natural one to consider from the point of view of knotted surfaces.
The permutohedral equation This equation was studied in and S S R R S S R R S S R R S S
. and each parallelogram is dual to a crossing point on the graph. Scott Carter and Masahico
Saito Fig. Each hexagon represents the tensor S and each parallelogram represents R. the
equation reduces to the variant of the tetrahedral equation given above. The permutohedral
equation . J. Finally. . and it acts as the identity on the other factors. A pictorial
representation is given in Fig. . observe that by setting R equal to the identity. The subscripts
indicate factors of vector spaces on which the tensor acts nontrivially. Each hexagon is dual
to a valence vertex. . respectively. and the numbers assigned to the boundary edges of
hexagons and parallelograms determine the subscripts of the tensor. The gure is the dual to
the CIM move on charts. This gure shows two sides of a permutohedron truncated
octohedron. as it expresses one of the braid movie moves. The equation expresses the
equality among tensors that are associated to the faces on the sides of the permutohedron.
Parallel edges receive the same number.
. AM M M M A stands for A M M M M A EndV .. and R EndV V satisfy the following
conditions AAA AAA. RM B BM R ARM M RA. . . they are listed explicitly as in eqns . BM M
M M B AM B BM A AM R RM A. M RB BRM M AR RAM ARB BRA. For example. BRA ARB
BAR RAB. . In case those equalities are used.. and Beyond Fig. . and resp. Braid Movies.
RAB BAR M RM M RM . Assigning tensors to the hexagons . BBB BBB M M A AM M. . . .
RHS of every relation has subscripts . .Knotted Surfaces. The subscripts indicate the factor
in the tensor product on which these operators are acting. . . We do not require the equalities
with the subscripts of the RHS and LHS switched. and and lives in EndV . B. and . M .
Solutions Suppose A. . . where the LHS resp. BM M M M B M M A AM M. .
J. . The computational technique . Scott Carter and Masahico Saito Fig.
.. we indicate that if A. and R satisfy YangBaxterlike relations. BBB BBB BRR RRB. . i /. B
be the universal Rmatrix in Uqi sl Uqi sl resp. In this way. we embedded quantum groups into
bigger quantum groups.Knotted Surfaces. Uq sln Uq sln. then go directly down to the second
row and read right to left. then we can get from one side of the permutohedron to the other.
Let R i R where i is xed. Now in the solution given in . and Beyond Proposition . i r is a basis
of i simple roots of the root system. .. where qi qi . travel left to right. . set A M R. Planar
versions Simplex equations can be represented by planar diagrams when the subscripts
consist of adjacent sequences of numbers. Furthermore. we were able to construct solutions.
then S A M B is a solution to the permutohedral equation. The proof is presented in detail in .
is the canonical inner product. The zigzag pattern continues. . To read this gure start at the
upper left. Then in Fig. B. . . If we take a representation f Uq sln EndV for a vector space V
over the complex numbers. There exists a canonical embedding i Uqi sl Uq sln
corresponding to the ith vertex of the Dynkin diagram. and r is the rank p. in we showed that
the remaining relations held between R and B. In this way we reinforce the diagrammatic
nature of the subject. In Fig. M . Then the list of the conditions reduces to RRR RRR. Here
we include the illustrations that led to the proof. Let R resp. hold. RRB BRR . we assign the
product ABM to each of the S tensors.. with the same subscript convention as in eqns . To
get Rmatrices that satisfy these conditions. We can use planar diagrams to solve such
equations. If the conditions stated in eqns . both f R and f B satisfy the quantum Yang Baxter
equation RRR RRR. . Braid Movies. This idea was inspired by Kapranov and Voevodskys
idea of using projections of quantum groups to obtain solutions to Zamolodchikovs equation .
BBB BBB since f R f i R is a representation of Uqi sl and R is a universal Rmatrix.
we found a continuous family of solutions to the above equation expressed as a linear
combination with variables in generators of the Temperley Lieb algebra. or they can be glued
along common objects.. and another class of solutions reected the noninjectivity of the Burau
representation. Scott Carter and Masahico Saito . Turaev described the category of tangles
and laid the foundations for his paper . is a morphism from n to m. Elements in the
TemperleyLieb algebra are represented by diagrams involving and . Such generalizations of
the Kauman bracket seem useful in general. Some of the solutions were noninvertible. For
any two objects in a category there is a set of morphisms. and morphisms between
morphisms called morphisms. there are objects. A category has objects. Playing with such
diagrams. r P R.. categories and the movie moves At the Isle of Thorns Conference in .
morphisms. Consequently. The morphisms can be composed in essentially two dierent ways
either they can be glued along common morphisms. there is hope that the yet to be found
higherdimensional generalizations can be found as representations of the category of tangles
that we will describe here. then. Such a tangle. All of the known generalizations of the Jones
polynomial can be understood as representations on this category. The objects in the
category are in onetoone correspondence with the nonnegative integers. . We found some
new and interesting solutions to this version by using the Burau representation of the braid
group. And the morphisms are isotopy classes of tangles joining n dots along a horizontal to
m dots along a parallel horizontal. . J. A week later in France similar ideas were being
discussed by Joyal . and these were presented in . Overview of categories In a category. We
regarded the corresponding planar diagram as the diagrams in the TemperleyLieb algebra.
Planar tetrahedral equations One such equation is S S S S S S S S . after multiplying by
permutation maps s P S. . The planar permutohedral equation It was observed by Lawrence
that the permutohedral equation becomes the following equation r s s r r s s s s r r s s r .
Equivalently. a morphism can be represented by a movie in which stills dier by at most an
EBC. consider the category in which the collection of objects consists of the natural numbers
. and the set of morphisms from n to n is the set of nstring braid diagrams or equivalently
words on the alphabet .Knotted Surfaces. the fundamental problem in a category is to
determine if two morphisms are the same. we can dene all the data various morphisms in
terms of braid movies such that the category of braid movies denes a braided monoidal
category. that is. In the middle of gures polytopes are depicted. by braiding n strings in front
of m strings. . Thus. . the integer n. . n . and polygons where poly means two or more for
morphisms. The set of morphisms is the set of nstring braid diagrams without an equivalence
relation of isotopy imposed. . by composing morphisms we obtain faces of polyhedra. Here
we give a description of the category associated to nstring braids. they can be glued only if
they have either a morphism or an object in common. . A tensor product is dened by addition
of integers. the braid diagram represented by the braid word n n nm n n nm m . arrows
between dots for morphisms. Assertion With the tensor product. . each number n is identied
with n dots arranged along a horizontal line. . More generally. These are two exam . a
morphism is a word in the letters . The geometric depiction of a category consists of dots for
objects. . The category of braid movies This section is an interpretation of Fischers work in
the braid movie scheme. . and braiding dened above. and this is identied with n points
arranged along a line. Equivalently. and an equality among morphisms is a solid bounded by
these faces. These are illustrations of the KapranovVoevodsky axioms as interpreted in our
setting. identity. Braid Movies. but for example two straight lines and a braid dia gram
represented by i i are distinguished. Just as the fundamental problem in an ordinary category
is to determine if two morphisms are the same. . Sketch of Proof In Figs and a set of movie
moves to ribbon diagrams are depicted. The set of morphisms from n to m when m n is
empty. We dene a braiding which is a morphism from n m to m n. . A morphism is a surface
braid that runs between two nstring braid diagrams. and Beyond Obviously.. Two diagrams
related by a levelpreserving isotopy of a disk which keeps the crossings are identied. . . and
acts as an identity for this product. A generating set of morphisms is the collection of EBCs.
There is one object. . n .
Scott Carter and Masahico Saito Fig. . A ribbon movie move . J.
Knotted Surfaces. and Beyond Fig. Braid Movies. Another ribbon movie move . .
he denes this category. Edges in these polytopes are morphisms. Alternatively. . and
changes between two ribbon diagrams. braided. Letters represent objects where tensor
product notation is abbreviated. rigid. Since moves CIM and CIM hold. Fischers result
Fischers dissertation gives an axiomatization of the category that is associated to regular
isotopy classes of knotted surface diagrams. is represented by ribbon diagrams to simplify
movie diagrams. These EBCs can be written in terms of charts. If R is thought of from R as
crossing the strings of A over those of B. They are in turn represented by braid words. In
addition. In particular.. or the right hand side. In ribbon diagrams. . Algebraic interpretations
of braid movie moves Our motivation in dening braid movies and their moves was to get a
more algebraic description of knotted surfaces. a braid word. these morphisms are
isomorphisms. Scott Carter and Masahico Saito ples of axioms in that are to be satised by
morphisms. The polytopes are relations satised by compositions of morphisms. Fischers
theorem means that if one has an RBSMcat then one automatically has constructed an
invariant of regular isotopy classes of knotted surface diagrams. we need a braiding R B A A
B that also gives a morphism R to the identity morphism on A B. Ribbons represent parallel
arcs as in the classical case. That is. then R crosses Bs strings over As. Then two ribbon
movies are equivalent if and only if the resulting charts are chart equivalent. it is hoped that
this description can be dened and studied in a more general setting . Thus we can check the
equality by means of chart moves. Therefore each ribbon diagram represents a composition
of edges in the polytopes. Then each morphism. J. semistrict monoidal category FRBSMcat
generated by a set.. this means that two ways of deforming ribbon diagrams have to be
equal. we can decompose the polytopes into simpler pieces that correspond to the polytopes
of the braid movie moves. Each frame in a movie can be written as a product of braid
generators while each morphism can be decomposed as a sequence of EBCs. to the
braiding R AB B A. Remark The existence of a braided monoidal category is not enough to
conclude that one has an invariant of knots. The ribbon diagrams surrounding the polytopes
represent braid movies. The conditions say that these two ways are equal. A morphism is a
face in the polytopes. There are two ways to deform by braid movie moves one the top
ribbon diagram to the other the bottom by going along the left hand side of the polytope. and
he shows that it is the free.
ri . . where F is the free group on X. Braid Movies. where details appear. we work towards
this goal. A Peier insertion inserts an adjacent pair a. and Beyond for example. . R. A
collection of disjoint oriented arcs labelled by elements of X that either form simple closed
loops or join points on the boundaries of two or possibly one of the . for any group. ui . . ri . A
presentation of G is a triple X.Knotted Surfaces. ui u wri ui . Also. .. and G is isomorphic to
F/N wR. we observe close relations between braid movie moves and various concepts in
algebra and algebraic topology. Let H be the free group on Y R F . it is desirable if there is an
algebraictopological description of braid movies generalizing the fact that a classical braid
represents an element of the fundamental group of the conguration space. w R F is a map.
Identities among relations Let G be a group. . .. ui . w such that R is a set. deletions.
Identities among relations and Peier elements In this section we recall denitions of various
concepts for group presentations following . An identity among relations is any element in the
kernel of . . ui . a into a Y sequence. an where ai ri . n. Pictures for presentations Let X. . .
where N is the normal closure. Peier transformations A Y sequence is a sequence of
elements y a . u u wru. . A picture over the presentation consists of A disk D with boundary
D. A Peier equivalence is a nite sequence of exchanges. . . let H F denote the
homomorphism r. ui ui wri ui or ri . R. i A Peier deletion deletes an adjacent pair a. a in a Y
sequence. ui is a generator or its inverse of H for i . In this section. and insertions to a Y
sequence performed in some order. . k of disjoint disks in the interior of D that are labelled by
elements of R. Then H N wR. The following operations are called Peier transformations A
Peier exchange replaces the successive terms ri . . A collection . . w denote a group
presentation. .. ri . ui with either ri . . .
From a picture a Y sequence can be obtained. . An incoming edge is interpreted as having a
positive exponent. The arcs are the inverse images of interior points on the edges of the
complex. starting from the base point and reading around the boundary of any in a
counterclockwise fashion. Pictures for group presentations A base point for each of the and
one for the big disk D. Furthermore. The dotted arcs indicate how to obtain a Y sequence
from a picture. J. the labels on the edges that are encountered should spell out either the
relation R or its inverse that is associated to the small disk. that is dened up to homotopy.
Finally. A picture determines a map of a disk into BG. the classifying space of the group with
given presentation. The base point on a is distinct from the endpoints of the connecting arcs.
and an outgoing edge has a negative exponent. These arcs do not intersect in their interiors.
for each disk a dotted arc is chosen that connects the base point of the small disk to that of
the larger disk. The Y sequence obtained from a spherical picture is an identity sequence in
the sense that the product a an in the free group. and by transversality if such a map is given
then there is a picture the little disks are the inverse images of regular neighborhoods of
points at the center of the disks that correspond to relators. Scott Carter and Masahico Saito
Fig. In a spherical picture is dened as a picture in which none of the edges touch the outer
boundary. Figure illustrates a spherical picture. Namely. the dotted arcs are chosen to be
transverse to the arcs that .
CIM. The correspondence is given by taking duals. where r R and u F . but corresponds to
CIM and CIR combined with CIM. Peier exchanges correspond to locality equivalences
because ordering the vertices is tantamount to choosing a height function. . CIR looks like
straightening up a cubic curve. This is a direct analogue of the CIM move on charts. they
dene deformations of pictures. . and this gives the ordering of the Y sequence. Moves
corresponding to Peier transformations Braid movie moves CIM and CIR are used to perform
Peier insertions and deletions as noted in the previous section.. are the dual cycles in the
Cayley complex. it is noted that an elementary Peier exchange corresponds to rechoosing a
pair of successive arcs from the base points of a pair of to the base point of D. Braid Movies.
the hexagonal prism. The element r is the intersection sequence on the boundary of read in a
counterclockwise fashion. Moves corresponding to cycles The chart moves CIR.
respectively. The rst type is to insert or delete small circles labelled by the generating set.
and Beyond interconnect the disks. surgery between arcs with compatible labels and
orientations is analogous to the CIM move. and what the others correspond to. but oppositely
oriented. They provide a diagrammatic tool for the study of groups from an
algebraictopological viewpoint. Moves on surface braids and identities among relations In
this subsection we observe which moves on surface braids correspond to Peier
transformations dened above. Other CI moves These correspond to rechoosing the dotted
arcs in the homotopy classes relative to the endpoints. and CIM are identities among
relations that correspond to nondegenerate cycles in the Cayley complex for the standard
presentation of the braid group.Knotted Surfaces. The cube. To each dotted arc we
associate a pair r . u.. Third is an insertion or deletion of a pair of small disks labelled with the
same relator. and the permutohedron. The collection of dotted arcs is ordered in a
counterclockwise fashion at the base point. For example. The element u is the word in the
free group corresponding to the intersection sequence of the dotted arc with connecting arcs.
.. Furthermore. There are three types of moves to pictures. . The exponent on a letter is
determined by an orientation convention. CIR. Thus if . In . Thus pictures can be used to
study Y sequences and their Peier equivalences. Secondly. joined together by arcs
corresponding to generators that appear in the relator. This is an isolated conguration disjoint
from other components of the graph in the picture.
In summary. Since BG . Consider a small disk B centered at a black vertex v. Note here that
S is trivial i. add cells to kill the second homotopy group of the skeleton thus constructed.
Proposition . S is an unlinked collection of n spheres. This generalizes Kamadas denition of
symmetric equivalence for charts. Furthermore. then this move corresponds to rechoosing a
dotted arc so that it intersects the edge with two less intersection points. CI moves of isotopy
of surface braids correspond to picture moves of BG .. braid movie moves correspond to
Peier transformations. J. . In particular. This is the same as a sequence of CI moves realizing
an isotopy of surface braids from S to the trivial surface braid whose chart is the empty
diagram. . In other words. Let G be a braid group Bn on n strings and BG its classifying
space constructed from the standard presentation. Symmetric equivalence and charts Fix a
base point on the boundary of the disk on which the chart of a surface braid lives. Pick a
point d on the boundary of B which misses the edge coming out of v. More precisely. the arc
misses the vertices and crossings of the chart and intersects the edges transversely.
Suppose there are black vertices. vertex. and continue killing higher homotopies to build BG.
Note that pictures are dened for maps of disks to BG as in the case of maps to the Cayley
complexes by transversality. such that the picture corresponding to f is the same isotopic as
a planar graph as C. attach cells corresponding to relators along the skeleton. Then there is
a map f from a disk to BG. Let C be a chart of a surface braid S without black vertices. this
deformation is interpreted as identities among relations. add n oriented edges corresponding
to generators. f D .e. BG is constructed as follows start with one vertex. . Connect d to the
base point by a simple arc which intersects the chart transversely. or rechoosing the dotted
arcs in pictures. this picture is deformed to the empty picture by a sequence of picture
moves. This rechoosing in turn corresponds to changing the words ui in the terms ri . or cells
of BG. Specically. ui of the Y sequences. we interpret charts without black vertices as null
homotopies of the second homotopy group of the classifying space. Other moves can be
interpreted similarly. since dotted arcs correspond to these words. Scott Carter and
Masahico Saito the xaxis in the gure is a part of a dotted arc. D BG. These moves
correspond to CII and CIII moves on charts which are the moves involving black vertices.
Symmetric equivalence and moves on surface braids In this section we consider moves on
group elements that are conjugates of generators.
y X are generators and V is a word in generators. Kamada dened symmetric equivalence
between words that are conjugates of generators in the braid group. Then go around v along
B counterclockwise. We indicated the word changes that correspond to CII and CIII moves.
Let A be the set of all the words in X i. Then before/after the CII move. where also denotes
and i j gt . Let G X R be a nite Wirtinger presentation. and Beyond Starting from the base
point. Dene symmetric equivalence on A as follows Two words Y xY and Z xZ are
symmetrically equivalent if Y and Z represent the same element in G. Either of these moves
corresponds to sliding the disk by homotopy across the polygon representing the given
relation. elements in the free group on X of the form w Y xY . He proved that two such words
are symmetrically equivalent i they represent the same element in the braid group. where i is
the label of the edge coming from v. Consider a chart with black vertices. we consider
Wirtinger presentations of groups. and then he used this theorem to prove that C moves
suce. and go back to the base point. Thus each of the relators is of the form r y V xV . This
gives a picture associated to a chart that has black vertices. . where i j . The equivalence
relation reects the changes above. where x. The chart moves CII and CIII have the following
interpretation as moves on pictures. travel along the arc and read the labels as it intersects
the edges. . Chart moves that involve black vertices Charts that have black vertices can be
interpreted as pictures in the sense of as follows. and . Braid Movies. and Y is the word we
read along the arc.e. Cut the disk open along the arcs dened above to obtain a picture in
which the boundary of the disk is mapped to the product of conjugates of generators. The
picture represents a map from the disk into BG where G is the braid group. the word thus
dened changes as follows Y i Y Y j i j Y. . Similarly a CIII move causes the word change Y i Y
Y j i j i i Y. Generalizations of Wirtinger presentations Generalizing Kamadas theorem.. . This
yields a word w in braid generators of the form of a conjugate of a generator w Y i Y . The
words on the boundary of the disk change as indicated above... where x X and Y is a word in
X. Kamadas denition of symmetric equivalence is dierent from the above. depends on the
orientation of this edge.Knotted Surfaces.
consider small disks centered at each univalent vertex and simple. our data consist of a
picture on a disk where univalent vertices are allowed. In terms of pictures. We generalize
pictures for group presentations to include univalent vertices. then we can diagrammatically
represent the above two changes in terms of pictures. Scott Carter and Masahico Saito Fig.
More precisely. J. if we allow univalent vertices in pictures. To make homotopytheoretic
sense out of such pictures. . a small disk centered at each univalent vertex with a specied
point on the boundary of each disk which misses the edge coming from the univalent vertex.
mutually disjoint arcs connecting them to a base point. and if we dene arc systems
connecting univalent vertices to a base point on the boundary of the disk. . They are depicted
in Fig. These arcs miss vertices of the picture and intersect the edges transversely.
Generalized chart move Two words Y yY and Y V xV Y are symmetrically equivalent if r y V
xV R where x. y X. Such a generalization of pictures will be discussed next. . Two words in A
are symmetrically equivalent i they are related by a nite sequence of the above changes. a
system of mutually disjoint simple arcs such that for each univalent vertex a simple arc from
this system connects the specied point on the boundary of the small disk to a xed base point
on the boundary of the big disk where the picture lives.
Go around the vertex along the boundary of the small disk and come back to the base point
to get a word w . . . This has the following meaning as explained in the case of braid groups
above. Given a generalized picture. . Here we remark that Kamadas theorem on symmetric
equivalence seems to be generalized to other groups. The moves in this case would consist
of the moves that correspond to Peier equivalences and the moves on pictures described in .
. This edge path represents the product w wn in G BG which was assumed to be the identity.
. Then we trace a up to an to get a sequence w . The transverse preimages of points in cells
resp. the group with which we started. Therefore the map on the boundary of the cutopen
disk extends to a map of the big disk. and Beyond Let a . Furthermore we can generalize
braid movie moves to such sequences for Wirtinger presented groups by examining moves
on pictures in the presence of height functions. we can dene sequences of words for
Wirtinger presented groups generalizing braid movies. . We further discussed potential
generalizations to Wirtinger presented groups. n. We conclude this section with remarks on
these aspects. charts and classifying spaces of braid groups. vertices of the rest of the
picture. Call this a generalized picture. . Now we require that the product w wn is in G. we do
not want to allow cancelling black vertices since if we did charts represent just null
homotopies in the case of braid groups and we instead want something highly nontrivial.
Kamada used Garsides results . Starting from the base point. where xi is the label of the
edge coming from the ith univalent vertex. . cells gives edges resp. . . By xing a height
function on the disk where pictures are dened. remove the interior of small disks around each
univalent vertex. to the skeleton of BG. Thus we need restrictions on homotopy classes of
maps from cutopen disks to classifying spaces to be able to dene nontrivial homotopy
classes to represent surface braids. More natural interpretations are expected in terms of
other standard topological invariants. . and Garside gave other groups to which his
arguments can be applied.Knotted Surfaces. Each word is in a conjugate form wi Yi xi Yi i .
the moves that correspond to cycles in the Cayley . Braid Movies. the moves that are
dependent on the choice of height functions. and null homotopy disks in such spaces. . an be
the system of arcs dened above. read the labels of the edges intersecting a when we trace it
towards the univalent vertex. . However. wn of words. Keeping base point information is such
an example. Concluding remarks We have indicated relations among braid movie moves.
Then dene a smooth map from the boundary big disk cut open along dotted arcs connecting
small disks to the base point. In the preceding section we discussed null homotopy disks in
classifying spaces in the presence of generalized black vertices.
Carter. . Knot Theory Ramications . Proc. Dennis McLaughlin. V. Dan Flath. and
Huebschmann. Acknowledgements We thank the following people for valuable conversations
and correspondence John Baez. Scott and Saito. Chiral Potts Models as a Descendant of
the SixVertex Model. J.. Preprint. Scott and Saito. The algebraic signicance of such
sequences of words and relations among them deserves further study. . Scott and Saito. . G.
London Math. YangBaxter Equation in Integrable Systems. Masahico. Seiichi Kamada. Phys.
Preprint. . AMS . Carter. Brown. . and Thickstun. in Brown. Masahico Reidemeister Moves
for Surface Isotopies and Their Interpretation as Moves to Movies. Syzygies among
Elementary String Interactions in Dimension . Knot Theory Ramications . Carter. Bazhanov.
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Their Skeins. Singapore.. Baxter. since this is a natural context in which the braid movie
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pp. J. Some of the material here was presented at a special session hosted by YongWu
Rong and at a conference hosted by David Yetter and Louis Crane. J. and Kunio Murasugi.
Math. Singapore. . Financial support came from Yetter and Crane and from John Luecke. M.
V. YangBaxter Equation in Integrable Systems. . . Commun. On Zamolodchikovs Solution of
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