gr-qc0303068.tex

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\begin{center}
\vskip3.7 cm

{\LARGE Brief comments on Jackiw-Teitelboim gravity}

\vskip0.3 cm

{\LARGE coupled to Liouville theory}

\vskip1.0 cm

{\large Gast\'{o}n E. Giribet}

\vskip1.0 cm

Instituto de Astronom\'{\i}a y F\'{\i}sica del Espacio, IAFE

C.C. 67 - Suc. 28, (1428) Buenos Aires, Argentina

and

Physics Department, University of Buenos Aires, FCEN-UBA

gaston@iafe.uba.ar

\vskip2.0 cm

\textbf{Abstract}
\end{center}

The Jackiw-Teitelboim gravity with non-vanishing cosmological constant
coupled to Liouville theory is considered as a non-critical string on $d$
dimensional flat space-time. It is discussed how the presence of
cosmological constant leads to consider additional constraints on the
parameters of the theory, even though the conformal anomaly is independent of
the cosmological constant. The constraints agree with the necessary conditions required
to ensure that the tachyon field turns out to be a primary prelogarithmic
operator within the context of the world-sheet conformal field theory. Thus,
the linearized tachyon field equation allows to impose the diagonal
condition for the interaction term.

We analyze the neutralization of the Liouville mode induced by the coupling
to the Jackiw-Teitelboim lagrangian. The standard free field prescription
leads to obtain explicit expressions for three-point functions for the case
of vanishing cosmological constant in terms of a product of Shapiro-Virasoro
integrals; this fact is a consequence of the mentioned neutralization effect.

\newpage

\vskip0.3 cm

\section{Introduction}

As it was early observed by Tseytlin, one of the principal motivations of
the understanding of the dependence of the effective action on the tachyon
field comes from the connection with studies of the two dimensional gravity 
\cite{tseytlin}. Consequently, in this note we study the linearized tachyon
field equation within the context of the description of the
Jackiw-Teitelboim gravity \cite{teitelboim} in terms of a non-linear $\sigma 
$-model describing the world-sheet dynamics of a non-critical string.

This class of analogies existing between two-dimensional models of
gravitation and non-critical string theories was previously discussed in the
literature. For instance, in reference \cite{chamseddine}, the action for
two-dimensional gravity coupled to matter fields was quantized in the
context of the conformal field theory method. In that reference, it was
argued that the quantization could be carried out without restricting the
dimension of the coupled matter. To do this, the author of \cite{chamseddine}
considered the case of vanishing cosmological constant $\Lambda =0$ argueing
that this case results suitable to take into account the general features of
the model since the conformal anomaly is independent of the value of $%
\Lambda $. Regarding this assertion, in this brief note we show how
additional restrictions really appear if a non-vanishing cosmological
constant $\Lambda $ is considered in the Jackiw-Teitelboim two dimensional
theory of gravitation, even though the central charge of the model actually
does not depend of the explicit value of $\Lambda $. This is achieved by the
interpretation of the model as a consistent non-critical string theory.

In reference \cite{mazzitelli}, the Jackiw-Teitelboim theory was also
interpreted as a noncritical string theory in presence of background fields.
Following this representation, we show that the additional restrictions on
the coupling constant and the parameters of the model arise by solving the
linearized tachyon field equation yielding from the mentioned interpretation
of the model as a non-critical string theory.

The central point in the argumentation is based in the fact that the
Jackiw-Teitelboim gravity coupled to Liouville theory and matter fields, as
mentioned, could be geometrically interpreted as a conformal field theory
describing the dynamic of a string on a $d$ dimensional Minkowski target
space-time. And, in the case of non-vanishing cosmological constant, the two
dimensional world-sheet theory is conformed by sum of two parts: the first,
a $d$-dimensional free field bosonic theory, and the second is an
interaction term, like a screening charge, which in the general case is
conformed by a logarithmic operator. Then, an additional constraint on the
parameters of the theory is required in order to impose the necessary
conditions to turn the interaction term into an integral of primary fields.
This constraint, which is absent if $\Lambda =0$ because it is automaticaly
satisfied in such case, can be translated into a restriction on the
functional form and value of the central charge of the theory.

It is important to mention that the coupling between two-dimensional
models
of gravity and Liouville theory was also studied in references \cite
{odintsov,mann}. 

In the following paragraphs we study the model within the context of the
conformal field theory formalism. We analyze the associated non-linear $%
\sigma $-model which can be recognized as the non-critical string theory on
the $d$-dimensional flat space-time coupled to linear dilaton background.
Secondly, we explicitly show how the expression for three-point correlation
functions in the case of vanishing cosmological constant can be written down
in terms of Virasoro-Shapiro integrals because of the neutralization of
Liouville mode induced by the constant curvature condition. This computation
follows the steps of the standard Coulomb gas prescription.

\section{Jackiw-Teitelboim gravity coupled to Liouville theory}

Let us begin by considering the Jackiw-Teitelboim gravitational action
coupled to Liouville theory and $d-2$ scalar fields representing the matter,
namely 
\begin{equation}
S_{eff}=S_{JT}+S_{L}+S_{M}+S_{ghost}  \label{sjt}
\end{equation}
being 
\begin{eqnarray*}
S_{JT} &=&\frac{b}{\pi }\int_{M}d^{2}x\sqrt{\hat{h}}\varphi \left( \tilde{R}+%
\tilde{\Lambda}\right) +\frac{\lambda }{4\pi }\int_{M}d^{2}x\sqrt{\hat{h}}%
\tilde{R} \\
S_{L} &=&\frac{1}{\pi }\int_{M}d^{2}x\sqrt{h}\left( ah^{\alpha \beta
}\partial _{\alpha }\sigma \partial _{\beta }\sigma +QR\sigma +\frac{\mu }{2}%
e^{\gamma \sigma }\right) \\
S_{M} &=&\frac{1}{4\pi }\int_{M}d^{2}x\sqrt{h}h^{\alpha \beta }\partial
_{\alpha }X^{i}\partial _{\beta }X^{j}\delta _{ij}
\end{eqnarray*}
where $a$ and $b$ are (positive) real parameters interpreted as coupling
constant of the Jackiw-Teitelboim lagrangian and the Liouville lagrangian
respectively; $\tilde{\Lambda}$ represents the target space cosmological
constant, while $\mu $ is the quoted Liouville cosmological constant; on his
turn, $R$ refers to the Ricci scalar corresponding to the two-dimensional
metric $h_{\alpha \beta }$ on the two-manifold $M$, while $\tilde{R}$
corresponds to the Ricci scalar constructed from the rescaled metric $\hat{h}%
_{\alpha \beta }$ \cite{david}, defined by 
\begin{equation}
\hat{h}_{\alpha \beta }=e^{\gamma \sigma }h_{\alpha \beta }  \label{gauge}
\end{equation}
where $(\alpha ,\beta )\in \{0,1\}.$

The field $\varphi $ is the Jackiw-Teitelboim scalar field, while $\sigma $
is the Liouville scalar field; there exist other $d-2$ scalar fields which
represent the matter, being denoted by $X^{i},$ $i\in \{2,3...d-1\}$. The
Hilbert term also appears in the action (\ref{sjt}) contributing with the
Euler characteristic number $\chi (h)$; and where $\lambda $ is a real
coupling constant which can be related with the dilaton zero-mode. On the
other hand, the ghost action $S_{ghost}$ needs to be included; it can be
expressed in terms of the $b$-$c$ system as 
\begin{equation}
S_{ghost}=\frac{1}{2\pi }\int_{M}d^{2}x\sqrt{h}h^{\alpha \beta }c^{\rho
}\nabla _{\alpha }b_{\beta \rho }
\end{equation}
Notice that the Jackiw-Teitelboim field $\varphi $ acts as a Lagrange
multiplier, which can be integrated out introducing the constant curvature
constraint 
\begin{equation*}
\tilde{R}+\tilde{\Lambda}=0.
\end{equation*}
As introduced before, this class of two-dimensional gravitational action
coupled to Liouville theory was treated in different contexts. Indeed, an
interesting analysis of its perturbative aspects was effectuated in \cite
{mazzitelli,mazzitelli2}; in reference \cite{chamseddine} this action was
quantized by using the path integral approach and conformal field methods
too; it was also studied in the context of $W_{3}$ gravity \cite{mohammedi},
and a careful analysis of its classical and quantum features was presented
in \cite{burwick}. In \cite{chamseddine,chamseddine2} it was shown that the
Teitelboim-Jackiw lagrangian appears as the metric version of a topological
construction based on the $SO(1,2)$ group. This realization relies in the
philosophy of treating the two-dimensional models of gravity as a kind of
projective reduction of three-dimensional Chern-Simon theory. See also \cite
{shapiro} for an interesting discussion.

Here, we shall consider the classical model defined by action (\ref{sjt}) as
the starting point; with this purpose, let us define the following variables 
\begin{eqnarray}
\sigma &=&\frac{\kappa }{\gamma }\left( X^{1}+X^{0}\right)  \label{var1} \\
\varphi &=&\frac{\kappa }{\gamma }\left( X^{1}+\xi X^{0}\right)  \label{var2}
\end{eqnarray}
where $\kappa =\frac{\gamma }{2\sqrt{a+\gamma b}}$ and $\xi =-1-\frac{2a}{%
\gamma b}$. These variables are well defined in terms of the original
coordinates $\varphi $ and $\sigma $ iif $a+b\gamma \neq 0$. In terms of the
new coordinates, the above action can be written as follows 
\begin{equation}
S_{eff}=S_{0}+S_{I}+\lambda \chi (h)  \label{sx}
\end{equation}
being 
\begin{eqnarray}
S_{0} &=&\frac{1}{4\pi }\int d^{2}z\left( \eta _{\mu \nu }\partial X^{\mu }%
\bar{\partial}X^{\nu }+RQ_{0}X^{0}+RQ_{1}X^{1}\right) ,  \label{sx1} \\
S_{I} &=&\frac{1}{4\pi }\int d^{2}z\left( \xi \Lambda X^{0}+\Lambda
X^{1}+2\mu \right) e^{\kappa (X^{0}+X^{1})}  \label{interaction}
\end{eqnarray}
Here, $\eta _{\mu \nu }$ denotes the $d$-dimensional Minkowski metric and
the following notation was introduced 
\begin{eqnarray*}
Q_{0} &=&2\frac{Q-b-2a/\gamma }{\sqrt{a+\gamma b}} \\
Q_{1} &=&2\frac{Q+b}{\sqrt{a+\gamma b}}
\end{eqnarray*}
In (\ref{sx1}) and (\ref{interaction}) $z$ and $\bar{z}$ denote the usual
world-sheet complex coordinates, being $\partial =\frac{\partial }{\partial z%
}$ and $\bar{\partial}=\frac{\partial }{\partial \bar{z}}$. Notice that the
cosmological constant $\Lambda $ was also rescaled by a constant factor,
namely $\Lambda =\frac{2b}{\sqrt{a+\gamma b}}\tilde{\Lambda}$. We are
considering greek indices which cover the $d$-dimensional space-time, $(\mu
,\nu )\in \{0,1,...d-1\}$.

Indeed, it would be also interesting to note that the constant curvature
constraint can be derived by combining the Euler-Lagrange equation for $%
X^{0} $ and $X^{1}$ fields, obtaining 
\begin{equation*}
e^{-\gamma \sigma }\left( R-\gamma \partial ^{2}\sigma \right) +\tilde{%
\Lambda}=0,
\end{equation*}
which can be written in the usual form $\tilde{R}+\tilde{\Lambda}=0$ if the
relation (\ref{gauge}) is taken into account.

Thus, we have written the two-dimensional action (\ref{sjt}) in terms of a
theory of auto-interacting scalar fields coupled to classical gravity in the
conformal gauge (\ref{gauge}); it will be a useful picture to work out the
features of the model.

\vskip0.3 cm

\section{The conformal field theory}

The above free action $S_{0}$ defines a two-dimensional conformal quantum
field theory, which could be analyzed by the usual techniques of 2D CFT.
Indeed, the holomorphic component of the stress-tensor corresponding to this
model is given by 
\begin{equation}
T(z)=-\frac{1}{2}\eta _{\mu \nu }\partial X^{\mu }\partial X^{\nu }+\frac{%
Q_{0}}{2}\partial ^{2}X^{0}+\frac{Q_{1}}{2}\partial ^{2}X^{1}  \label{t}
\end{equation}
with free field bosonic correlators, 
\begin{equation}
\left\langle X^{\mu }(z)X^{\nu }(w)\right\rangle =-\eta ^{\mu \nu }\ln (z-w)
\label{correlators}
\end{equation}
Thus, it is easy to show that the following operator product expansion holds 
\begin{equation*}
T(z)T(w)=\frac{\frac{d}{2}+\frac{3}{2}(Q_{1}^{2}-Q_{0}^{2})}{(z-w)^{4}}+%
\frac{T(w)}{(z-w)^{2}}+\frac{\partial T(w)}{(z-w)}+...
\end{equation*}
From this expansion, we obtain that the central charge of the model is given
by 
\begin{equation}
c=d+48\left( \frac{Q}{\gamma }-\frac{a}{\gamma ^{2}}\right)  \label{c}
\end{equation}
which does not include the ghost contribution $c_{ghost}=-26$.

On the other hand, the interaction term $S_{I}$ is not an integral of
primary fields as it occurs in the usual treatement of two-dimensional
models with conformal invariance. In fact, the operator 
\begin{equation}
\mathcal{T}_{\Lambda }(z)=\left( \xi \Lambda X^{0}(z)+\Lambda X^{1}(z)+2\mu
\right) e^{\kappa X^{0}(z)+\kappa X^{1}(z)}  \label{taq}
\end{equation}
turns out to be a logarithmic operator as it can be shown by the emergence
of non-diagonal terms in the operator product expansion with the
stress-tensor of the free field theory, namely 
\begin{equation}
T(z)\mathcal{T}_{\Lambda }(w)=h\frac{\mathcal{T}_{\Lambda }(w)}{(z-w)^{2}}%
+\varepsilon \frac{\mathcal{T}_{\Lambda =0}(w)}{(z-w)^{2}}+\frac{\partial 
\mathcal{T}_{\Lambda }(w)}{(z-w)}+...  \label{ope1}
\end{equation}
\begin{equation}
T(z)\mathcal{T}_{\Lambda =0}(w)=h\frac{\mathcal{T}_{\Lambda =0}(w)}{(z-w)^{2}%
}+\frac{\partial \mathcal{T}_{\Lambda =0}(w)}{(z-w)}+...  \label{ope2}
\end{equation}
being 
\begin{equation*}
\mathcal{T}_{\Lambda =0}(z)=2\mu e^{\kappa (X^{0}+X^{1})}
\end{equation*}
which is actually a primary field with conformal dimension $h$; being 
\begin{eqnarray}
h &=&-\frac{\kappa }{2}\left( Q_{0}-Q_{1}\right)  \label{peso} \\
\varepsilon &=&\frac{\Lambda }{2\mu }\left( \kappa \left( \xi -1\right) -%
\frac{\xi }{2}Q_{0}+\frac{1}{2}Q_{1}\right)  \label{log}
\end{eqnarray}
Then, in order to realize the theory as a 2D CFT representing a non-critical
string theory, we have to impose the condition $h=1$, which is directly
satisfied by the original definition of the Liouville cosmological term,
being proportional to $e^{\kappa (X^{0}+X^{1})}$.

An question could appear in this point:\ it could be possible to ask whether
the constraints $h=1$ and $\varepsilon =0$ represent additional restrictions
which need to be imposed on the theory indeed; in fact, these constraints
are necessary conditions required by the stringy interpretation of the model
as a consistent non-critical theory. The point is that, since our further
analysis involving the $\beta $-function equations will permit to recover
the equations (\ref{peso}) and (\ref{log}) as necessary conditions for
conformal invariance of the model, we see these equations as the necessary
constraints to be impossed in order to obtain a quantum conformal field
theory. In the next section, we will remark this point with more detail.

On the other hand, in order to make the charge $\mathcal{T}_{\Lambda }(w)$ a
primary field, we also need to impose the constraint $\varepsilon =0$ which
turns the operator (\ref{taq}) into a diagonal (prelogarithmic) operator.
Thus, we obtain the following condition 
\begin{equation}
2\Lambda \kappa \left( \xi -1\right) =\Lambda \left( \xi Q_{0}-Q_{1}\right)
\end{equation}
which can be written in terms of the original parameters as follows 
\begin{equation}
\tilde{\Lambda}\left( a+b\gamma \right) \left( \gamma ^{2}-2Q\gamma
+4a\right) =0  \label{a}
\end{equation}
This equation, moreover the trivial case $\Lambda =0$, has solutions of the
form 
\begin{equation}
\gamma =Q\pm \sqrt{Q^{2}-4a}  \label{result}
\end{equation}
This is an additional restriction on the value of central charge, which
would permit to realize the interaction term in the context of the CFT
method. Note that $\xi =1$ also solves the equation (\ref{a}); however, it
is not an allowed value since in such case the transformation (\ref{var1})-(%
\ref{var2}) becomes singular and the equivalence between the original
gravity action and the linear dilaton background (\ref{sx}) breakes down.

The introduction of the contraint (\ref{result}) is not a minor detail;
notice that if it is replaced in (\ref{c}) the central charge of the model
becomes 
\begin{equation}
c=d+24+\frac{48a}{\gamma ^{2}}  \label{cc}
\end{equation}
which can not be set to $26$ for the case $d\geq 2$, $a>0$. On the other
hand, this constraint excludes particular cases like $Q=a$, $\gamma =2$,
which were considered in the $\Lambda =0$ case as examples of a critical
string theory \cite{mazzitelli}. On his turn, the $Q=\frac{3}{2}$ model
receives particular interest in the context of studies of $W_{3}$ gravity 
\cite{mohammedi}. Notice that condition (\ref{result}) is not present in the 
$\Lambda =0$ case since the interaction term becomes primary in this point.
Of course, (\ref{result}) can be obtained directly by using the original
variables, for instance by solving the linearized tachyon equation in the
case of non-vanishing $\Lambda $ in the coordinates adopted in reference 
\cite{mazzitelli}. On his turn, in terms of the nomenclature of reference 
\cite{chamseddine} this condition can not be satisfied because of the
particular fixation of the value of the background charge $Q$.

Then, we are compeled to state that\textit{\ }an additional condition
impossed on the central charge arises if cosmological constant is included
in the Jackiw-Teitelboim model coupled to Liouville theory. And, this new
restriction is independet of the explicit value of $\Lambda $.

This proposal needs to be contrasted with the \textit{healing} proposed in
reference \cite{chamseddine2}, where, instead imposing condition (\ref
{result}), it was suggested to change the tachyon field by adding a new term
proportional to $\sigma e^{\gamma \sigma }$, which could appear in certain
unknown step in the renormalization procedure applied to the non-local
Polyakov action.

It would be important to mention that similar operators $\sim \varphi
e^{\gamma \sigma }$ appear in other treatements of two-dimensional gravity;
for instance, see \cite{andreev}.

\bigskip \vskip0.3 cm

\section{Linearized tachyon field equation}

On the other hand, we can analyze the above restriction on the central
charge within the context of the $\beta $-function equations for this model.
In fact, as it was observed early \cite{chamseddine,mazzitelli}, the whole
action $S_{0}+S_{I}$ can be interpreted as a world-sheet string action
formulated on Minkowski target space-time in presence of non-trivial
background fields $\Phi (X)$ (the dilaton) and $\mathcal{T}(X)$ (the
tachyon), whose classical configurations are given by 
\begin{eqnarray}
\Phi (X) &=&\frac{\lambda }{2}+\frac{1}{2}Q_{\mu }X^{\mu }  \label{solu} \\
\mathcal{T}_{\Lambda }(X) &=&\left( \Lambda \zeta _{\mu }X^{\mu }+2\mu
\right) e^{\kappa _{\nu }X^{\nu }}  \label{solution}
\end{eqnarray}
Notice that the functional form (\ref{solution}) is not a particular case
within the context of the theory of gravity defined by action $S_{eff}$;
moreover, this form is a more general \textit{ansatz} which includes the
particular case of the tachyonic configuration arising in (\ref{interaction}%
).

The $\beta $-function equations $\beta ^{\Phi }=\beta _{\mu \nu }^{G}=\beta
_{\mu \nu }^{B}=0$ are immediately satisfied at the lowest order in $\alpha
^{\prime }$ by the condition $c=26$. However, the linearized tachyon field
equation $\beta ^{\mathcal{T}}=0$ needs to be imposed by means of additional
functional relations between the parameters, namely 
\begin{equation}
\beta ^{\mathcal{T}}=-\frac{1}{2}\nabla _{\mu }\nabla ^{\mu }\mathcal{T}%
_{\Lambda }+\nabla _{\mu }\Phi \nabla ^{\mu }\mathcal{T}_{\Lambda }-\frac{2}{%
\alpha ^{\prime }}\mathcal{T}_{\Lambda }+...=0  \label{beta}
\end{equation}
where we neglected highest orders in the $\alpha ^{\prime }$ expansion and,
then, we can adopt the convention $\alpha ^{\prime }=2$ (note that our
convention will be consequent with $\alpha ^{\prime }=2$. For instance, see
eqs. (2.5.1a), (3.7.6) and (9.9.2) of reference \cite{polchinski}. Similarly
to the convention adopted in \cite{mazzitelli}, our notation is congruent
with the one used in \cite{tseytlin}).

The configuration (\ref{solution}) represents a small tachyon, since we are
considering perturbation theory in $\alpha ^{\prime }$. The leading terms in
the tachyon effective action are linear in $\mathcal{T}_{\Lambda }$;
neglecting the higher powers of tachyonic field is consequent with dealing
with this field as it would not be perturbating the background as a source
of geometry.

Then, it is necessary to solve the linearized equation 
\begin{equation}
-\partial _{\mu }\partial ^{\mu }\mathcal{T}_{\Lambda }+Q_{\mu }\partial
^{\mu }\mathcal{T}_{\Lambda }-2\mathcal{T}_{\Lambda }=0
\end{equation}
which admits a solution of the form (\ref{solution}) if the following
constraints hold 
\begin{eqnarray}
\kappa _{\mu }(Q^{\mu }-\kappa ^{\mu }) &=&2  \label{w1} \\
\zeta _{\mu }\left( Q^{\mu }-2\kappa ^{\mu }\right) &=&0  \label{w2}
\end{eqnarray}
Clearly, from (\ref{w1}) we can see that in the absence of the linear
dilaton field ($Q_{\mu }=0$) the mass-shell condition for the tachyon field $%
\kappa _{\mu }\kappa ^{\mu }=-2$ is recovered. Here, we have $%
c_{gravity}=2+3Q_{\mu }Q^{\mu }$.

In our case, we are dealing with the particular configuration given by 
\begin{eqnarray*}
\zeta _{\mu }\zeta ^{\mu } &=&\Lambda ^{2}\frac{4a}{b\gamma }\left( \frac{4a%
}{b\gamma }+1\right) \\
Q_{\mu }Q^{\mu } &=&\frac{16}{\gamma }\left( Q-\frac{a}{\gamma }\right) \\
\kappa _{\mu }\kappa ^{\mu } &=&0
\end{eqnarray*}
$i.e.$ in the coordinates defined above we have $\zeta _{\mu }=\Lambda (\xi
,1,0,0...0)$, $Q_{\mu }=(Q_{0},Q_{1},0,0,...0)$ and $\kappa ^{\mu }=(-\kappa
,\kappa ,0,0,...0).$ Then we can note that, in terms of the original
parameters, the constraint (\ref{w2}) demands 
\begin{equation}
\tilde{\Lambda}\left( a+\gamma b\right) \left( \gamma ^{2}-2Q\gamma
+4a\right) =0,  \label{tuco}
\end{equation}
which exactly agrees with the condition (\ref{a}), which we obtained above
by considerations about the prelogarithmic condition $\varepsilon =0$ for
the interaction term in the CFT action. Of course, the agreement between (%
\ref{tuco}) and (\ref{a}) has not to be a surprise since the linearized
tachyon field equation manifestly represents the requirements of conformal
invariance for the interaction term \cite{tseytlin,becker,witten}.

Thus, we find that the $\beta $-function equation for the tachyon field, $%
\beta ^{\mathcal{T}}=0,$ allows to obtain the prelogarithmic condition $%
\varepsilon =0$ for the operator $\mathcal{T}_{\Lambda }$ in the world-sheet
conformal field theory$.$ Moreover, the logarithmic part of the operator
product expansion $T(z)\mathcal{T}_{\Lambda }(w)$ is parametrized by the
values ($Q,\gamma $), and these diagonalize the $L_{0}$ operator in the
critical point satisfying (\ref{result}). Notice that this argumentation is
independent of the explicit value of the non-vanishing cosmological constant 
$\Lambda $.

Hence, a new restriction would appear if cosmological constant is
considered. In the general case, it is not possible to interpret the theory
as a critical string theory as it was done in the $\Lambda =0$\ case, which
would appear as a disctintive particular case.

As in the case of the exponential functional form of $\mathcal{T}_{\Lambda
=0}$, the above solution for the tachyon field in the linear dilaton
background solves the lowest orders in $\alpha ^{\prime }$ of $\beta $%
-function equations only for the lowest order in powers of the field $%
\mathcal{T}_{\Lambda }$.

The solution (\ref{solution}) is a simple generalization of the
David-Distler-Kawai background \cite{david,distler}. From (\ref{c}), (\ref
{w1}) and (\ref{w2}), we can write

\begin{equation}
\kappa _{\mu }\kappa ^{\mu }-\kappa _{\mu }Q^{\mu }+\frac{1}{4}Q_{\mu
}Q^{\mu }+\frac{d-2}{12}=0
\end{equation}
With the intention to analyze the general aspects of the solutions of such
equation of motion we can see that in the case of $\kappa ^{\mu }$ being a
time-like vector of the particular form $\kappa ^{\mu }=(\kappa ,0,0,...0)$,
the solution for the linearized tachyon field takes the form $\kappa ^{\mu
}=-\frac{1}{2}\left( Q_{0}\pm \sqrt{Q_{1}^{2}+\frac{d-2}{6}}\right) \delta
^{0\mu },$ whose qualitative behaviour depends on the relative value of $d$
and $Q_{1\text{.}}$ On his turn, in the case of $\kappa ^{\mu }$ being a
space-like vector, $v.g.$ the case $\kappa ^{\mu }=(0,\kappa ,0,...0)$, we
have $\kappa ^{\mu }=\frac{1}{2}\left( Q_{1}\pm \sqrt{Q_{0}^{2}-\frac{d-2}{6}%
}\right) \delta ^{1\mu }.$ The physical aspects of such solutions were
extensively studied within the context of non-critical string theory where
it was discussed how the tachyon field acts as a barrier of potential
competeing with the action of the linear dilaton background, see for
instance \cite{david,polchinski,distler,becker,witten}.

Moreover, the case we are interested on is the analogous to the
Jackiw-Teitelboim model coupled to Liouville theory, which corresponds to
the light-like case $\kappa _{\mu }\kappa ^{\mu }=0$. In this case, the
solutions fall into the region described by the constraint 
\begin{equation}
\kappa _{\mu }Q^{\mu }=2  \label{n29}
\end{equation}
which is precisely the mass shell condition $h=1$ for the tachyon field in
this dilatonic configuration. Here, the null condition $\kappa _{\mu }\kappa
^{\mu }=0$ appears as the manifestation of the neutralization effect of
Liouville mode induced by the constant curvature constraint, which was
studied in reference \cite{chamseddine} within the context of the path
integral approach. The mentioned light-like condition (\ref{n29}) can be
seen as a change in the propagator of Liouville mode \cite
{chamseddine,mazzitelli,david} and it leads to the linear functional form
for the conformal dimension of the operators with the form $\sim e^{\kappa
(X^{0}+X^{1})}$, instead the quadratic dependence which leads to the
definition of two-diferent screening charges, Weyl-like reflection relations
and the existence of conjugated pictures in the case of similar conformal
models. Let us postpone some comments about this for the next section.

Here, the intention here was to explore the general features of including a
non-vanishing cosmological constant $\Lambda $ in the context of the stringy
interpretation of the Jackiw-Teitelboim gravity coupled to Liouville theory.

\vskip0.3 cm

\section{Three-point correlation functions}

Now, by using this CFT description of the model, it is feasible to follow
the steps of the Coulomb gas prescription with the purpose to write down
explicit expressions for the integral representation of correlation
functions in the theory defined by the action (\ref{sx}). For instance, in
reference \cite{chamseddine} it was pointed out the importance of extending
the CFT\ analysis with the purpose to represent correlation functions in
this model; on the other hand, in \cite{dotsenko}, a similar treatement of
three-point functions was presented for the case of minimal models coupled
to two-dimensional gravity. Indeed, it is possible to compute the
three-point functions in the Jackiw-Teitelboim gravity coupled to Liouville
theory for the case $\Lambda =0$ by using the quoted free field realization.
In fact, the $N$-point correlation functions of primary fields $\Phi
_{k}(z)=\exp ik_{\mu }X^{\mu }(z)$ on the sphere are described by the
following form 
\begin{equation*}
\left\langle \prod_{i=1}^{N}\Phi _{\alpha ^{(i)},\beta
^{(i)},k^{(i)}}(z_{i})\right\rangle _{S_{eff}}=e^{-2\lambda }\mu ^{t}\Gamma
(-t)\prod_{r=1}^{t}\int d^{2}w_{r}\left\langle \prod_{i=1}^{N}\Phi _{\alpha
^{(i)},\beta ^{(i)},k^{(i)}}(z_{i})\prod_{r=1}^{t}\mathcal{T}_{\Lambda
=0}(w_{r})\right\rangle _{S_{0}}
\end{equation*}
where standard formulae of the path intergal realization is involved (see,
for instance \cite{becker,goulian,francesco,dotsenko2,dotsenko}) and we have
defined $\alpha =ik_{0}$, $\beta =ik_{1}$ . By using the projective
invariance, three different inserting points of vertex operators can be
fixed on the world-sheet in order to cancellate the volume of the $SL(2,C)$
group, $v.g.$ being $\left( z_{1},z_{2},z_{3}\right) =\left( 0,1,\infty
\right) .$ Thus, in the usual way, we can write the $N$-point correlation
functions $G^{N}(z_{1},z_{2},...z_{N})$ as 
\begin{equation*}
G_{k_{\mu _{1}}^{(1)},k_{\mu _{2}}^{(2)},...k_{\mu
_{N}}^{(N)}}^{N}(z_{1},z_{2},...z_{N})=\left\langle \prod_{i=1}^{N}\Phi
_{\alpha ^{(i)},\beta ^{(i)},k^{(i)}}(z_{i})\right\rangle _{S_{eff}}=
\end{equation*}
\begin{eqnarray*}
&=&\frac{(2\pi )^{d-1}(2\mu )^{t}e^{-2\lambda }}{\kappa }\Gamma
(-t)\prod_{i>j}^{N,N-1}\left| z_{i}-z_{j}\right| ^{2\left(
k^{(i)}k^{(j)}+\alpha ^{(i)}\alpha ^{(j)}-\beta ^{(i)}\beta ^{(j)}\right)
}\times \\
&&\times \prod_{r=1}^{t}\int d^{2}w_{r}\prod_{r=1}^{t}\prod_{i=1}^{N}\left|
z_{i}-w_{r}\right| ^{2\kappa (\alpha ^{(i)}-\beta ^{(i)})}\times \\
&&\delta \left( \sum_{i=1}^{N}(\alpha ^{(i)}-\beta
^{(i)})+Q_{0}-Q_{1}\right) \delta ^{(d-2)}\left( \sum_{i=1}^{N}k^{(i)}\right)
\end{eqnarray*}
where we denoted $t=\frac{1}{2\kappa }\left(
Q_{0}+Q_{1}-\sum_{i=1}^{N}(\alpha ^{(i)}+\beta ^{(i)})\right) $. The delta
functions and the global factor $\frac{(2\pi )^{d-1}(2\mu )^{t}}{\kappa }$
yield from the integration over the zero-modes of $X^{\mu }$ fields, while
the factor $e^{-2\lambda }$ stands by the Einstein-Hilbert term in the genus
zero contribution of the functional measure $e^{-S_{eff}}$. In order to make
the correlation functions non-vanishing we get$\ \sum_{i=1}^{N}\left( \alpha
^{(i)}-\beta ^{(i)}\right) =Q_{0}-Q_{1}$. Thus, we can write the
conservation laws as 
\begin{equation}
\sum_{i=1}^{N}\alpha ^{(i)}\pm \sum_{i=1}^{N}\beta ^{(i)}=\left(
Q_{0}-t\kappa \right) \pm (Q_{1}-t\kappa )  \label{numeralon}
\end{equation}
with $t$ being a non-negative integer.

Notice that mixed terms of the form $\left| w_{r}-w_{r^{\prime }}\right|
^{2\delta }$ do not exist because of the particular configuration $\delta
=-\kappa _{\mu }\kappa ^{\mu }=0$. This fact makes that the integration over
the different screening insertions turns out to be simplified leading to
obtain a direct product of separated integrals. This is a manifestation of
the neutralization effect of the Liouville mode pointed out before; $i.e.$
in the case of three-point functions, the Dotsenko-Fateev type integrals 
\cite{goulian,francesco,dotsenko2,dotsenko} become a product of
Virasoro-Shapiro integrals because of the condition $\kappa _{\mu }\kappa
^{\mu }=0$.

Then, we can explicitly compute the three-point correlators as follows 
\begin{equation*}
G_{k_{\mu _{1}}^{(1)},k_{\mu _{2}}^{(2)},k_{\mu _{3}}^{(3)}}^{3}(0,1,\infty
)=\left\langle \Phi _{\alpha ^{(1)},\beta ^{(1)},k^{(1)}}(0)\Phi _{\alpha
^{(2)},\beta ^{(2)},k^{(2)}}(1)\Phi _{\alpha ^{(3)},\beta
^{(3)},k^{(3)}}(\infty )\right\rangle =
\end{equation*}
\begin{eqnarray*}
&=&\frac{(2\pi )^{d-1}\left( 2\mu \right) ^{t}e^{-2\lambda }}{\kappa }\Gamma
(-t)\prod_{r=1}^{t}\int d^{2}w_{r}\prod_{r=1}^{t}\left| w_{r}\right|
^{2\kappa (\alpha ^{(1)}-\beta ^{(1)})}\left| 1-w_{r}\right| ^{2\kappa
(\alpha ^{(2)}-\beta ^{(2)})}\times \\
&&\times \delta \left( \sum_{i=1}^{3}(\alpha ^{(i)}-\beta
^{(i)})+Q_{0}-Q_{1}\right) \delta ^{(d-2)}\left( \sum_{i=1}^{3}k^{(i)}\right)
\end{eqnarray*}
obtaining

\begin{equation*}
G_{k_{\mu _{1}}^{(1)},k_{\mu _{2}}^{(2)},k_{\mu _{3}}^{(3)}}^{3}=\frac{(2\pi
)^{d-1}e^{-2\lambda }}{\kappa }\Gamma \left( \frac{1}{\kappa }\left( \alpha
^{(1)}+\alpha ^{(2)}+\alpha ^{(3)}-Q_{0}\right) \right) \times
\end{equation*}
\begin{equation*}
\times \left( 2\pi \mu \frac{\gamma (1+\kappa (\alpha ^{(1)}-\beta
^{(1)}))\gamma (1+\kappa (\alpha ^{(2)}-\beta ^{(2)}))}{\gamma (2+\kappa
(\alpha ^{(1)}+\alpha ^{(2)}-\beta ^{(1)}-\beta ^{(2)}))}\right) ^{\frac{1}{%
\kappa }\left( Q_{0}-\sum_{i=1}^{3}\alpha ^{(i)}\right) }
\end{equation*}
where $\gamma (x)=\frac{\Gamma (x)}{\Gamma (1-x)}$. By considering the
condition $Q_{1}-Q_{0}=\frac{2}{\kappa }$ and noting that $\gamma
^{-1}(x)=\gamma (1-x),$ we can write $\prod_{i=1}^{3}\gamma ^{-1}(\kappa
(\beta ^{(i)}-\alpha ^{(i)}))=\frac{1}{2}\prod_{i\neq j}\gamma (\kappa
(\beta ^{(i)}+\beta ^{(j)}-\alpha ^{(i)}-\alpha ^{(j)})-1)$; then, the above
expression can be rewritten as

\begin{equation}
G_{k_{\mu _{1}}^{(1)},k_{\mu _{2}}^{(2)},k_{\mu _{3}}^{(3)}}^{3}=\frac{(2\pi
)^{d-1}e^{-2\lambda }}{\kappa }\Gamma \left( \sum_{i=1}^{3}\frac{\alpha
^{(i)}}{\kappa }-\frac{Q_{0}}{\kappa }\right) \left( 2\pi \mu \prod_{i=1}^{3}%
\frac{\Gamma \left( 1+\kappa (\alpha ^{(i)}-\beta ^{(i)})\right) }{\Gamma
\left( \kappa (\beta ^{(i)}-\alpha ^{(i)})\right) }\right) ^{\frac{1}{\kappa 
}\left( Q_{0}-\sum_{i=1}^{3}\alpha ^{(i)}\right) }  \label{yupi}
\end{equation}
satisfying (\ref{numeralon}). These would be the general form for
three-point functions if the standard Coulomb gas prescription is adopted
for computing the correlators.

Let us make a last comment about the role of $\varphi $ field. Indeed, it is
usual to consider an additional dynamical term for $\varphi $ (see for
instance \cite{mazzitelli}), namely 
\begin{equation}
S_{\varphi }=\frac{g}{4\pi }\int d^{2}z\sqrt{h}h^{\alpha \beta }\partial
_{\alpha }\varphi \partial _{\beta }\varphi  \label{din}
\end{equation}
Notice that if such a term is included, the case $\xi \rightarrow \xi =-%
\frac{2\gamma a+b}{2g+b}$ needs to be considered in (\ref{var2}) in order to
diagonalize the metric of target space. In this case, we would obtain a
non-vanishing correlator $\left\langle \sigma (z)\sigma (w)\right\rangle $
for the Liouville mode ($i.e.$ $\kappa _{\mu }\kappa ^{\mu }\neq 0$). Then,
the inclusion of a dynamical term for $\varphi $ field restores the
influence of the $\sigma $ field, which became harmless because of the
presence of the constant curvature constraint. On his turn, if (\ref{din})
is included, the three-point correlators are given in terms of
Dotsenko-Fateev type integrals $\prod_{r=1}^{t}\int
d^{2}w_{r}\prod_{r=1}^{t}\left| w_{r}\right| ^{2(\kappa _{0}\alpha
^{(1)}-\kappa _{1}\beta ^{(1)})}\left| 1-w_{r}\right| ^{2(\kappa _{0}\alpha
^{(2)}-\kappa _{1}\beta ^{(2)})}\prod_{r^{\prime }<r}^{t-1,t}\left|
w_{r^{\prime }}-w_{r}\right| ^{2(\kappa _{0}^{2}-\kappa _{1}^{2})}$ which
can be performed by using the result of reference \cite{dotsenko2}
(specificaly, see eq. (B.9) of this paper). In such case, a global factor of
the form $\Gamma (t+1)\prod_{r=0}^{t}\frac{\gamma (r(\kappa _{0}^{2}-\kappa
_{1}^{2})/2)}{\gamma ((\kappa _{0}^{2}-\kappa _{1}^{2})/2)}$ stands in the
expression for three-point correlators; it is easy to see that in the $%
\kappa _{\mu }\kappa ^{\mu }\rightarrow 0$ limit, where the Liouville mode
becomes harmless, the expression (\ref{yupi}) is recovered.

\vskip0.3 cm

\section{Conclusions}

The implications of considering a non-vanishing cosmological constant in
Jackiw-Teitelboim two-dimensional theory of gravity coupled to Liouville
theory have been discussed. This was achieved by the treatement of the model
as a non-critical string theory. Within the context of this stringy
interpretation, we enphasize that the presence of a non-zero value for $%
\Lambda $ demands a non trivial treatement of the coupling of both actions.
Indeed, additional restrictions on the value of the central charge of the
model need to be considered even though the conformal anomaly is independent
of the explicit value of $\Lambda $. The constraint, which realize the
additional restriction, agrees with the condition necessary to turn the
tachyon field operator $\mathcal{T}_{\Lambda }(z)$ into a primary
(prelogarithmic) operator \cite{kogan} in the context of the world-sheet
conformal field theory. As mentioned, the linearized tachyon field equation
leads to impose the diagonal condition for the operator product expansion $%
T(z)\mathcal{T}_{\Lambda }(w)$.

We have seen how the coupling between two-dimensional gravity and Liouville
theory leads to the neutralization of the Liouville mode as it was pointed
out in reference \cite{chamseddine}, where it was explained how the constant
curvature constraint renders the Liouville action \textit{harmless} by 
\textit{trivializating the} $\sigma $ \textit{dependence}. This
neutralization effect is reflected, within the context of the conformal
field theory description, by means of the null condition $\kappa _{\mu
}\kappa ^{\mu }=0$, which would permit to realize the correlation functions
in terms of a direct product of multiple integrals in the whole complex
plane. Then, explicit expression for three-point functions were obtained in
terms of Virasoro-Shapiro integrals which yield from the neutralization
condition imposed on the Dotsenko-Fateev type realization. These results
have been obtained by means of the description of the model in the context
of the 2D CFT formalism, which, once again, turns out to be a fruitful
method to study several features of two-dimensional quantum gravity.

\bigskip \vskip0.5 cm

I wish to thank Noureddine Mohammedi and Mariano Galvagno. This work was
supported by Consejo Nacional de Investigaciones Cient\'{i}ficas y
T\'{e}cnicas (CONICET) and University of Buenos Aires (UBA).

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gr-qc0303068.extracted_bib
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\bibitem{mazzitelli}  D.F. Mazzitelli and N. Mohammedi, Nucl.Phys. \textbf{%
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\bibitem{odintsov}  S.D. Odintsov and I.L. Shapiro, Phys.Lett. \textbf{B263}%
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575 (1991). S.D. Odintsov and I.L. Shapiro, Int.J.Mod.Phys. \textbf{D1}, 571
(1993). S.Nojiri and S.D. Odintsov, Int.J.Mod.Phys. \textbf{A16}, 1015
(2001). E.Elizalde, S. Naftulin and S.D. Odintsov, Int.J.Mod.Phys. \textbf{A9%
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\bibitem{mazzitelli2}  D.F. Mazzitelli and N. Mohammedi, Phys.Lett. \textbf{%
B268}, 12 (1991)

\bibitem{mohammedi}  N. Mohammedi, \textit{Constant curvature and
non-perturbative W3 gravity}, arXiv:hep-th/9109031

\bibitem{burwick}  T.T. Burwick and A.H. Camseddine, Nucl.Phys. \textbf{B384}%
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\bibitem{chamseddine2}  A.H. Camseddine, Nucl.Phys. \textbf{B368}, 98 (1992)

\bibitem{shapiro}  I.L. Shapiro, \textit{Perturbative approach to the
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\bibitem{distler}  Distler and Kawai, Nucl.Phys. \textbf{B321,} 509 (1989)

\bibitem{becker}  K. Becker, \textit{Strings, Black Holes and Conformal
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\bibitem{witten}  E. Witten, Phys.Rev. \textbf{D44}, 314 (1991)

\bibitem{goulian}  M. Goulian and M. Li, Phys.Rev.Lett. \textbf{66}, 2051
(1991)

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