Download Quantum Moduli Spaces 1 Introduction

Quantum Moduli Spaces
L. Chekhov
Steklov Mathematical Institute,
Gubkina 8, 117966, GSP{1, Moscow, Russia
Abstract
Possible scenario for quantizing the moduli spaces of Riemann
curves is considered. The proper quantum observables are quantum
geodesics that are invariant with respect to a quantum modular group
and satisfy the quantum algebra.
1 Introduction
A problem of constructing an appropriate quantum analogue of moduli spaces
of various structures related to the higher genus Riemann surfaces drew much
attention. In 1], the gauge theory on graphs were associated with the moduli
space of at connections. Amazingly, the moduli spaces of complex structures
of Riemann surfaces had not been adequately quantized. The main trouble is
the huge discrete symmetry group (the modular group) of the moduli space.
So, the moduli spaces are orbifolds of very involved structures. A convenient
way to deal with these symmetry groups is to consider a triangulation of the
moduli space a la Penner and Kontsevich 2, 3]. This triangulation admits the
description in terms of graphs, which we extensively use in the present paper.
Besides, in classical case, the Kontsevich matrix model was constructed 3]
that generate all observables for 2D quantum gravity with matter.
The question of physical content of such quantum systems remains open.
Indeed, already classical cohomology theory of moduli spaces of Riemann
E-mail: [email protected].
1
surfaces provides a description of quantum topological 2D gravity therefore,
the extra quantization procedure could be related to quantizing the 2D space{
time itself.
2 Classical moduli spaces
Recall briey a classical description of Teichmuller spaces of complex structures on Riemann surfaces with holes (punctures).
Teichmuller space T h is a space of complex structures on S modulo dieomorphisms homotopy equivalent to identity. This complex structure inambiguously determines a metric on the Riemann surface, while, in this metric,
a neighborhood of each hole corresponds to the exponential mapping from
the punctured disc to a half-innite cylinder.
An oriented 2D surface can be continuously conformally transformed to
the constant curvature surface. The Poincare uniformization theorem claims
that any complex surface S of a constant negative curvature (equal ;1 in
what follows) is a quotient of the upper half-plane H+, endowed with the hyperbolic metric, by a discrete Fuchsian subgroup (S ) of the automorphism
group PSL(2 R),
S = H+=(S ):
In the hyperbolic metric, geodesics are either half-innite circles with endpoints at the real line R or vertical half-lines all points of the boundary R
are at innite distance from each other and from any interior point.
Any hyperbolic homotopy class of closed curves contains a unique closed
geodesic of the length l( ) = jlog 1=2 j, where 1 and 2 are (dierent)
eigenvalues of the element of PSL(2 R) that corresponds to .
2.1 Teichmuller space T H
The graph technique
(
S)
of surfaces with holes.
The central point of the construction is a description of the moduli space
T H (S ) in terms of oriented graphs.
Let a fat graph (oriented graph) ; be a graph with the given cyclic ordering of edges entering each (usually, three-valent) vertex. Then, to each
(unoriented) edge we put into the correspondence the real number Z 2 R
2
the set of all such numbers is fZj 2 E (;)g where E (;) is the set of all
(unoriented) edges of the graph ;.
Theorem 1 For a given three-valent graph ; of the given genus g and num-
ber of punctures n, there exists a one-to-one correspondence between the set
of points of T H (S ) and the set R# edges of edges of this graph supplied with
real numbers (lengths).
We propose the explicit way to construct the Fuchsian group (S ) PSL(2 R) such that S = H+=(S ).
To each edge we associate the matrix Xz 2 PSL(2 R) of the Mobius
transform
Z =2 !
0
;
e
Xz = e ;Z =2
:
(2.1)
0
In order to parametrize a path over edges of the graph, we introduce the
matrices of the \right" and \left" turns
!
!
1
1
0
1
2
R = ;1 0 L R = ;1 ;1 :
(2.2)
We introduce now the notion of geodesic on the graph.
Let a closed path in the graph ; be any oriented path, which starts
and terminates at some oriented edge (point) of the corresponding graph.
To each such path we put into the correspondence the product of matrices
Pz1zn = LXzn LXzn;1 RXzn;2 RXz2 LXz1 , where the matrices L or R are
inserted depending on which turn|left or right|the path is going on the
corresponding step.
Proposition 1 6] 1. There is a one-to-one correspondence between the
set of all oriented paths fPz zn g and closed (oriented ) geodesics f g on the
1
moduli space. Moreover, the length L( ) of a geodesic can be determined as
follows:
G( ) 2 cosh L( )=2 = tr Pz1 zn :
(2.3)
2. Considering the universal covering T; of the graph ;, ; = T;=; , where
; is the Fuchsian group of the graph ;, we obtain that there is a one-toone correspondence between the set of all closed geodesics and the primitive
conjugacy classes fi g ; .
3
Summing up, we have the following chain of the one-to-one correspondences:
8
9 8
9 8
9
>
>
conjugacy
<
= >
< closed >
= >
< closed paths >
=
classes
of
geodesics
in
(any
of)
$
$
:
>
>
>
>
>
>
: ( )
:
:
(2 R)
on
the graph ;( )
S
P SL
S
S
2.2 Weil{Peterson forms
A canonical Poisson structure called the Weil{Peterson structure exists on
the set of coordinates fZ g. This structure is degenerate, and the Casimir
functions are just the lengths of geodesics surrounding holes. Fixing all these
lengths, we obtain a symlpectic leaf. On every such leaf, the Poisson structure
can be inverted as to give a symplectic structure. The resulting degenerate
two-form is also called the Weil{Peterson form 2].
Denote (1) the oriented edge that is obtained from an oriented edge
by rotating by the angle 2=3 clockwise around the vertex these edges
are starting with and denote ;1 the oriented edge that are opposite to an
edge . Then, the Weil{Peterson Poisson brackets are
8
1 if 1(1) = or ;1
and 1 6= (1)
>
>
1
;1
>
(1) or (1) and 1 6= (1)
< ;1 if 1 = 1
fZ Z g = > 2 if = (1) or 1(1)
>
;2 if 1(1) = 1 or 1
>
: 0
otherwise:
(2.4)
2.3 Classical modular transformations
In 6], the graph transformations that preserve Poisson structure (2.4) were
obtained. There exists a natural operation called the ip or Whitehead move,
which corresponds to transitions between graphs. We must determine such
a transformation of the variables Z that, rst, preserve the Poisson structure and, second, after a series of ips that transform a graph to itself, the
resulting transformation should be the identity. This implies the presence of
another relation called the pentagon identity. It appears that the transformations depicted in Fig. 1 (2.5) satisfy both the demands.
4
In the classical case 6],
fA B C D Z g ! fA + (Z ) B ; (;Z ) C + (Z ) D ; (;Z ) ;Z g
e C
e D
f Ze g
fAe B
(2.5)
Z
(Z ) = log( e + 1):
Lemma 1 Transformation (2.5) preserves the products over paths, so the
classical geodesic length is a modular-invariant function.
A
B
HH
H
H
Z
HH
H
H
D
C
A + (Z )
A
A
-
AA
D ; (;Z )
B ; (;Z ) ;Z
A
A
AA
C + (Z )
Fig. 1
2.4 Poisson algebra of geodesics
The functions fG g (2.3) were studied in 7]. They generate a Poisson algebra
(w.r.t. the multiplication and the Weil-Petersson Poisson bracket) over Z. (It
means that a product and a Poisson bracket of two such functions is a linear
combination of such functions with integer coe cients.)
Consider now the Poisson structure of geodesics. Two nonintersecting
geodesics have trivial bracket and, due to the linearity property, we may
consider only \simple" intersections of two geodesics G1 and G2 of the form
G1 = tr 1 : : : XC1 R1XZ1 L1XA1 : : : (2.6)
2
2
2
2
2
2
2
G = tr : : : XB L XZ R XD : : : (2.7)
respectively. (Superscripts 1 and 2 label the matrix spaces.)
A Poisson bracket of geodesics is
fG1 G2g = 12 ( tr GH ; tr GI )
(2.8)
5
where GI is the geodesic that is obtained by erasing the edge Z and joining together the edges \A" and \D" as well as \B " and \C " in a natural way the geodesic GH passes over the edge Z twice, so it has the form
tr : : : XC RZ RD : : : XB LZ LA : : :. These relations were obtained in 7] in the
continuous parameterization (the classical Turaev{Viro algebra), and introducing an additional factor|the total number of geodesics #G|we can uniformly present the classical skein relation as
(;1)#G
@
;
@ ;
@
; @
;
@
$' &%
'$
%&
+ (;1)#G
+ (;1)#G
3 Quantization
= 0: (2.9)
Once the graph ; is chosen, the corresponding Teichmuller space T H (S )
can be easily quantized. Consider the -algebra T h(;) generated by real
generators fZhj 2 E (;)g, (Zh) = Zh with the dening relations
Zh Zh] = 2ih$ fZ Z g
cf. (2.4):
(3.1)
obvious
n This algebra has an
o center generated by the perimeters of faces
P j 2 F (;) P = P2 Zh . One can easily describe all irreducible representations of this algebra using the Stone{von Neumann theorem. An
irreducible representaion is unambiguously xed by the values of the operators P . For example one can represent all operators Zh in L2 (Rn), where
n = 21 (E (;) ; F (;)), by linear combinations with rational coe cients standing by constants and the operators xi and i @x@ i , where fxiji = 1 : : : ng is a
standard coordinate system on Rn. Now our task is to identify the -algebras
constructed using dierent graphs ; and ;0 corresponding to a given surface
S . In order to make this identication we just construct a -homomorphism
K (; ;0) : T h(;) ! T h(;0 ) of the -algebra generated by fZh0 j0 2 E (;0)g
to the algebra generated by fZhj 2 E (;)g. We require this homomorphism
to have the following properties:
1. Classical limit. We demand that the algebra homomorphism should
tend to the classical homomorphim of the algebras fo function on T H when
the parameter h$ tends to zero.
6
2. Path independence. We demand that if we have three graphs ;, ;0 and
00
; then the homomorphisms should satisfy the condition K (;00 ;0)K (;0 ;) =
K (;00 ;).
One can check the latter condition only for one distinguished sequence of
ips since others are just compositions of this one. Consider two edges having
exactly one common vertex. Then, a sequence of ve ips of these edges (such
that we never ip the same edge twice consequtively) does not change the
graph. This becomed more geometrically transparent if we consider the dual
graph where the two edges correspond to two edges separating three triangles
forming a pentagon. A pentagon can be cut into three triangles in only ve
possible ways which are related by ips (Fig. 2)
H
C
B HHB4
4 B
HH
H
B
Q
A Q
B 4
A Q
B
4 A Q4Q B 4
Q
I
@
A
@
B QQ
;
HH
H
;
A
B
B HH 3
0
0
3 HH
HH
B
4
0
H
H
H
B
Q
A Q
A
B 3
3
A Q
A
B
0
3
0 A Q0Q
3A B
Q
A
A B
QQ A
A
B
X
D
E
A
Y
X
Y
D
C
A
E
C
1
1 A
Y
E
D1
Fig. 2
E
X
B
B0
AAU
HH A
B
1 H 1
HH
X
1
H
A
A
1A
A
A
Y
D
C3
C
H
2 HH 2
HH
H
A
2
A
- 2A
2
A 2 A
D
E
Y
X
A2
B
(3.2)
The quantum transforms of the edge variables are as in Fig. 1 with the
7
function
Z
;ipz
h
$
(z) = ; 2 sinh(pe) sinh(h$ p) dp
(3.3)
and the contour % goes along the real axis bypassing the origin from above.
The function (3.3) and F (x h$ ),
Z
;ipx
1
F (x h$ ) := exp(; 4 p sinh(pe ) sinh(h$ p) dp)
(3.4)
rst appeared in 8] possess the following properties.
1. hlim
(x h$ ) = log(ex + 1) (classical limit).
(3.5)
!0
@ log F (x h$ ) = (x h$ ):
2. 2ih$ @x
(3.6)
3. F (x + ih$ h$ ) = F (x ; ih$ h$ )(1 + ex ):
(3.7)
1x
(3.8)
F (x + i) = F (x ; i
)(1 + e h ):
4. (x h$ ) = h$ h$1 x h$1 :
(3.9)
Lemma 2 4] Quantum transformations (2.5) satisfy the pentagon identity.1
3.1 Geodesic length operators
The aim of this paragraph is to imbed the algebra of geodesics (2.3) into a
suitable completion of the constructed algebra T h(S ).
The function G (2.3) can be expressed for any in terms of graph
coordinates on T H ,
X 12 P 2E(;) mj ()z
(3.10)
G tr PZ1Zn = e
j 2J
where mj ( ) are certain integer numbers and J is a nite set of indices.
In order to nd the quantum analogues of these functions, we denote by Tb h
a completion of the algebra T h containing exZ for any real x. Let for any
closed path on S , the operator Gh 2 T^ h be
X 21 P 2E(;)(mj ()Zh +2ihcj ())
h
G tr Pz :::z =
e
:
(3.11)
1
1
n
j 2J
2fj g
This result was independently obtained by R. Kashaev 9].
8
Here some quantum ordering is assumed. The main problem is to nd
the ordering that satisfy all of the conditions below. In (3.11), the numbers
mj ( ) are the same as in (3.10) and integer coe cients c
j( ) are to be
determined from the procedure of1 the quantum ordering.
Note that the operators fGh g can be considered as belonging to the
algebra T^ h. In terms of the generators of T^ h they are
1
X 1P
h
Gh = e 2h 2E(;)(mj ()Z +2icj ()) (3.12)
j 2J
Now let us formulate the dening properties of quantum geodesics.
1. Modular invariance. The modular group (S ) (2.5) preserves the set
fGhg, i.e., for any 2 (S ) and any closed path , we have (Gh) = Gh .
2. Geodesic algebra. The product of two quantum geodesics is a linear
combination of quantum geodesics governed by the skein relation 10].
3. Unorientness. Quantum traces of direct and inverse geodesic operators
coincide.
4. Exponents of geodesics. Being raised to any power n, a quantum
geodesics G = 2 cosh L( )=2 admits the expansion into the linear combination of quantum geodesics Gk , k = 1 : : : n the (binomial) coe cients are
the same as in the (classical) relation
n2 ] k !
X
n
2
cosh
(
n
;
2
k
)
L
(
)
=
2
:
(2 cosh L( )=2) =
n
k=0
5. For any and 0, the operators Gh and Gh0 commute.
6. If two closed paths and 0 do not intersect, then the operators Gh
1
and Gh0 commute.
Let a simple geodesic be a geodesic that passes through each edge of a
graph no more than once. We denote the Weyl ordering by a usual normal
ordering symbol : : : : :, i.e.,
: e a1 e a2 e an : e a1 ++an for any faig:
Lemma 3 4] If the quantum transformations (2.5) transform a simple geodesic
G into the simple geodesic Ge , than, for these geodesics, the quantum ordering
is the Weyl ordering and all c
( ) 0 in (3.11).
j
9
Then we can consider modular transformations that do not preserve the
simplicity property and compare the result with the one obtained from the
algebraic relations.
The question is which modication of the Weyl ordering produces the
proper modular-invariant set of geodesics (3.11)? We partially answer this
question in the next section.
3.2 Algebra of quantum geodesics
Let G1 and G2 be two simple geodesics with nontrivial intersection. So, for
G1 and G2, formula (3.11) implies, by virtue of Lemma 3, the mere Weyl
ordering.
After some algebra we have (cf. (2.8))
G1 G2 = e ;ih=2 GZ + e ih=2 Ge Z (3.13)
where
GZ = tr 1 tr 2 b(1ca)b2(bd)e111 e222 + e122 e211 ; e112 e221 ; e121 e212 ] (3.14)
just like GI in the classical case (2.8), whereas Ge Z contains the quantum
correction term,
Ge Z = tr 1 tr 2 b(1ca)b2(bd)(e1ij e2ji)XZ1 XZ2 ] = : tr 1 tr 2 b(1ca)b2(bd)(e1ij e2ji)XZ1 XZ2 + 2(1 ; cos h$ )e111 e222 ]: :
Here e1ij e2ji is the standard r-matrix that permutes the spaces \1" and \2,"
so, as a result, the \skein" relation of the form (2.8) appears. Locally, this
relation has exactly the form proposed by Turaev 10], i.e., for intersecting
simple geodesics, we have the dening relation
G1
G2
$' &%
'$
%&
GZ
@
;
@ ;
@
; @
;
@
GfZ
+e ih=2
= e ;ih=2
(3.15)
10
(The order of crossing G1 and G2 depends on which geodesic occupies the
rst place in the product.) Note, however, that assuming the geodesics G1
and G2 to be simple, we may turn the geodesics Ge Z again into the simple
geodesics Ge 0Z by performing the quantum ip w.r.t. the edge Z .
If we now compare two inambiguously dened exrpessions: Ge 0Z , which
must be Weyl ordered, and Ge Z obtained from the geodesic algebra, we nd
that Ge Z = Ge 0Z .
Lemma 4 The quantum geodesic ordering : : : generated by the geodesic
algebra is consistent with the quantum modular transformations (2.5), i.e.,
the quantum geodesic algebra is modular invariant.
4 Conclusion
In this paper, we briey described how one may quantize the algebra of
geodesics on moduli spaces of complex structures of Riemann surfaces. One
may construct all the variety of classical as well as quantum geodesics from
a nite number of rather simple objects|the lengths of edges of the graph.
I am grateful to the organizers of the International Seminar \Quarks'98"
for the hospitality. The work was partially supported by the RFFI Grants
No. 96-02-19085.
References
1] V. V. Fock and A. Rosly, Poisson structures on moduli of at connections
on Riemann surfaces and r-matrices, Preprint ITEP 72{92 (1992) Flat
connections and Poluybles, Theor. Math. Phys., 95, (1993), 228.
2] R. C. Penner, The decorated Teichmuller space of Riemann surfaces,
Commun. Math. Phys., 113, (1988), 299.
3] M. Kontsevich, Intersection theory on the moduli space of curves and
the matrix Airy function, Commun. Math. Phys., 147, (1992), 1.
4] L. Chekhov and V. Fock, talk on May, 25 at St. Petersburg Meeting on
Selected Topics in Mathematical Physics, LOMI, 26{29 May, 1997 the
printed version is in preparation.
5] K. Strebel, Quadratic Dierentials, Springer, Berlin{Heidelberg{New
York 1984.
11
6] V. V. Fock, Combinatorial description of the moduli space of projective
structures, hepth/9312193.
7] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian
ows of surface group representations, Invent. Math., 85, (1986), 263.
8] L. D. Faddeev, Discrete Heisenberg{Weyl group and modular group,
Lett. Math. Phys., 34, (1995), 249{254.
9] R. M. Kashaev, Quantization of Teichmuller spaces and the quantum
dilogarithm, preprint q-alg/9705021.
10] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Scient. Ec. Norm. Sup., Ser. 4, 24, (1991), 635.
12