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%%%    A Quantum Group Apprach to C>1 Liouville Gravity 


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\makeatletter
\begin{document}


\begin{titlepage}
% \rightline{YITP/U-95-07}
\rightline{September  1996}
\vspace{1.0cm}
\begin{center}
\LARGE{A Note on Quantum  Liouville Theory via 
Quantum Group} \\[.3em]
\Large ------- An Approach to Strong Coupling Liouville Theory ------- \\[1cm]
\large{  Takashi Suzuki$^\ast$ } \\
\normalsize    Department of Electronics \\             
\normalsize    Hiroshima Institute of Technology            \\
\normalsize       Saeki, Hiroshima 731-51 \\
                                 JAPAN 
\end{center}           

\pagestyle{plain}

\vspace{.3cm}

\begin{abstract}
Quantum Liouville theory is annualized in terms of the infinite 
dimensional representations of $\qslc$ with $q$ a root of unity. 
Making full use of characteristic features of the representations, 
we show that vertex operators in this Liouville theory are 
factorized into {\em classical} vertex operators and 
those which are constructed from the  
finite dimensional representations of $\qslc$. 
We further show explicitly that fusion rules in this model also 
enjoys such a factorization. 
Upon the conjecture that the Liouville action effectively 
decouples into the classical Liouville action and that of a quantum 
theory, correlation functions and transition amplitudes are discussed, 
especially an intimate relation between our model 
and geometric quantization 
of the moduli space of Riemann surfaces is suggested. 
The most important result is that our Liouville theory is 
in the strong coupling region, \ie, the central charge $c_L$ 
satisfies $1<c_L<25$. 
An interpretation of quantum space-time is also given within this 
formulation.  
\end{abstract}

\vspace{3.5cm}
-----------------------

{}$^\ast$ e-mail address: stakashi@cc.it-hiroshima.ac.jp


\end{titlepage}
\makeatother

\newpage


\section{Introduction and Setup}

Liouville theory, classically, has a deep connection to the 
geometry of Riemann surfaces. 
Indeed, the solution to the Liouville equation yields Poincar\'e 
metric on the upper-half-plane or the Poincar\'e 
unit disk on which Riemann surfaces are uniformized \cite{Li}. 
In the light of this fact it is natural to expect that quantum Liouville 
theory gives some insights into quantum geometry of surfaces. 
Physically speaking, this problem is nothing but the quantum gravity 
of $2D$ space-time. 
The appearance of the Liouville theory as a theory of quantum 
gravity, so-called Liouville gravity, 
was first recognized by Polyakov in the study of 
non-critical string theory \cite{Po1}. 
In the string theory embedded in the $D$ dimensional target space, 
the partition function is a function $Z[g]$ of the surface metric $g$,  
and is invariant under the group of diffeomorphism acting on the
metric $g$, while it transforms covariantly under local rescalings  
of the surface metric as, 
\be 
Z[e^\Phi \hat g] = e^{\frac{C}{48\pi}S_L(\Phi:\hat g)}\, Z[\hat g]. 
\label{eq:start}
\ee
Here $S_L(\Phi:\hat g)$ is the Liouville action with the background metric 
$\hat g$, and is written as  

\be
S_L(\Phi:\hat g)=\int_\Sigma d^2z 
\sqrt{\hat g}\left\{ \half 
 \hat{g}^{ab}\partial_a \Phi \partial_b \Phi + \Lambda
e^{\Phi(z,\bz)} + R_{\hat g}\Phi(z,\bz) \right\},  
\label{eq:action}
\ee
%
where $C$ is the central charge of the total system, 
the string coordinates and the reparametrization ghosts. 
The metric on the Riemann surface $\Sigma$ parameterized by  
the complex coordinate $(z,\bz)$ is given by 
$ds^2=  e^{\Phi(z,\bar{z})} {\hat g}_{z\bz} dz d\bar{z}$.  
We denote by $\Lambda$ the cosmological constant. 
$R_{\hat g}$ is the Gaussian curvature measured with the background 
metric $\hat g$ and is given by 
$R_{\hat g}=-2 {\hat g}^{z\bz}\partial_z \partial_{\bz}\log\,
\sqrt{\hat{g}}$.  
Up to now, a number of works (there are too many works to cite here, 
see, e.g., Refs.\cite{Po2}--\cite{DK} and Ref.\cite{Se} for a review) 
have looked at the quantum Liouville gravity and revealed 
many remarkable results. 

One of the important features of the quantum Liouville theory 
is that it possesses quantum group structure of $U_q\slc$ implicitly. 
Precisely, vertex operators in the theory can be 
expressed in terms of the highest weight representations 
of $U_q\slc$. 
We should now remember that, as in the classical algebra $\slc$, 
there are two kinds of highest weight representations 
of $U_q\slc$, \ie, of finite dimension and of infinite dimension, 
and they are completely different from each other.  
Owing to this fact, one can expect that there are two 
entirely different versions of the quantum Liouville theory.  
In one version either finite or infinite dimensional
representations are well-defined and 
we will show that which type of representations 
appears depends on the charges of the vertex operators. 
The quantum Liouville gravity investigated so far are mainly 
associated with the finite dimensional representations of 
$U_q\slc$\cite{Gerv}. 
In this case, however, we have a strong restriction on the central
charge of the Liouville gravity, $c_L\geq 25$, so-called $D=1$
barrier. 
Such gravity is often called the Liouville gravity 
in the weak coupling regime, or for short, 
weak coupling Liouville gravity.  

On the other hand, one expects that the quantum Liouville gravity 
associated with the infinite dimensional representations  
is completely different from the previous one 
and that it can get rid of the barrier, that is, 
the central charge may be in the region $1 < c_L < 25$.  
The Liouville gravity whose central charge is in this region is often 
called the strong coupling Liouville gravity. 
One of the progresses for the strong coupling Liouville gravity 
has been made in Refs.\cite{ST,Ge}.    
They have shown that, upon using infinite dimensional representations of 
$\qslc$, consistent gravity theories can be constructed  
if and only if the central charge takes the special values 
$7,\, 13$ and $19$. 
This result makes us confident that  the Liouville gravity associated with 
the infinite dimensional  representation is in the strong coupling regime. 
However, as will be explained in Section 3,  
the deformation parameter $q$ of the algebra $\qslc$ 
dealt with in Ref.\cite{Ge} is not a root of unity.  
It should be emphasized here that  the infinite dimensional 
representations of $\qslc$ when $q$ is a root of unity are 
drastically different from those with generic $q$ \cite{MS}. 
In particular, a new and remarkable feature is that 
every irreducible infinite dimensional representations 
necessarily factorizes into a representation of the {\it classical} 
algebra $\slr$ and a {\it finite} dimensional one of $\qslc$. 
Along this line, it is worthwhile to investigate the Liouville gravity
associated with the infinite dimensional representations with $q$ at a
root of unity and observe how the characteristic feature of the 
representations works in the theory of Liouville gravity. 
Motivated by this, the aim of this paper is to investigate 
the quantum Liouville gravity via such representations 
with hoping that such a theory would lead us to a different 
strong coupling Liouville gravity from \cite{Ge}. 

Hereafter, let us use the notations, $\ncom$ and $\gncom$ 
the infinite dimensional representations of $\qslc$ 
{\it with $q$ at a root of unity} and {\it generic} $q$, respectively, 
and $\com$ the finite dimensional ones. 

We will see that the fields $\Phi(z,\bz)$ of the Liouville theory 
based on $\ncom$ should be expanded around the classical 
Liouville field $\cf$ as 
$\Phi = \cp \oplus \phi$, $\phi$ representing quantum fluctuations 
around the classical field (see also Ref.\cite{TS}). 
In other words, the metric with which our quantum Liouville theory is 
constructed should be chosen as $ds^2=e^{\kappa\phi(z,\bz)} \hat g_P$ 
where $\kappa$ is some constant and $\hat g_P$ is the Poincar\'e metric 
$\hat g_P =e^{\cf}dz d\bz$ playing the role of the background metric. 
In view of this, it will turn out that our Liouville theory is effectively 
composed of two Liouville theories, one is the classical Liouville 
theory $S^{cl}(\cp)$ and the other is the quantum Liouville theory 
$S^q(\phi:\hat g_P)$ with respect to the field $\phi(z,\bz)$. 
The sector $S^q(\phi:\hat g_P)$ represents quantum fluctuations 
around $S^{cl}(\cp)$ and is the Liouville theory associated with $\com$. 
We will explicitly show that our Liouville gravity  
is actually in the  strongly coupled regime, \ie, $1<c_L<25$.  

% Although the explicit relation between the quantum Liouville gravity 
% of this paper and that of Takhtajan is yet to be discussed, 
% one can expect an intimate relation between the two. 

The organization of this paper is as follows: 
Section 2 looks at the classical Liouville theory in some
details, especially from the viewpoints of geometric and algebraic 
structures of the theory. 
The discussions of the quantum Liouville gravity 
are given in Section 3.  
We will review in \S 3-1 the weak coupling Liouville theory briefly, 
and other sections, \S 3.2--\S 3.5 are devoted to our main concern,  
\ie, the Liouville theory based on $\ncom$.   
We will show the decomposition of our theory into the 
classical theory and the quantum one,  
and give a concept of {\em quantum space-time}. 
We further discuss about fusion rules and transition amplitude. 
 It will turn out that our formulation of the quantum Liouville 
theory admits a nice interpretation in the context of the 
geometric quantization of K\"ahler geometry of moduli space. 
Here the classical sector, whose appearance is the peculiarity of 
our formulation, of correlation function plays an important role, 
\ie, the Hermitian metric of a line bundle over the moduli space. 
In section 4, some discussions are given. 
Appendix reviews  the infinite dimensional representations of 
$\qslc$ with $q$ at a root of unity. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Classical Liouville Theory} 

In this section, we will introduce some backgrounds of the classical 
Liouville theory and Riemann surfaces for our later use, 
especially algebraic and geometric aspects of the theory. 

It is known that the classical Liouville theory describes 
hyperbolic geometry of Riemann surfaces. 
Let $\Sigma_{g,N}$ be a Riemann surface with genus $g$ and 
$N$ branch points $\{z_i\}$ of orders $k_i\in \rm{I}\!\rm{N}_{\geq 2}$. 
The equation $\delta S_L=0$ yields the Liouville equation 
of motion 
%
\be 
\partial \bp \Phi(z,\bz) = \frac{\Lambda}{2}
 e^{\Phi(z,\bz)}. 
\label{eq:equation}
\ee
Here we have made a specific choice of the background metric as 
${\hat g}_{ab}=\delta_{z\bz}$. 
In that case, the Gaussian curvature is given by 
$R_g= -\Lambda$, \ie, 
constant curvature. 
According to the Gauss-Bonnet theorem, 
$\Sigma_{g,N}$ admits metrics $\,g\,$ with constant negative curvature, called 
the Poincar\'e metric, 
for the negative Euler characteristic $\chi(\Sigma_{g,N})<0$.  
Noticing that, for any metric $\tilde g$ on $\Sigma_{g,N}$, there 
exists a scaling factor $e^\lambda$ such that 
$g= e^{-\lambda}\tilde g$ has constant negative curvature 
$R_g=-1$, we will set $\Lambda=1$ hereafter. 
Note however that such a setting cannot be allowed for the 
quantum case. 

Let us explain the connection between the Liouville theory and 
the Fuchsian uniformization of Riemann surfaces. 
The uniformization theorem states that every Riemann surface 
with negative constant curvature is conformally equivalent to 
the quotient of the unit disk $D=\{w \in \C \vert 
\, \vert w\vert <1\}$ by the action of a finitely generated 
Fuchsian group $\Gamma$, \ie, $\Sigma_{g,N} \cong D/\Gamma$. 
In terms of the uniformization map, 
$J_\Sigma\,:\, D \rightarrow \Sigma_{g,N}$ 
a solution to the Liouville equation (\ref{eq:equation}) 
is written as  
\be 
e^{\cf} = 
\frac{\vert \partial_z J_\Sigma^{-1}(z)\vert^2}
{(1- \vert J_{\Sigma}^{-1}(z){\vert^2}){}^2}. 
\label{eq:csolution}
\ee    
Upon rewriting $J_\Sigma^{-1}(z) = w$, the coordinate on $D$, 
the solution (\ref{eq:csolution}) gives the Poincar\'e metric on $D$, 
\be 
ds^2 = \frac{dw \wedge d\bar{w}}{(1-w\bar{w})^2}. 
\ee
Note that the $PSL(2,\rm{I}\!\rm{R})$ fractional transformations 
for $J_\Sigma^{-1}(z)$ leave the metric invariant. 
% This is the evidence of the fact that the classical Liouville theory 
% possesses the $\SLC$ symmetry. 

The classical Liouville theory is a conformally invariant theory. 
This is due to the fact that the energy-momentum (EM) tensor defined 
by $T_{ab}=2\pi/\sqrt{\hat g}(\delta S_L/\delta\hat{g}_{ab})$ 
is traceless, \ie, $T^{cl}_{z\bz}=0$. 
The $(2,0)$-component is given by 
\be
T^{cl}_{zz}=\frac{1}{\gamma^2}\left[ -\half \partial_z\cp\partial_z\cp 
       + \partial_z^2\cp \right], \label{eq:emtensor} 
\ee 
and is conserved, \ie, $\partial_\bz T^{cl}_{zz}=0$. 
Here it should be emphasized that the second term in 
eq.(\ref{eq:emtensor}) coming from the last term of the action 
(\ref{eq:action}) is indispensable in order for the EM-tensor 
to be traceless and satisfy the correct transformation law of a 
projective connection under holomorphic coordinate change 
$z=f(\tilde z)$, 
\begin{eqnarray}
\tilde T^{cl}_{\tilde z \tilde z}(d\tilde z)^2 
&=& \left(\frac{df}{d\tilde z}\right)^2 T^{cl}_{zz} (dz)^2 + 
\frac{1}{\gamma^2}\{f,\tilde z\}_S (d\tilde z)^2, \nonumber \\
\{f,\tilde z\}_S &=& \frac{f'''}{f'}
-\frac{3}{2}\left(\frac{f''}{f'}\right)^2.  
\label{eq:projective}
\end{eqnarray}
Indeed, using the fact that the metric $e^{\Phi(z,\bz)}$ 
is a primary field of dimension $(1,1)$, or equivalently, 
it transforms under the coordinate change as 
\be
e^{\tilde\cp(\tilde z,\bar{\tilde z})}d\tilde z d\bar{\tilde z} 
=\left\vert\, \frac{d\tilde z}{dz}\,\right\vert^2 \,e^{\cp(z,\bz)}dz d\bz,
\ee 
one can check that the EM-tensor satisfies 
the transformation law (\ref{eq:projective}). 
This transformation law of EM-tensor indicates that the 
central charge $c_0$ of the classical theory is 
\be 
c_0=\frac{12}{\gamma^2}. \label{eq:classicalcentral}
\ee

We next discuss the algebraic structure of the classical Liouville 
theory. 
To see this, let us take the $\alpha$-th 
power of the metric (\ref{eq:csolution}) at a special point $P$. 
Geometrically it corresponds to a branch point $P$ on the surface, and 
$\alpha$ is related to the order $k$ of the branch point as 
\be 
\alpha_i=\frac{1-k^{-2}}{2\gamma^2}. 
\label{charge}
\ee
One immediately finds there are two entirely different 
regions of $\alpha$, namely, 
(I) $\alpha=-j\leq0,\, (j\in \Z/2)$ and  (II) $\alpha=h>0$. 
It lies in this fact that there are two kinds of quantum Liouville 
gravity, \ie, of strong coupling and weak coupling regimes. 
In order to see this more explicitly, 
we calculate the $\alpha$-th power of eq.(\ref{eq:csolution}),  
%
\begin{eqnarray}
({\rm I}) \quad &e^{-j\cp(z,\bz)}& = 
\left(\frac{1-\vert J_\Sigma^{-1}\vert^2}
{\sqrt{\partial_z J_\Sigma^{-1} \partial_{\bz}
\overline{J_\Sigma^{-1}}}}
\right)^{2j}=
\sum_{m=-j}^j N^j_m \psi^j_m(z)\psi^{j,m}(\bz)  \label{eq:I} \\
({\rm II}) \quad &e^{h\cp(z,\bz)}& = 
\left(\frac{\sqrt{\partial_z J_\Sigma^{-1} \partial_\bz J_\Sigma^{-1}}}
{1-\vert J_\Sigma^{-1}\vert^2}
\right)^{2h}= \sum_{r=0}^\infty N^h_r \lambda^h_r(z)\lambda^{h,r}(\bz), 
\label{eq:II}
\end{eqnarray}
where $N^j_m,\,N^h_r$ are binomial coefficients. 
Although we can represent the functions $\psi^j_m(z)$ and 
$\lambda^h_k(z)$ in terms of a free field, we are not interested 
in the explicit expressions within the latter discussions. 
The crucial fact is that $\psi^j_m(z)$ and $\lambda^h_r(z)$ form, 
respectively, the finite and the infinite dimensional representations,  
$V^{cl}_j$ and $V^{cl}_h$, of $\slc$:  
Denoting by  $E_+, E_-, H$ the generators of $\slc$ 
satisfying the relations $[E_+,E_-]=2H, [H, E_\pm]=\pm E_\pm$, 
these representations are, 
%
\begin{eqnarray}
V^{cl}_j &=& \{ \psi^j_m(z) \, \vert\, E_+\psi^j_j = E_-\psi^j_{-j}=0, 
H\psi^j_m=m\psi^j_m, \, -j\leq m \leq j \}, \nonumber \\ [.1cm] 
V^{cl}_h &=& \{ \lambda^h_r (z)  \,\vert\,  E_-\lambda^h_0=0, 
H\lambda^h_r =(h+r) \lambda^h_r, \, r=0, 1, \cdots \}.  
\end{eqnarray} 
%
One can further show that the chiral sector $\psi^j_m(z)$ and  
$\lambda^h_r(z)$ satisfy the Poisson-Lie relations of the algebra
$\slc$\cite{GN,FT}. 
\be
\{\psi^{j_1}_{m_1}(z_1)\stackrel{\otimes}{,}\psi^{j_2}_{m_2}(z_2)\}
=-\pi\gamma^2(r^{j_1j_2})^{m_1'm_2'}_{m_1m_2}
\psi^{j_2}_{m'_2}(z_2)\psi^{j_1}_{m'_1}(z_1), \label{eq:poisson}
\ee
where $r^{j_1j_2}$ is the $\slc\otimes\slc$ valued 
classical $r$-matrix, $r=H\otimes H+E_+\otimes E_-$. 
The same relations hold also for $\lambda^h_r(z)$. 
Upon the general philosophy that 
the quantization of the Poisson-Lie algebra ${\bf {\rm g}}$ 
yields the quantum universal enveloping algebra $U_q{\bf {\rm g}}$ 
endowed with Yang-Baxter relation, 
these algebraic structures will be essential later.  

As the last comment, it is necessary for the later discussions to 
summarize the geometrical aspects of the classical Liouville action. 
Since our main concern will be on the $N$ punctured 
sphere, \ie, a sphere with $N$ branch points of orders infinity, 
we will confine ourselves to the surface $\Sigma_{0, N}$ where 
punctures are located at $\{z_1, z_2, \cdots, z_N=\infty\}$. 
Two Riemann surfaces  of this type are isomorphic if and only if 
they are related by an element of the group $PSL(2,\C)$ $-$ 
a group of all automorphisms of ${\bf P}^1$. 
Using this freedom we can normalize such Riemann surfaces by setting 
$z_{N-2}=0, z_{N-1}=1, z_N=\infty$, then 
$\Sigma_{0,N}=\C\backslash \{z_1, \cdots, z_{N-3}, 0, 1\}$. 
Defining the space of punctures as 
${\cal T}_N=\{(z_1, \cdots, z_{N-3})\in \C^{\otimes N-3}
\vert z_i\neq z_j, \,  {\rm for}\, i\neq j\}$, 
one sees that a point in ${\cal T}_N$ represents a Riemann surface 
of the type $\Sigma_{0,N}$. 
% ${\cal T}_N$ is nothing but the Teichm\"uler space of the surfaces 
% $\Sigma_{0,N}$ and ${\rm dim}\,{\cal T}_N= N-3$. 
Moreover, if two $\Sigma_{0,N}$'s are connected by an action of the
symmetric group ${\rm Symm}(N)$, 
they should be regarded as the same with each other. 
Hence, we get the moduli space of Riemann surfaces of the type 
$\Sigma_{0,N}$ as,  
\be
{\cal M}_{0,N} = {\cal T}_N / {\rm Symm}(N). \label{eq:moduli}
\ee

For the surface $\Sigma_{0,N}$, the Liouville field $\cp(z,\bz)$ 
has the following asymptotics near the punctures, 
\be 
\cp(z,\bz) \stackrel{z\rightarrow z_i}{\longrightarrow} \left\{ 
\begin{array}{ll} 
-2\log\,\epsilon_i -2\log \vert \log\,\epsilon_i\vert,  \quad\,\, 
\epsilon_1\equiv\vert z-z_i\vert, \quad & {\rm for}\quad i\neq N, \\[.15cm]
-2\log\,\vert z\vert -2\log\,\vert \log\vert z\vert\vert, 
\quad & {\rm for} \quad i=N.  
\end{array}\right. \label{eq:asymp}
\ee 
Of course, such asymptotics enjoy the Liouville equation 
(\ref{eq:equation}). 
Denote by $\Im$ a class of fields on $\Sigma_{0,N}$ satisfying
asymptotics (\ref{eq:asymp}). 
Due to the asymptotics, the action (\ref{eq:action}) diverges 
for $\cp\in \Im$ and a regularized action has been obtained by 
Takhtajan and Zograf \cite{ZT} in a reparametrization invariant 
manner as 
%
\be
\overline{S}_L(\cp) =\lim_{\epsilon\rightarrow 0}
\left\{
\int_{\Sigma_\epsilon} d^2z\left( \partial_z\cp\partial_{\bz}\cp
+ e^{\cp(z,\bz)}\right)+ 2\pi N\log\,\epsilon+4\pi(N-2)
\log\vert\log\,\epsilon\vert\right\}, 
\label{eq:Action}
\ee
where $\Sigma_\epsilon=\Sigma\setminus \cup_{i=1}^{N-1}
\{\vert z-z_i\vert <\epsilon\}\cup\{\vert z\vert >1/\epsilon\}$. 
The Euler-Lagrange equation $\delta \overline{S}_L =0$ derives 
again the Liouville equation (\ref{eq:equation}). 
Owing to the regularization, however, the classical action 
$\overline{S}_L(\cp)$ is no longer invariant under the action of 
${\rm Symm}(N)$ on ${\cal T}_N$. 
According to Ref.\cite{Zo}, one can find, for a real constant $k$, 
1-cocycle $f^k_\sigma$ of ${\rm Symm}(N)$ satisfying 
\be
\exp\left(\frac{k}{\pi}\overline{S}_L(\cp)\circ \sigma\right)
\vert f^k_\sigma\vert^2 =
\exp\left(\frac{k}{\pi}\overline{S}_L(\cp)\right). 
\label{eq:moduleranomaly}
\ee
Here ${\rm Symm}(N)$ acts on the trivial bundle 
${\cal T}_N \times {\C} \rightarrow {\cal T}_N$ as 
$(t, z)\mapsto (\sigma t, f^k_\sigma(t) z)$, where 
$ t\in {\cal T}_N,\, z\in {\C}$ and $\sigma\in{\rm Symm}(N)$.  
It follows that the cocycle $f^k_\sigma$ defines a Hermitian 
line bundle ${\cal L}_k={\cal T}_N \times {\C} / {\rm Symm}(N)$ 
over the moduli space ${\cal M}_{0,N}$ and the function 
$\exp(\frac{k}{\pi}\overline{S}_L^{\cp})$ can be interpreted as a 
Hermitian metric in the line bundle ${\cal L}_k$. 
This fact will play an important role in the next section. 





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Quantum Liouville Gravity}

As we have seen in the previous section, the classical 
Liouville theory describes the hyperbolic geometry of 
Riemann surfaces $\Sigma$. 
Indeed, the classical Liouville field $\cf$  
is a function on $\Sigma$ defining  the Poincar\'e  metric 
in terms of the Fuchsian uniformization map. 
On the contrary, in quantum theory, the Liouville field 
$\Phi(z,\bz)$ expresses quantum fluctuations of the metric 
and so does the uniformization map $J_\Sigma^{-1}(z)$.  
In other words, the coordinates $(w, \bar{w})$ on the unit disk 
are not ordinary complex numbers but in some sense quantum objects. 
% In this fact, we can glance at quantum fluctuations of 
% metrics of the surface. 
This is the reason why we regard the quantum Liouville theory as 
a theory of  quantum 2$D$ gravity or quantum geometry of surfaces. 

\subsection{Introduction of Quantum Liouville Action}

We start by summarizing how the quantum Liouville theory appears 
in the string theory. 
The original definition of the string partition function is 
\be
Z=\int d\tau\, [d\Phi]_g[dX]_g [d(gh)]_g
\large{e^{-S_X(X;g)-S_{gh}(b,c;g)-\Lambda_0\int\sqrt{g}}},
\label{string}
\ee
with $\tau$ the Teichm\"uler parameter, $X=\{X^\mu\}$ the string 
coordinates embedded in the $D$ dimensional target space and 
$(gh)$ stands for the ghost coordinates $\{b,c\}$ associated with 
the diffeomorphism invariance. 
Choosing a metric slice $g=e^{\Phi}\hat{g}$ gives the following relation 
for the path integral measures, 
\be
[d\Phi]_{e^{\Phi}\hat{g}}[dX]_{e^{\Phi}\hat{g}} [d(gh)]_{e^{\Phi}\hat{g}}
=J(\Phi;{\hat{g}})\, 
[d\Phi]_{\hat{g}}[dX]_{\hat{g}}[d(gh)]_{\hat{g}}
\ee
where $J(\Phi;{\hat{g}})$ is the Jacobian. 
The contributions to the Jacobian from $[dX]$ and $[d(gh)]$ was 
obtained by Polyakov \cite{Po1}, and that from 
$[d\Phi]$ was postulated \cite{DK} so that the partition function 
(\ref{string}) finally had the following form,  
%
\begin{eqnarray}
Z &=& \int d\tau\, [d\Phi]_{\hat{g}}[dX]_{\hat{g}} [d(gh)]_{\hat{g}}\,
\large{e^{-S_X(X;\hat{g})-S_{gh}(b,c;\hat{g})} }
\label{string2} \\[.15cm] 
 &&\qquad\qquad\qquad\quad\times\, 
\large{e^{-\int d^2\xi \sqrt{\hat{g}}
\left( \frac{A}{2}\hat{g}^{ab}
\partial_a\Phi \partial_b\Phi + BR_{\hat{g}}\Phi +\Lambda 
e^{C\Phi}\right)}}. \nonumber
\end{eqnarray}
The constants $A, B$ are determined by calculating the responses 
of the Weyl rescaling $\hat{g}\,\rightarrow\,e^{\sigma}\hat{g}$ and 
demanding the invariance of $Z$, one finds $A=B=\frac{25-D}{48\pi}$. 
Upon replacing $\Phi\,\rightarrow\sqrt{\frac{12}{25-D}}\Phi$, 
we obtain the Liouville action 
\be
\frac{1}{4\pi}S_L(\Phi:\hat g)=\frac{1}{4\pi}
\int_\Sigma d^2\xi \sqrt{\hat g}\left\{ \half
 \hat{g}^{ab}\partial_a \Phi \partial_b \Phi
+ Q_0R_{\hat g}\Phi+ \Lambda e^{\gamma\Phi}
 \right\},  \label{waction}
\ee
where 
\be
Q_0=\sqrt{\frac{25-D}{12}}. 
\ee
On the other hand, the constant $C$ plays the role of 
renormalized coupling constant. 
In eq.(\ref{waction}), we have replaced $OCC$ by $\gamma$, 
since the field $\Phi$ has been rescaled.  and $\gamma$ 
will be related to the deformation parameter $q$ of quantum group 
$\qslc$. 

% This Liouville action is invariant under the Weyl transformation 
% together with the shift of the Liouville field; 
% \be
% \hat{g}\,\rightarrow\,e^\sigma \hat{g}, \quad 
% \gamma \Phi\,\rightarrow\,\gamma \Phi-Q_0\sigma. \label{Weyl}
% \ee
% Due to the invariance, we can freely choose the background metric. 

The energy-momentum  tensor of the Liouville sector is obtained  as  
\be
T_{zz} = 
-\half (\partial_z \Phi)^2 +  Q_0 \partial^2_z\Phi,
\quad\, T_{z\bz}=0. 
\label{eq:qEM}
\ee
The second equation is derived upon the equation of motion. 
The central charge of the quantum Liouville theory is calculated as 
\be
c_L=1+12Q_0^2.   \label{eq:central}
\ee
Notice that the total conformal anomaly vanishes, 
\be
c_X + c_L +c_{gh} =D+(26-D)-26=0,
\ee
which is consistent with the requirement of the conformal invariance. 

For the time being, we will concentrate only on the Liouville sector. 
The correlation function of $N$ vertices are given by 
%
\be
\left\langle V_{\alpha_1}(z_1,\bz_1) V_{\alpha_2}(z_2,\bz_2) \cdots 
V_{\alpha_N}(z_N,\bz_N) 
\right\rangle  = \int_{\Im} [d\Phi] \, 
e^{-\frac{1}{4\pi}S_L(\Phi:\hat{g})}\,
\prod_{i=1}^N V_{\alpha_i}(z_i,\bz_i), 
\label{eq:correlation1}
\ee
where the Liouville vertex operator with charge $\alpha_i$ is given by 
\be
V_{\alpha_i}(z_i,\bz_i)=e^{\alpha_i\gamma\Phi(z_1,\bz_i)}. 
\ee
Notice that, in this definition of the correlation function, 
the base manifold on which the vertices live is not 
a manifold with  branch points but just the sphere ${\bf P}^1$. 
Then the functional integral is performed over the space 
$\Im$ of all smooth metrics $\Phi(z,\bz)$ on ${\bf P}^1$. 
Since puncture corresponds to the branch point of order infinity, 
all the punctures on $\Sigma_{0,N}$ correspond to the vertex operators
with charges $\alpha_i=1/2\gamma^2$ (see eq.(\ref{charge})). 
As in the classical theory,  (\ref{eq:I}) and (\ref{eq:II}), 
there are two distinct regions according to the value of the charge 
$\alpha$, \ie, (I) $\alpha\leq 0$ and (II) $\alpha>0$. 
We will refer these two regions as (I) weak coupling region and 
(II) strong coupling region.
\footnote{According to the standard convention, 
the Liouville theories which are defined in the spacetime of dimension 
$D\leq 1$ and $1<D<25$ are called, respectively, weak coupling 
theory and string coupling theory. 
The reason why we use here the terms weak and strong is that 
it will be turned out later that vertices defined 
in the weak (strong) coupling theory carry negative(positive) charges.} 
%   
In quantum theory, however, the difference between them becomes 
more sever than the classical theory. 
There is a big gap known as the $D=1$ barrier between two regions.  
The weak coupling region, studied first by KPZ and DDK,  
has been investigated by many authors.  
On the other hand, the quantum Liouville theory in the strong 
coupling region is a long-standing problem. 
Our concern in this article is on this region and 
we will investigate it by using quantum group methods. 
Before doing this, it is instructive to review the Liouville 
theory in the weak coupling region. 


\subsection{Brief Review of the Weak Coupling Region}

The standard approach to the weak coupling Liouville theory 
begins  with the action (\ref{waction}). 
Till now remarkable progresses have been done by many authors.   
Let us, in this section, observe  
the quantum group aspects of the weak coupling Liouville theory. 
The quantum group structure appears through the vertex operators, 
and has been studied extensively in \cite{Gerv,ST}. 
A crucial fact is that, in this region,  
the finite dimensional representations of $\qslc$ appear.  
To be precise, writing the vertex operator with charge $-j$ as 
\be 
e^{-j\gamma\Phi(z,\bz)}=\sum_{m=-j}^j {\cal N}^j_m 
\Psi^j_m(z)\overline{\Psi}^{j,m}(\bz), 
\ee
they have shown that $\Psi^j_m(z),\, m=-j, \cdots, j$ 
form the $2j+1$ dimensional representation of $\qslc$ with 
\be
q=e^{\pi i \frac{\gamma^2}{2}},  \label{defq}
\ee 
and satisfy the braiding-commutation relations, 
\be
\Psi^{j_1}_{m_1}(z_1)\otimes \Psi^{j_2}_{m_2}(z_2)
= \left(R^{j_1 j_2}\right)^{m_1' m_2'}_{m_1 m_2}
\Psi^{j_2}_{m'_2}(z_2)\otimes \Psi^{j_1}_{m'_1}(z_1), 
\label{eq:braid}
\ee
where $R^{j_1j_2}$ is the universal $R$-matrix of $\qslc$. 
The factorization property of the quantum vertices is a natural guess 
from the classical result (\ref{eq:I}). 
The braiding-commutation relation (\ref{eq:braid}) 
reduces to the Poisson-Lie relation (\ref{eq:poisson}) 
in the classical limit, $\gamma\rightarrow0$. 
This fact is in agreement with the general concept of quantum groups
as the quantization of Poisson-Lie algebras of classical Lie groups. 

An important fact of the region (I) arises in 
the relation between $Q_0$ and the renormalized coupling constant $\gamma$,  
that is, 
\be
Q_0=\frac{1}{\gamma} +\frac{\gamma}{2}.  \label{eq:QW}  
\ee
% This relation is derived by calculating anormalous dimension of a 
% vertex operator. 
% It is well-known that anormalous dimension can be obtained by 
% analizing braiding relation, which is deeply related to 
% the $q$-$6j$ symmbol. 
% In other words, the relation (\ref{eq:QW}) can be obtained 
% only from quantum group structure. 
Substituting (\ref{eq:QW}) into the general expression of the 
central charge (\ref{eq:central}), one obtains 
\be
c_L=13+6\left(\frac{\gamma^2}{2} + \frac{2}{\gamma^2}\right)  
\label{eq:weakcentral}
\ee
and finds that the central charge always satisfies $c_L\geq 25\,(\leq 1)$ 
for a real (imaginary) $\gamma$. 
Notice that $\gamma\Phi$ should be real since $e^{\gamma\Phi}$ is 
the metric in this model. 
The very origin that restricts the theory to the weak coupling 
region is 
the relation (\ref{eq:QW}). 
This was first obtained 
in Ref.\cite{CT} by requiring that the EM tensor (\ref{eq:qEM}) 
satisfies Virasoro algebra on the cylindrical basis $S^1\times \R$.  
Once Virasoro structure is found, one can apply the Coulomb gas 
formulation of the minimal CFT to the Liouville theory, 
although the Liouville field is no longer a free field. 
Let us explain briefly. 
Notice  that the parameter $Q_0$ corresponds to $i\alpha_0$, 
$\alpha_0$ being the background charge in the Coulomb gas formulation 
and the highest weight vector $\Psi^j_j(z)$ can be related to  the
primary field of the type $V_{2j+1, 1}(z)$ in the Kac's table, 
where $V_{n,m}(z)= :e^{i\alpha_{n,m}\hat\phi(z)}:$ with the charge 
$\alpha_{n,m}=\{(1-n)\alpha_++(1-m)\alpha_-\}/2$ and a free field 
$\hat\phi(z)$.  
Upon comparing eq.(\ref{eq:QW}) and the relations 
$\alpha_0=(\alpha_+ +\alpha_-)/2,\, \alpha_+\alpha_-=-2$, 
one immediately sees that the renormalized coupling constant 
$\gamma$ is related to one of the screening charges, 
say, $\alpha_+$ as $\gamma=i\alpha_+$.  
With the above connections at hand, 
the correspondence between the metric $e^{\gamma\Phi(z,\bz)}$ and 
the screening operator $e^{i\alpha_+\hat\phi(z)}$ which has 
conformal dimension $1$ is now clear. 
Thus we obtain the correct dimension $(1,1)$ for  the metric 
$e^{\gamma\Phi(z,\bz)}$ \cite{DK}. 
Furthermore, the relation (\ref{eq:QW}) is  consistent with 
the quantum group structure of $\qslc$ when the representation is 
of finite dimension. 
Indeed, the braiding commutation relation which is deeply related 
to the $q$-$6j$ symbols indicates that the operator $e^{-j\gamma\Phi}$ 
has dimension $-j-\frac{\gamma^2}{2}j(j+1)$. 
By comparing with the result of CFT that the dimension of the operator 
is given as $-\frac{1}{2}j\gamma(j\gamma+2Q_0)$, 
we then obtain (\ref{eq:QW}). 
Thus in the weak coupling Liouville theory, the relation (\ref{eq:QW}) 
works well and plays an important role, although 
it is the origin of the severe problem, the $D=1$ barrier. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%                         3.1
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The Strong Coupling Region}
Now we are at the stage to enter into  the mysterious world 
of the strong coupling Liouville theory. 
At the beginning, we propose a Liouville action for the 
strong coupling region, 
%
\be
\widehat{S}_L(\Phi:\hat g) = S_L +S', \quad\, \mbox{with} \quad\, 
S'=\lambda \int d^2z \sqrt{\hat  g}e^{\cp(z,\bz)}, 
\label{eq:qaction}
\ee
where $S_L$ is the Liouville action (\ref{waction}).  
We have introduced the additional cosmological term $S'$ whose necessity 
will be  understood later. 
Note here that, owing to this term, 
the potential reduces to the classical metric $e^{\cp(z,\bz)}$ 
in the limit $\Phi\rightarrow -\infty$, while, without this term, 
the potential vanishes in the limit.  
% The holomorphic sector of the energy-momentum tensor is exactly same as 
% eq.(\ref{eq:qEM}), and so  
% the central charge is also given by eq.(\ref{eq:central}). 
% on the other hand, the trace part does not disappear even uopn the 
% equation of motion for $\Phi$. 
% Owing to the additional cosmological term $S_1$, we have 
% \be 
% T_{z\bz} = \Lambda_1 e^{\cp(z,\bz)}.
% \ee
% The central charge is also given by eq.(\ref{eq:central}). 
% Moreover the action is not invariant under the Weyl rescaling 
% with $\Phi$ translation (\ref{Weyl}). 
% It recovers if the classical field $\cp$ is shifted 
% $\cp\rightarrow\cp-\sigma$ simltaneously.  
We will work, however, not  with this action 
but with the representations of the quantum group $\qslc$ 
in the following discussions. 

Let us turn to the quantum group structure. 
To do this, we introduce vertex operator $e^{h\gamma\Phi(z,\bz)}$   
denoting the charge $h$ instead of $\alpha$ in this region. 
By definition the charge $h$ is always positive. 
As in the weak coupling case, the vertex operator allows 
holomorphic decomposition as, 
%
\be
e^{h\gamma\Phi(z,\bz)}=\sum_{r=0}^\infty {\cal N}_q^{h,r} \Lambda^h_r(z) 
\overline{\Lambda}^{h,r}(\bz). \label{eq:strongvertex}
\ee  
It can be shown that the vectors $\Lambda^h_r(z), r=0, 1, \cdots$ 
form an infinite dimensional highest weight representation 
$V_h$ of $\qslc$ with (\ref{defq}). 
Thus in this region, infinite dimensional  representations 
appear instead of finite ones in the weak-coupling region. 
The question here is the relation between $Q_0$ and the 
renormalized coupling constant $\gamma$. 
If we require $c_L>1$ with maintaining the relation 
(\ref{eq:QW}), the constant $\gamma$ becomes complex. 
This fact means that, as suggested in the previous section 3.1, 
the screening charges become complex and, therefore, 
the dimensions of primaries are not real in general. 
Gervais and his collaborators \cite{Ge} have 
discussed the strong coupling Liouville theory 
under the situation. 
Their solution to the problem is summarized in \lq\lq 
truncation theorem'', which guarantees that 
only in the special central charges $c_L=7, 13$ and $19$, 
the chiral components of the Liouville vertices form 
Virasoro Verma modules with real highest weights.  
Upon the relation (\ref{eq:QW}) together with the central charge  
(\ref{eq:central}), these central charges 
$c_L=7, 13$ and $19$ give the values of the constants $\gamma$, 
respectively, $\gamma^2=-1+i\sqrt{3},\, 2i$ and $\gamma^2=1+i\sqrt3$.  
Notice here that, with the definition eq.(\ref{defq}), 
the parameter $q$ of the quantum group $\qslc$ is generic. 
In such a situation the relation (\ref{eq:QW}) is actually consistent. 
As in the weak coupling region, the braiding commutation relation 
in the case of infinite dimensional representations 
shows that the dimension of the operator $e^{h\gamma\Phi}$ 
is calculated as $h-\frac{\gamma^2}{2}h(h-1)$. 
By comparing this dimension with that obtained from the CFT method, 
that is, $-\frac{1}{2}h{\gamma}(h\gamma-2Q_0)$, we again obtain 
(\ref{eq:QW}). 

Now we would like to ask a question, what will happen if 
we take the parameter $q$ being a root of unity. 
With the idea that the infinite dimensional representations 
$\ncom$ and $gncom$ are drastically different from each other, 
it is quite natural to expect that the quantum Liouville theory 
associated with $\ncom$ has big difference from that based on $\gncom$. 
Furthermore, since the $q$-$6j$ symbols for the representations $\ncom$ 
is completely different from those for $\gncom$, 
we could get rid of the relation (\ref{eq:QW}) and, therefore,  
have any other values of the central charge. 
In the following discussions, 
the only essential assumption  is that our Liouville theory 
possesses the structure of $\ncom$. 

Let $q$ be a $p$-th root of unity, 
\footnote{ 
In this paper, as a matter of convention, 
$q$ such that $q^p=-1$ for ${}^\exists p\in \N$ 
is also called a root of unity. }, 
\ie,   
\be
 q=e^{\pi i \qvalue} \quad \quad {\rm \ie}, \quad 
\frac{\gamma^2}{2}=\qvalue, \label{eq:gvalue}
\ee
with $p, p'\in \N$, coprime with each other. 
Such representations are reported in the Appendix. 
Once the parameter $q$ is set to the value (\ref{eq:gvalue}), 
the infinite dimensional 
highest weight representations are parameterized by two integers 
$\mu$ and $\nu$ such that $h\rightarrow h_{\mu\nu}=\zeta p-j$ where 
$\,\zeta=\nu/2p', \,j=(\mu-1)/2$. 
We denote by $V_{\mu,\nu}$ the module on the highest weight
state of weight $h_{\mu\nu}$ and by $\Lambda^{\mu\nu}_r(z)$ 
the $r$-th weight vector in the module $V_{\mu,\nu}$, namely,  
it corresponds to the $r$-th holomorphic sector  
in the decomposition of the vertex operator $e^{h_{\mu\nu}\Phi(z,\bz)}$. 
One immediately sees that the terms $\Lambda^{\mu\nu}_{kp+\tilde s}(z)
\overline{\Lambda}^{\mu\nu, kp+\tilde s}(\bz)$,  $k=0, 1, \cdots, 
\tilde{s}=\mu, \cdots, p-1$, disappear from the vertex operator 
$e^{h_{\mu\nu}\Phi(z,\bz)}$ because, as shown in 
eq.(\ref{eq:null}), $\Lambda^{\mu\nu}_{kp+\tilde{s}}$ 
are the elements in ${\cal X}_{\mu,\nu}$, 
the space of zero norm states, and so 
the vertex operator is given by means of the summation only of 
$\Lambda^{\mu\nu}_{kp+s}(z)\overline{\Lambda}^{\mu\nu,kp+s}(\bz)$, 
$s= 0,\cdots,\mu-1$, namely, the elements 
in the irreducible highest weight module 
$V^{irr}_{\mu,\nu}= V_{\mu,\nu}\backslash {\cal X}_{\mu_\nu}$. 
Therefore we can now use the important theorem in the Appendix, 
more concretely, the relation (\ref{eq:essential}): 
The irreducible highest weight module $V^{irr}_{\mu,\nu}$ 
is isomorphic to the tensor product 
$V_{\zeta}^{cl}\bigotimes{\cal V}_j $  
where $V_{\zeta}^{cl}$ and ${\cal V}_j$ are, respectively, 
the highest weight representation of 
the classical algebra $\slr$ with highest weight $\zeta$ 
and the $(2j+1)$-dimensional representation of $\qslc$. 
Owing to the isomorphism, the holomorphic sector factorizes  as 
\be
\Lambda^{\mu\nu}_{kp+s} (z)\mapsto \lambda^{\zeta}_k(z) 
\bigotimes \Psi^j_m(z),  \label{eq:iso}
\ee
and also does the anti-holomorphic sector.  
 
Keeping this remarkable  facts of $\ncom$ in mind, let us look at  the 
$N$-point correlation function in the Liouville theory 
associated with $\ncom$, 
\begin{eqnarray}
Z^S[{\sf m}:\mu_i,\nu_i] &:=& 
\langle V_{\mu_1\nu_1}(z_1,\bz_1)\cdots V_{\mu_N \nu_N}(z_N,\bz_N)
\rangle  \nonumber \\ 
{} &=& \int[d\Phi]e^{-\frac{1}{4\pi}\widehat{S}_L(\Phi;\hat{g})}
\prod_{i=1}^N V_{\mu_i \nu_i}(z_i,\bz_i), 
\label{eq:defcorr}  
\end{eqnarray}
where $V_{\mu_i \nu_i}(z_i,\bz_i) =e^{h_{\mu_i \nu_i}\gamma\Phi(z_i,\bz_i)}$,  
and ${\sf m}=(\sl{m},\overline{\sl m})$ 
stands for the moduli parameters of the surface. 
We will come back later to the discussions of correlation function and 
transition amplitude as well. 
Before that let us enjoy our model with the remarkable nature of $\ncom$. 
Thanks to the factorizations (\ref{eq:iso}) 
of the holomorphic sectors and the similar factorization 
of the anti-holomorphic sector,  
the vertex operators in (\ref{eq:defcorr}) are factorized as, 
\begin{eqnarray}
V_{\mu_i\nu_i}(z,\bz) &=& \sum_{k=0}^\infty N^{\zeta_i}_k 
\lambda^{\zeta_i}_k(z_i) 
\lambda^{\zeta_i,k}(\bz_i)\bigotimes\sum_{m=-j_i}^{j_i} {\cal N}^{j_i}_m 
\Psi^{j_i}_m(z_i)\Psi^{j_i,m}(\bz_i) \nonumber \\ 
 {} &=& e^{\zeta_i\cp(z_i,\bz_i)}\bigotimes 
e^{-j_i\tilde\gamma\phi(z_i,\bz_i)} 
\label{eq:separation}
\end{eqnarray}
where  $\cp(z,\bz)$ and $\phi(z,\bz)$ are, respectively, 
the classical and the quantum Liouville fields. 
With the remarkable decompositions of the vertex operators at hand, 
we can now  conjecture that the action also separates effectively as  
\be
\widehat{S}_L(\Phi;1\!\!\rm{I})\,\rightarrow\, 
S^{cl}(\cp;1\!\!\rm{I}) + S^q(\phi;\hat{g}_P).  
\label{eq:saction}
\ee
Here the actions $S^{cl}(\cp;1\!\!\rm{I})$ and $S^q(\phi;\hat{g}_P)$ 
are the Liouville actions with respect to the fields $\cf$ 
and $\phi(z,\bz)$, respectively, and are explicitly given by 
%
\begin{eqnarray}
S^{cl}(\cp;1\!\!\rm{I}) &=& \frac{1}{\beta^2}\int d^2z\left(
\partial_z\cp\partial_{\bz}\cp + \lambda  e^{\cp(z,\bz)}\right), 
\label{eq:caction}\\
S^q(\phi;\hat{g}_P) &=& \int d^2z 
\sqrt{\hat{g}_P}\left( 
\nabla\phi\overline{\nabla}\phi + QR_{\hat{g}}\phi(z,\bz) + 
\Lambda e^{\tilde\gamma\phi(z,\bz)}\right), 
\label{eq:waction}
\end{eqnarray}
%
where $\beta$ is some constant and 
$\hat{g}_P=e^{\tau\cp}dzd\bz$ is chosen as the background metric 
for  the quantum sector. 
When $\tau=1$ in the exponent of $\hat{g}_P$, the background 
metric is just the Poincar\'e metric. 
The reason why we have introduced new coupling constant 
$\beta$ is that, at this stage, we have no idea how the 
coupling constant of the classical sector should be.   
On the contrary, remembering that (see Appendix) the finite 
dimensional representations ${\cal V}_j$ appeared by the decomposition  
has the same deformation parameter $q$ as that of $V^{irr}_{\mu\nu}$,  
the renormalized coupling constant $\tilde\gamma$ of 
the quantum sector should satisfy, 
\be
\frac{\tilde\gamma^2}{2}\,=\, \frac{\gamma^2}{2}+2n, \,\quad\quad 
n=0,\,\pm1,\,\pm2,\cdots
\label{tgam}
\ee 
% However the relative strength $\kappa^2$ between the classical and 
% quantum sectors has been introduced and can be renormalized into 
% the coupling constant $\gamma$, 
% $\gamma\rightarrow \tilde\gamma+\gamma/\kappa$. 
It should be emphasized that, upon the condition 
$\gamma Q_0=\tilde\gamma Q=1$, the conjecture (\ref{eq:saction}) 
together with (\ref{eq:caction}),(\ref{eq:waction}) 
is exactly true under the substitution 
$\gamma\Phi=\tau \cp +\tilde\gamma\phi$ with $\tau=\gamma/\beta$.  

Some remarks are now in order. 
The above observations seems to suggest that, in the 
quantum Liouville theory via $\ncom$, the Liouville field 
$\Phi(z,\bz)$  should be expanded as 
$\gamma\Phi=\tau\cp+\tilde\gamma\phi$ and,  
therefore, one can  interpret the field $\phi$ as the quantum fluctuation 
around $\tau\cp$. 
In other words, the metric $ds^2=e^{\gamma\Phi(z,\bz)} dz d\bz$ 
is to be written as $ds^2=e^{\tilde\gamma\phi(z,\bz)}{\hat g}_{P}$  
and $e^{\tilde\gamma\phi(z,\bz)}$ represents quantum fluctuation of metrics 
around the classical background metric ${\hat g}_{P}=e^{\tau\cp} dz d\bz$.  
If $\tau$ is set to be 1, the background metric ${\hat g}_{P}$ 
becomes the Poincar\'e metric. 
Now we can give a possible interpretation of quantum $2D$ manifold     
according to the above observations: 
{\em Quantum manifold  with metric $e^{\gamma\Phi(z,\bz)}$ should be 
considered as the total system of 
$2D$ classical manifold  with the metric $e^{\tau\cp} dz d\bz$ 
and quantum fluctuations around the classical surface}. 
This interpretation matches quite well to the general concept 
of quantum object.   
% From the viewpoint of eq.(\ref{eq:start}), 
% the action $S^q(\phi;\hat{g}_P)$ in (\ref{eq:waction}) has arisen in the way 
% $Z[e^{\kappa\phi} {\hat g}_P]=\exp (-const\cdot S^q(\phi))Z[{\hat g}_P]$. 
Note also that the action $S^q(\phi)$ governing the quantum 
fluctuations is again the quantum Liouville theory 
associated with $\com$. 

It is worthwhile to comment here the difference between our 
formulation and the standard weak coupling Liouville theory 
\cite{Po2}--\cite{DK}.  
In the standard approaches 
whose action is given in (\ref{waction}),  
the theory also has $\com$ structure as shown in section 3.1. 
But unlike the quantum sector $S^q(\phi;\hat{g}_P)$ of our model, 
the background metric can be freely chosen together with 
the shift of Liouville field $\Phi(z,\bz)$.  
For example, in Refs.\cite{Po2,KPZ},  
the metric in the light-cone gauge $ds^2=\vert dx_+ + \mu dx_-\vert^2$,  
which is conformally equivalent to the metric in the conformal gauge 
$ds^2=e^{\tilde\gamma\tilde\phi(z,\bz)}dz d\bz$ 
was chosen. 
These choices have the flat background metric. 
On the contrary, in our formulation, the background metric is 
automatically selected as $\hat{g}_P$. 
Thus the appearance of the classical sector in addition to the 
quantum theory associated with $\com$ is the characteristic 
feature of our formulation. 
We will recognize  the importance of the classical sector 
from the viewpoints of geometric quantization of moduli space 
in section 3.5. 


\subsection{Fusion Rules}

Before going to the discussion of the correlation function, 
it is interesting to look at fusion rules in our model. 
First, we have to investigate Clebsh-Gordan (CG) decomposition rule 
for the tensor product of two irreducible infinite dimensional 
representations of $\qslc$,
%
\begin{equation}
V_1 \otimes V_2\;\longrightarrow\; V_3,
\end{equation}
%
where $V_i:=V_{\mu_i, \nu_i}^{irr}\cong V_{\zeta_i}^{cl}\otimes 
{\cal V}_{j_i}$. 
The quantum CG coefficient, known as the $q$-$3j$ symbol, 
for the infinite dimensional representations of $\qslc$ 
is given when $q$ is $not$ a root of unity as follows,  
%
\begin{eqnarray}
&&\left[ \begin{array}{ccc} h_1 & h_2 & h_3 \\
                             r_1 & r_2 & r_3 \end{array}\right]^{\qslc}_q
=C(q)\delta_{r_1+r_2,\,r_3}\tilde{\Delta}(h_1, h_2, h_3)  \nonumber \\
&&\times \left\{
\frac{[2j-1][r_3-h_3]![r_1-h_1]![r_2-h_2]![r_1+h_1-1]![r_2+h_2-1]!}
      {[r_3+h_3-1]!}\right\}^{1/2} \nonumber \\
&&\times\sum_{R\ge 0}(-)^R q^{\frac{R}{2}(r_3+h_3-1)}
                \frac{1}{[R]![r_3-h_3-R]![r_1-h_1-R]![r_1+h_1-R-1]!}
\nonumber\\
&&\quad\cdot\frac{1}{[h_3-h_1-r_1+R]![h_3+h_2-r_1+R-1]!},
\label{CG}
\end{eqnarray}
%
where $C(q)$ is a factor which is not important for our analysis below and
\begin{eqnarray}
&{}&\tilde{\Delta}(h_1, h_2, h_3)  \\
&{}&\quad =\{[h_3-h_1-h_2]![h_3-h_1+h_2-1]!
                        [h_3+h_1-h_2-1]![h_1+h_2+h_3-2]!\}^{1/2}. \nonumber
\end{eqnarray}
%
The notations $h_i:=h_{\mu_i\,\nu_i}=\zeta_i p-j_i, \;
r_i=(\zeta_i+k_i)p-m_i$ have been used.
Of course for our case, \ie, $q=\exp(\pi i \frac{p'}{p})$, 
the CG coefficient is not necessarily well-defined 
due to the factor $[p]=0$.
What we have to do is only to find conditions which give  
finite CG coefficients. 
Since the calculation is lengthy and our interest is not in the details, 
we describe here only the quite interesting result;  
finite CG coefficients exist if and only if
%
\begin{equation}
\begin{array}{l}
\zeta_1 + \zeta_2 \le \zeta_3, \\
{}\\
\vert j_1 - j_2 \vert -1 < j_3 \le {\rm min}\,(j_1+j_2, p-2-j_1-j_2).
\end{array}
\end{equation}
It should be noticed that, on the modules $V^{cl}$ and ${\cal V}$, 
the coproducts of the 
operators $K=q^H$ and $L_0$ are $\Delta(K)=K\otimes K$ and 
$\Delta(L_0)=L_0\otimes 1 +1\otimes L_0$, respectively. 
The coproduct of the Cartan operator yields the conservation
law of the highest weights, physically speaking, the conservation of the
spin or angular-momentum along the $z$-axis.
Now, from the above coproducts, we have the conservation laws
$(\zeta_1+k_1) + (\zeta_2+k_2) = (\zeta_3+k_3)$ and $m_1 + m_2 = m_3$
(mod $p$). 
Therefore the minimum value of $j_3$ is just $\vert j_1-j_2 \vert$ because
the difference between $\vert j_1-j_2 \vert$ and $j_3$ is always integer.
We therefore obtain the following decomposition rule of the tensor product of
two infinite dimensional representations of $\qslc$ at a root of unity,
%
\begin{equation}
\left( V_{\zeta_1}^{cl}\otimes {\cal V}_{j_1}\right) \bigotimes
\left( V_{\zeta_2}^{cl}\otimes {\cal V}_{j_2}\right)
= \left( \bigoplus_{\zeta_1+\zeta_2\le\zeta_3}V_{\zeta_3}^{cl}\right)
\bigotimes \left(\bigoplus_{j_3=\vert j_1-j_2\vert}^{{\rm min}\,
\{j_1+j_2, p-2-J_1-j_2\}}{\cal V}_{j_3}\right).
\end{equation}
%
The decomposition rules for the tensor products of $V_\zeta^{cl}$ 
and of ${\cal V}_j$ are the same as those for the tensor products 
of the $\slr$ and of the finite dimensional 
representation of $\qslc$, respectively. 

Applying this decomposition rule to our model, 
fusion rule is 
\be
\left[e^{h_1\gamma\Phi}\right]\times \left[e^{h_2\gamma\Phi}\right]
=\left(\sum_{\zeta_1+\zeta_2\le\zeta_3}\left[e^{\zeta_3\cp}\right]
\right)\bigotimes \left(
\sum_{j_3=\vert j_1-j_2\vert}^{{\rm min}\,
\{j_1+j_2, p-2-J_1-j_2\}}\left[e^{-j_3\tilde\gamma\phi}\right)\right), 
\ee
namely, exactly the same as the tensor product the fusion rules 
of the classical theory and those of the weak coupling theory. 
This result means that the classical sector and the quantum sector never 
mix with each other. 

\subsection{Correlation Functions and Amplitudes}

Let us return to the discussion of the correlation functions 
in our model. 
The original definition was given in eq.(\ref{eq:defcorr}). 
What we will see in this section is how the correlation function 
and amplitude can be written by the decompositions of vertices and actions. 
Since, as we have seen, the Liouville field $\Phi$ can be expanded around 
the classical solution, the functional measure is  
$[d\Phi]=[d\phi]$ up to a constant.  
The correlation function (\ref{eq:defcorr}) is expected finally 
to be factorized into classical sector and quantum sector, 
\ie, up to some constant, it can be written as 
\be
Z^S[{\sf m}:\{\mu,\nu\}] = 
Z^{cl}[{\sf m}:\{\zeta\}]Z^q[{\sf m}:\{j\}].    
\label{eq:correlation}
\ee
Here the quantum sector is given by 
\be
Z^q[{\sf m}:\{j\}]:=\int [d\phi] 
\,e^{-\frac{1}{4\pi} S^q(\phi;\hat{g}_P)}\,
\prod_{i=1}^N  e^{-j_i\phi(z_i,\bz_i)},  
\label{eq:qpartition}
\ee
and the classical sector is  
\be
Z^{cl}[{\sf m}:\{\zeta\}]=e^{-\frac{c_0}{48\pi}\overline{S}^{cl}(\cp)},
\label{eq:classical}
\ee
where $c_0=12/\beta^2$ is the central charge of 
the classical Liouville theory. 
The action $\overline{S}^{cl}(\cp)$ 
denotes the classical action defined on the surface with 
$N$ branch points and  regularized by subtracting the singularities 
near the branch points. 
In particular, in the case when the topology is the 
$N$-punctured sphere,  \ie, 
all the vertices carry the charges 
$\zeta_i=1/2\beta^2$, $\overline{S}^{cl}(\cp)$ is 
given by eq.(\ref{eq:Action}). 
Let us discuss in more detail.  
As stated above, the action $S^q(\phi;\hat{g}_P)$ corresponds to 
the Liouville action in the conformal gauge 
$ds^2=e^{\tilde\gamma\phi(z,\bz)}{({\hat g}_P)}_{z\bz}dz d\bz$ with 
$\hat{g}_P$ being the classical background. 
% The central charge coming from this sector is 
% \be
% c_q=1+12Q^2.   \label{eq:weakcenter}
% \ee
Suppose that the correlation function $Z^q[{\sl m},\{j\}]$ 
admits holomorphic factorization as, 
\be
Z^q[{\sl m}:\{j\}] =\sum_{I,J}N^{I,J}\overline{\Psi}_I[\overline{m}:\{j\}]
\Psi_J[m:\{j\}],  \label{eq:holomorphic}
\ee
where $N^{IJ}$ is some constant matrix. 
Here $\Psi_I[m:\{j\}]$ is expected to be a holomorphic section of 
a line bundle over the moduli space ${\cal M}_{0,N}$. 
We will see later that this expectation is acceptable. 

Now we are at the stage to discuss transition amplitudes 
of our Liouville theory. 
In order to obtain a transition amplitude, we have to integrate 
the correlation function $Z[{\sf m}]$ over the moduli space of surface 
metric with the Weil-Peterson metric. 
Combining (\ref{eq:classical}) and (\ref{eq:holomorphic}) 
and integrating over the moduli space, $N$-point transition amplitude 
${\cal A}_N(\{\zeta\},\{j\})$ has the following form,   
\be
{\cal A}_N(\{\zeta\},\{j\}) = \sum_{I,J}N^{I,J}
\int_{{\cal M}_{0,N}} d(WP) e^{-\frac{c_0}{48\pi}\overline{S}^{cl}}
\overline{\Psi}_I[\overline{m};\{j\}]\,\Psi_J[m:\{j\}], 
\label{eq:amplitude}
\ee
%
where $d(WP)$ stands for the Weil-Peterson measure on the moduli space. 
Here let us see the geometrical meaning of the classical sector. 
For the topology of the $N$-punctured sphere, 
there are quite remarkable facts shown by Zograf and Takhtajan \cite{ZT}  
about the connection between the classical Liouville theory 
and K\"ahler geometry of the moduli space of complex structure. 
The important facts are as follows: 
First the Liouville action evaluated on the classical solution is 
just the K\"ahler potential of the Weil-Peterson symplectic structure, 
precisely, $\omega_{WP}=i\overline{\partial}\partial \overline{S}^{cl}/2$.  
Second the accessory parameters $c_i, i=1\sim N$ are written as 
$-2\pi c_i=\partial \overline{S}^{cl}/\partial z_i$.\footnote{This 
relation between the classical Liouville action and the accessory
parameter which is associated with every puncture was first 
conjectured by Polyakov.\cite{Po3}} 
>From these facts, one can show that the functions $c_i$ are in 
involution \cite{Ta3}, \ie, $\{c_i, z_j\}_{WP}=i\delta_{ij}$, 
where $\{\,\,,\,\,\}_{WP}$ 
is the Poisson bracket with respect to the Weil-Peterson symplectic 
2-form $\omega_{WP}$. 
Thus the classical Liouville theory can be regarded as the 
K\"ahler geometry of the moduli space of surfaces. 

Due to the relation, it is natural to expect the deep connection 
between our quantum Liouville theory and the geometric quantization 
of the moduli space which corresponds to 
the phase space of the classical geometry. 
With this hope, we should try to observe our result 
from the viewpoints of geometric quantization. 
Before doing this, it is helpful to summarize the basic 
facts about the geometric quantization of a classical theory.  
Consider a K\"ahler manifold ${\cal M}$ equipped with a symplectic 
structure $\omega$ which is written in terms of the K\"ahler
potential $K$ as $\omega=i\partial\overline{\partial} K$.  
The K\"ahler manifold plays the role of the phase space 
of the classical theory. 
Geometric quantization is performed by building a line bundle 
${\cal L}\rightarrow {\cal M}$ over the $2N$ dimensional 
manifold ${\cal M}$ with the curvature two form $F= -i\omega$. 
Let us parameterize ${\cal M}$ by $(q_i, p_i),\, i=1\cdots N$, 
and denote the section as $\psi(q_i, p_i)$. 
The final manipulation to complete the geometric quantization 
is to impose on ${\cal L}$ the condition that 
the section is annihilated by the derivatives of 
half of the variables, which is called a polarization.  
In other words, by the  choice of a polarization, 
sections are represented by $N$ variables.  
Let us, for example, choose the polarization as 
$\partial_{p_i}\psi=0,\, i=0,\cdots,N$. 
Then the Hilbert space ${\cal H}$ is the space of sections with the 
polarization, 
\be
{\cal H}=\{\psi\,\vert\, \partial_{p_i}\psi=0\}
\ee
The inner product on ${\cal H}$ is
introduced with the Hermitian metric $e^{-K}$, \ie, 
for $\psi_1, \psi_2 \in {\cal H}$, it is  
$\langle\psi_1, \psi_2\rangle = \int 
e^{-K}\overline{\psi}_1\cdot \psi_2$,   
where the measure is that defined by $\omega$. 
Putting our expression (\ref{eq:amplitude}) together with 
the fact that, as we have seen in the last part of Section 2, 
a line bundle with the Hermitian metric 
$\exp(\frac{k}{\pi}\overline{S}^{cl})$ 
can be constructed on the moduli space 
${\cal M}_{0,N}=({\sl m}_i, c_i)$,  
the holomorphic part $\Psi_I[m]$ of the quantum sector can be 
regarded as a holomorphic section of the  
Hermitian line bundle ${\cal L}_{c_0}\rightarrow {\cal M}_{0,N}$ 
with the polarization $\partial_{c_i}\Psi=0$ and 
the curvature is $\frac{c_0}{12\pi}\omega_{WP}$. 
On the other hand, the classical correlation function 
(\ref{eq:classical}) corresponds to a Hermitian metric 
defining an inner product $\langle\,\,,\,\,\rangle_{c_0}$  
on the Hilbert space which is the space of sections $\Psi[m]$. 
Hence, at least for the topology of the $N$-punctured sphere, 
the amplitude can be written as 
${\cal A}=\sum N^{I,J}\langle \Psi_I , \Psi_J\rangle_{c_0}$. 

Thus the quantum Liouville theory associated with $\ncom$ 
fits well with the geometric quantization of moduli space. 
The factorization property into the classical sector and 
the quantum sector plays an important role here and is just the special 
feature appearing only in the Liouville theory associated with
$\ncom$. 


\subsection{Central Charge and Some Discussions}

Finally we give some discussions about our Liouville theory.   
What we have understood in the above discussions is that 
the quantum Liouville theory based on the infinite dimensional 
representations of $\qslc$ at a root of unity factorizes 
into the classical Liouville theory and  the quantum Liouille theory 
based on the finite dimensional reprsentations of $\qslc$, 
and the latter governs quantum fluctuations around the 
classical surface.  
% Not only vertex operators but also fusion rules 
% can be considered as the tensor products 
% of those in the classical and weak coupling theories.  
The following discussions are on this observation, namely, 
we start from the action 
\be
\frac{1}{4\pi}\widehat{S}_L=\frac{1}{4\pi\beta^2}\int d^2z \left( \partial \cp 
\overline{\partial}\cp + e^{\cp}\right) +
\frac{1}{4\pi}\int d^2z \sqrt{\hat{g_P}} \left(\nabla \vp
\overline{\nabla}\vp+ Q R_{\hat{g_P}}\vp \right), 
\label{totaction}
\ee
rather than the original one (\ref{eq:qaction}). 
Here we have set $\Lambda=0$ for the convenience. 
It is important to notice that,  since  $e^{\gamma\Phi}$ is a 
Riemannian metric, the Liouville field $\gamma\Phi$ should be real,  
and, therefore, $\tilde\gamma \phi$ is real as well. 
Recall the relation (\ref{tgam}) and notice that $\tilde\gamma^2$ 
can be negative. 
When $\tilde\gamma$ is pure imaginary, the quantum filed $\vp$ is 
also imaginary and the second term in (\ref{totaction}) has 
wrong sign, while if $\tilde\gamma$ is real, $\vp$ is a real field 
and the quantum sector has correct sign. 
In the following discussions, 
we would like to impose another assumption that 
$\beta=\gamma$, equivalently, $\tau=1$, in order for the background 
metric $\hat{g}_P$ to be the poincar\'e metric, 
  
It is easy to calculate EM tensor from the total action, 
and one finds, 
\begin{eqnarray}
T^{tot}_{zz}&=&T^{cl}(z) + T^q(z) + T^{\rm{mix}}(z), \label{emtot} \\ 
&& T^{cl}(z)=\frac{1}{\gamma^2}\left(-\frac{1}{2}\partial \cp \partial 
\cp+ \partial^2\cp\right),  \nonumber \\[.1cm]
&&T^q(z)=\lim_{w\rightarrow z}\left[\left(-\frac{1}{2}
\partial_z \vp \partial_w \vp + Q\,\partial^2 \vp\right)
-\frac{1}{(z-w)^2}\right],\nonumber \\[.1cm] 
&&T^{\rm{mix}}(z)=Q\partial \cp \partial \vp, \nonumber \\[.1cm]
T^{tot}_{z\bz}&=& 0. \label{totEM}  
\end{eqnarray}
The term $T^{\rm{mix}}$ comes from the curvature term 
in (\ref{totaction}) where the classical and quantum 
Liouville fields $\cp,\, \vp$ interact with each other. 
The second line (\ref{totEM}) guarantees that the total system is 
again conformally invariant. 

It is the time to estimate the central charge to confirm thet 
our model is actually a  strong coupling theory. 
To do this, let us observe how the EM tensor $T(z)\equiv T^{tot}_{zz}$ 
transfroms under the change of variable $z\rightarrow f(z)$. 
Since the EM tensor is a second rank tensor with central extension, 
it should satisfy the following transformation law, 
\be 
\hat{T}(z)=\left(\frac{df}{dz}\right)^2T(f)+\frac{c_L}{12}
\{f,z\}_S. \label{EMlaw}
\ee
However, one sees that eq.(\ref{emtot}) satisfies (\ref{EMlaw}) 
if and only if $Q=0$, and the central harge is given by 
\be
c_L=1+\frac{12}{\gamma^2}. \label{cofstrong}
\ee
The first term 1 arises from the quantum sector $T^q(z)$, precisely, 
from the subtraction of the singularity $1/(z-w)^2$, \footnote{
The author thanks N. Ano for a very useful discussion on this}   
and the second term $12/\gamma^2$ comes from the classical sector. 
Puttig together the central charge (\ref{cofstrong}) with eq.(3.8), 
one finds  
\be
\frac{1}{\gamma} = \sqrt{\frac{25-D}{12}}=Q_0. 
\label{gam}
\ee
Notice that the original coupling constant $\gamma$ is real 
for the dimension $D\leq25$, and that 
the central charge of our Liouville theory 
can be actually in the strong coupling region, 
$1<c_L<25$ for $\gamma>1/\sqrt{2}$.  
It is interesting to give some of the allowed 
dimension $D$ of the target space in our model. 
Recalling the assignment $\gamma^2/2=p'/p$ with integers 
$p,p'$ being coprime with each other and (\ref{gam}), 
one finds \\[.05cm]
\begin{eqnarray}
&({\rm i})&  \mbox{when}\,\,p'=1, \quad D=1, 7, 13, 19, 
\nonumber \\[.1cm]
&({\rm ii})&  \mbox{when}\,\,p'=2, \quad D=4, 10, 16, 22, 
\label{dim} \\[.1cm]
&({\rm iii})&  \mbox{when}\,\,p'=3, \quad D=3,5,9,15,17,21,23. 
\nonumber \\
&&\quad\quad\quad \cdots\cdots  \nonumber
\end{eqnarray}
We can obtain fractional dimensions as well. 
If another matter whose central charge is fractional couples 
to the string theory, such a fractional 
dimension will become crucial. 
In the case (i), the dimensions are completely the same as those given 
in \cite{Gerv}. 
This is a strange coincidence, since we are dealing with 
the case when $q$ is a root of unity, whereas, in \cite{Gerv}, 
the parameter $q$ is not a root of unity. 

This central charge (\ref{cofstrong}) 
coincides with that obtained by Takhtajan in 
Refs.\cite{Ta,Ta2}, where a manifestly geometrical approach 
was used. 
Along the Polyakov's original formulation of quantum Liouville 
theory, he started with conformal Ward identities via functional 
integral. 
Upon perturbation expansions around the classical solution 
corresponding to the Poincar\'e metric, he obtained 
the central charge (\ref{cofstrong}). 
It is worth mentioning that in both approaches, 
the quantum Liouville theory is expanded around the classical solution, 
\ie, the Poincar\'e metric. 
In other words, a quantum Riemann surface is treated  as a 
total of a classical surface endowed with the 
hyperbolic geometry and quantum corrections around it. 
Indeed, in Takhtajan's approach, the classical limit recovers 
the underlying hyperbolic geometry of Riemann surface. 
In our case, owing to the condition $Q=0$,  
the total action (\ref{totaction}) is just the sum of the 
clasical Liouville action and a quantum action 
without any mixing terms.   
The quabtu field $\vp(z,\bz)$ can be interpreted as the 
$D+1$-th component of the string, and we obtain 
the $D+1$ dimnsional spacetime.  
As suggested previously, the signature the spacetime has 
depends on the sign of $\tilde\gamma^2$.  
  
Next we turn our attention to the conformal dimension. 
In also this discussion, we are on the conjecture $Q=0$. 
The vertex operator of the string dressed by gravity is written as 
\be
V(z,\bz)=\int d^2z \sqrt{g}:{\cal O}_X :\quad \cong \quad 
\int d^2z e^{\tau\cp}: e^{\tilde\gamma\vp}{\cal O}_X: 
\ee
where ${\cal O}_X$ stands for a pure string vertex operator. 
Let $\Delta_0, \Delta_X$ be the dimensions of the classical Liouville 
exponential  and the string vertex, respectively, and 
one finds the following relation to the conformal dimensions
% 
\be
\Delta_0-\frac{\tilde\gamma^2}{2}+\Delta_X=1. 
\label{relation1}
\ee
The middle term in the left hand side of (\ref{relation1}) is the  
conformal dimension of the exponential $e^{\tilde\gamma\vp}$. 
Since we have chosen  $\tau=1$, 
the background metric $\hat{g}_P$ is  the 
Poincar\'e metric and so $\Delta_0=1$.  
Then, together with (\ref{tgam}), one finds  
\be
\frac{\tilde\gamma^2}{2}=\Delta_X=\frac{\gamma^2}{2}+2n, 
\qquad n=0,\,\pm1,\,\cdots.
\label{Xdim}
\ee
This equation suggests that the matters (string) which couple  
to the Liouville theory form a discrete series in conformal dimension 
and that the constant $\tilde\gamma$ can be imaginary.  


In summary, our Liouville theory can be 
interpreted as follows: 
It is the total system of the classical Liouville theory plus the 
fluctuation around the classical manifold, 
which becomes the  $D+1$-th component of the string. 
The same situation occurs in the weak coupling Liouville theory, 
although in that case the classical Liouville sector does not appear.  
In the weak coupling theory when the dimension of the target space 
is $D=25$, the quantum Liouville field is 
regarded as the time component of the string with wrong sign 
and,  as the result, the spacetime becomes the 
26 dimensional Minkowski space. 
On the contrary, in our case, we have more choices for the 
dimension $D$ as listed in (\ref{dim}), especially,  
for $\gamma^2=\frac{6}{11}$, the dimension of the 
target space is $D=3$ and, therefore,  with the field $\vp$ 
as the time component, our space-time is the $4$ dimensional 
Minkowski or Euclid space according to the sign of the quantum 
sector $S^q$. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Discussions}

We have developed a new  approach to the quantum Liouville theory  
via the infinite dimensional representations 
of $\qslc$ when $q=\exp (\pi i\, p'/ p)$. 
In this Liouville theory we have dealt with, 
only the vertex operators with positive charges 
$\alpha$ can be defined. 
The characteristic feature of $\ncom$ is that every irreducible 
highest weight module $V^{irr}_{\mu,\nu}$ necessarily factorizes 
into the highest weight module $V_\zeta^{cl}$ of the classical 
algebra $\slr$ and ${\cal V}_j$, the $(2j+1)$-dimensional 
representation of $\qslc$. 
Our investigations in this article have been performed by making full 
use of this feature. 
Owing to this fact, we observed that the vertex operators with positive 
charge $\alpha=h_{\mu\nu}$ factorized into the {\em classical} vertex 
operator with charge $\zeta$ and the vertex operator with negative 
charge $-j$ as in eq.(\ref{eq:separation}), 
where the relation $h_{\mu\nu}=\zeta p -j$ was understood. 
We further conjectured that the Liouville action 
$\widehat{S}_L(\Phi)$ should also 
decompose into the classical Liouville action $S^{cl}(\cp)$ and 
$S^q(\phi;\hat{g}_P)$ with the classical background 
$\hat{g}_P=e^{\tau\cf}$, where $e^{\cp}$ defines the Poincar\'e metric. 
Since the field  $\phi(z,\bz)$ can be interpreted as the Liouville 
field which measures the quantum fluctuations of the metric 
around the classical metric $e^{\tau\cp}$, 
the Liouville theory $S^q(\phi;\hat{g}_P)$ governs the theory of 
quantum fluctuations of quantum Riemann surfaces around 
the classical surfaces. 
Namely in our formulation, the quantum Liouville theory 
$\widehat{S}_L(\Phi)$ 
describes quantum $2D$ space-time as the total system 
of the classical space-time and 
the quantum fluctuations around it. 
We found that the Liouville theory governed by the action 
$S^q(\phi;\hat{g}_P)$ was associated with the {\em finite} dimensional 
representations $\com$. 
We remarked the difference between our Liouville theory 
and the standard Liouville theory  \cite{Po2}--\cite{DK}  
which is confined to the weak coupling region. 
Although, in both theories, quantum fluctuations of metric are 
deeply related to $\com$,  
our theory contains the classical Liouville theory yielding 
the underlying K\"ahler geometry, whereas  the latter does not. 
It is quite important to emphasize that,  
our  quantum Liouville theory is certainly in the strong coupling regime.  
% and in that case the quantum 
% fluctuation around the classical space-time yields matter and, 
% therefore,  $S_L(\Phi)$ can be interpreted as the unified action 
% of matter and classical gravity. 
 
Another interesting result was that our Liouville theory  
fitted well with the concepts of geometric quantization of the 
moduli space of metrics. 
First of all, we have noted that the classical Liouville theory is 
in agreement with the K\"ahler geometry of the moduli space, where 
the Liouville action evaluated on the classical solution $\cp$  
is nothing but the K\"ahler potential of the Weil-Peterson metric 
$\omega_{WP}$. 
Second, we have observed that 
the Hermitian line bundle ${\cal L}_{c_0}$ can be built 
over the moduli space ${\cal M}_{0,N}$ with the Hermitian metric 
$\exp\,(-\frac{c_0}{48\pi}\overline{S}^{l})$. 
The role of holomorphic sections of the line bundle ${\cal L}_{c_0}$ 
is played by the holomorphic part $\Psi[m;\{j\}]$ of the quantum 
sector $Z^q[{\sl m}:\{j\}]$. 
Thus, as we have seen in eq.(\ref{eq:amplitude}), 
the transition amplitude in our theory agrees  with 
the inner product of two wave functions corresponding to the 
sections of the line bundle ${\cal L}_{c_0}$. 
Remembering that the quantum Liouville theory can be regarded as 
a quantum geometry of Riemann surfaces, our observation of our quantum 
Liouville  theory from the viewpoints of geometric quantization of 
moduli space seems to be quite natural. 

At this stage, it is worthwhile to compare our quantum gravity with  
that formulated in Ref.\cite{Ver} where $c>1$ two-dimensional 
quantum gravity was treated via geometric quantization of 
moduli space. 
Riemann surfaces in his consideration are compact and have 
genus $g>1$. 
There, the transition amplitude between some initial state 
$\Psi_I$ and some final state $\Psi_F$ is given by the inner product 
(see eq.(5.7) in Ref.\cite{Ver})  
\be
\langle \Psi_I, \Psi_F\rangle =\int d(WP) Z_L[{\sl m}] 
\overline{\Psi}_I[\overline{\sl m}] \Psi_F[{\sl m}], \label{eq:Verlinde}
\ee
where $Z_L[{\sl m}]$ 
is the Liouville partition function (let $\sigma(z,\bz)$ be the 
Liouville field). 
$\Psi[m]$ is a holomorphic section of a line bundle over the moduli 
space and can be identified with the conformal block obtained by 
solving the conformal Ward identity of Polyakov \cite{Po2}. 
Therefore $\Psi[m]$ is considered as the holomorphic sector 
of the partition function in the weakly coupled Liouville gravity. 
In that paper, the metric on the Riemann surface is parameterized as 
$ds^2=e^{\sigma(z,\bz)}\vert dz + \mu d\bz\vert^2$. 
Now it is easy to find intimate relations  between the formulation 
of Verlinde and our Liouville gravity except only one big difference. 
In both theories, a quantum Riemann surface is composed of  two 
sectors, one is the Riemann surfaces as a background and the other 
corresponds to quantum fluctuations around the background surface. 
In the Verlinde's formulation, the quantum fluctuation is 
parameterized by the Beltrami differential $\mu$. 
The difference arises in the choice of background surfaces; 
the background Riemann surface in our theory is just 
the classical surface with the Poincar\'e metric $e^{\cf}$, 
while in the Verlinde's formulation it is again a quantum surface 
with the metric $e^{\sigma(z,\bz)}$. 
Because $\sigma(z,\bz)$ is not a classical field, the partition 
function $Z_L[{\sl m}]$ in eq.(\ref{eq:Verlinde}) 
cannot be written as $Z^{cl}[{\sl m}]$ in our theory. 
Note again that the holomorphic sections $\Psi[m]$ in both theories 
are related only to the quantum fluctuations. 

\vspace{.2cm}

Although we have observed some remarkable features of the quantum 
Liouville theory associated with $\ncom$ and obtained a natural 
concept of quantum $2D$ space-time within the framework of our 
formulation, 
there still remain some important problems to be investigated.  
I list some of them below. 
First, and maybe most important problem is the explicit 
relation between our quantum Liouville theory  
and Takhtajan's one. 
These two models give the same central charge.   
Moreover, in both approaches, the classical geometry 
appears as a background, namely, the quantum metrics 
are to be expanded around the classical metric of Riemann surface, 
the Poincar\'e metric.  
Inspired by the agreements, it is quite interesting and 
important to find the relation, although at first glance 
these two approaches are completely different, \ie, Takhtajan's approach 
is fully  geometric and ours is algebraic.   
Second, Virasoro structure of this model is an interesting problem. 
Finding this structure will allow more explicit discussions 
of our Liouville theory, especially correlation functions. 
This also maybe sheds light on the problem of $c>1$ discrete 
series of the Virasoro algebra. 
It is well-known that representations of the Virasoro algebra 
with $c>1$ always form continuous series. 
However, since the highest weight representations $\ncom$ form a 
discrete series and our model can have central charges greater than 1, 
it can be expected to find  such a discrete series. 
These discussions will appear elsewhere.\cite{AS} 


\vspace{.5cm}
\noindent 
{\bf Acknowledgment :}\,\, I wish to thank  Profs. L. D. Faddeev and 
L. Takhtajan for communications on strongly coupled Liouville gravity. 
It is also a pleasure to acknowledge Profs. M. Ninomiya,  R. Sasaki 
and Dr. N.Ano for discussions and comments.  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  APPENDIX  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix 
\section{Infinite Dimensional Representations with $q$ at a root of unity} 

The Appendix gives a brief review of Ref.\cite{MS} 
where the infinite dimensional highest weight representations of 
$\qslc$ is examined for the case 
when the deformation parameter $q$ is a root of unity.  
Let $q=\exp\,\pi i/p$. \footnote{In the previous sections, 
we used the choice $q=\exp\,\pi i \frac{p'}{p}$.  
For the brevity of our discussions, we will choose $p'=1$ in the 
appendix.  
The essential parts of our discussion is independent of 
the choice for $p'$. }
The most essential feature which we have made full use of in our 
discussions is stated in the following theorem; 

\vspace{.3cm}
\noindent
{\bf Theorem}. \hspace{.3cm} 
Every {\em infinite} dimensional irreducible highest weight
representation, denoted as $V^{\qslc}_I$ 
is necessarily isomorphic to the tensor product 
of two highest weight representations as  
\be 
V^{\qslc}_I \cong V^{\slr} \bigotimes V^{\qslc}_F, 
\label{eq:theorem}
\ee
where $V^{\slr}$ is a representation of the {\it classical} algebra 
$su(1,1)\cong\slr$ and $V^{\qslc}_F$ is a {\em finite} dimensional 
representation of $\qslc$. 

\vspace{.3cm}
\noindent
Below we will prove this theorem (see Refs.\cite{MS} for the detailed
discussions). 

Let $X_+, \, X_-, K=q^H$ be the  generators of the quantum universal 
enveloping algebra $\qslc$ satisfying the relations 

\be
[X_+, X_-]=\frac{K^2-K^{-2}}{q-q^{-1}}, 
\quad K X_{\pm}=q^{\pm 1}X_\pm K. 
\label{eq:qslcrelation}
\ee
To get infinite dimensional representations, 
we define hermitian conjugations as  
$X^\dagger_\pm = -X_\mp, \, K^\dagger = K^{-1}$. 
As in the classical case, we represent $\qslc$ with these 
conjugations by constructing the highest 
weight module $V_h$ on the highest weight state $\Lambda^h_0$ such as 
$X_-\Lambda^h_0 =0, \,  K\Lambda^h_0=q^h\Lambda^h_0$,  
\be
V_h=\{\Lambda^h_r\, \vert \,\Lambda^h_r :=\frac{X_+^r}{[r]!}
\Lambda^h_0, \,\, r=0, 1,\cdots \,\}. 
\ee
%
Upon using the hermitian conjugations and the relations 
(\ref{eq:qslcrelation}), 
the norm of the state $\Lambda^h_r$ is easily calculated as  
\be
\parallel \Lambda^h_r \parallel^2 = \left[ \begin{array}{c} 
                                         2h + r - 1 \\ r 
                                  \end{array} \right]_q
\label{eq:norm}
\ee 
with the normalization $\parallel \Lambda^h_0 \parallel^2=1$. 

The first problem we come across when $q$ is a $p$-th root of unity is
that the norm of the state $\Lambda^h_p$ diverges owing to the factor 
$[p]=0$ in the denominator. 
Thus, for arbitrary highest weight $h$, 
the highest weight module $V_h$ is not necessarily well-defined. 
The only way to avoid this undesirable situation is to require that   
there exist two integers $\mu\in\{0, 1, \cdots, p\}$ and 
$\nu\in {\N}$, such that the highest weight is given by, 
\be 
h=h_{\mu\nu}:=\half(p\nu -\mu +1).    
\label{eq:highest}
\ee
For the highest weight $h_{\mu\nu}$, 
the factor  $[2h_{\mu\nu}+\mu-1]=0$ appearing in the numerator of the 
right hand side of the norm (\ref{eq:norm}) makes 
the $p$-th state a finite-norm state, and so the module 
$V_{h_{\mu\nu}}$ is well-defined.  
However, due to the zero, there appears the set of zero-norm states 
\be
{\cal X}_{\mu\nu}=\bigoplus_{k=0}^\infty 
\{\Lambda^h_{kp+\mu}, \Lambda^h_{kp+\mu+1}, \cdots,
\Lambda^h_{(k+1)p-1}\}.
\label{eq:null}
\ee
One can immediately show that, upon the relation 
$[kp+x]=(-)^k[x]$, $\parallel \chi \parallel^2=0$ for 
$\chi\in {\cal X}_{\mu,\nu}$. 
It is interesting to notice that the $V_{h_{\mu\nu}}$ actually 
form a discrete series parameterized by the two integers $\mu$ and
$\nu$. 
This is a characteristic feature of $\ncom$. 
Indeed this is not the case for the infinite dimensional 
representations of the classical algebra $\slc$ and of $\qslc$ 
with generic $q$ as well. 
It will be turned out that one of the parameter, say, $\nu$  
parameterizes the classical sector $V^{\slr}$  and the other, $\mu$,
is concerned with $V^{\qslc}_F$. 

Let $V_{\mu,\nu}:=V_{h_{\mu\nu}}$ and 
$\Lambda^{\mu\nu}_r:=\Lambda^{h_{\mu\nu}}_r$. 
The second problem we encounter in the construction of an irreducible 
highest weight module is that $X_\pm^p$ are nilpotent on the module 
$V_{\mu\nu}$, \ie, 
$(X_\pm)^p \Lambda =0$ for ${}^\forall \Lambda\in V_{\mu,\nu}$. 
Therefore one cannot move from a state to another state by acting 
$X_+$ or $X_-$ successively. 
Moreover, the relation $K^{2N}=1$ on $V_{\mu,\nu}$ 
indicates that one cannot measure the weight of a state completely. 
In order to remedy these situations, we should change the definition 
of $\qslc$ by adding new  generators, 
%
\be 
L_1:= -\frac{(-X_-)^p}{[p]!}, \quad L_{-1} := \frac{X_+^p}{[p]!}, 
\quad L_0 := \half \left[ \begin{array}{c} 2H+p-1 \\ p 
                          \end{array}\right]_q 
\ee
to the original ones, $X_+, X_-$ and $K=q^H$. 
The complete highest weight module $V_{\mu,\nu}$ is constructed by 
acting $X_+$ and $L_{-1}$ on the highest weight state 
$\Lambda^{\mu\nu}_0$ which is defined by 
the relations, $X_-\Lambda^{\mu\nu}_0=L_1\Lambda^{\mu\nu}_0=0$. 

The third problem, which we do not encounter in the classical case 
and the case when $q$ is not a root of unity, is that 
there is an infinite chain of submodules in $V_{\mu,\nu}$. 
That is to say, $V_{\mu,\nu}$ is no longer irreducible. 
Let us observe this characteristic feature briefly. 
First of all, one sees that the state $\Lambda^{\mu\nu}_\mu$ is 
a highest weight state because both $X_-$ and $L_1$ annihilate this state. 
Therefore a submodule exists on the state $\Lambda^{\mu\nu}_\mu$. 
Since this state has weight $h_{\mu\mu}+\mu=h_{-\mu\nu}$, 
the submodule  can be regarded as $V_{-\mu,\nu}$. 
Next one again finds a submodule $V_{\mu,\nu+2}$ on 
$\Lambda^{-\mu\nu}_{p-\mu}\in V_{-\mu,\nu}$. 
Repeating this procedure, one obtains the following chain of 
submodules in the original module $V_{\mu,\nu}$, 
%
\be 
V_{\mu,\nu}\supset V_{-\mu,\nu}\supset V_{\mu,\nu+2}\supset 
V_{-\mu,\nu+2}\supset\cdots \supset V_{\mu,\nu+2k}\supset 
V_{-\mu,\nu+2k}\supset\cdots . 
\ee
Then, irreducible highest weight module on the highest weight state 
$\Lambda^{\mu\nu}_0$, denoted by $V^{irr}_{\mu,\nu}$, 
is obtained by subtracting all the submodules, and it is obtained as 
%
\be 
V^{irr}_{\mu,\nu}=\bigoplus_{k=0}^\infty V^{(k)}_{\mu,\nu}, \quad 
V^{(k)}_{\mu,\nu}:= \{\Lambda^{\mu\nu}_{kp+s} \,\vert \,
s=0, 1, \cdots, \mu-1\}. \label{eq:irreducible}
\ee
It should be noticed that all the states subtracted in the procedure 
are the elements in the set ${\cal X}_{\mu,\nu}$ and, therefore, 
no zero norm state exists in $V^{irr}_{\mu,\nu}$, 
namely, 
\be
V_{\mu,\nu}^{irr}=V_{\mu,\nu}\backslash {\cal X}_{\mu,\nu}. 
\ee
We have now obtained  irreducible highest weight modules.  
Notice that every irreducible infinite dimensional highest weight 
representation is composed of an infinite number of blocks 
$V^{(k)}, k=0, 1, \cdots$, each of which contains finite number 
of states. 
This feature is the very origin of the fact stated in the
Theorem (\ref{eq:theorem}). To see this is the next task. 

Now we are the last stage to prove the Theorem. 
To complete the proof, we make some observations. 
In the following we will denote the level of a state by 
$kp+s$ where $s$ runs from $0$ to $\mu-1$ and $k=0, 1, \cdots$, 
instead of $r$, $r=0, 1, \cdots$.  

\vspace{.2cm}
\noindent
{\sc Observation} 1. \hspace{.2cm}On $V^{irr}_{\mu,\nu}$ 
\be 
\frac{X_\pm^{kp+s}}{[kp+s]!}=(-)^{\half k(k-1)p+ks}\frac{L_{\mp1}^k}{k!}
\frac{X_\pm^r}{[\,s\,]!}. 
\ee

\vspace{.2cm}
\noindent
{\sc Observation} 2. \hspace{.2cm}The sets of generators 
$\{X_+, X_-, K\}$ and $\{L_1, L_{-1}, L_0\}$ are mutually commutable 
on the irreducible highest weight module $V^{irr}_{\mu,\nu}$. 

\vspace{.2cm}
\noindent
These observations indicate that there exists a map 
$\rho\,:\, V^{irr}_{\mu,\nu}\rightarrow V^{inf}\otimes {\cal V}$, 
where ${\cal V}$ is the finite dimensional space composed of 
$\mu$ states. 
Further $\rho$ induces another map $\hat{\rho}\,:\, 
\qslc\rightarrow U^{inf}\otimes {\cal U}$, such that 
$\rho({\cal O}\Lambda)=\hat{\rho}({\cal O})\rho(\Lambda)$. 
In the second paper in Ref.\cite{MS}, 
such the isomorphisms $\rho$ and $\hat{\rho}$ have been obtained. 
In the following, we shall restrict the module $V^{irr}_{\mu,\nu}$ 
to the unitary irreducible representation. 
In that case, 
\be 
\rho(\Lambda^{\mu\nu}_{kp+s}) = (-)^{\half k(k-1)p+ks}
\lambda^\zeta_k \bigotimes \Psi^j_m, 
\label{eq:isomorphism}
\ee
and 
\begin{eqnarray}
\hat{\rho}(X_\pm) = {\bf 1}\otimes (\pm{\cal J}_\pm), &\quad&
\hat{\rho}(K) = {\bf 1} \otimes {\cal K}, \\
\hat{\rho}(L_{\pm1}) = (\mp G_{\pm1}) \otimes {\bf 1}, &\quad&  
\hat{\rho}(L_0) = G_0 \otimes {\bf 1},
\end{eqnarray}
where $\zeta=\nu/2, \, j=(\mu-1)/2$ and $m=-j+s$. 

\vspace{.2cm}
\noindent
{\sc Observation} 3. \hspace{.2cm} 
The actions of $\{{\cal J}_\pm, {\cal K}\}\in {\cal U}$ on the state 
$\Psi^j_m\in {\cal V}$ are calculated as  
\be 
{\cal J}_\pm \Psi^j_m = [j \pm m +1] \Psi^j_{m\pm1}, \quad 
{\cal K}\Psi^j_m = q^m \Psi^j_m, 
\ee 

\vspace{.2cm}
\noindent
{\sc Observation} 4. \hspace{.2cm}
The actions of $G_n, (n=\pm1, 0)\in U^{inf}$ 
on $\lambda^\zeta_k \in V^{inf}$ are as follows,  
\be
G_1 \lambda^\zeta_k = (2\zeta +k -1) \lambda^\zeta_{k-1}, 
\quad G_{-1} \lambda^\zeta_k = (k+1) \lambda^\zeta_{k+1}, 
\quad G_0 \lambda^\zeta_k = \left( \half\zeta+k\right)
\lambda^\zeta_k. 
\ee

\hspace{.2cm}
\noindent
The {\sc Observation} 3 shows that 
${\cal J}_\pm, {\cal K}$ satisfy the relations of 
$\qslc$, \ie, ${\cal U}=\qslc$  with the hermitian conjugations 
${\cal J}_\pm^\dagger = {\cal J}_\mp, {\cal K}^\dagger =
{\cal K}^{-1}$. 
Therefore, taking it into account that ${}^\forall\Psi^j_m$ 
has positive norm, ${\cal V}$ is a unitary finite dimensional 
representation of $\qslc$ with the highest weight $j$. 
We rewrite such the representation as ${\cal V}_j$, \ie, 
${\rm dim}.\,{\cal V}_j=2j+1$. 
On the other hand, {\sc Observation} 4 leads us to the following  
relations among $G_n, \, n=0, \pm1$, 
\be
[G_n. G_m]=(n-m) G_{n+m}, 
\ee 
and hermitian conjugations $G_{\pm1}^\dagger = G_{\mp1}, 
G_0^\dagger = G_0$. 
Therefore we can conclude that $U^{inf}$ is just the classical 
universal enveloping algebra of ${\rm su}(1,1)\cong\slr$, 
\ie, $U^{inf}=U\slr$ and $V^{inf}$ 
is the unitary highest weight representation of $\slr$ with highest 
weight $\zeta$. We denote the representation by $V^{cl}_\zeta$. 
We have now finished the proof of the theorem and 
obtained the important structure of $\ncom$ as 

\be 
V^{irr}_{\mu,\nu}\cong V^{cl}_\zeta \bigotimes {\cal V}_j.  
\label{eq:essential}
\ee 
In the Theorem, $V^{irr}_{\mu,\nu}$, $V^{cl}_\zeta$ and 
${\cal V}_j$ are denoted by $V^{\qslc}_I$, $V^{\slr}$ and 
$V^{\qslc}_F$, respectively. 





%%%%%%%%%%%%%%%%                BIBLIOGRAPHY                 %%%%%%%%%%%%%%%%%%


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\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
However we should recall here that, as discussed in the above, 
the condition (\ref{eq:QW}) prevents us from obtaining other 
strongly coupled Liouville gravities than those with central charges 
$c_L=$7, 13 and 19. 
Therefore, we have to get rid of the relation (\ref{eq:QW}), 
although without the relation EM tensor 
no longer satisfies the Virasoro algebra. 
Fortunately quantum group structure has nothing to do with 
whether Virasoro structure can be found or not in a theory.  
Before going to our discussion, it is interesting to mention about 
a quantum Liouville gravity in which the relation (\ref{eq:QW}) 
does not hold. 
Quite recently, Takhtajan has investigated quantum Liouville gravity 
in a geometrical manner \cite{Ta}, where 
the value of $Q_0$ is chosen not as (\ref{eq:QW}) but as the 
classical one, \ie, $Q_0=Q_{cl}=1/\gamma$. 
Upon a perturbation expansion of functional integral around 
the classical theory, the central charge was obtained as 
\be 
c_L=1+\frac{12}{\gamma^2}.  \label{eq:strongcentral}
\ee
This is consistent with the general expression of the central charge 
(\ref{eq:central}) under the choice $Q_0=1/\gamma$.  
The central charge (\ref{eq:strongcentral}) 
is always greater than 1 including the strongly coupled 
region $1 < c_L < 25$. 
Although his formulation is geometrically well-established 
and probably is a successful candidate for the strongly coupled 
Liouville gravity, the quantum group structure is still unclear.   



% Dear Dr. Takashi Suzuki,

% Here are my answers (rather incomplete) to the questions you have raised
% in your letter.

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{\bf 1}. Certainly I consider my version of the Liouville gravity as a 
conformal field theory. The reason is the following. Relation
\beq\label{1}
Q_0=\frac{1}{\gamma}+\frac{\gamma}{2},
\eeq

derived by Curtright and Thorn, is valid only when one uses a theory on a 
cylinder ($\R \times S^1$) and is doing canonical quantization (i.e.~using 
free fields)! This is a perfectly legitimate quantum field theory and results 
of C-T and other people on correlation functions, dimensions, etc., are 
perfectly OK. However, as I argue, they are not immediately applicable to 
2-dim quantum gravity in Euclidean formulation. Indeed, in the gravity we have 
a fundamental condition that 

(i) $e^{\phi}$ is not a scalar but a density of a $(1,1)$-form;

(ii)$e^{\phi}$ is positive.

In other words, 
$$e^{\phi}|dz|^2$$
is a Riemannian metric on a two-dimensional manifold (say a Riemann surface).
This fact is build in into my approach. However, it is almost impossible (or
very difficult) to impose it in the standard approach using free fields. 

To summarize, equation \ref{1} is not a necessary and sufficient condition for
a theory to be conformal; it is a result of a special approach using free fields
and disregarding geometric constraints. 

{\bf 2.} I am not using results from Friedan-Schenker work; just the opposite, 
in the framework of the functional integral formulation I effectively have 
proved the properties that Friedan-Schenker have postulated to be a geometric 
manifestation of conformal invariance. In that sense I have proved (using
perturbation theory, of course) that my approach to the Liouville gravity is
a conformal theory! Thus my results provide Friedan-Schenker ``program'' with 
new meaningful example, in addition to minimal models and WZW type of models.

These are the things I was trying to articulate during recent years. I didn't
publish them exactly in this form  because I (and everybody) was unable to 
explain what kind of connection (if any!) exists between standard approach to 
Liouville theory and my geometric approach.

I've read your paper on $c_L>1$ Liouville. 
It looks very interesting, though a lot remains to be done! 
Below please find my (minor) remarks and possible 
suggestions. 

(1) I guess than when you are mentioning $sl(2,\C)$ you really mean
$sl(2,\R)$, etc. (or sometimes it is not so relevant). However, when 
discussing formula (2.3) it should be, of course, ${\rm PSL}(2,\R)$ and
not ${\rm SL}(2,\C)$.

(2) You mentioned at the top of page 2 that infinite dimensional 
representations of quantum group for Liouville were proposed in [10]. 
A little bit earlier they were introduced by Smirnov and me in [9].

(3) On page 5: Teichm\"{u}ller space for the Riemann surfaces of the
type $(0,n)$ is not the space of punctures ${\cal T}_N$. The space 
${\cal T}_N$ is finite-dimensional cover of the moduli space of these 
Riemann surfaces; Teichm\"{u}ller space is different from ${\cal T}_N$ 
and you do not actually use it anyway.

(4) The passage from the standard definition of the correlation functions
by the functional integral (3.3) to the functional integral (3.5) is highly
nontrivial one! I was advocating this trick (and even introduced the notation 
$<X>$ )in my Cargese lectures (see in: New Symmetry Principles in Quantum 
Field Theory, Eds. J. Fr\"{o}lich et al, Plenum Press, New York-London, 1992) 
and in my Varenna lectures  ``Topics in Quantum Geometry of Riemann Surfaces: 
Two-Dimensional Quantum Gravity'', hep-th/9409088, to appear elsewhere. Thus 
the second interpretation is not a definition but rather is a proposal which 
I certain is a correct one! In other words, it is a kind of a ``theorem''.  

(5) In formula (3.8) it is tacitly assumed that one uses free fields; I found
discussion at the bottom of the page 8 rather vague. It seems that the main 
issue: whether 
$$e^{\Phi(z,\bar{z})}$$
is positive, should be addressed.

(6) Remark on the top of page 9 that ``...as long as the relation (3.8) 
holds...'' uses free fields again and, therefore, is not necessary for 
a theory to be conformal.

(7) In a same spirit: even if (3.8) does not hold, $T(z)$ still may satisfy
Virasoro algebra in the form of Ward identities (for $<T(z)X>,~<T(z)T(w)X>$,
etc) as I have shown (perturbatively).

(8) I don't quite understand formula (3.20): what is $[d\phi_{cl}]$? There
is no integration ``domain'' in this case, since you are already ``sitting''
onn the classical solution. If you make a passage from smooth to singular
$\phi$'s (as in remark (4)), then it is just the value (3.22), as you said. 

(9) The fact that accessory parameters are in involution with respect to the
Weil-Peterson Poisson bracket was first mentioned (and proved) in my paper
in: Proceedings Simposia in Pure Mathematics, v. 49, p.581, 1989.

(10) It was difficult to follow your discussion in the second half of page
14.

(11) Concerning discussion on page 17. First one should actually define
a theory based on your proposal in order to compare it with the geometric 
approach. By this I mean that one should determine operator contend of the 
theory, compute braiding, OPE (fusion rules) and find dimensions and the 
central charge. The latter can be obtained from OPE  containing the 
stress-energy tensor. Needless to say that Ward Identities should be always 
valid; what will not valid (and that is good!) is that there are no degenerate 
fields in the theory. It would be great if you can get (some) results in this 
direction! It will be extremely interesting, since you will actually 
construct the theory. 

\end{document}
****************************************************************************** 
You may wish to talk to Prof. Fedor Smirnov with whom we wrote [9], and who is 
now your neighbor:  he is visiting RIMS for a period of three months!


I hope that these comments may help. Please keep me informed of your future 
developments.

With best regards,

Leon Takhtajan


------- End of Forwarded Message

%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  3-3  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Representation of $SL(2,{\bf R}) \otimes U_q(su(2))$ }

Through this section we suppose that the $\ncom$-module $\V$ is unitary.
Namely, the integers $\mu$ and $\nu$ take their values in accordance with
proposition 3.2.
Now we have found the classical sector $\slr\simeq su(1,1)$ in $U_q(su(1,1))$
and, therefore,  we can construct a representation of
$\SL\otimes U_q(su(2))$ via the representations we have obtained.

The basic strategy we will follow is to represent the $\SL$ sector,
roughly speaking,  as follows:
Let $G$ be a semi-simple Lie group, and $T$ a maximal torus.
The homogeneous space  $G/T$ has a complex homogeneous structure,
$i.e.$, the group $G$ acts on $G/T$ by means of holomorphic
transformation.
Then we can interpret the unitary irreducible representations of
$G$ as spaces of holomorphic sections of holomorphic line bundles
over $G/T$.  In our case, the group is $G=\SL$ and $T=U(1)$.
The homogeneous space $D=\SL/U(1)$ can be identified with
the complex upper half plane or alternatively with the Poincar\'e disk
$\vert w \vert < 1$.
Now we wish to obtain a representation not only of $\SL$
but also of $U_q(su(2))$, that is to say, the holomorphic sections are also
the representations of the quantum Lie algebra $U_q(su(2))$.
This means that each section should have the index with respect to
$U_q(su(2))$ as well.

Let $L$ be a line bundle over $\SL/U(1)$, and define
$L_j:=L\otimes \mho_j$.
We then construct $\Psi_{j,m}^\zeta\in L_j$, that is, $\Psi_{j,m}^\zeta$
is a holomorphic section of $L$ and also a vector in $\mho_j$.
Let ${\cal H}$ be the Hilbert space spanned by $\zk$.
Having an element $\langle \Psi \vert \in {\cal H}^\dagger$,
we give  the section by,
\begin{equation}
\Psi_{j,m}^\zeta(w):=\langle \Psi \vert \sum_{k=0}^\infty w^k
\vert \zeta; k \rangle
\otimes \vert j; m \rangle,
\end{equation}
%
Under the action of $\SL$, $w\,\rightarrow\, w'=  (aw+b)/(cw+d)$,
$\fun$ transforms as
\begin{equation}
\fun\;\longrightarrow\; {\Psi'}^\zeta_{j,m}(w')=
\left(\frac{1}{cw+d}\right)^{2\zeta}
\Psi^\zeta_{j,m}\left( \frac{aw+b}{cw+d} \right)  \label{eq:sectrans}
\end{equation}
%%
Note that the action of $t\in U(1)$ is
$t\cdot\fun=\pi(t)\fun$ with $\pi(t)\in {\Bbb C}$.
It is well known that, on $L$,  we can pick a hermitian  metric
$(f, g)_\zeta= e^\eta f^*\cdot g$ by choosing the symplectic (K\"{a}hlar)
potential $\eta=2\zeta\log(1-\vert w\vert^2)$, $i.e.$, the curvature form on
$L$ is given by $w=-i \bar{\partial}\partial \eta(w^*,w)$.
With this hermitian metric,
the inner product on the space of sections of $L$ is given by
%%
\begin{equation}
\langle f, g \rangle_\zeta = \int_{\vert w\vert < 1}
d\mu\,e^{\eta(w^*,w)} f^*\cdot g.
\end{equation}
where $d\mu =(2\zeta - 1)/\pi dw^* dw(1-\vert w\vert^2)^{-2}$ is the $\SL$
invariant< measure.
Hence the inner product on $L_i$ is
%
\begin{eqnarray}
&{}&\langle \Psi^\zeta_{j,m}, \Psi^\zeta_{j,m'} \rangle_\zeta \nonumber \\
&{}&=\delta_{m,m'}\,
        \left[\begin{array}{c} 2j \\ j-m \end{array}\right]_q
  \frac{2\zeta -1}{\pi} \int_{\vert w\vert < 1} dw^* dw
(1-\vert w \vert^2)^{2\zeta-2}\psi_\zeta^*(w)\cdot \psi_\zeta(w),
\end{eqnarray}
where $\psi^\zeta(w)=\langle\Psi\vert \sum_{k=0}^\infty w^k \zk$.
It is worthwhile to notice that from equaton (\ref{eq:sectrans}),
the group $\SL$ now acts in a single valued way because $2\zeta\in{\Bbb N}$.
In the classical theory of representation, the condition that the $\SL$-spin
$\zeta$ should be a half-integer stems from the requirment of the group action
to be single-valued, while in the representation theory of
$U_q(su(1,1))$ at a root of unity,  the condition originates from the
requirment that all states have definite norms.
The action of the Lie algebra $\slr$ on $\fun$ is obtained upon
defining the action of $g\in\slr$ on the sections of $L$
as $g\cdot\psi^\zeta(w)=\langle\Psi\vert \sum_{k=0}^\infty w^k g\zk$.
By using $G_n$ actions (\ref{eq:deflp})-(\ref{eq:deflz}), we obtain

\begin{equation}
\hat G_n \fun= \left( w^{n+1} \partial_w + \zeta(n+1)w^n \right)\fun,
\quad\quad n= 0, \pm1, \label{eq:hatgaction}
\end{equation}

\noindent
where we have denoted the $\slr$-actions on the space of holomorphic
vectors as $\hat G_n$.
The right hand side of (\ref{eq:hatgaction}) spans infinitesimal
transformations of $\SL$, $w\rightarrow w+\epsilon(w)$
with $\epsilon(w)=\alpha w^2+\beta w +\gamma$.

On the other hand, the holomorphic vector $\Psi^\zeta_{j,m}$ transforms
under $\com$ as follows;
%
\begin{equation}
\begin{array}{l}
     J_\pm \Psi_{j,m}^\zeta(w) =
                    [j \pm m + 1] \Psi_{j,m\pm 1}^\zeta(w), \\
{}\\
    {\cal K} \Psi_{j,m}^\zeta(w) = q^m \Psi_{j,m}^\zeta(w),
\end{array}
\end{equation}
with  $ J_+\Psi_{j,j}^\zeta(w) = J_-\Psi_{j,-j}^\zeta(w) = 0$.
In the above we have given only the highest weight representations for
the $\com$ sector. However, it is possible to represent this sector
in other ways which are more useful for physical applications.
In particular, the construction given in \cite{GS,RRR} of the representation
space of $\com$ is important for the further investigations,
especially the connections with 2$D$ conformal field theories.

Let us turn our attention to  another remarkable feature of $\ncom$ at a
root of unity, that is, the connection with the generalized supersymmetry.
Of course we can easily guess this connection from (\ref{eq:ldef}) together
with the fact that $L_{0,\pm1}$ generate $\slr$ when the representation is
unitary.
In the following we will explicitly show that the operators $L_{0,\pm1}$ in
(\ref{eq:ldef}) can be written as the infinitesimal transformations of the
Poincar\'e disk and therefore the operators $X_\pm$ can be interpreted as the
$N$-th roots of the transformations.
Our discussion proceeds as follows.
First of all, let us find suitable functions on the disk so that
the operators $L_{0,\pm1}$ act on them as the infinitesimal transformations.
In the previous part of this section, we constructed the holomorphic vector
$\fun$ by summing $\zk\otimes\jm$ over the level $k$ in the $\slr$ sector.
Noticing that on $\fun$, $\hat G_n$ rather than $L_n$ played the roles of
such transformations, we should change our standing point and obtain other
holomorphic vectors on which $L_n$ naturally act.
Therefore, in this case, we have to construct the vectors by means of the
original states $\hkr\in V_{\mu,\nu}$ instead of $\zk\otimes\jm$.
Notice here that we do not restrict the highest weight modules to the
irreducible modules and so the level $r$ runs from $0$ to $N-1$.
Let us define the holomorphic vectors as follows;

\begin{equation}
\pfun_r(w) :=\langle\Phi\vert \sum_{k \in {\Bbb Z}_+}(-)^{\frac{1}{2}k(k-1)+kr}
          w^k \hkr,   \label{eq:defphi}
\end{equation}

\noindent
where $\vert w\vert < 1$. $\pfun_r(w)$ also behaves as a holomorphic section
of the line bundle over $\SL/U(1)$.
Now we have $N$ holomorphic functions
$\pfun_0(w),\,\pfun_1(w),\cdots, \pfun_{N-1}(w)$.
On the contrary, we had $2j+1$ functions $\Psi^\zeta_{jj}, \Psi^\zeta_{jj-1},
\cdots, \Psi^\zeta_{j-j}$ in the previous case.
{}From eq.(\ref{eq:qxpaction}) one can easily calculate the actions of
$X_+$, denoted as $\hat X_+$, on these functions as follows;
%
\begin{equation}
\hxp^\vp\pfun_r(w)=\left\{\begin{array}{ll}
       {[r+1]} \pfun_{r+1}(w), & \quad{\rm for}\quad 0\leq r \leq N-2 \\
   \rule{0mm}{.7cm}
       {[N]}_\vp\partial_w \pfun_0(w), & \quad{\rm for}\quad r=N-1.
                   \end{array}\right.
\label{eq:hxpact}
\end{equation}

\noindent
Similarly upon using eq.(\ref{eq:qxmaction}) we obtain
%
\begin{equation}
-\hxm^\vp\pfun_r(w)=\left\{\begin{array}{ll}
  w{[\mu]} \pfun_{N-1}(w), & \quad{\rm for}\quad r=0 \\
     \rule{0mm}{.5cm}
  {[\mu-r]} \pfun_{r-1}(w), & \quad{\rm for}\quad 1\leq r\leq \mu-1 \\
     \rule{0mm}{.5cm}
  {[N]}_\vp(w\partial_w +2\zeta)\pfun_{\mu-1}(w),
                                          &\quad{\rm for}\quad r=\mu \\
     \rule{0mm}{.5cm}
  {[N+\mu-r]} \pfun_{r-1}(w), & \quad{\rm for}\quad \mu+1\leq r\leq N-1.
                   \end{array}\right.
\label{eq:hxmact}
\end{equation}

\noindent
Here we have introduced the symbols $[N]_\vp$ and $\hxpm^\vp$
such that $[N]_\vp \neq 0$ and $\lim_{\vp\rightarrow 0} [N]_\vp = [N]=0$,
and $\lim_{\vp\rightarrow 0} \hxpm^\vp = \hxpm$.
Now let us calculate the $N$-th powers of $\hxpm^\vp$. We obtain

\begin{eqnarray}
 \lim_{\vp\rightarrow0}\,  \frac{(\hxp^\vp)^N}{[N]_\vp!} \pfun_r(w) &=&
   \partial_w\pfun_r(w)  \label{eq:XNpact} \\
 \lim_{\vp\rightarrow0}\,  \frac{(-\hxm^\vp)^N}{[N]_\vp!} \pfun_r(w) &=&
  \left\{\begin{array}{lc}
      \left(w^2\partial_w +2\zeta w\right)\pfun_r(w), & 0\leq r\leq \mu-1 \\
     \rule{0mm}{.5cm}
      \left(w^2\partial_w +2(\zeta+\frac{1}{2}) w\right)\pfun_r(w), &
                                                        \mu\leq r\leq N-1
        \end{array}\right.
     \label{eq:XNmact}
\end{eqnarray}

\newcommand{\wt}{\widetilde}

\noindent
Thus we  have shown that
$\hxpm^\vp$, which move the function $\pfun_r(w)$ to $\pfun_{r\pm 1}(w)$
accoeding to (\ref{eq:hxpact},\ref{eq:hxmact}), are related to the
$N$-th roots of infinitesimal transformations of the Poincar\'e disk.
Furthermore, eq.(\ref{eq:XNmact}) tells us that the functions $\pfun_r(w)$
for $0\leq r\leq \mu-1$ have dimensions $\zeta$ and the functions $\pfun_r(w)$
for $\mu \leq r\leq N-1$ have dimensions $\zeta+\frac{1}{2}$.
Let $\wt{\Phi}_r^\zeta(w)$ be the former functions, {\em i.e.}, $\pfun_r$
with dimensions $\zeta$  and $\Xi_r^{\zeta+\frac{1}{2}}(w)$ be the
latter with dimensions $\zeta+\frac{1}{2}$.
In the above, we examined $\hxpm$ and the $N$-powers of them only.
Of course we can also obtain
%
\begin{equation}
\lim_{\vp\rightarrow0}\, \frac{1}{2}
      \left[ \begin{array}{c} 2\hat{H}+N-1 \\ N \end{array} \right]_q
      = (w\partial_w + h ). \label{eq:znm}
\end{equation}

\noindent
where $\hat K=q^{\hat H}$ and $h$ is the dimension, {\em i.e.}
$h=\zeta$ for the functions $\wt{\Phi}^\zeta_r(w)$ and $h=\zeta+\frac{1}{2}$
for the functions $\Xi^{\zeta+\frac{1}{2}}_r(w)$.
Thus we have obtained all the generators which span the infinitesimal
holomorphic transformations of the homogeneous space $\SL/U(1)$
and obtained two kinds of functions.  One is the set of
functions whose dimensions are $\zeta$ and the other is the set of functions
whose dimensions are $\zeta+\frac{1}{2}$.
Moreover the generators $\hxpm$ mix between them.
Now we can conclude that $\ncom$ with the deformation parameter $\q$
may be viewed in terms of a $Z_N$-graded supersymmetry with the
upper half plane or Poincar\'e disk  interpreted as an external space.


We have constructed holomorphic vectors over the Poincar\'e disk
in two ways and obtained two sets, $\fun$ and
$\pfun_r(w)=\{\wt{\Phi}_r^\zeta(w), \Xi^{\zeta+\frac{1}{2}}_r(w)\}$.
We end this section with the discussion of the connection between them.
Upon the substitution $m=-j+r$, the actions of $\hxpm$ on
$\wt{\Phi}^\zeta_r(w)$ coincide with those of $J_\pm$ on $\fun$ except
the actions $\hxp \wt{\Phi}^\zeta_\mu$ and $\hxm \wt{\Phi}^\zeta_0$
corresponding to $J_+\Psi^\zeta_{jj}$ and $J_-\Psi^\zeta_{j-j}$,
respectively.
The latter vanish because  $\Psi^\zeta_{jj}$ is the
highest weight vector and $\Psi^\zeta_{j-j}$ is the lowest weight vector
with respect to $\com$.
However $\hxp \wt{\Phi}^\zeta_\mu$ and $\hxm \wt{\Phi}^\zeta_0$ do not
vanish but yield the functions $\Xi^{\zeta+\frac{1}{2}}_{\mu+1}$ and
$\Xi^{\zeta+\frac{1}{2}}_{N-1}$.
In other words, through these two actions the two classses of functions,
$\wt{\Phi}^\zeta_r$ and $\Xi^{\zeta+\frac{1}{2}}_r$, mix with each other after
taking the limit $\vp\rightarrow0$ in eqs.(\ref{eq:hxpact},\ref{eq:hxmact}).
Noticing that the functions $\Xi^{\zeta+\frac{1}{2}}_r$ have zero norms
because they correspond to the states lying between the $(kN+\mu)$-th level
to the $(kN+N-1)$-th level in the original module $V_{\mu,\nu}$,
we see that they are proportional to
$\sqrt{[N]_\vp}$ and may rescale them as
$\Xi^{\zeta+\frac{1}{2}}_r=\sqrt{[N]_\vp}\,\wt{\Xi}^{\zeta+\frac{1}{2}}_r$.
By the rescaling,  the set of functions $\wt{\Xi}^{\zeta+\frac{1}{2}}_r$
completely decouples from the set $\wt{\Phi}^\zeta_r$
after taking the limit $\vp\rightarrow0$.
We can then identify $\wt{\Phi}^\zeta_r(w)$ with $\fun$, and we have
another set of functions $\wt{\Xi}^{\zeta+\frac{1}{2}}_r$ whose dimensions
differ by $1/2$ from those of $\wt{\Phi}^\zeta_r$.


\vspace{0.7cm}

%%%%%%%%%%%%%%%%%%%%%%       SECTION 4       %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{$Osp(1\vert 2)$ and $U_q(su(1,1))$ with $q^2=-1$}

In this section we devote ourselves only to the case $N=2$, that is,
$[2]=0$ and find that $Osp(1\vert 2)$ can be represented in terms
of the representation of $U_q(su(1,1))$.
To this end it is convenient to introduce operators $\lp$ and $\lm$,
which are related to $X_\pm$ and $K$ by the relations
%
\begin{equation}
\lp = iq^{-\frac{1}{2}}KX_-, \quad\quad \lm = iq^{-\frac{1}{2}}X_+K.
\end{equation}
%
Further we define vectors $\phi_r^\zeta(w), \; r= 0, 1$ by means of the
highest weight representations of $U_q(su(1,1))$ by
%
\begin{equation}
\phi_r^\zeta(w) = \sum_{k=0}^\infty w^k \vert h_{\mu\nu};
            2k+r ), \quad\quad r=0, 1
\end{equation}
where we have introduced new weight vectors
\begin{equation}
  \vert h_{\mu\nu}; r ) := \frac{\lm^r}{\langle r \rangle !}
  \vert h_{\mu\nu}; 0 \rangle,
\end{equation}
%
with ${\langle r \rangle }=q^{r-1}{[r]}$.
The new weight vector $\vert h_{\mu\nu}; r )$ coincides with the original
one $\vert h_{\mu\nu}; r\rangle$ up to a phase factor.
We can, therefore, deal with the highest weight modules spanned by the new
weight vectors in the same fashion as  $V_{\mu,\nu}$.
In particular, the operator $\lp$ and $\lm$ act as
%
\begin{equation}
    \begin{array}{l}
      \lp \vert h_{\mu\nu}; r ) =  {\langle 2h_{\mu\nu}+r-1 \rangle }
      \vert h_{\mu\nu}; r-1 ), \\
  \rule{0mm}{.5cm}
      \lm \vert h_{\mu\nu}; r ) =  {\langle r+1 \rangle }
      \vert h_{\mu\nu}; r+1 ).
    \end{array}
\end{equation}
%
{}From these actions it is easily seen that the $\mu$-th state  has
zero norm as in the previous case.
Indeed, in $N=2$ case, the highest weight is given by (see eq.(\ref{eq:hmn})),
\begin{equation}
h_{\mu\nu}=\frac{1}{2}(2\nu - \mu +1). \label{eq:hmnt}
\end{equation}
%
Because $1\le\mu\le N$, there are two cases, $\mu=1$ and $\mu=2$;
in the case when $\mu=1$, $\po$ has zero norm, while neither $\pz$ nor
$\po$ has zero norm in the case when $\mu=2$.
We treat these cases separately.

We first examine the case when $\mu=1$.
The highest weight is given by $h_{1\nu}=\nu$.
Upon using the relation  $\langle 2n \rangle=\langle 2 \rangle n$
for an integer $n$,
the actions of $\lm$  on the vectors $\pz, \; \po$
are easily obtained,
%
\begin{equation}
  \lm\pz = \po, \quad\quad
  \lm\po = \langle 2 \rangle \partial_w \pz, \label{eq:ospp}
\end{equation}
%
and $\lp$ acts on them as
\begin{equation}
  \lp\pz = w \po, \quad\quad
  \lp\po = \langle 2 \rangle ( \nu+ w\partial_w) \pz. \label{eq:ospm}
\end{equation}
%
At first sight of eqs.(\ref{eq:ospp}) and (\ref{eq:ospm}) we might suspect
that $\pz$ and $\po$ are superpartner with each other and ${\cal L}_{\pm1}$
are the generators of supersymmetry transformations.
However we must be more cautious because the actions of
${\cal L}_{\pm1}$ on $\po$ are zero-actions due to the factor
$\langle2\rangle$.
Note that the vector $\po$ must be proportional
to $\sqrt{\langle 2 \rangle }$ since the norm of the vector is proportional to
$\langle 2 \rangle $.
Fortunately we can remedy the situation by scaling the operators and the
vector $\po$ as follows;
%
\begin{equation}
\begin{array}{l}
\phi^\zeta(w) = \pz,\quad\quad  \sqrt{\langle 2 \rangle }\,
\psi^\zeta(w) = \po, \\
\sqrt{\langle 2 \rangle }\,{\cal G}_{-1}
= \lm, \quad\quad \sqrt{\langle 2 \rangle }\,{\cal G}_1 = \lp.
\end{array}
\label{eq:scaling}
\end{equation}
%
The actions of ${\cal G}_\pm$ on $\phi^\zeta$ and $\psi^\zeta$ are
now non-vanishing and $\psi^\zeta$ has definite norm.

We are ready to discuss the connection between two-dimensional
supersymmetry and $U_q(su(1,1))$.
Let us define an infinitesimal transformation $\de$ as
\begin{equation}
\de:= a\,{\cal G}_1 + b\,{\cal G}_{-1},
\end{equation}
%
where $a, b$ are infinitesimal Grassmann numbers.
Under the transformation the fields $\ph$ and $\ps$  transform into
each other according to
\begin{equation}
   \begin{array}{l}
     \de\ph = \varepsilon(w)\ps, \\
  \rule{0mm}{.5cm}
     \de\ps = \left( \nu(\partial_w \vp(w)) + \vp(w) \partial_w \right) \ph,
   \end{array}
\end{equation}
%
where  $\vp(w)=aw +b$ is an anticommuting analytic function which
parametrises infinitesimal holomorphic transformation.
The commutation relations of two transformations
$\deo$ and $\dew$ are
\begin{equation}
   \begin{array}{l}
      {[\deo, \dew]}\ph = \left( \zeta (\partial_w \xi(w))
        +\xi(w)\partial_w \right) \ph,  \\
  \rule{0mm}{.5cm}
    {[\deo, \dew]}\ps = \left( (\zeta+\frac{1}{2})(\partial_w \xi(w))
        +\xi(w)\partial_w \right) \ps,
   \end{array}
\label{eq:sltr}
\end{equation}
%
with $\xi(w)=2\vp_1(w)\vp_2(w)$.
The right hand sides of equations (\ref{eq:sltr}) are just the
transformations of the fields $\ph$ and $\ps$ having dimensions $\zeta$ and
$\zeta+\frac{1}{2}$, respectively, under the the infinitesimal transformation
of $\SL$,  $w\,\rightarrow\,w+\xi(w)$.
We can therefore conclude that the infinitesimal transformation
$\de$ which is written in terms of generators of $U_q(su(1,1))$ is
just the \lq square root' of infinitesimal $SL(2,{\bf R})$
transformation, that is to say, $\de$ is an infinitesimal supersymmetry
transformation.
Further the fields $\ph$ and $\ps$ which are constructed in terms of
the highest weight representations of $U_q(su(1,1))$ can be regarded as
superpartners with each other.
Finally, $Osp(1\vert 2)$ algebra is obtained as follows:
Let $L_{\pm1}=({\cal G}_{\pm1})^2$ and $F_{\pm\frac{1}{2}}={\cal G}_{\pm1}$,
then the following commutation relation are easily checked on the fields
$\phi^\zeta(w)$ and $\psi^\zeta(w)$ to be
%
\begin{equation}
     \begin{array}{ll}
    {[L_n, L_m]}=(n-m)L_{n+m},  & n, m=0, \pm1, \\
    {[L_n, F_r]}=\left( \frac{1}{2}n-r \right)F_{n+r},
                                & r=\frac{1}{2}, -\frac{1}{2}, \\
    {\{F_r, F_s\}}= 2L_{r+s},     &  r, s =\frac{1}{2}, -\frac{1}{2}.
     \end{array}
\end{equation}
%
Thus we have succeeded in building the super-algebra $Osp(1\vert 2)$
and its representation in terms of the representation of
$U_q(su(1,1))$ when $q^2=-1$ and $\mu=1$, $i.e.$, all states at the
$(2{\Bbb N}-1)$-th levels are zero-norm states.

Next we turn to the $\mu=2$ case. By eq.(\ref{eq:hmnt}),
the highest weight is given by $\zeta-\frac{1}{2}$.
On the contrary to the case when $\mu=1$, no zero-norm state appears.
The actions of  $\lp$ and $\lm$ on $\phi_{0,1}$ are as follows;
\begin{equation}
\begin{array}{l}
  \lp\pz = \langle 2 \rangle (\nu w + w^2\partial_w)\po, \quad
  \lp\po = \pz,  \\  {}  \\
  \lm\pz = \po, \quad
  \lm\po = \langle 2 \rangle \partial_w \pz.
\end{array}
\end{equation}
%
Unfortunately, we cannot find any consistent ways to remove the factor
$\langle 2 \rangle$ as in the previous case (\ref{eq:scaling}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%    SECTION 5     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Discussion}

In this article the highest weight representations of $\ncom$ when
$\q$ has been  investigated in detail.
We have shown that the highest weight module $\V$ is isomorphic to the tensor
product of two highest weight modules $V_\zeta^{cl}$ and $\mho_j$.
This fact played a key role of this work, and novel features of $\ncom$
originated from this structure of $\V$. The module $\vc$ is a
classical non-compact $\slc$-module, while $\vq$ is a
$(2j+1)$-dimensional module of the quantum universal envelopping algebra
$U_q(\slc)$.  Theorem 3.1 states what $\vc$ and $\vq$  are.
In particular, when the original $\ncom$-module $\V$ is unitary,
$V_\zeta^{cl}$ is the unitary highest weight module of
$su(1,1)\simeq \slr$ and $\vq$ is the unitary highest weight $\com$-module.
In the following we restrict our discussions to this case, {\it i.e.},
$\V$ is unitary.

We summarize here the novel features of $\ncom$ when $\q$:
First we should notice that the non-compact nature appears only through the
classical module $\vc$, and the effects of $q$-deformation arise only from the
compact sector $\vq$.
Since the non-compact sector $\vc$ is classical, a representation of the Lie
group $\SL$ is naturally induced.
Indeed, we gave a representation of $\SL\otimes\com$ by means of the
holomorphic vector $\fun=\psi_\zeta(w)\otimes\vert j; m\rangle$.
Here we used holomorphic sections $\psi_\zeta(w)$ of a line bundle over the
homogeneous space $\SL/U(1)$ in order to represent the $\SL$ sector and
$\vert j; m\rangle\in\vq$ is a weight vector with respect to $\com$.
With our deformation parameter $q$, $i.e.$, $\q$, we have shown that
the value of the highest weight $j$ lies in ${\cal A}=\{0, \frac{1}{2}, 1,
\cdots, \frac{N-2}{2}\}$.
Notice that this finiteness of the number of
the highest weight states for the $\com$ sector comes from the
condition that the original highest weight representations $\V$ of
$\ncom$ be well-defined, that is, every state in them has finite norm.
The representation $\fun$ says that every point on the homogeneous space,
({\it i.e.}, the upper half plane or the Poincar\'e disk)
has the representation space of $\com$.
In this sense, we suggest that the non-compact homogeneous
space can be viewed as a base space or an external space and the
representation space of $\com$ as an internal space.

We have also discussed the connection between $\ncom$ with $\q$ and
$Z_N$-graded supersymmetry by presenting $N$ holomorphic vectors,
denoted as $\pfun_r(w),\,r=0, 1, \cdots, N-1$, in another way.
The generators, $X_\pm, K$ of $\ncom$ act on them and map $\pfun_r$ to
$\pfun_{r\pm1}$.
On the other hand, the operator $L_n$ which are related to the $N$-th
powers of $X_\pm, K$ by the relations (\ref{eq:ldef}) generate the holomorphic
transformations of the functions under the infinitesimal transformations
of the homogeneous space $\SL/U(1)$.
In this sense, we may say that generators of $\ncom$ give rise to
$Z_N$-graded supersymmetry transformations and the $N$-th powers of them
are related to the infinitesimal transformations with respect
to the external space.
Furthermore, by observing the transformations under $L_n$, we have shown
that these $N$ functions separate into two classes.
One of them is the set of functions,
$\wt{\Phi}^\zeta_r(w)$, $r=0\sim 2j$, with dimensions $\zeta$
and the other is the set of functions, $\Xi_r^{\zeta+\frac{1}{2}}(w)$,
$r=2j+1\sim N-1$, which have dimensions $\zeta+\frac{1}{2}$ and have
zero norms.
That is to say, the functions $\wt{\Phi}^\zeta_r$ and
$\Xi_r^{\zeta+\frac{1}{2}}$ behave as the covariant vectors with dimensions
$\zeta$ and $\zeta+\frac{1}{2}$, respectively, under $\slr$.
In particular, we have shown the explicit realization of two-dimensional
supersymmetry $Osp(1\vert2)$ via the representation of $\ncom$ when the
deformation parameter satisfies $q^2=-1$.

We have also discussed the Clebsch-Gordan decomposition for the tensor
product of two irreducible highest weight modules and found that the
decomposition rules for the two sectors $V_\zeta^{cl}$ and $\mho_j$ coincide
with those for the classical non-compact representations of $\slr$ and the
representations of $\com$.

\vspace{0.5cm}

Finally, we would like to future issues to be investigated.
As mentioned in chapter 1, it is quite interesting to expect the relationship
between $\ncom$ and $2D$ quantum gravity coupled with RCFT.
In order to make this expectation come true, we have to find a good
representation space of $\ncom$ for which such a physical theory is
associated \cite{MS2}.
The $Z_N$ graded supersymmetry implies that the internal symmetry, $\com$,
is not independent of the base manifold $\Sigma_g$ but yields the deformation
of the metric of $\Sigma_g$ through the $N$-th powers of the action.

Second, it is also interesting to investigate geometical aspect of our result.
As for the geometical viewpoint of quantum groups,
it is widely expected that quantum groups will shed light on the concept
of ``quantum'' space-time.
In particular, quantum groups in the sense of $A_q(G)$,
the $q$-deformation of the functional ring over the group $G$,
rather than $U_q({\Got{g}})$ play the central role in the noncommutative
geometry initiated by Manin\cite{Ma}, and Wess and Zumino \cite{WZ}.
Further Wess and Zumino have studied a $q$-deformed quantum mechanics
in terms of the noncommutative differential geometry based on $A_q(G)$.
The phenomena observed in this paper suggest that by the quantization of the
Poincar\'e disk, a certain ``$q$-deformed space'' appears as a
($q$-deformed) fiber at each point on the disk which remains classical.
This observation is reminiscent of the result obtained in Ref.\cite{Su}.
Actually, in order to construct $q$-deformed mechanics,
a $q$-deformed phase space was introduced in \cite{Su} by attaching
an internal space at each point on the phase space of the classical
mechanics and all effects of $q$-deformation stemed only from the internal
space.



\vspace{1.0cm}

\noindent
I am grateful to Dr. T. Matsuzaki for fruitful collaborations and
discussions. I would also like to thank Dr. H. W. Braden for
carefully reading this manuscript and valuable comments.




\newpage



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\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the weak-coupling case, where $c_0=\delta=0$ with $\kappa=1$, 
the branch $\gamma_-$ is chosen so that the model has classical limit 
$\gamma=0$. 
Suppose that we also take the $\tilde\gamma_-$ branch. 





There are three phases for the quantum sector $S^q(\phi)$ according to 
the values of $Q$. 
(i) $D+c_0 > 25$,  (ii) $D+c_0 < 25$ and the boundary case   
(iii) $D+c_0 = 25$.   
Before going to the discussions of each phase, it is convenient to rewrite 
the action of the quantum sector (\ref{eq:waction})  times 
$\kappa^2/2\pi\gamma^2$ as 


In the following, we will use the notations $D'=D+c_0$. 
Let us see these phases separately.  

\vspace{0.2cm}
\noindent
(i) The case when $D'=D+c_0>25$. 
In this case, $Q$ is a pure imaginary number. 
Supposing $\Lambda_2=0$ and denoting $Q=i\alpha_0$,  
the quantum sector is nothing but the 
minimal CFT with the central charge $c_q=1-12\alpha_0^2$. 
The screening charges are given by 
$\alpha_\pm= (\sqrt{D'-25} \mp \sqrt{D'-1})/\sqrt{12}$. 
The vertex operator $e^{-j\tilde{\gamma}\vp} = 
e^{-ij\alpha_+\vp}$ corresponds to the highest weight state 
in the $(2j+1)\times (2j+1)$ dimensional representation 
of $\qslc\otimes\overline{\qslc}$ and the section $\Psi[m,j_i]$ 
can be regarded as the conformal block of the Virasoro minimal series. 
Thus, in the region $D'>25$, our quantum Liouville gravity  
is by itself the minimal CFT theory coupled to the classical Liouville 
gravity. 
The physical interpretation of the quantum surface, \ie, 
quantum space-time in this region is the following: 
Every point on the space-time fluctuates by the quantum effect, 
but the quantum fluctuation separates from the classical space-time  
completely and yields minimal matter\,!

\vspace{0.2cm}
\noindent
(ii) The case when $D'<25$. 
In this case, $Q$ is a real number and so one should think  
that, unlike the previous case (i),  the quantum sector represents 
again quantum Liouville gravity based on $\com$ rather than matter.  
However, further investigations are required before we understand  
this region precisely. 

\vspace{0.2cm}
\noindent
(iii) The case when $D'=25$. 
Since in this case $Q=0$,  
the quantum sector contributes 1 to the central charge 
$c_L$ of the quantum Liouville gravity. 
In this case, the field $\vp(z,\bz)$ can be interpreted as the 
$(D+1)$-th component of the string coordinates, \ie, 
$\vp(z,\bz)=X^{D+1}(z,\bz)$. 
Recall that, in the weakly coupled Liouville gravity when $D=25$, 
the Liouville field is regarded as the time component of string 
in the 26-dimensional Minkowski space. 
In our case, however, the same situation can happen 
for a different dimension $D$ by the classical central charge $c_0$. 

\vspace{.2cm}
\noindent
In the cases (i) and (iii) and when $\beta > 1/\sqrt{2}$, 
our Liouville theory can be a strongly coupled Liouville gravity, 
\ie, $1 < c_L < 25$,  since in these cases  
the maximal value of $c_q$ is 1. 
On the contrary, some further discussions are required 
before the complete understanding of the case (ii), 
for example, the relation $Q$ and $\tilde\gamma$. 
In the next section, we will discuss about the remaining tasks 
briefly. 
%%%%%%%%%%%%%%%%%%%%%%%






the relation between the quantum sector $S^q(\phi;\hat{g}_P)$ 
corresponding to the quantum fluctuations around the classical gravity 
and the weakly coupled Liouville gravity should be clarified.  
What we know at present is just the fact that they both are 
the theories associated with $\com$ but  are different in the choices 
of background metrics. 
In order to answer these questions, 
it is quite important to understand the quantum sector $S^q(\phi)$ 
in more detail, for example, 
what kind of equation governs for the correlation functions 
$Z^q[{\sl m}:\{j\}]$. 
Of course, in the standard weakly coupled Liouville theory, 
the equation is the conformal Ward identity. 
Remembering that our quantization of Liouville theory can be related 
to the geometric quantization, we expect that such equation has 
something to do with polarization for the section of a line bundle. 
Furthermore we need to know the relation between $Q$ and 
$\tilde\gamma$. 
To know this is to know the relation between $Q_0$ and the original 
coupling constant $\gamma$. 
Third, does the metric $e^{\Phi(z,\bz)}$ actually have the 
conformal dimension $(1,1)$? 
We have not considered the conformal dimension of operators  
in our model at all. 
In order to see this in the framework of quantum groups, 
we have to examine the braiding matrix in terms of the $q$-$6j$ 
symbol of the case $\ncom$. 
Thus the key to the complete understanding of our Liouville gravity 
is the quantum sector $S^q(\phi)$. 




























































