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Geometrie Aspects of Quantum Field Theory
Graeme Segal
Mathematical Institute, 24-29 St. Giles, Oxford OX1 3LB, England
Quantumfieldtheory has been the basic tool of particle physics for more than half
a century, but unlike earlier such tools it has not been accompanied by a satisfying
mathematical theory. Recently this has begun to change. One reason is that the
ideas of quantum field theory have turned out to shed light on purely mathematical
questions. These applications are my subject today. So far, nevertheless, the field
theory has played either a heuristic or an explanatory role in the mathematics, and
the actual theorems can, and often must, be proved by other means. I hope that
this will be less true as the mathematics of field theory becomes better developed.
Meanwhile I shall just indicate some areas where field theory and geometry have
come together, trying to illustrate the point of view rather than formulate theorems.
For the most part I shall be summarizing other people's work, predominantly
Witten's.
§1. The Framework
In his address to the Berkeley ICM Witten described d + 1 dimensional quantum
field theory as follows. One considers "fields" defined on some class of oriented
rf -h 1 dimensional manifolds M. A "field" might mean a map from M to some
auxiliary manifold X, or a section of some natural fibre bundle on M, or even an
equivalence class of such sections. In any case one has a space F(M) of fields for
each compact manifold M with boundary. Afield/ e F(M) has a boundary value
f\dM which belongs to some space F0(dM) of fields on the boundary. We also
suppose given an "action" functional S : F(M) -> R, defined uniformly for all M.
Then field theory is the study of the functions XPM on F0(dM) of the form
»V(/o)=[
e-™af,
(1.1)
JF(Mi/o)
where F(M;f0) = {fe F(M) \f\dM = f0}. More generally, if the boundary dM =
£0 II £t consists of an incoming part £0 and an outgoing part Z\ we are interested
in operators WM : HEQ -• HSi9 where HZi is a space of functions on F0(Lt)9 on the form
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
© The Mathematical Society of Japan, 1991
1388
Graeme Segal
KM{fo,fi)+ifoWo,
(**V)(/i) =
F0(£0)
where
e~™®f.
^MC/OJ/I) =
(1.2)
F(M; fo.fi)
(A boundary component is 'outgoing' or 'incoming' according as its orientation
agrees or not with that of M; and E0 denotes E0 with reversed orientation.)
The preceding formulae are only schematic, and so far it has proved impossible
to develop an integration theory of the type needed. But let us at least try to abstract
the essential structure. It comprises
(i) a vector space HE for each closed oriented d-dimensional manifold with
whatever structure is appropriate;
(ii) a bilinear pairing H^ x HE-* <C;
(iii) an element *FM e HdM for each d + 1 dimensional manifold M with appropriate structure.
The most obvious properties these data should have are
(a)
HziUz2 = HLi®Hz2
and
^MiUMj
=
y M i ® ^M 2 i
(in particular Hz = (C when E = 0 , and so WM e <C when M is closed.)
(b) if two components Ex and E2 of the boundary of M are sewn together by
an orientation-reversing diffeomorphism to form a new manifold M such that
dM = E1UZ2UdM then the map HdM-+HdM defined by the bilinear pairing
takes WM to ÎP^.
In particular, when dM = 0 and WM is regarded as an operator HE^ -> HSj9
property (b) asserts that
t r a c e d ) =¥^e<C.
I do not know how far an axiomatization of this kind is appropriate or helpful
in traditional quantumfieldtheory, but with some especially simple kinds of theory
it works well and is a useful tool in geometry, rather like a new kind of cohomology
theory. I shall mention some limitations of the framework in §§ 5 and 6 below.
A feature of each of the examples I shall describe is that either the phase space
isfinitedimensional because of the presence of a large group of gauge symmetries,
or else the path integral (1.1) reduces to a finite dimensional integral because the
integrand is an exact differential form outside afinitedimensional submanifold of
the space of fields (i.e. "the stationary phase calculation is exact"). One might take
this to mean that genuine quantum field theory is not involved. A more optimistic
moral, however, is that one can sometimes best studyfinitedimensional problems
by the infinite dimensional methods offieldtheory.
Geometrie Aspects of Quantum Field Theory
1389
§ 2. Index Theory and the Elliptic Genus
A particle moving in a Riemannian manifold X affords the simplest example of the
path integral idea, and can be regarded as a 0 + 1 dimensional field theory. The
action for a path y : [0, T] -• X is S(y) = i J J ||y'(r)ll2 dt. For a point P the vector
space HP is L2(X), and to the 1-manifold [0, T] is associated the heat operator e~TA
in Hp, where A is the Laplacian of X. The formula (1.2) is then the usual path integral
representation of the heat kernel; and if we replace [0, T] by a circle ST of length
T we have a formula for trace(e _7M ) as an integral over the loop space ££TX =
Map(S r ; X). This path integral does not reduce to a finite dimensional integral.
The position is different and more relevant if we replace the action S : 3?TX -• IR
with the inhomogeneous differential form S = S + co, where œ is the 2-form on 3?TX
which to two deformations <!;, q of a loop y assigns the number
T
co(y; & ri) =
<«*), ij'(t)> A.
(Here rç'(t) is the covariant derivative.) (*) Witten observed (see [2], [15]) that the
action S corresponds to the 0 + 1 dimensional field theory for which HP is the mod 2
graded space of L 2 spinor fields on X, while the operator associated to [0, T] is the
spinorial heat operator e~Tâ. The graded trace (or "supertrace") tr(e~TA) is now
independent of T, and is the index of the Dirac operator on X. This is a topological
invariant of X called its/-genus. On the other hand the top degree component of
the differential form e~s on 1£TX is exact outside the finite dimensional manifold
of point loops, so by Stokes's theorem the path integral can be reduced to an integral
over X (identified with the point loops). The outcome is the Atiyah-Singer formula
for the index of the Dirac operator. The elaboration of this idea was described by
Bismut [8] at the Berkeley ICM.
So far we have been dealing with well-known material. But we can go on to
consider a 1 + 1 dimensional theory whose action is a differential form on the space
F(E) of maps from a surface E to X. Then the vector space Hs associated to a circle
S will be the space of L2 spinors on the loop space SâX. When E is a torus the path
integral over F(E) is called the elliptic genus eE(X) oiX. (In fact there are a number
of variants, applying in slightly different situations.) The 0 -h 1 dimensional result
that the supertrace of e~TA was independent of T has the analogue that e£(X)
depends only on the conformai structure of E, i.e. for each X it is a modular function
on the upper ^-plane. The elliptic genus can be interpreted formally as the equivariant index of a version of the Dirac operator on the manifold X. As with the
/4-genus the path integral defining eL(X) collapses to an integral over X, and this
expresses it in terms of the characteristic numbers of X already familiar in algebraic
topology. Nevertheless the elliptic genus has striking and unexpected properties,
especially in connection with the topology of circle actions, and it stimulated the
discovery of elliptic cohomology, a new theory whose true nature remains obscure.
(An account of this subject can be found in [22]. Cf. also [1], [26]).
* More accurately, we replace e S(y)@y by e s. The integral of an inhomogeneous form means
the integral of its component of top degree.
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Graeme Segal
§ 3. Topological Field Theories
Afieldtheory is topological if it is defined for smooth manifolds with no additional
structure (apart from a question of fixing projective multipliers which I shall
suppress in this talk.) The vector spaces HE must then be finite dimensional. A
discussion of the formal properties can be found in [4].
(a) 1 + 1 dimensional Theories. These are completely described by giving a
commutative ring A with a F together with a linear map 0 : A -> C such that the
bilinear form (a, b)\-*9(ab) is non-degenerate. In fact A = Hsi, and the product
A (g) A -> A is WM, where M is a disc with two holes.
(b) 2 + 1 dimensional Theories. These are by far the most studied, and the
structure is much richer: it appears to be roughly equivalent to a quantum group.
A theory gives us an invariant for each closed 3-manifold, and a representation of
the mapping class group of each closed surface. If we choose an element Ç e i/ s i xS i
we get an invariant fa(K9 M) e C for each knot K in a 3-manifold M by defining
^(X,M) = <^,'PM_t/>,
where U is a tubular neighbourhood of K. In Reshetikhin's, Turaev's, and Feigin's
talks at this Congress we heard how the same output arises from a quantum group.
The relation between the two approaches does not seem completely understood,
but I shall say a little more about it in § 4 below.
2 + 1 dimensional theories are important because there is a supply of natural
examples which lead to the knot invariants of Vaughan Jones and others. There is
a theory for each compact Lie group G and choice of "level" k. The level is an element
k e H4(BG; Z), i.e. an integer if G is simple and simply connected. Regardingfcas a
characteristic class for G-bundles there corresponds to it a secondary Chern-Simons
characteristic class Sk with values in R/2TCZ which is defined on the space F(M) of
isomorphism classes of G-bundles with connection on a 3-manifold M. This, or
rather iSk, is the action defining the theory [33]. But the theory can be constructed
without mentioning path-integrals in the following way.
The vector space HE associated to a surface E is the "quantization" of the
symplectic manifold ME of flat G-bundles on E. (This is the symplectic quotient [5]
of the space of all connections on E by the action of the gauge group.) The symplectic
structure of JiE depends on the level: its class is the image offcunder the transgression HAr(BG) -> H2(JtE). To obtain a definite quantization one method is to
(i) choose a complex structure on E,
(ii) identify ME with the moduli space of stable holomorphic G-bundles on E
by the Narasimhan-Seshadri theorem [23], thereby giving JiE a Kahler structure,
(iii) represent the symplectic form as the curvature of a holomorphic line bundle
L on JtE, and
(iv) define HE as the space of holomorphic sections of L.
One must show that HE is essentially independent of the complex structure chosen.
Even after that one must construct the vectors WM associated to 3-manifolds. No
natural way of doing this is known, though in general terms one can say that if
Geometrie Aspects of Quantum Field Theory
1391
E = dM then the boundaries of flat G-bundles on M form a Lagrangian submanifold in JfE, and this should define a vector in the quantization HE. But the
connection of the spaces Jt E with 3-manifolds was a great surprise, for they arise
more obviously from 1 + 1 dimensional conformai theories, as we shall see below.
The application of this theory to the study of knots and 3-manifolds is discussed
elsewhere at this Congress, so here I shall just emphasize that it has led to many
new results about the geometry of the spaces J4E, notably Verlinde's beautiful
formula [30] for the dimension of the space HE. By applying quantum field theory
to JtE in a slightly different way Witten has recently been led to conjecture a formula
for the volume of Jt.'E in terms of ÇG(2g — 2), where g is the genus of E, and
Us) = 27(dim V)~s,
the sum being over the irreducible representations V of G.
(c) 3 + 1 dimensional Theories. For each compact group G there is an important
3 + 1 dimensional theory [3, 13, 17] which assigns to a closed 4-manifold W its
Donaldson invariant, i.e. (roughly) the number of "instantons" on. W. (An instanton
is a solution of the self-dual Yang-Mills equations.) This theory was described in
Floer's talk at this Congress. The vector space HM for a 3-manifold M is the Floer
cohomology group defined by applying infinite dimensional Morse theory to the
space F(M) of isomorphism classes of G-connections on M, the Morse function
being the circle-valued Chern-Simons form already mentioned. A 4-manifold W
with boundary M has a relative Donaldson invariant in HM. Unfortunately field
theory, although strikingly exemplified here, has not so far helped much with the
geometry, except insofar as it is a field-theoretic idea to study the instanton moduli
spaces in terms of the space of all connections.
I should say a word about Floer cohomology. The infinite dimensional manifolds F which arise in field theory are usually polarized, in the sense that their tangent
spaces are roughly decomposed into positive- and negative-energy halves. (Cf. [26]
§ 4.) Floer's Morse function defines a decomposition of this kind, into the positive
and negative eigenspaces of the Hessian. For such a manifold F one expects to be
able to define "middle dimensional cohomology", by considering infinite dimensional cycles whose tangent spaces roughly fill the negative half of the tangent spaces
to F. This idea goes back, of course, to Dirac's treatment of electrodynamics in terms
of a sea of negative energy electrons. The same idea has been formalized by Feigin
in his "semi-infinite" cohomology of Lie algebras [14]. Apart from the space of
connections above, Floer cohomology has also been applied to the loop space of a
symplectic manifold [12], and there too it arises as the state space of a field theory,
the 1 + 1 dimensional topological <7-model of [32].
§ 4. 1 + 1 Dimensional Conformai Field Theory
Conformai field theory is akin to topological field theory in the sense that, up to
isomorphism, a compact surface has only a finite dimensional space of conformai
structures. Conformai theories can be axiomatized in the same way as topological
ones [27, 28]. They have been much studied since the influential paper [7], partly
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Graeme Segai
for their relevance to string theory, but also because of their role in at least three
areas of mathematics:
(i) the representation theory of loop groups and of Diff^1),
(ii) the study of the moduli spaces of Riemann surfaces and holomorphic
bundles, and
(iii) the construction of the monster simple group and its representations.
For the third of these areas I refer to [15]. A slightly more conventional
approach to conformai field theory is summarized in [18].
A conformai theory consists of a vector space H naturally associated to the
standard circle S1, together with an operator WE : H®m -> H®n for each Riemann
surface E with m incoming and n outgoing parametrized boundary circles. Thus
Diffus1) acts (projectively) on H, and so does the semigroup sé of surfaces which are
topologically cylinders. (The composition-law is sewing end-to-end.) The semigroup
sé has twice the dimension of Diff^S1), and is a complex manifold. One of the
important ideas of the theory is that sé plays the role of a complexification of the
group Diffus1): more precisely, the relation between them is the same as that between
the unitary group Un and the semigroup {g e GL„((C) : \\g\\ < 1} of contraction
operators. (Cf. [28] and also Neretin [24].)
Let us recall that the loop group S£G of a compact group G has an interesting
class of irreducible projective representations {HktV} - the positive energy representations - which are parametrized by their levelfce H*(BG; Z), which describes
the projective multiplier of the representation, and an irreducible representation V
of G. (The image offcin H2(J£G) is the class of the circle bundle defined by the
central extension.) For a given levelfconly afiniteset of representations Fcan occur.
An important fact about the representations Hkv is that they possess a canonical
projective action of Diff^1) intertwining with that of «JSfG. This action extends to more accurately, is the boundary value of- an action of sé by trace-class operators
i
FA:H-^H. In fact WA is characterized by intertwining with the group GA of
holomorphic maps A -» Gc, which acts on the source and target of WA via restriction
to the two ends of A. The remarkable fact is that the irreducible representations of
a given level constitute something very close to a conformai field theory. To state
this precisely one needs the concept of a modular functor [27,28]. (In the literature
modular functors are usually referred to as "conformai blocks" [7] or "solutions of
the Knizhnik-Zamolodchikov equations". For the latter, see Varchenko's talk at
this Congress.)
A modular functor has a finite set 0 of labels. It assigns a finite dimensional
vector space EE to each Riemann surface with boundary where each boundary circle
is labelled with an element of #. The axioms are
(i) EE = C when E is the Riemann sphere,
(ii)
EElUEi^EEi®EE2,
(iii) E è = ® EEt<£ where (27, (j)) is obtained from È by cutting it along a simple
closed curve and giving both new boundary circles the label <j>.
For the application 0 is the set of irreducible representations of £t?G of a given
level. There is a modular functor E such that when E is a Riemann surface with m
Geometrie Aspects of Quantum Field Theory
1393
incoming and n outgoing boundary circles labelled with representations
Ham and Hßi,..., Hßn there is an operator
Hai,...,
¥E^.Hai®--®Ham^Hßi®"-®Hßn
for each £, e EE which intertwines with the action of the group GE of holomorphic
maps E -> Gç. (In fact EE can be defined as the space of such intertwining operators:
then the point to establish is property (iii) above, which amounts to a version of the
Peter-Weyl theorem for loop groups.) The first complete proof of this result is in
[29], (Cf. Tsuchiya's talk at this Congress.)
One of the advantages of the field-theoretic viewpoint in the representation
theory of loop groups is to make plain the otherwise mysterious modularity properties of the characters: in field theory the values of the characters are naturally
associated to complex tori.
Witten realized [33] that the modular functor EE just described is essentially
independent of the complex structure of E, and is the state space of the corresponding 2 + 1 dimensional topological theory based on the Chern-Simons action.
More recently Kontsevich [21] has sketched an argument to show, still more
surprisingly, that the concepts of modular functor and 2 + 1 dimensional topological theory are exactly equivalent.
A "topological" modular functor is closely related to a quantum group, for the
quantum deformation of G amounts essentially to a way of defining an exotic tensor
product on the category of representations of G. For a modular functor E we can
define
Vi®EV2 =
w
®EEtVuVi,w®W,
where W runs through the irreducible representations of G, and E is a disc with two
holes whose boundary components are labelled Vx, V2 (incoming) and W (outgoing).
It is easy to relate the modular functor EE to the space HE = r(JtE, L) of
holomorphic sections described in § 3. Let us decompose the closed surface E as
E1 u E2 by a simple closed curve S. A holomorphic bundle on E is automatically
trivial on Ex and E2, so it can be described by a clutching function on S, i.e.
by an element of J^G^. The set of isomorphism classes of bundles on E - essentially the same as JtE - is therefore the double coset space GEi\^,G(C/GE2, and
the space r(JtE, L) is the GEi-invariant part of H = F(^£G^jGE2, n*L), where
7i : JSfGc/G^ -> JtE. If we now take E2 to be a standard disc then H is the basic
representation of JSfG of levelfc,constructed by the Borei-Weil method [25]. Finally,
it is easy to see that when the boundary of Ex is labelled with H we have EE = EEi,
and so
EE s HG*> £ r(J/E; L).
The preceding argument, which shows how representations of i?G define functions on the moduli space of G-bundles, also shows how representations of Dif^S1)
give functions on the moduli space (€E of complex structures on a smooth surface E.
For (€E behaves like a double coset space of the semigroup sé: if we write E =
ExuE2, and choose fixed complex structures on E1 and E2, then E1 u A u E2 runs
through an open set of <ßE as A runs through sé. The representation theory of
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Graeme Segal
Diff^S1) allows us, for example, to identify and classify holomorphic line bundles
on #£ much more simply than does conventional algebraic geometry. (Cf. [6,28].)
§ 5. Zamolodchikov's c-Theorem
The point of view of this talk is successful with topological and conformai field
theories, but so far it has never been taken seriously in a wider context. At present
the only general definition of a field theory is the classical one in terms of the vacuum
expectation values of a class of operators varying from theory to theory. This is not
sufficiently manageable for one to be able to speak, for instance, of the "space 5" of
all 1 + 1 dimensional theories". Nevertheless one of the most interesting recent
developments has been the following result of Zamolodchikov [36], which is framed
in terms of the space 0~.
Whatever may be the definition of a theory, it will presumably be true that from
any theory T one can derive a 1-parameter family of theories Tx (for X e R x ) simply
by multiplying all lengths by X. The resulting flow on 0' is the renormalization group
flow. Conformai theories are fixed points of this flow. Zamolodchikov's idea is to
define a Riemannian metric on 0' and a smooth function c : 0~ -» R such that
(i) the renormalization group flow is the gradient flow of c, and
(ii) c(T) is equal to the central charge if Tis a conformai theory.
The central charge of a conformai theory is the number describing the central
extension of Diffi^S1) which acts on the state space of the theory.
It is fairly straightforward to calculate the possible non-conformal infinitesimal
deformations of a conformai theory, so Zamolodchikov's theorem suggests that one
could in principle discover the global topology of the space 0~ by Morse theory.
Vafa and others have made some steps in this direction.
Zamolodchikov's argument is based on perturbation theory. I think it is a
fascinating challenge to put it in a better mathematical setting.
§ 6. 1 + 1 Dimensional Quantum Gravity
The most dramatic recent development in quantum field theory has been a breakthrough in 2-dimensional quantum gravity. It has suddenly appeared possible to
perform integrals over the space of all metrics on a surface and get very explicit
answers. The main success has come from the technique of random matrices [9,11,
19], and I cannot discuss it here. One very unexpected outcome is to link the theory
with the classical completely integrable systems of non-linear partial differential
equations such as the KdV equation. So far the situation has not really been
assimilated mathematically, but the results seem to describe the algebraic topology
of the space of metrics on a surface, or - equivalently - the moduli spaces of complex
structures.
From ordinary algebraic geometry one knows (see Morita's talk at this Congress) a ring of stable cohomology classes on the moduli spaces Jiq of surfaces of
genus g ("stable" means that they are defined independently of g), and Witten has
Geometrie Aspects of Quantum Field Theory
1395
claimed that the field-theoretic results are the integrals of these classes over the
spaces J4.g, i.e. the characteristic numbers of J4r This has led him to the striking
conjecture [34] that the generating function for the characteristic numbers is a
certain specific solution of the KdV hierarchy. His lectures [35] give an excellent
account of the present state of the subject.
From the point of view of this talk it is interesting that quantum gravity does
not fit directly into the framework of § 2 above, but nevertheless seems likely to be
axiomatizable along related but more subtle lines. The crucial point that distinguishes the gauge-theory situations to which §2 applies from the gravitational ones
to which it does not is simply that an automorphism of a bundle P on Mx u M2 is
just a pair of automorphisms of P\M1 and P\M2, but a diffeomorphism of Mi u M2
cannot be broken into two diffeomorphisms.
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