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\title{Liouville Field Theory and 2d Gravity}

\tocauthor{G. Jorjadze}
\author{George Jorjadze\thanks{e-mail: jorj@rmi.acnet.ge}\\
{\small\it Dept. of Theoretical Physics, Razmadze Mathematical
Institute}\\
\small\it Tbilisi, Georgia}
\date{}
\maketitle


%\end{center}

\begin{abstract}
We investigate character of physical degrees of freedom
for the models of 2d gravity and study it's dependence
on the topology of space-time manifold.
\end{abstract}


\section{Introduction}

The Einstein-Hilbert action in 2-dimensions
does not lead to dynamical equations for the metric tensor
$g_{ab}(X)$ ($X:=(x^0,x^1)$; ($a$ and $b= (0,1)$) \cite{D-N-F}.
This degeneracy is related to the fact that
the corresponding Lagrangian ${\cal {L}}_0 = \sqrt{-g}~R(g_{ab})$
is a `total derivative' and the action is expressed
only by the `surface terms'.

For the conformal gauge \cite{Jackiw}
\begin{equation}
 g_{ab}(X) = \exp{q(X)}\;\;\left( \begin{array}{cr}
 1 & 0 \\ 0 & -1
 \end{array} \right).
 \end{equation}
the scalar curvature $R(X)$ takes the form
\begin{equation}
R(X)=e^{- q (X)}(\partial ^2_1- \partial ^2_0)q (X).
\end{equation}
and the degeneracy of the Einstein-Hilbert Lagrangian becomes
apparent.

One can specify a model of 2d gravity by requiring that the
field $q(X)$ satisfies some  dynamical equation.
The Liouville field equation \cite{Liouville}
\begin{equation}
(\partial ^2_0 - \partial ^2_1)\;q(X)  + \Lambda e^{q(X)} = 0
\end{equation}
(where $\Lambda$ is a non-zero constant) is usually considered
as a model of 2-dimensional general relativity \cite{Jackiw}, \cite{Seiberg}.
The field $q(X)$
satisfying (1.3) leads to the manifold with a constant curvature $R(X)
=\Lambda\;\;$ (see (1.2)).
The Liouville equation
arises in a large variety of problems of
mathematical physics due to the conformal invariance
of this equation. General solution of (1.3) has the form \cite{Liouville}
\begin{equation}
q(x^+,x^-) = \log \frac{8A^{+~\prime}(x^+)A^{-~\prime}(x^-)}{|\Lambda |
\left [A^+(x^+)+\epsilon A^-(x^-)\right ]^2}
\end{equation}
where  $x^\pm=x^1\pm x^0$ are the light cone coordinates, $\epsilon$
is a sign of $\Lambda$ ($\epsilon =\Lambda/|\Lambda|$) and
$A^\pm$ are any functions with $A^{\pm~\prime}>0$.

The standard action for the scalar field in Minkowski space, which leads
to the Liouville equation (1.3)
has a non-covariant form from the point of view of general relativity.
To obtain the Liouville equation
(as the equation for the metric tensor in the conformal gauge)
from the covariant action, it is necessary
to introduce some additional auxiliary fields.
For a model with a reparametrization invariant
action and auxiliary fields, usually
it is difficult to guess what kind of physical
degrees of freedom the system has and what are its physical variables
\cite{Dirac, Henn-T}.
Construction of gauge invariant physical variables is the most
important problem of 4-dimensional Einstein general relativity as well.
In this note we study this problem for the simple models of 2d gravity,
using the conformal invariance of the Liouville theory.

\setcounter{equation}{0}


\section{2d gravity with the dilaton field}

We start with the action \cite{Seiberg}
\begin{equation}
S=\int d^2x \sqrt{-\hat g}~\left [
\frac{1}{2}\hat g^{ab}\partial_a\phi\partial_b\phi
 +\phi R(\hat g_{ab})-\Lambda e^\phi \right ],
\end {equation}
where $\Lambda$ is a `cosmological constant',
$\hat g:=det~\hat g_{ab}$,
$\hat g^{ab}$ is the inverse matrix to  the fiducial metric tensor
$\hat g_{ab}$ , $ R(\hat g_{ab})$ denotes the
corresponding scalar curvature  and $\phi$ is a scalar filed, which is
known as the Liouville mode, or the dilaton field.

The physical metric tensor $g_{ab}$
is related to $\hat g_{ab}$ by
\begin{equation}
g_{ab}=e^\phi\hat g_{ab}
\end {equation}
and the action (2.1) takes the form
\begin{equation}
S=\int d^2x \sqrt{-g}~\left [
- \frac{1}{2}g^{ab}\partial_a\phi\partial_b\phi
+\phi R(g_{ab}) -\Lambda  \right ],
\end {equation}
where  we have used the relation of scalar curvatures
in 2-dimensions for conformally related metrics (2.2)
\begin{equation}
R(\hat g_{ab})=e^\phi ~[\nabla^2\phi +R(g_{ab})]
\end {equation}
and we have neglected the surface term
$\partial_a(\phi \sqrt{-g}~g^{ab}\partial_b\phi)$, using the form of
the Beltrami operator $\nabla^2$ for the metric $g_{ab}$.

Action (2.3) defines the following `dynamical' equations
\begin{equation}
\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta \phi} =
\nabla^2\phi +R(g_{ab})=0,
\end {equation}
\begin{equation}
-\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{ab}} =
\partial_a\phi\partial_b\phi + g_{ab}\left
(-\frac{1}{2}g^{cd}\partial_c\phi\partial_d\phi -\Lambda\right )
-2(g_{ab} \nabla^2 -\nabla_a\nabla_b)\phi = 0.
\end {equation}

Taking trace of (2.6) and using (2.5) we get
that the dynamical system (2.3) describes a space-time manifold
with a constant curvature $R(g_{ab})=\Lambda$.

For the light cone coordinates $x^\pm=x^1\pm x^0$
the conformal metric (1.1) takes the form:
$$ g_{+,+} = g_{-,-}= 0, ~~~~~~~~~g_{+,-} =g_{-,+}= -\frac{1}{2}e^q$$
and from (2.5)-(2.6) we obtain
\begin{equation}
\partial_{+,-}^2 \varphi =0,~~~~~~~~~~
4\partial_{+,-}^2 q = {\Lambda}e^q,
\end{equation}
\begin{equation}
\left (\partial_+\varphi \right)^2 -2\partial^2_{+,+}\varphi
=\left (\partial_+ q \right)^2 -2\partial^2_{+,+} q,~~~~~~~
\left (\partial_-\varphi \right)^2 -2\partial^2_{-,-}\varphi
=\left (\partial_- q \right)^2 -2\partial^2_{-,-} q,
\end{equation}
where $ \varphi = q -\phi$.

As it was expected
the field $q(X)$ satisfies the Liouville equation (1.3), while
$\varphi (X)$ is a free field. According (2.8),
the traceless energy-momentum
tensors  of these two fields are equal to each other
\cite{Jackiw, J-P-P, J85}.

One can integrate equations (2.8) and express the Liouville field
$q(X)$ as a functional of the free field $\varphi (X)$.
After integration we get
\begin{equation}
q(x^+,x^-)=\log\frac{8F_+^\prime(x^+)F_-^\prime(x^-)}
{|\Lambda |\left [\alpha F_+(x^+)F_-(x^-) +\beta_+F_+(x^+) +
\beta_-F_-(x^-) +\gamma\right ]^2},
\end{equation}
where
\begin{equation}
F_{\pm}(x^\pm )=\int_{x_0^\pm}^{x^\pm}d\xi^\pm \exp \left
(\int_{x_0^\pm}^{x^\pm}d\eta^\pm\partial_\pm\varphi (\eta^\pm \right ),
\end{equation}
$(x_0^+,x_0^-)$ are the coordinates of a fixed point on the space-time,
and the integration constants $(\alpha  , \beta_\pm , \gamma)$ satisfy
the condition
\begin{equation}
\beta_+\beta_- -\alpha \gamma =\epsilon ~~~~~~~~~
(\epsilon := \Lambda /|\Lambda|) .
\end{equation}
The latter defines the $SL(2.R)$ group manifold,
and the freedom of the Liouville field $q(X)$
(for a given free field $\varphi (X)$)
is described only by three parameters of $SL(2.R)$ group.

The solution (2.9) for $\alpha \neq 0$ can be written in the form
\begin{equation}
q(x^+,x^-)=\log\frac{8B_+^\prime(x^+)B_-^\prime(x^-)}
{|\Lambda |\left [B_+(x^+)B_-(x^-) -\epsilon (\Lambda)\right ]^2},
~~~~~\mbox {where}~~~ B_\pm =\alpha F_\pm +\beta_\mp
\end{equation}
The case $\alpha =0$ is a degenerated one and the corresponding
relation between the Liouville and free fields arises for the Backlund
transformation \cite{Lamb} (see also \cite{J85}).

Note that (2.12) reduces to (1.4) after the substitution
$B_\pm \mapsto \pm A_\pm^{\pm 1}$.
Thus, (2.9) reproduces the general solution (1.4).

Integrating  equations (2.8) with respect to the free field,
one can derive the map from the Liouville
fields to the free fields
\begin{equation}
\varphi (x^+,x^-)=\log\frac{8G_+^\prime(x^+)G_-^\prime(x^-)}
{|\Lambda |[(G_+(x^+) +a_+)(G_-(x^-) +a_-)]^2} + b,
\end{equation}
which is also
characterized by three parameters ($a_\pm$, $b$) and the functions
$G_\pm$ have the form
\begin{equation}
G_{+}(x^+)=\int_{x_0^+}^{x^+}d\xi^+ e^{q(\xi^+, x_0^+)}, ~~~~~~
G_{-}(x^-)=\int_{x_0^-}^{x^-}d\xi^- e^{q(x_0^+, \xi^-)}.
\end{equation}

The conformal form of the metric tensor (1.1) fixes the gauge freedom
only up to the conformal transformations
\begin{equation}
x^\pm \mapsto y^\pm =
f^\pm(x^\pm)
\end{equation}
The corresponding transformation of the field $q$ is given by
\begin{equation}
q(x^+,x^-) \mapsto \tilde q(x^+,x^-)=q (f^+(x^+), f^-(x^-) )
+\log f^{+~\prime} (x^+) f^{-~\prime} (x^-).
\end{equation}

it is easy to check that the general solution (1.4) can be
obtained by the conformal transformations
of the function
\begin{equation}
q(x^+,x^-) = \log \frac{8}{|\Lambda |
(x^++\epsilon x^-)^2},
\end{equation}
which is a solution of (1.3).

Therefore, all Liouville fields (with fixed $\Lambda$) are related
to (2.17) by the conformal transformations (2.16), and hence,
by suitable choice of local coordinates one can
fix the field $q(X)$.
After fixing $q(X)$ the dynamical freedom for the free field $\varphi (X)$
is described only by the parameters $a_\pm , b$ (see (2.13)).
As a result, the full gage fixing leads
to the finite dimensional system
rather than the field theory.

It should be noted that all our construction of this section
(as well as the conjecture about the conformal form of
the metric tensor (1.1)) is valid only locally.

The maps (2.9) and (2.13) contain singularities when the corresponding
denominators are equal to zero. The domain of regularity
of $q$ and $\varphi$ fields can be considered as a patches of the global
space-time manifold.

For the global description of the system one has to glue the Liouville
(and free) fields given on different patches.
In this way we can conclude that the system (2.1) has a finite number
of physical degrees of freedom.
The number and dynamical character of these
parameters are related to the global properties of the
space-time manifold.

\setcounter{equation}{0}


\section{Hamiltonian reduction of 2d gravity}

In this section we analyze the same problem
using the Hamiltonian approach. This approach is non-covariant
and we have to separate the  space-time coordinates.
We assume that the space-time manifold $\cal M$ can be represented
in the form ${\cal {M}}= {\cal {T}}\otimes{\cal {X}}$,
where ${\cal {T}}$ is isomorphic to
${\bf {R}}^1$ and ${\cal {X}}$
is a one dimensional manifold.
The coordinates $x^0$ ($x^0\in {\cal {T}}$) and $x^1$ ($x^1\in
{\cal {X}}$) we interpret as the time and the space coordinates respectively.
We also use the notations $x^0:=t$, $x^1:=x$ and the corresponding
derivatives we denote by `dot'
(e.g. $\partial_t \phi :=\dot \phi$) and `prime'
(e.g. $\partial_x \phi :=\phi^\prime$). We assume
$g_{00} > 0$ and $g_{11} < 0$.

Due to the reparametrization invariance the Lagrangian of (2.3) is singular
and in the Hamiltonian description leads to a constrained
dynamical system \cite{Dirac, Henn-T}.
For the reduction to the physical (gauge invariant) variables
we use the Faddeev-Jackiw's method \cite{Fadd-J}
(see also \cite{C-J-K}).

One can check by direct calculation
that the above mentioned degeneracy of the Einstein-Hilbert Lagrangian
in 2-dimensions can be written in the form
\begin{equation}
\sqrt{-g}R(g_{ab}) =\partial_t\left[\frac{1}{G}
\left (\dot C-2B^\prime +B\frac{C^\prime}{C}\right )\right ]
+\partial_x\left[\frac{1}{{G}}
\left (A^\prime -B\frac{\dot C}{C}\right )\right ],
\end{equation}
where $A, B$ and $C$ are the components of the metric tensor $g_{ab}$
\begin{equation}
g_{ab}= \left( \begin{array}{cr}
 A & B \\ B & C
 \end{array} \right)
\end{equation}
and
\begin{equation}
G:=\sqrt{-g} = \sqrt{B^2-AC}.
\end{equation}
Then, up to the `surface terms', the Lagrangian of (2.3) takes the form
\begin{eqnarray}
\lefteqn{{\cal {L}}=}\nonumber\\
&&\textstyle \frac{1}{2 {G}}\left[C\dot\phi^2 \!-\!2B\dot\phi\phi^\prime
\!+\!\frac{B^2}{C}\phi^{\prime ~2}\! +\! 2\dot\phi\left(\!2 B^\prime
-\dot C- B\frac{C^\prime}{C}\!\right) +
 2\phi^\prime B\frac{\dot C}{C}-
 2\phi^\prime\left (\frac{B^2}{C}\right )^\prime ~\right ] \nonumber \\
&&\textstyle - {G}\left (\Lambda +\frac{2\phi^{\prime\prime}C-
\phi^\prime C^\prime}{C^2} +\frac{\phi^{\prime ~2}}{2C}\right ),
%\nonumber\\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray}
where we have excluded the variable $A$, using  relation  (3.3).
Since the Lagrangian (3.4) does not contain the time derivatives of
$G$ and $B$ `fields', these variables can play a role of Lagrange
multipliers.

According to \cite{Fadd-J} the equivalent system
can be described by the action
\begin{equation}
S=\int dx~dt \left [\Pi_\phi (\dot\phi -V_\phi) +\Pi_C (\dot C -V_C)
+ {\cal {L}}(\phi , V_\phi,...)\right ],
\end{equation}
where the $\Pi_\phi,\Pi_C, V_\phi$ and $V_C$ are auxiliary fields
and the function  ${\cal {L}}(\phi , V_\phi,...)$  is constructed
from Lagrangian (3.4), substituting the time derivatives $\dot \phi$
and $\dot C$ by the `velocities' $V_\phi$ and $V_C$  respectively.

One can easily eliminate the variables  $V_\phi$ and $V_C$ (taking
the corresponding variations) and we arrive at
\begin{equation}
S=\int dx~dt~ \left [\Pi_\phi \dot\phi  +\Pi_C \dot C +
\frac{ {G}}{C}~\Phi_0
+\frac{B}{C}\Phi_1\right ],
\end{equation}
with
\begin{eqnarray}
\Phi_0 = C\Pi_C\Pi_\phi +
\frac{1}{2}C^2\Pi_C^2 -\Lambda C -\frac{1}{2}\phi^{\prime ~2}
-2\phi^{\prime\prime} +
\frac{\phi^\prime C^\prime}{C},  \nonumber \\
\Phi_1 =
\Pi_C C^\prime - \Pi_\phi \phi^\prime +
2C\Pi_C^\prime .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray}
The `fields' $G$ and $B$  are indeed Lagrange multipliers
and the corresponding variations give two constraints
$\Phi_0 = 0$ and $\Phi_1 = 0$.
To analyze this constraint surface
it is convenient to introduce the new set of canonically
conjugated variables
\begin{eqnarray}
q:=\log (-C),~~~~~~\varphi :=\log (-C) -\phi ,~~~~~
{\mbox {(assuming $C<0$) }} \nonumber \\
p:=C\Pi_C +\Pi_\phi,~~~~~~~\pi := -\Pi_\phi .
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray}
Here we use the notations of Sec.II,
taking into account that
the field $\log (-C)$ in the conformal gauge (1.1) coincides with
the field $q$ of the previous section.
Note that the conformal gauge (1.1) corresponds to the
choice of Lagrange multipliers ($G=-C,  B=0$),
which  provides the Liouville and the free
equations for $q$ and $\varphi$ fields respectively.

For the new variables we find
\begin{equation}
\Phi_0 = U_0 -V_0 =0,~~~~~~~~\Phi_1 = U_1 + V_1 =0,
\end{equation}
where
\begin{eqnarray}
U_0
 = \frac{1}{2}(p^2 + q^{\prime ~2})+\Lambda e^q - 2q^{\prime\prime},
~~~~~ V_0
 = \frac{1}{2}(\pi^2 + \varphi^{\prime ~2})
  - 2\varphi^{\prime\prime} , \nonumber \\
U_1 = -pq^{\prime} + 2p^{\prime},~~~~~~~~~~~~~~~~~~~~~~~~
V_1 = -\pi \varphi^{\prime} + 2\pi^{\prime}~~~~~~~~~~~
\end{eqnarray}

Note that the functions $U_0$ and $U_1$  are
$T_{00}$ and $T_{01}$ components of the traceless energy-momentum tensor
of the Liouville theory
\cite{Jackiw, J-P-P, J85}, while $V_0$ and $V_1$ are the same
components for the free theory.

 The next step of the reduction procedure \cite{Fadd-J} is a calculation of
the canonical 1-form
\begin{equation}
\theta= \int_{\cal X} dx \left [p(x)~ dq(x) + \pi (x)~d\varphi (x)  \right ]
\end{equation}
on the constrained surface (3.9).
This surface can be represented in the form
$\Phi_\pm  := U_{\pm} - V_{\pm} = 0$, where
\begin{eqnarray}
U_{\pm}:=\frac{1}{2}\left (U_{0}\mp U_{1}\right ) =
\frac{1}{4}(q^{\prime} \pm p)^2
- (q^{\prime} \pm p)^\prime +\frac{1}{2}\Lambda e^q, \nonumber \\
V_{\pm}:= \frac{1}{2}\left (V_{0}\pm V_{1}\right )
= \frac{1}{4}(\varphi^{\prime} \mp \pi)^2
- (\varphi^{\prime} \mp \pi)^\prime .~~~~~~~~~~
\end{eqnarray}

The canonical 1-form (3.11) defines the canonical commutation relations
\begin{equation}
 \{ p(x), q(y)\}=\delta (x-y),~~~ \{\pi (x), \varphi (y) \} =\delta (x-y),
 \end{equation}
and  we obtain the Magri brackets \cite{Magri}
for the functions   $U_\pm$  and  $V_\pm$
\begin{eqnarray}
\{ U_\pm (x), U_\pm (y)\} =
\mp \left [\left (U_\pm (x) + U_\pm (y)\right )
\delta^\prime (x-y) -2 \delta^{\prime\prime\prime} (x-y)\right ],
\nonumber \\
\{V_\pm (x), V_\pm (y)\} = \pm \left [\left (V_\pm (x) + V_\pm (y)\right )
\delta^\prime (x-y) -2 \delta^{\prime\prime\prime} (x-y)\right ] ~~
\end{eqnarray}
and also
\begin{equation}
\{ U_+ (x), U_- (y)\} = 0,      ~~~~~~~
\{V_+ (x), V_- (y)\} = 0.
\end{equation}
As a result, the functions $\Phi_\pm$ are the first class constraints,
since they
satisfy  the commutation relations of the following Lie algebra
\begin{equation}
\{ \Phi_\pm (x), \Phi_\pm (y)\} =
\mp \left [\Phi_\pm (x) + \Phi_\pm (y)\right ]
\delta^\prime (x-y) .
\end{equation}

The two first class constraints $\Phi_\pm (x)$ usually
need two gauge fixing conditions
for each point $x$ \cite{Henn-T}. Since, at this stage, we have only
four fields $(p,q;\pi ,\varphi)$, it is expected that the
system has no physical
degrees of freedom  for the field variables. Then, after the full Hamiltonian
reduction, the system with only a finite number of degrees of freedom
can remain. Below we  demonstrate this fact explicitly using again
the method \cite{Fadd-J}.

Here it is useful to introduce the functions
\begin{equation}
g_\pm =
\left ( q^\prime \mp p -\varphi^\prime \mp \pi \right )
e^{-(q \pm \rho)/2},~~~~~~~{\mbox {with}}~~~ \rho (x)=
\int_{x_0}^{x}d\eta~ \pi (\eta),
\end{equation}
where $x_0$ is a fixed point from $\cal X$ ($x_0\in \cal X$).
Up to the total derivative, we obtain
\begin{equation}
p\dot q +\pi\dot \varphi  \simeq g_- \partial_t e^{(q - \rho)/2}
-g_+ \partial_t e^{(q + \rho)/2} \simeq e^{(q + \rho)/2}~\dot g_+
- e^{(q - \rho)/2}~\dot g_-.
\end{equation}

For the restriction of (3.18) on the constrained surface
(3.9)
we integrate these equations with respect to
$p$ and $q$ fields. After integration we get a result, which is similar to
(2.9):
\begin{eqnarray}
p=\frac{f_+^{\prime\prime}}{f_+^\prime}- \frac{f_-^{\prime\prime}}{f_-^\prime}
-2\frac{\alpha f_-f_+^\prime -\alpha f_+f_-^\prime +\beta_+f_+^\prime
-\beta_-f_-^\prime}{D},~~~~~~~~~~~~ \nonumber \\
 q =\log\frac{8f_+^\prime f_-^\prime}
{|\Lambda |D^2},~~~~~{\mbox {with}}~~~~~~
D:= \alpha f_+f_- +\beta_+f_+
+ \beta_-f_- +\gamma ,
\end{eqnarray}
where
\begin{equation}
f_{\pm}(x)=\int_{x_0}^{x}d\xi~ \exp \frac{1}{2}\left
[\varphi(\xi ) \mp \int_{x_0}^{\xi}d\eta~ \pi (\eta) \right ],
\end{equation}
and the integration constants
($\alpha ,\beta_\pm , \gamma$) satisfy the condition (2.11).

Using (3.17) - (3.20), we find that the restriction of the functions $g_\pm$
on the constrained surface (3.9) is given by
\begin{equation}
 g_\pm =-\pm \sqrt {2|\Lambda |}(\alpha f_\pm +\beta_\mp )
\end{equation}
and (3.18) takes the form
\begin{equation}
p\dot q +\pi\dot \varphi  \simeq \frac{4}{D}\left [
f_+^\prime(\alpha \dot f_-+
\dot\alpha f_-  +\dot\beta_+)-f_-^\prime(\alpha \dot f_+
+\dot \alpha f_++\dot\beta_- )\right ].
\end{equation}
After integration over $\cal X$,
the coefficients of $\dot \alpha$ and $\dot\beta_\pm$ give the corresponding
canonical conjugated variables. The only `field theory' term is
\begin{equation}
\frac{4\alpha (f_+^\prime \dot f_- -f_-^\prime \dot f_+)}{D(f_+,f_-)},
\end{equation}
but it again gives the `surface terms', since it is related
to the closed 2-form in $f_\pm$ space.

Thus, we get that after the full Hamiltonian reduction there are
no degrees of freedom for the field variables. In this way we
reproduce the result of Sec.II. To analyze the character of finite
dimensional system we need the global description of the model.
Such description can be related to the gluing of Liouville
fields with singularities (see, for example, \cite{J-P-P} and \cite{J85})
and we are going to discuss this approach elsewhere.

%\vspace{5mm}
\newpage

{\large\bf {Acknowledgments}}

One of the authors (G.J.) is very grateful to
the organizers of the seminar `q-98' for the invitation
and kind hospitality.

This work was supported by the grants from:
INTAS (96-0482), RFBR (96-01-00344) and
the Georgian Academy of Sciences.


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\end{document}



