%Paper: hep-th/9208056
%From: tanii@th.phy.saitama-u.ac.jp
%Date: Sun, 23 Aug 92 12:58:48 +0900

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\begin{flushright}
STUPP--92--130 \\
August, 1992
\end{flushright}
\vspace{15mm}
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\begin{center}
{\large{\bf{Semiclassical Quantization of \\[2mm]
Two-Dimensional Dilaton Gravity}}} \\[25mm]
%
{\sc Yoshiaki Tanii} \\[15mm]
{\it Physics Department, Saitama University \\[2mm]
Urawa, Saitama 338, Japan} \\[25mm]
%
{\bf Abstract}\\[1cm]
%
{\parbox{13cm}{\hspace{5mm}
Quantization of the dilaton gravity in two dimensions is discussed
by a semiclassical approximation. We compute the fixed-area
partition function to one-loop order and obtain the string
susceptibility on Riemann surfaces of arbitrary genus. Our result
is consistent with the approach using techniques of conformal
field theories.}}
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\end{center}
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%%%%%%%%%  1. Introduction  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
A two-dimensional metric-dilaton system coupled to matter fields
(dilaton gravity) was proposed in refs.\ \cite{CGHS,VERLINDE}
as a simple model to discuss the quantum theory of black holes.
Although this model is exactly solvable at the classical level,
the quantum theory is not yet fully understood.
\par
%
Recently, quantization of this model was discussed in
refs.\ \cite{STR,ABC} using techniques of conformal field theories.
(Quantization was also discussed using other methods in ref.\
\cite{RT}.) The authors in refs.\ \cite{STR,ABC} made an ansatz
about the functional measures of path integrals following a
procedure applied to ordinary two-dimensional gravity in
ref.\ \cite{DDK}.
In the case of ordinary gravity the ansatz was justified by
comparing its results with those of the matrix models \cite{KM}.
The ansatz was also checked by other approaches such as
semiclassical analyses \cite{ZAM,CKT}, the light-cone gauge
quantization \cite{KPZ} and direct calculations of the functional
measures \cite{MMDK}. Since matrix models for the dilaton gravity
are not known at present, it is important to study other approaches
and compare their results with those of refs.\ \cite{STR,ABC}.
\par
%
The purpose of the present paper is to study the dilaton gravity
by a semiclassical approximation, which becomes exact for a matter
central charge $c \rightarrow - \infty$.
We find a saddle point of the path integral with a fixed `area' and
quantize fluctuations of fields around it to one-loop order.
In particular, we compute the string susceptibility on Riemann
surfaces of arbitrary genus. Our result is consistent with that of
refs.\ \cite{STR,ABC}. We also study a case in which the functional
measures are modified as in ref.\ \cite{STR}.
\par
%
%%%%%%%%%  2. Model  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
We consider a conformal field theory with a central charge $c$
coupled to a metric-dilaton system on a compact closed surface.
The metric is chosen to have the Euclidean signature.
The classical action is
\be
S = {1 \over 2\pi} \int d^2\xi \sqrt{g} \e^{-2\phi}
\left( R_g + 2\mu + 4 g^{\alpha\beta} \partial_\alpha \phi
\partial_\beta \phi \right) + S_{\rm M}[g_{\alpha\beta}, f],
\label{action}
\ee
where $\phi$ is the dilaton field, $R_g$ is the scalar curvature of
the metric $g_{\alpha\beta}$ and $S_{\rm M}$ is the action for the
matter field $f$. The parameter $\mu$ is a generalized cosmological
constant. We have chosen the sign of the
metric-dilaton terms in eq.\ (\ref{action}) opposite to that of
ref.\ \cite{CGHS}. This choice of the sign is convenient for a
semiclassical analysis in the limit $c \rightarrow - \infty$.
The partition function on the genus $h$ Riemann surface is given
by a path integral
\be
Z_\chi(\mu) = \int {\D_g \, g_{\alpha\beta} \D_g \phi \,
\D_g f \over V_{\rm gauge}} \e^{- S}
\label{partition}
\ee
where $\chi = 2 - 2h$ is the Euler number of the surface and
$V_{\rm gauge}$ is the volume of the group of diffeomorphisms.
The functional measures are defined in a diffeomorphism invariant
way using the metric $g_{\alpha\beta}$. In particular, the
measures of the metric and the dilaton are defined by the
norms \cite{POLYAKOV}
\ba
|| \delta g_{\alpha\beta} ||_g^2 \A = \A \int d^2 \xi \sqrt{g} \,
(g^{\alpha\gamma} g^{\beta\delta}
+ u g^{\alpha\beta} g^{\gamma\delta})
\delta g_{\alpha\beta} \delta g_{\gamma\delta} \quad
\left( u > -{1 \over 2} \right), \nonu
|| \delta \phi ||_g^2 \A = \A \int d^2 \xi \sqrt{g} \,
(\delta \phi)^2.
\label{measure}
\ea
In ref.\ \cite{STR} a different choice of the measures was used.
We will discuss such modifications of the measures at the end of
the paper. For a semiclassical analysis it is more convenient to
consider the partition function with a fixed `area' $A$
\be
\tilde Z_\chi(A) = \int {\D_g g_{\alpha\beta} \, \D_g \phi \,
\D_g f \over V_{\rm gauge}} \,
\delta \left( \int d^2 \xi \sqrt{g} e^{-2\phi} - A \right)
\e^{- S}.
\label{apartition}
\ee
The partition function (\ref{partition}) is obtained from
eq.\ (\ref{apartition}) by integrating over $A$.
\par
%
To fix the diffeomorphism invariance we choose the conformal gauge
\cite{POLYAKOV}
\be
g_{\alpha\beta}(\xi)
= \e^{2\rho(\xi)} \hat g_{\alpha\beta}(\xi; \tau),
\label{gauge}
\ee
where $\hat g_{\alpha\beta}$ is a reference metric which
depends on the moduli $\tau$ of the Riemann surface.
After the gauge fixing and an integration of the matter field
we obtain
\be
\tilde Z_\chi(A)
= \int {[d\tau] \over V_{\rm CKV}} \int \D_g \rho \D_g \phi \,
\delta \left( \int d^2 \xi \sqrt{\hat g} e^{2(\rho-\phi)}
- A \right) \e^{- S_{\rm eff}},
\label{gaugefixed}
\ee
where $V_{\rm CKV}$ is the volume of the group generated by the
conformal Killing vectors, which exist for genera $h = 0, 1$.
The effective action is given by
\be
S_{\rm eff} = {1 \over 2\pi} \int d^2\xi \sqrt{\hat g} \, \left[
\e^{-2\phi} \left( R_{\hat g} - 4 \hat g^{\alpha\beta}
\partial_\alpha \phi \partial_\alpha \rho
+ 4 \hat g^{\alpha\beta} \partial_\alpha \phi \partial_\beta \phi
\right) \right] + S_{\rm L} + {\mu \over \pi} A,
\label{effaction}
\ee
where we have used the fixed area condition. The Liouville action
\be
S_{\rm L}[\hat g_{\alpha\beta}, \rho]
= {\gamma \over 2\pi} \int d^2 \xi \sqrt{\hat g}
\left( \hat g^{\alpha\beta} \partial_\alpha \rho
\partial_\beta \rho + R_{\hat g} \, \rho
+ 2 \mu' \e^{2\rho} \right), \quad
\gamma = {26-c \over 12}
\label{liouville}
\ee
is a result of the Weyl anomaly of the matter and the
Faddeev-Popov ghost fields \cite{POLYAKOV}. The parameter $\mu'$
is regularization dependent and we choose it to be zero for
simplicity.
\par
%
The functional measures of $\rho$ and $\phi$ in
eq.\ (\ref{gaugefixed}) are defined by the norms induced from
eq.\ (\ref{measure})
\ba
|| \delta \rho ||_g^2 \A = \A \int d^2 \xi \sqrt{\hat g}
\e^{2\rho} (\delta \rho)^2, \nonu
|| \delta \phi ||_g^2 \A = \A \int d^2 \xi \sqrt{\hat g}
\e^{2\rho} (\delta \phi)^2.
\label{rhomeasure}
\ea
Due to the factor $\e^{2\rho}$ in these norms it is not obvious how
to evaluate the functional integral.
In refs.\ \cite{STR,ABC} a relation of these measures to those without
the factor $\e^{2\rho}$ was given as an ansatz following
the procedure in ref.\ \cite{DDK}.
We will check this ansatz by evaluating the functional integral
(\ref{gaugefixed}) in a semiclassical approximation,
which becomes exact for $c \rightarrow - \infty$.
\par
%
%%%%%%%%%%  3. classical solution  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
As a first step of the semiclassical quantization let us find
a saddle point of the functional integral (\ref{gaugefixed}).
We have to find a minimum of the exponent $-S_{\rm eff}$
under the condition of fixed area.
Introducing the Lagrange multiplier $\lambda \in \R$
it can be found by the variational principle of $S_{\rm eff}
+ {\lambda \over \pi} (\int d^2 \xi \sqrt{\hat g}
e^{2(\rho-\phi)} - A)$. The Euler-Lagrange equations are
\ba
2 \Delta_{\hat g} \rho - R_{\hat g} - 2 \lambda \e^{2\rho}
- 4 ( \Delta_{\hat g} \phi - \hat g^{\alpha\beta}
\partial_\alpha \phi \partial_\beta \phi ) \A = \A 0, \nonu
2 \Delta_{\hat g} \rho - R_{\hat g} - {4 \lambda \over \gamma}
\e^{2(\rho-\phi)} - {4 \over \gamma} ( \Delta_{\hat g} \phi
- \hat g^{\alpha\beta} \partial_\alpha \phi \partial_\beta \phi )
e^{-2 \phi} \A = \A 0, \nonu
\int d^2 \xi \sqrt{\hat g} e^{2(\rho-\phi)} - A \A = \A 0.
\label{eleq}
\ea
If we choose the dilaton field to be a constant
\be
\phi_{\rm cl} = -{1 \over 2} \ln {\gamma \over 2},
\label{classphi}
\ee
then the first two equations of eq.\ (\ref{eleq}) are reduced to
a single equation
\be
R_{\bar g} = - 2 \lambda, \quad
\bar g_{\alpha\beta} \equiv \e^{2 \rho_{\rm cl}} \,
\hat g_{\alpha\beta},
\label{classmetric}
\ee
which expresses the fact that the metric $\bar g_{\alpha\beta}$
has a constant curvature.
The multiplier $\lambda$ can be fixed by integrating
eq.\ (\ref{classmetric}) over the surface and using the third
equation of eq.\ (\ref{eleq})
\be
\lambda = - {\pi \gamma \chi \over A}.
\label{lambdavalue}
\ee
It is a mathematical theorem that there exists a unique metric
with a constant curvature on any Riemann surface.
Thus eq.\ (\ref{classmetric}) with eq.\ (\ref{lambdavalue})
determines $\rho_{\rm cl}$.
It is easy to see that the $A$-dependence of $\rho_{\rm cl}$ is
\be
\rho_{\rm cl} = \rho_{\rm cl}^{A=1} + {1 \over 2} \ln A.
\label{adepofrho}
\ee
\par
%
If we considered the partition function (\ref{partition}) instead
of the area-fixed one (\ref{apartition}), the saddle point
condition would be the first two equations of eq.\ (\ref{eleq})
with $\lambda$ replaced by the cosmological constant $\mu$. Then,
as one can easily see by integrating the equation corresponding
to eq.\ (\ref{classmetric}), a solution of the form
(\ref{classphi}), (\ref{classmetric}) is possible only for either
$\chi > 0$ or $\chi < 0$ depending on the sign of $\mu$.
This is the reason why we have considered the area-fixed
partition function (\ref{apartition}).
\par
%
%%%%%%%%%%  4. semiclassical analysis  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Next we define quantum fluctuations around the above configuration
\be
\tilde \phi = \phi - \phi_{\rm cl}, \quad
\tilde \rho = \rho - \rho_{\rm cl}
\label{fluc}
\ee
and expand the action $S_{\rm eff}$ in these fluctuations.
The action up to quadratic terms is
\ba
S_{\rm eff} \A = \A {\mu \over \pi} A + \gamma\chi
+ S_{\rm L}[\hat g_{\alpha\beta}, \rho^{A=1}_{\rm cl}]
+ \gamma \chi \ln A
+ {\gamma \over 2\pi} \int d^2\xi \sqrt{\bar g} \, \Bigl[ \,
\bar g^{\alpha\beta} \partial_\alpha \tilde \phi
\partial_\beta \tilde \phi \nonu
\A\A + \, {2\pi\gamma\chi \over A} \, \tilde \phi^2
+ \bar g^{\alpha\beta} \partial_\alpha
(\tilde \rho - \tilde \phi) \partial_\beta
(\tilde \rho - \tilde \phi) - {2\pi\gamma\chi \over A} \,
(\tilde \rho - \tilde \phi)^2 \, \Bigr].
\label{expansion}
\ea
To obtain eq.\ (\ref{expansion}) we have used the fixed area
condition
\ba
0 \A = \A \int d^2 \xi \sqrt{\hat g} \e^{2(\rho-\phi)} - A \nonu
\A = \A \gamma \int d^2 \xi \sqrt{\bar g} \left[ \,
(\tilde\rho - \tilde\phi) + (\tilde\rho - \tilde\phi)^2
+ \cdots \, \right]
\label{linear}
\ea
to eliminate linear terms in the expansion.
\par
%
We now substitute eq.\ (\ref{expansion}) into
eq.\ (\ref{gaugefixed}) and evaluate the integrals.
We decompose the fields into zero (constant) modes
$\rho_0,\ \phi_0$ and nonzero modes $\rho',\ \phi'$
($\tilde\rho = \rho_0 + \rho',\ \tilde\phi = \phi_0 + \phi'$)
with respect to the Laplacian $\Delta_{\bar g}$.
To the one-loop order the norm of $\rho$ (\ref{rhomeasure})
can be approximated as \cite{ZAM,CKT}
\ba
|| \delta \rho ||_g^2 \A = \A \int d^2 \xi \sqrt{\bar g}
\e^{2\tilde\rho} (\delta \tilde\rho)^2, \nonu
\A \cong \A \int d^2 \xi \sqrt{\bar g} \,
(\delta \tilde\rho)^2, \nonu
\A = \A (\delta \rho_0)^2 \int d^2 \xi \sqrt{\bar g}
+ \int d^2 \xi \sqrt{\bar g} \, (\delta \rho')^2.
\label{oneloopmeasure}
\ea
Therefore the functional measure becomes
\be
\D_g \rho \cong d \rho_0 \D_{\bar g} \rho' \left( \int d^2 \xi
\sqrt{\bar g} \right)^{1 \over 2}
= d \rho_0 \D_{\bar g} \rho' \left( {2A \over \gamma}
\right)^{1 \over 2}.
\label{decompmeasure}
\ee
A similar formula holds for the measure of $\phi$.
The $\rho_0$ integral can be evaluated as
\be
\int d \rho_0 \, \delta \left( \int d^2 \xi \sqrt{\hat g}
\e^{2(\rho-\phi)} - A \right) \e^{2\gamma\chi (\rho_0 - \phi_0)^2}
\cong {1 \over 2A},
\label{zeromodeint}
\ee
while the $\phi_0$ integral gives an $A$-independent constant.
The integrals of the nonzero modes $\rho',\ \phi'$ give functional
determinants of differential operators
\be
{\rm det}' \left( \Delta_{\bar g}
- {2\pi\gamma\chi \over A} \right)^{-{1 \over 2}}
{\rm det}' \left( \Delta_{\bar g}
+ {2\pi\gamma\chi \over A} \right)^{-{1 \over 2}},
\label{determinant}
\ee
where the primes denote an exclusion of the zero mode of
$\Delta_{\bar g}$. They are the same type of operators as those
appearing in the semiclassical analysis of the ordinary Liouville
theory \cite{ZAM,CKT}. The $A$-dependence is given
by \cite{ZAM,CKT}
\be
A^{1 \over 2} {\rm det}' \left( \Delta_{\bar g}
+ {4\pi\eta\chi \over A} \right)^{-{1 \over 2}}
= {\rm const.} \times A^{{1 \over 12}\chi
+ {1 \over 2}\eta\chi} \e^{-KA},
\label{det}
\ee
where $K$ is a regularization dependent constant, which we choose
to be zero.
\par
%
Collecting the above results we finally obtain
the $A$-dependence of the area-fixed partition function
\be
\tilde Z_\chi(A) = {\rm const.} \times
A^{\Gamma(\chi) - 3} \e^{- {\mu \over \pi} A},
\label{adep}
\ee
where the string susceptibility is given by
\be
\Gamma(\chi) = {c - 24 \over 12} \, \chi + 2 + O((-c)^{-1}).
\label{susceptibility}
\ee
The $\mu$-dependence of the partition function (\ref{partition})
is obtained from eq.\ (\ref{adep}) by integrating over $A$
\be
Z_\chi(\mu) = {\rm const.} \times \mu^{-\Gamma(\chi) + 2}.
\label{mudep}
\ee
\par
%
%%%%%%%%%%  5. comparison with other approaches  %%%%%%%%%%%%%%%%%%%
%
Let us compare this result with that of ref.\ \cite{ABC}.
It was argued in ref.\ \cite{ABC} that the functional
integral (\ref{partition}) can be reduced to
\be
Z_\chi(\mu) = \int {[d\tau] \over V_{\rm CKV}}
\int \D_{\hat g} \, \chi \D_{\hat g} \Omega \, \e^{- S_{\rm CFT}},
\label{abc}
\ee
where the action is given by
\be
S_{\rm CFT} = {1 \over 8\pi} \int d^2\xi \sqrt{\hat g} \left(
\hat g^{\alpha\beta} \partial_\alpha \chi \partial_\beta \chi
+ 2 \sqrt{\kappa} R_{\hat g} \, \chi - \hat g^{\alpha\beta}
\partial_\alpha \Omega \partial_\beta \Omega + 2 \mu \kappa
\e^{{1 \over \sqrt{\kappa}}(\chi - \Omega)-1} \right).
\label{abcaction}
\ee
The parameter $\kappa$ is determined by the conformal invariance
as $\kappa = {24-c \over 12}$. To make the integral of $\Omega$
convergent one has to integrate it along the imaginary direction
$\Omega = i \bar \Omega$ ($\bar \Omega \in \R$).
The $\mu$-dependence of the partition function can be found by
shifting the field as $\chi \rightarrow \chi
- \sqrt{\kappa}\ln\mu$ in eq.\ (\ref{abc}).
Thus we find the string susceptibility in this approach
\be
\Gamma_{\rm CFT}(\chi) = {c - 24 \over 12} \, \chi + 2,
\label{abcmudep}
\ee
which is consistent with our semiclassical result
(\ref{susceptibility}).
\par
%
%%%%%%%%%%  6. other choices of measures  %%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Finally let us discuss other choices of the functional measures
in the path integral. In ref.\ \cite{STR} the measures of the
metric and the dilaton were defined using the modified metric
$\e^{-2\phi} g_{\alpha\beta}$, while that of the matter field was
defined by $g_{\alpha\beta}$.
More generally, one may consider also a modification of the matter
measure. Here, we consider the following general situation.
Suppose that among the matter-ghost system contributing to the
Weyl anomaly proportional to $\gamma$, a fraction
${\gamma_i \over \gamma}$ ($i = 1,\ 2,\ 3,\ \cdots,\ \sum_i
\gamma_i = \gamma$) has a measure defined by a modified metric
$\e^{2 \alpha_i \phi} g_{\alpha\beta} = \e^{2(\rho+\alpha_i \phi)}
\hat g_{\alpha\beta}$.
The Liouville action (\ref{liouville}) is then replaced by
\ba
S'_{\rm L} \A = \A \sum_i {\gamma_i \over \gamma} \,
S_{\rm L}[\hat g_{\alpha\beta}, \rho + \alpha_i \phi] \nonu
\A = \A {1 \over 2\pi} \int d^2 \xi \sqrt{\hat g}
\Bigl( \gamma \, \hat g^{\alpha\beta} \partial_\alpha \rho
\partial_\beta \rho + \gamma R_{\hat g} \, \rho
+ a \, \hat g^{\alpha\beta} \partial_\alpha \phi
\partial_\beta \phi \nonu
\A\A + \, 2 b \, \hat g^{\alpha\beta} \partial_\alpha \phi
\partial_\beta \rho + b \, R_{\hat g} \, \phi \Bigr),
\label{mliouville}
\ea
where $a = \sum_i \gamma_i \alpha_i^2,\
b = \sum_i \gamma_i \alpha_i$.
The case we have considered so far corresponds to $a = b = 0$,
while the choice in ref.\ \cite{STR} corresponds to
$a = -b = {13 \over 6}$. The measures of $\rho$ and $\phi$
(\ref{rhomeasure}) may also be modified. However, it does not
change the result of the string susceptibility in the
one-loop approximation.
\par
%
Using eq.\ (\ref{mliouville}) instead of eq.\ (\ref{liouville})
we can repeat the above semiclassical analysis.
The saddle point is found to be
\be
\phi_{\rm cl} = -{1 \over 2} \ln {\gamma + b \over 2}, \quad
R_{\bar g} = {2 \pi (\gamma+b) \chi \over A},
\label{mclassfield}
\ee
where $\bar g_{\alpha\beta}$ is defined in
eq.\ (\ref{classmetric}).
Integrating the fluctuations around the configuration
(\ref{mclassfield}) we find that the area-fixed partition function
is given by the same expression as before except that the
functional determinants (\ref{determinant}) are replaced by
\be
{\rm det}' \left( \Delta_{\bar g} - {2\pi(\gamma+b)^2\chi \over
(\gamma + a + 2b) A} \right)^{-{1 \over 2}}
{\rm det}' \left( \Delta_{\bar g} + {2\pi(\gamma+b)\chi \over A}
\right)^{-{1 \over 2}}.
\label{mdeterminant}
\ee
Thus we find the string susceptibility
\be
\Gamma(\chi) = \left[ {c - 24 \over 12}
+ {(a+b) (26-c+12b) \over 4 (26-c+12a+24b)} \right] \chi
+ 2 + O((-c)^{-1}).
\label{msusceptibility}
\ee
The choice in ref.\ \cite{STR} $a = -b = {13 \over 6}$ gives the
same susceptibility as eq.\ (\ref{susceptibility}),
which is consistent with the result of refs.\ \cite{STR,ABC}.
It is interesting to study the model with other choices of $a$
and $b$ using the method of refs.\ \cite{STR,ABC} and compare its
result with eq.\ (\ref{msusceptibility}).
\par
%
\vspace{5mm}
%
The author would like to thank S. Yamaguchi for a collaboration in
an early stage of this work. He would also like to thank N. Sakai,
Y. Matsumura and T. Uchino for useful discussions on the dilaton
gravity.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newpage
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%
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%
\end{thebibliography}
%
\end{document}

