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%\begin{center}
\title{Quantum Moduli Spaces}
%\vspace{4mm}
%\end{center}
%\begin{center}
%\vspace{4mm}
\tocauthor{L. Chekhov}
\author{L. Chekhov\footnote{E-mail: chekhov@genesis.mi.ras.ru.}\\
  {\small\it Steklov Mathematical Institute,}\\
  {\small\it  Gubkina 8, 117966, GSP--1, Moscow, Russia}}
\date{}
\maketitle
%\end{center}
%\vskip 0.9 cm
\begin{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.
\end{abstract}

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\newsection{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~\cite{FockRosly},
the gauge theory on graphs were associated with the moduli space of 
flat 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~\cite{Penn1,Konts}. 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~\cite{Konts} that generate all observables for
2D quantum gravity with matter.

The question of physical 
content of such quantum systems
remains open. Indeed, 
already {\it classical} cohomology 
theory of moduli spaces of Riemann surfaces provides a description of 
{\it quantum} topological 2D gravity; therefore, 
the extra quantization procedure could 
be related to quantizing the 2D space--time itself.

\newsection{Classical moduli spaces}

Recall briefly a classical description of
Teichm\"uller spaces of complex structures
on Riemann surfaces with holes (punctures).

Teichm\"uller space ${\cal T}^h$ is a space of
complex structures on~$S$ modulo diffeomorphisms
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-infinite cylinder.

An oriented 2D surface can be continuously
conformally transformed to the constant curvature surface.
The Poincar\'e 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 ${\HH}_+$,
endowed with the hyperbolic metric,
by a discrete Fuchsian subgroup~$\Delta(S)$ 
of the automorphism group $PSL(2,\RR)$,
$$
S={\HH}_+/\Delta(S).
$$
In the hyperbolic metric, geodesics are either
half-infinite circles with endpoints
at the real line $\RR$ or vertical half-lines; all points of the 
boundary $\RR$ are at infinite distance from each other and from any 
interior point.

Any hyperbolic homotopy class of closed curves $\gamma$
contains a unique {\it closed
geodesic} of the length $l(\gamma)=\left|\log
{\lambda_1}/{\lambda_2}\right|$,
where $\lambda_1$ and $\lambda_2$ are (different) 
eigenvalues of the element of $PSL(2,\RR)$ that corresponds to~${\gamma}$. 

\subsection{Teichm\"uller space ${\cal T}^H(S)$ of surfaces with holes.
The graph technique}

The central point of the construction is 
a description of the moduli space ${\cal T}^H(S)$
in terms of oriented graphs. 

Let a {\it fat graph} (oriented graph) $\Gamma$ 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 $Z\in\RR$; 
the set of all such numbers is $\{Z_\alpha|\alpha\in E(\Gamma)\}$
where $E(\Gamma)$ is
the set of all (unoriented) edges of the graph $\Gamma$. 

\begin{theorem}
%{\rm (Strebel~\cite{Streb1}).} 
For a given three-valent graph $\Gamma$
of the given genus~$g$ and number of punctures~$n$, 
there exists a one-to-one
correspondence between the
set of points of ${\cal T}^H(S)$ and the set 
${\bf R}^{\hbox{\small \#\,edges}}$
of edges of this
graph supplied with real numbers
{\rm(}lengths{\rm)}. 
\end{theorem}


We propose the explicit way to construct the
Fuchsian group $\Delta(S)\subset PSL(2,\RR)$ such that 
$S=\HH_+/\Delta(S)$.

To each edge~$\alpha$ we
associate the matrix $X_{z_\alpha}\in PSL(2,\RR)$ of the M\"obius transform
\be
\label{XZ}
X_{z_\alpha}=\left(
\begin{array}{cc} 0 & -\e^{Z_\alpha/2}\\
                \e^{-Z_\alpha/2} & 0\end{array}\right).
\ee
In order to parametrize a {\it path} over edges of the graph, we introduce
the matrices of the ``right'' and ``left'' turns
\be
\label{R}
R=\left(\begin{array}{cc} 1 & 1\\ -1 & 0\end{array}\right), \qquad
L\equiv R^2=\left(\begin{array}{cc} 0 & 1\\ -1 &
-1\end{array}\right).
\ee
We introduce now the notion of {\it geodesic} on the graph. 

Let a {\it closed path} in the graph~$\Gamma$ 
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
$P_{z_1\cdots z_n}=
LX_{z_n}LX_{z_{n-1}}RX_{z_{n-2}}\cdots RX_{z_2}LX_{z_1}$, where the
matrices $L$ or $R$ are inserted depending on
which turn---left or right---the path is going on the corresponding step.

\begin{proposition}\label{prop1}~{\rm\cite{Fock1}}
{\rm1.} There is a one-to-one correspondence between the set of all
oriented paths $\{P_{z_1\cdots z_n}\}$
and closed {\rm(}oriented\/{\rm)} geodesics~$\{\gamma\}$ on the moduli
space.  Moreover, the length~$L(\gamma)$ of a geodesic can be determined as
follows:
\be
\label{geod}
G(\gamma)\equiv 2\cosh L(\gamma)/2=\tr P_{z_1\cdots z_n}.
\ee
{\rm2.} Considering the universal covering $T_{\Gamma}$ of the
graph~$\Gamma$, \ ${\Gamma}=T_{\Gamma}/\Delta_{\Gamma}$,
where $\Delta_{\Gamma}$ is the Fuchsian
group of the graph~${\Gamma}$, we obtain that there is a one-to-one
correspondence between the set of all closed geodesics and the primitive
conjugacy classes $\{\sigma_i\}\subset \Delta_{\Gamma}$.
\end{proposition}

Summing up, we have the following chain of the one-to-one correspondences:
$$
\left\{\begin{array}{c}
\hbox{\small conjugacy}\\
\hbox{\small classes\ of}\\
\hbox{\small $\Delta(S)\subset PSL(2,\RR)$}
\end{array}
\right\}
\leftrightarrow
\left\{\begin{array}{c}
\hbox{\small closed}\\
\hbox{\small geodesics}\\
\hbox{\small on $S$}
\end{array}
\right\}
\leftrightarrow
\left\{\begin{array}{c}
\hbox{\small closed paths}\\
\hbox{\small in\ (any\ of)}\\
\hbox{\small the\ graph\ $\Gamma(S)$}
\end{array}
\right\}.
$$

\subsection{Weil--Peterson forms}

A canonical Poisson structure called the Weil--Peterson
structure exists on the set of coordinates $\{Z_\alpha\}$. 
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 {\it Weil--Peterson form}~\cite{Penn1}.

Denote $\alpha(1)$ the oriented edge that is obtained from an oriented edge
$\alpha$ by
rotating by the angle $2\pi/3$ clockwise around the vertex these
edges are starting with and
denote $\alpha^{-1}$ the oriented edge that are opposite to an
edge~$\alpha$. Then,
the Weil--Peterson Poisson brackets are
\be
\label{WP-PB}
\{Z_\alpha,Z_\beta\}=\left\{
\begin{array}{rlll}
1&\ \mbox{if}\ &\alpha^{\pm1}(1)=\beta\ \mbox{or}\ \beta^{-1}\ \
&\mbox{and}\ \ \alpha^{\mp1}\ne\beta,\beta(1),\\
-1&\ \mbox{if}\ &\alpha^{\pm1}=\beta(1)\ \mbox{or}\
\beta^{-1}(1)\ \ &\mbox{and}\ \ \alpha^{\mp1}\ne\beta,\beta(1),\\
2&\ \mbox{if}\ &\alpha^{\pm1}=\beta^{\pm1}(1)\ \mbox{or}\
\beta^{\mp1}(1),&\\
-2&\ \mbox{if}\ &\alpha^{\pm1}(1)=\beta^{\pm1}\ \mbox{or}\
\beta^{\mp1},&\\
0&&\mbox{otherwise}.&
\end{array}
\right.
\ee

\subsection{Classical modular transformations}

In~\cite{Fock1}, the graph transformations that
preserve Poisson structure (\ref{WP-PB}) were obtained.
There exists a natural operation called the
{\it flip} or {Whitehead move}, which corresponds to transitions between
graphs. We must determine such a transformation of the variables
$Z_\alpha$ that, first, preserve the Poisson structure 
and, second, after a series of
flips that transform a graph to itself, the resulting transformation should
be the identity. This implies the presence of another relation called the
{\it pentagon identity}.
It appears that the transformations
depicted in Fig.~1 (\ref{abc}) satisfy both the demands.

In the classical case~\cite{Fock1},
\bea
\nonumber
\{A,B,C,D,Z\}&\to&\{A+\phi(Z), B-\phi(-Z), C+\phi(Z), D-\phi(-Z), -Z\}
\\
\label{abc}
&\equiv&\{\wtd A,\wtd B,\wtd C,\wtd D,\wtd Z\},
\\
\nonumber
\phi(Z)&=&\log(\e^Z+1).
\eea

\begin{lemma} \label{lem1}
Transformation~{\rm(\ref{abc})} preserves
the products over paths, so
the classical geodesic length is a~modular-invariant function.
\end{lemma}

\vspace{10pt}

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\centerline{\bf Fig.~1}

\hspace{10pt}

\subsection{Poisson algebra of geodesics}

The functions $\{G_\gamma\}$ (\ref{geod})
were studied in~\cite{Gold}.
They generate a Poisson algebra (w.r.t.\ the multiplication and the
Weil-Petersson Poisson bracket) over ${\Bbb Z}$. (It means that a product and
a Poisson bracket of two such functions is a linear combination of such
functions with integer coefficients.) 


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
$G^1$ and $G^2$ of the form
\bea\label{G1}
G^1&=&\tr^1\dots X_C^1R^1X_Z^1L^1X_A^1\dots,
\\
\label{G2}
G^2&=&\tr^2\dots X_B^2 L^2 X_Z^2R^2X_D^2\dots,
\eea
respectively. (Superscripts $1$ and~$2$ label the matrix spaces.)

A Poisson bracket of geodesics is
\be
\label{Goldman}
\{G_1,G_2\}=\frac{1}{2}(\tr G_{\mbox{H}}-\tr G_{\mbox{I}}),
\ee
where $G_{\mbox{I}}$ 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 $G_{\mbox{H}}$
passes over the edge~$Z$ twice, so it has the form
$\tr \dots X_CR_ZR_D\dots X_BL_ZL_A\dots$. These 
relations were obtained in~\cite{Gold} 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
\be
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\put(95,48){\oval(20,20)[t]}
\put(95,72){\oval(20,20)[b]}
\put(110,60){\makebox(0,0){$=0.$}}
%%%%%%%%%%%
\end{picture}
\label{skeinclass}
\ee


\newsection{Quantization}  
Once the graph $\Gamma$ 
is chosen, the corresponding Teichm\" uller space ${\cal T}^H(S)$ can be
easily quantized. Consider the $*$-algebra ${\cal T}^\hbar(\Gamma)$ generated
by real generators $\{Z^\hbar_\alpha | \alpha \in E(\Gamma)\}$, \  
$(Z^\hbar_\alpha)^* = Z^\hbar_\alpha$ with the defining relations
\begin{equation} \label{comm}
[Z^\hbar_\alpha, Z^\hbar_\beta ] = 2\pi i\hbar\{Z_\alpha, Z_\beta\}
\qquad \hbox{cf.\ (\ref{WP-PB})}.
\end{equation}

This algebra has an obvious center generated by the perimeters of faces
$\left\{P_\gamma| \gamma \in F(\Gamma),\
P_{\gamma} = \sum_{\alpha \in \gamma} Z^\hbar_\alpha\right\}$.
One can easily describe all irreducible $*$-representations
of this algebra using 
the Stone--von Neumann theorem. An irreducible representaion
is unambiguously fixed by the values of the operators $P_\gamma$.
For example one can represent
all operators $Z^\hbar_\alpha$ in $L^2({\Bbb R}^n)$, where
$n = \frac{1}{2}(E(\Gamma) - F(\Gamma))$, by linear
combinations with rational coefficients 
standing by constants and the operators $x_i$
and $i\frac{\partial}{\partial x_i}$, where
$\{x_i|i=1,\ldots,n\}$ is a standard coordinate system on ${\Bbb R}^n$.
Now our task is to identify the $*$-algebras constructed using different
graphs $\Gamma$ and $\Gamma^\prime$ corresponding to a given surface $S$.
In order to make this identification we just construct a $*$-homomorphism
$K(\Gamma, \Gamma^\prime): {\cal T}^\hbar(\Gamma) \rightarrow {\cal
T}^\hbar(\Gamma^\prime)$ of the $*$-algebra generated by
$\{Z^\hbar_{\alpha^\prime}|\alpha^\prime \in E(\Gamma^ \prime)\}$ to the 
algebra generated by  $\{Z^\hbar_\alpha|\alpha \in E(\Gamma )\}$.  We 
require this homomorphism to have the following properties:

1. {\em Classical limit}. We demand that the algebra homomorphism should
tend to the classical homomorphim of the algebras fo function on ${\cal
T}^H$ when the parameter $\hbar$ tends to zero.

2. {\em Path independence}. We demand that if we have three graphs
$\Gamma$, $\Gamma^\prime$ and $\Gamma^{\prime \prime}$ then the
homomorphisms should satisfy the condition
$K(\Gamma^{\prime \prime}, \Gamma^\prime)K(\Gamma^\prime, \Gamma) =
K(\Gamma^{\prime \prime}, \Gamma)$.

One can check the latter condition  only for one 
distinguished sequence of
flips since others  are just compositions of this one.  
Consider  two edges
having exactly one common vertex. Then, a sequence of
five flips of these edges (such that we never flip the same edge twice
consequtively) does not change  the graph. This becomed more 
geometrically transparent if we consider
the {\it 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 five possible ways which are related by
flips (Fig.~2)

\begin{equation}
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\label{pent1}
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\centerline{\bf Fig.~2}

\hspace{10pt}


The quantum transforms of the edge variables are as in Fig.~1 with
the function 
\begin{equation} \label{phi}
\phi(z) =
-\frac{\pi\hbar}{2}\int_{\Omega} \frac{e^{-ipz}}{\sinh(\pi p)\sinh(\pi \hbar
p)}dp,
\end{equation}
and the contour $\Omega$ goes along
the real axis bypassing the origin from above.

The function (\ref{phi}) and $F(x,\hbar)$,
\begin{equation}
F(x,\hbar) := \exp( -\frac{1}{4}\int_{\Omega}
\frac{e^{-ipx}}{p\sinh(\pi p)\sinh(\pi \hbar p)}dp),
\end{equation}
first appeared in~\cite{Faddeev} possess the
following properties.
\bea
&{}&\hbox{\bf 1.}\qquad
\lim_{h \rightarrow 0} \phi(x,\hbar) = \log(e^x + 1)
\hbox{\ (classical\ limit).}\\
&{}&\hbox{\bf 2.}\qquad
\label{prop9}
2\pi i \hbar \frac{\partial}{\partial x} \log F(x,\hbar) = 
\phi(x,\hbar).\\
&{}&\hbox{\bf 3.}\qquad
\label{prop10a}
F(x+i\pi \hbar,\hbar) = F(x-i\pi\hbar,\hbar)(1+e^x).\\
&{}&\phantom{\hbox{\bf 5.}}\qquad
\label{prop10b}
F(x+i\pi) = F(x-i\pi)(1+e^{\frac{1}{\hbar}x}).\\
&{}&\hbox{\bf 4.}\qquad
\phi(x,\hbar ) =
\hbar\phi\left(\frac{1}{\hbar}x,\frac{1}{\hbar}\right).
\eea

\begin{lemma}\label{lem4} \cite{ChF}
Quantum transformations {\rm(\ref{abc})} satisfy the pentagon 
identity.\footnote{This result was independently obtained by
R.~Kashaev~\cite{kashaev}.}
\end{lemma}


\subsection{Geodesic length operators}

The aim of this paragraph is to imbed the algebra of geodesics
(\ref{geod}) into a suitable
completion of the constructed algebra ${\cal T}^\hbar(S)$.

The function $G_\gamma$ (\ref{geod})
can be expressed for any $\gamma$ in terms of
graph coordinates on ${\cal T}^H$, 
\begin{equation}
G_\gamma \equiv \tr P_{Z_1\cdots Z_n}=
\sum_{j\in J}e^{\frac{1}{2} \sum_{\alpha \in E(\Gamma)}
m_j(\gamma,\alpha) z_\alpha},  \label{clen}
\end{equation}
where $m_j(\gamma,\alpha)$ are
certain {\it integer} numbers and $J$ is a finite set of indices.
In order to find the
{\it quantum analogues} of these functions, we
denote by $\widehat{\cal T}^\hbar$ a completion of the algebra ${\cal
T}^\hbar$ containing $e^{xZ_\alpha}$ for any real $x$.
Let for any closed path $\gamma$ on $S$, the operator $G^\hbar_\gamma \in
\hat{\cal T}^\hbar$ be
\begin{equation} \label{qlen}
G^\hbar_\gamma \equiv
\ORD{\tr P_{z_1\dots z_n}}=
\sum_{{j\in J\atop \kappa\in\{j\}}}
e^{\frac{1}{2} \sum_{\alpha \in E(\Gamma)}
\bigl(m_j(\gamma,\alpha) Z^\hbar_\alpha
+2\pi i\hbar 
c_j^\kappa(\gamma,\alpha) 
\bigr)}.
\end{equation} 
Here some quantum ordering $\ORD{\cdot}$ is assumed. The main 
problem is to find the ordering that
satisfy all of the conditions below. In (\ref{qlen}),
the numbers $m_j(\gamma,\alpha)$ are the same as in (\ref{clen})
and integer coefficients  $c_j^\kappa(\gamma,\alpha) $ are to be determined
from the procedure of the quantum ordering.

Note that the operators $\{G^\frac{1}{\hbar}_\gamma\}$ can be considered as
belonging to the algebra $\hat{\cal T}^\hbar$.
In terms of the generators of $\hat{\cal T}^\hbar$ they are 
\begin{equation} \label{dlen}
G^\frac{1}{\hbar}_\gamma = \sum_{j\in J}e^{\frac{1}{2\hbar} \sum_{\alpha \in
E(\Gamma)} \bigl(m_j(\gamma,\alpha) Z^\hbar_\alpha
+2\pi i c_j^\kappa(\gamma,\alpha) \bigr)},
\end{equation}


Now let us formulate the defining properties of quantum geodesics.



{\bf1.} {\it Modular invariance}. 
The modular group $\Delta(S)$ (\ref{abc}) preserves the set
$\{G^\hbar_\gamma\}$, i.e., for any $\delta \in \Delta(S)$ and
any closed path $\gamma$, we have 
$\delta(G^\hbar_\gamma) = G^\hbar_{\delta \gamma}$.

{\bf2.} {\it Geodesic algebra}. The product of two quantum geodesics 
is a linear combination of quantum geodesics governed by the skein 
relation \cite{Turaev}.

{\bf3.} {\it Unorientness.} Quantum traces of
direct and inverse geodesic operators coincide.

{\bf4.} {\it Exponents of geodesics.} Being raised to any power~$n$,
a quantum geodesics $G_\gamma=2\cosh L(\gamma)/2$ admits the expansion
into the linear combination of quantum geodesics $G_{k\gamma}$, 
$k=1,\dots, n$; the (binomial) coefficients are 
the same as in the (classical) relation 
$$
(2\cosh L(\gamma)/2)^n=\sum_{k=0}^{\bigl[\frac{n}{2}\bigr]}
\left({k\atop n}\right)2\cosh\Bigl((n-2k)L(\gamma)/2\Bigr).
$$

{\bf5.} For any $\gamma$ and $\gamma^\prime$, the operators 
$G^\hbar_\gamma$ and  $G^\frac{1}{\hbar}_{\gamma^\prime}$ commute.

{\bf6.} If two closed paths $\gamma$ and $\gamma^\prime$ do not 
intersect, then the operators $G^\hbar_\gamma$ and  
$G^\hbar_{\gamma^\prime}$ commute.

Let a {\it 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 
${:}\dots{:}$, i.e.,
$$
{:}\e^{a_1}\e^{a_2}\cdots\e^{a_n}{:}\equiv
\e^{a_1+\cdots+a_n}\quad\hbox{for\ any}\quad \{a_i\}.
$$

\begin{lemma}\label{lem31} \cite{ChF}
If the quantum transformations {\rm(\ref{abc})} transform a 
simple geodesic $G$ into the simple geodesic~$\wtd G$, 
than, for these geodesics, the quantum ordering $\ORD{\cdot}$ 
is the Weyl ordering and
all $c_j^\kappa(\gamma,\alpha)\equiv0$ in {\rm(\ref{qlen})}.
\end{lemma}
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 modification of the Weyl ordering produces
the proper modular-invariant set of geodesics 
(\ref{qlen})?  We partially answer this question in the next section.

\subsection{Algebra of quantum geodesics}
Let $G_1$ and $G_2$ be two simple geodesics with nontrivial
intersection.
So, for $G_1$ and $G_2$, formula (\ref{qlen}) 
implies, by virtue of Lemma~\ref{lem31}, the mere Weyl ordering.

After some algebra we have (cf. (\ref{Goldman}))
\be
\label{AG4}
G_1G_2=\e^{-i\pi\hbar/2}G_Z+\e^{i\pi\hbar/2}\wtd G_Z,
\ee
where 
\be
\label{AG5}
G_Z=\tr_1\tr_2 \widehat\xi^1_{(ca)}\widehat\xi^2{(bd)}
\bigl[e^1_{11}\otimes e^2_{22}+e^1_{22}\otimes e^2_{11}
-e^1_{12}\otimes e^2_{21}-e^1_{21}\otimes e^2_{12}\bigr]
\ee
just like $G_{\mbox{I}}$ 
in the classical case (\ref{Goldman}), whereas $\wtd G_Z$
contains the quantum correction term,
\bea
\label{AG6}
\wtd G_Z&=&\ORD{\tr_1\tr_2 \widehat\xi^1_{(ca)}\widehat\xi^2{(bd)}
(e^1_{ij}\otimes e^2_{ji}) [X^1_Z\otimes X^2_Z]}
\nonumber\\
&=&{:}\tr_1\tr_2 \widehat\xi^1_{(ca)}\widehat\xi^2{(bd)}
(e^1_{ij}\otimes e^2_{ji}) \bigl[X^1_Z\otimes X^2_Z
+2(1-\cos\pi\hbar)e^1_{11}\otimes e^2_{22}\bigr]{:}\ .\nonumber
\eea
Here $e^1_{ij}\otimes e^2_{ji}$ is the standard $r$-matrix that permutes 
the spaces ``$1$'' and ``$2$,'' so, as a result, the ``skein'' 
relation of the form (\ref{Goldman}) appears. Locally, this relation 
has exactly the form proposed by Turaev~\cite{Turaev}, i.e., for 
intersecting simple geodesics, we have the defining relation
\be
\setlength{\unitlength}{.8mm}%
\begin{picture}(90,40)(0,45)
\thicklines
\put(-10,50){\line(1, 1){9}}
\put( 1, 61){\line(1, 1){9}}
\put(-10,70){\line(1,-1){20}}
\put(18,60){\makebox(0,0){$=$}}
\put(30,61){\makebox(0,0){$\e^{-i\pi\hbar/2}$}}
\put(40,60){\oval(20,20)[r]}
\put(63,60){\oval(20,20)[l]}
\put(70,60){\makebox(0,0){$+$}}
\put(78,61){\makebox(0,0){$\e^{i\pi\hbar/2}$}}
\put(95,48){\oval(20,20)[t]}
\put(95,72){\oval(20,20)[b]}
%%%%%%%%%%%
\put(-11,68){\makebox(0,0)[rb]{\Large$G_1$}}
\put(-11,52){\makebox(0,0)[rt]{\Large$G_2$}}
\put(52,74){\makebox(0,0)[cb]{\Large$G_Z$}}
\put(95,74){\makebox(0,0)[cb]{\Large$\wtd G_Z$}}
\end{picture}
\label{skein}
\ee
(The order of crossing $G_1$ and $G_2$ depends on which geodesic
occupies the first place in the product.)
Note, however, that assuming the geodesics 
$G_1$ and $G_2$ to be simple, we may turn the geodesics 
$\wtd G_Z$ again into the simple geodesics $\wtd G'_Z$
by performing the quantum flip w.r.t.\ the edge $Z$.

If we now compare two inambiguously defined exrpessions:
$\wtd G'_Z$, which must be Weyl ordered, and $\wtd G_Z$ obtained from
the geodesic algebra, we find that $\wtd G_Z=\wtd G'_Z$.

\begin{lemma}\label{lem34}
The quantum geodesic ordering $\ORD{\dots}$ generated by the geodesic
algebra is consistent with the quantum modular transformations 
{\rm(\ref{abc})}, i.e., the quantum geodesic algebra is modular 
invariant.  
\end{lemma}

\newsection{Conclusion}

In this paper, we briefly 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 finite 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.

\begin{thebibliography}{99}
\bibitem{FockRosly}
V.~V.~Fock and A.~Rosly, {\it Poisson structures on moduli of flat
connections on Riemann surfaces and $r$-matrices}, Preprint ITEP
72--92 (1992); {\it Flat connections and Poluybles}, 
{\sl Theor. Math. Phys.}, {\bf95}, (1993), 228.
\vspace{-2.5mm}
\bibitem{Penn1}
R.~C.~Penner, {\it The decorated Teichm\"uller space of Riemann surfaces},
{\sl Commun. Math. Phys.}, {\bf113}, (1988), 299.
\vspace{-2.5mm}
\bibitem{Konts}
M.~Kontsevich, {\it Intersection theory on the moduli space of curves
and the matrix Airy function}, {\sl Commun. Math. Phys.}, {\bf147}, 
(1992), 1. 
\vspace{-2.5mm}
\bibitem{ChF}
L.~Chekhov and V.~Fock, talk on May, 25 at {\em St.~Petersburg 
Meeting on Selected Topics in Mathematical Physics}, LOMI,
26--29 May, 1997; the printed version is in preparation.
\vspace{-2.5mm}
\bibitem{Streb1} K.~Strebel, {\em Quadratic 
Differentials}, Springer, Berlin--Heidelberg--New York 1984.  
\vspace{-2.5mm}
\bibitem{Fock1}
V.~V.~Fock, {\it Combinatorial description of the moduli space of
projective structures}, hepth/9312193.
\vspace{-2.5mm}
\bibitem{Gold}
W.~M.~Goldman, {\it Invariant functions on Lie groups and Hamiltonian
flows of surface group representations}, {\sl Invent. Math.}, {\bf85},
(1986), 263.
\vspace{-2.5mm}
\bibitem{Faddeev}
L.~D.~Faddeev, {\it Discrete Heisenberg--Weyl group and modular group},
{\sl Lett. Math. Phys.}, {\bf34}, (1995), 249--254.
\vspace{-2.5mm}
\bibitem{kashaev}
R.~M.~Kashaev, {\it Quantization of Teichm\"uller spaces and the 
quantum dilogarithm}, preprint q-alg/9705021.
\vspace{-2.5mm}
\bibitem{Turaev} V.~G.~Turaev, {\it Skein 
quantization of Poisson algebras of loops on surfaces}, {\sl Ann. 
Scient. \'Ec. Norm. Sup.}, Ser.~4, {\bf24}, (1991), 635.  
\end{thebibliography}

\end{document}


