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\title{Properties}

\begin{document}

\begin{titlepage}

\begin{flushright}
                LPT Orsay 03/09\\
                February 2003  \\
                hep-th/0302211
\end{flushright}
\bigskip

\begin{center}

{\LARGE \bf Non-critical string \\
Liouville theory \\
and \\
geometric bootstrap hypothesis \\}


\end{center}
\bigskip

\begin{center}
    {\bf
    Leszek Hadasz}\footnote{e-mail: hadasz@th.u-psud.fr} \\
    Laboratoire de Physique Th\`{e}orique, B{\^a}t. 210,\\
    Universit{\'e} Paris-Sud,  91405 Orsay, France \\
        {\em and }\\
    M. Smoluchowski Institute of Physics, \\
    Jagiellonian University, 
    Reymonta 4, 30-059 Krak\'ow, Poland \\
\vskip 5mm
    {\bf
    Zbigniew Jask\'{o}lski}\footnote{e-mail: Z.Jaskolski@if.uz.zgora.pl }\\
    Physics Institute\\
    University of Zielona G\'{o}ra\\
    ul. Szafrana 4a,
    65-069 Zielona G\'{o}ra, Poland



\end{center}

\vskip 1cm

\begin{abstract}
Basing on the standard construction of critical string amplitudes we
analyze properties of the longitudinal sector of the non-critical
Nambu--Goto string. We demonstrate that it cannot be described by
standard (in the sense of BPZ) conformal field theory. As an alternative
we propose a new version of the geometric approach to
Liouville theory and formulate its basic consistency condition --- the
geometric bootstrap equation.
\end{abstract}
\bigskip\bigskip



\thispagestyle{empty}
\end{titlepage}


\section{Introduction}
\setcounter{equation}{0}
It has been well known since early days of string theory that the
covariant quantization of the free Nambu-Goto string
\cite{brower72}  leads in non-critical dimensions to
consistent free quantum models with longitudinal excitations. However
in spite of numerous attempts no consistent theory of the longitudinal
dynamics has been found. 
The folklore evolved to the negative conclusion that the joining--splitting interaction of 
Nambu-Goto  string
is inconsistent in non-critical dimensions. The aim of this paper is to
reconsider the problem 
from the today perspective and analyze under what assumptions 
this statement can be turned into a ``no go'' theorem. 

We restrict ourselves to non-critical strings with the space of asymptotic free 
states   coinciding with the space of states of the free non-critical Nambu-Goto string.
This rules out all non-critical strings obtained by compactifications of critical string  
and all models with the Liouville sector describing one of the target space coordinates. 

Our strategy is to follow the standard construction of critical string
amplitudes and to analyze what 
properties of the theory in the longitudinal sector are indispensable for
Lorentz covariance and unitarity. This can be done either in the covariant or in the 
light-cone formulation. We have chosen the latter one as it is
more suitable for discussing the unitarity. The additional advantage is 
 that the Lorentz covariance  can be
analyzed by the light-cone methods known from the critical string 
\cite{Mandelstam73,Mandelstam74,Mandelstam86,Sin88,kisa}. 

% 

The paper is organized as follows. In Sect.2 we briefly recall the light-cone formulation of 
the free non-critical Nambu--Goto string. In Sect.3 we analyze those properties of the longitudinal
dynamics which are necessary for the Lorentz covariance of the tree light-cone amplitudes.
In Sect.4 we show that the Liouville theory
regarded as the standard (in the sense of BPZ \cite{Belavin:1984vu})
conformal field theory with continuous spectrum 
does not provide an appropriate model of the Nambu-Goto string longitudinal excitations. 
The axiom of the standard CFT which has to be lifted is the operator product expansion 
and the standard bootstrap equation. 
Using general properties of the geometric approach to Liouville theory we formulate in Sect.5
a new dynamical principle --- the geometric bootstrap equation. 
Finally Sect.6 contains conclusions and brief discussion of open problems.

\section{Light-cone formulation}
\setcounter{equation}{0}

In the light-cone gauge the closed free non-critical Nambu-Goto string can be formulated as a 
representation of the algebra of zero modes
$$
[P^{i},x^j] \;  =\;  -  i\delta^{ij}\ ,
\hskip 5mm
[P^+,x^-] \;  =\;  i\ ,
$$
along with the  algebra of the transverse
$$
\begin{array}{rlllll}
{[a_m^i,a_n^j]} &=&  m\delta^{ij}\delta_{m,-n}\ ,&\\
{[\widetilde{a}_m^i,\widetilde{a}_n^j]}&=&
m\delta^{ij}\delta_{m,-n} \ , &\;\;\;m,n\in \mathbb{Z}
\setminus\{0\}
\end{array}
$$
and the longitudinal
$$
\begin{array}{rll}
{[L^{\rm \scriptscriptstyle L}_m,L^{\rm \scriptscriptstyle L}_n]}
&=& (m-n) L^{\rm \scriptscriptstyle L}_{m+n} + {c^{\rm \scriptscriptstyle L}\over 12} (m^3 - m) \delta_{m,-n}\ ,
\\ [6pt]
{[\widetilde{L}^{\rm \scriptscriptstyle L}_m, \widetilde{L}^{\rm
\scriptscriptstyle L}_n]} &=& (m-n) \widetilde{L}^{\rm
\scriptscriptstyle L}_{m+n} + {\bar c^{\rm \scriptscriptstyle L}\over12} (m^3 - m) \delta_{m,-n}\ , 
\end{array}
$$
excitations $(1<d<25)$ \cite{dahaja98}. We denote by $ {\cal V}_h,\widetilde{{\cal V}_h}$ the Verma modules of the left and the right
Virasoro algebras of longitudinal excitations with the central charges $c^{\rm \scriptscriptstyle L}=\bar c^{\rm \scriptscriptstyle L}$
and the highest weights $h=\bar h$ related to the dimension of the target space by
$$
c^{\rm \scriptscriptstyle L}= 1+6Q^2 \;=\; 26-d \ ,\;\;\;\;h\;=\;{Q^2\over 4}\;=\; 
{25-d\over 24}\ .
$$
For the central charge  in the range $1\leqslant c^{\rm \scriptscriptstyle L}\leqslant 25$, 
the left Verma module $ {\cal V}_h$ can be 
realized as the Fock space ${\cal F}^{\rm \scriptscriptstyle L}$ generated
by the oscillators 
$$
\begin{array}{rlllll}
{[c_m,c_n]} &=&  m\delta_{m,-n}\ ,
 &\;\;\; m,n\in \mathbb{Z} \setminus\{0\}
\end{array}
$$
out of the vacuum state $\omega$. In this realization 
the left Virasoro generators take the form \cite{Fairlie}
\begin{eqnarray}
L_{n}^{\rm \scriptscriptstyle L} &=&
 {\textstyle\frac{1}{2}}\sum_{k\neq 0,-n}\!
 :\!c_{-k}c_{n+k}\!: \;+\;i{Q\over \sqrt{2}} n c_n \;+\;
 {Q^2\over 4}\delta_{n,0}
 \nonumber \ .
\end{eqnarray}
The same construction yields the realization of $\widetilde{{\cal V}_h}$ in terms
of the right Fock space $\widetilde{{\cal F}^{\rm \scriptscriptstyle L}}$.
The  space of states is given by
$$
{\cal H}=
\int {dp_+ \over | p^+|}\,
 d^{\,d-2}\overline{p}\;
 {\cal F}^{\rm \scriptscriptstyle T}(p^+,\overline p)\otimes
 {\cal F}^{\rm \scriptscriptstyle L}
\otimes \widetilde{{\cal F}^{\rm \scriptscriptstyle L}}\ ,
$$
where $\overline{p}= (p^1,\ldots,p^{d-2})$,
and ${\cal F}^{\rm \scriptscriptstyle T}(p^+,\overline{p})$ denotes  the Fock space
generated by the left and right transverse  excitations 
out of the unique vacuum state 
$|\,p^+,\overline{p}\,\rangle^{\rm \scriptscriptstyle T}$ satisfying 
$$
P^i\,|\,p^+,\overline{p}\,\rangle^{\rm \scriptscriptstyle T} = 
\,p^i \,|\,p^+,\overline{p}\,\rangle^{\rm \scriptscriptstyle T}\;\;\;,\;\;\;
P^+\,|\,p^+,\overline{p}\,\rangle^{\rm \scriptscriptstyle T} = 
p^+  \,|\,p^+,\overline{p}\,\rangle^{\rm \scriptscriptstyle T}
\ . 
$$
${\cal H}$ carries a unitary, anomaly free representation of the Poincare algebra:
\begin{eqnarray}
{P}^- &=&
{\frac {2\alpha}{ P^+}} (L_0 +
\widetilde{L}_0 - 2) \ ,\;\;\;P^+\ ,\;\;\; P^i\ ,
\nonumber\\
{M}^{ij}_{\rm lc} & = & {P}^i{x}^j-{P}^j{x}^i + i\sum_{n\geqslant
1} {\frac{1}{n}} \left (a_{-n}^ia_{n}^j - a_{-n}^ja_{n}^i +
\widetilde a_{-n}^i\widetilde a_{n}^j - \widetilde
a_{-n}^j\widetilde a_{n}^i\right) \ ,
\nonumber \\
{M}^{i+}_{\rm lc} & = & {P}^+ {x}^i \ ,
\label{poincare}\\
{M}^{+-}_{\rm lc} &=& {\textstyle\frac{1}{2}}({P}^+{x}^-+x^-P^+) \ ,
\nonumber \\
{M}^{i-}_{\rm lc} &=& {\textstyle\frac{1}{2}}( {x}^i P^-
 + P^- {x}^i ) - {P}^i{x}^- -
 i{\frac{ 2\sqrt{\alpha}}{P^+}} \sum_{n\geqslant 1} {\frac{1}{n}}
\left( a_{-n}^i L_n
- L_{-n} a_n^i
+ \widetilde a_{-n}^i \widetilde L_n
- \widetilde L_{-n} \widetilde a_n^i
\right)\ ,  \nonumber
\end{eqnarray}
where 
$ 
L_{n}=L^{\rm \scriptscriptstyle T}_n+L^{\rm \scriptscriptstyle L}_n,\ 
$ 
$
L_{n}^{\rm \scriptscriptstyle T} =
 {\textstyle\frac{1}{2}}\sum_{k= -\infty}^{+\infty}
 \sum_{i=1}^{d-2}
 :\!a_{-k}^ia_{n+k}^i\!:\ ,
$ 
$
a^i_0=\sqrt{\alpha'\over 2} P^i,
$ 
and the corresponding formulae for the right sector are assumed.
 
The algebra of non-zero modes is by construction
isomorphic to the (diagonalized)
algebra of the DDF operators of the covariant approach
\cite{brower72}.   In particular,  $L^{\rm
\scriptscriptstyle L}_m$ correspond to
the shifted Brower longitudinal operators. In consequence the space of states
and the representation of the Poincare algebra are isomorphic to those obtained by the
covariant quantization of the Nambu--Goto string \cite{dahaja98}.
%\newpage

\section{General properties of the longitudinal dynamics}
\setcounter{equation}{0}


In the light-cone formulation string amplitudes 
are constructed in terms of 2-dim field theory on  light-cone diagrams describing time ordered sequences of 
elementary splitting and 
joining processes \cite{Mandelstam73,Mandelstam74,Mandelstam86}. 
The problem of introducing interactions can be seen as a problem of extending 
the theory from the cylinder, where it is completely determined by the
free string theory at hand, 
to an arbitrary light-cone diagram. 
In the case of the Nambu--Goto non-critical string the extension in the transverse sector
is  given by the tensor product of $d-2$ copies of the scalar CFT. 
Searching for possible solution in  the longitudinal sector one can try
to extract a minimal set of properties  
any  possible extension should satisfy in order to achieve covariant and
unitary string amplitudes. 


According to our assumptions concerning the free theory
the space of states in the longitudinal sector 
is a tensor product of single left and single right Verma
module. 
The only ground state 
$|\,0\,\rangle^{\rm \scriptscriptstyle L}=\omega\otimes\widetilde{\omega}$  
is not $PSL(2,\mathbb{C})$-invariant and 
the energy momentum-tensor\footnote{The antyholomorphic counterparts of
the formulae are assumed.} 
\begin{eqnarray}
\label{tensor}
{T^{\rm \scriptscriptstyle L}(z)}
&=&
\sum\limits_{n}L_n^{\rm \scriptscriptstyle L}z^{-n-2}
\end{eqnarray}
is singular in the limit $z\to 0,$
$$
T^{\rm \scriptscriptstyle L}(z)
|\,0\,\rangle^{\rm \scriptscriptstyle L}
= {Q^2 \over 4 z^2}
|\,0\,\rangle^{\rm \scriptscriptstyle L}
+ {1\over z}L^{\rm \scriptscriptstyle L}_{-1}
|\,0\,\rangle^{\rm \scriptscriptstyle L}
+ {\rm regular \; terms} \ .
$$
The ground state
applied to the free end of a semi-infinite cylinder thus
corresponds to a puncture on the complex plane with prescribed singularity structure of
the energy--momentum tensor at it.

In the case of tree light-cone diagrams with $N$ arbitrary external states
the longitudinal sector is described by correlation
functions of appropriate number of the energy--momentum tensor
insertions on the Riemann sphere $S^2(z_1,\dots,z_N)$ with $N$
punctures:
\begin{equation}
\label{badnotation}
\left\langle\, \prod_j T^{\rm \scriptscriptstyle L}(w_j)
\prod_k \widetilde T^{\rm \scriptscriptstyle L}(\bar w_k)
\,\right\rangle_{S^2(z_1,\dots,z_N)} \ .
\end{equation}
Such formulation may seem strange from the point of view of the critical string.
Indeed  if we had assumed the operator--state correspondence we could have reduced
the correlation functions on $N$-punctured sphere  to
correlation functions on the sphere with no punctures.
The space of states in the longitudinal sector
is however ``too small'' to accommodate such construction. 
Even though the operator--state correspondence does not hold it is convenient to replace  
 the adequate but clumsy notation
(\ref{badnotation}) by 
\begin{equation}
\label{Lcorrelators}
\left\langle \,\prod_j T^{\rm \scriptscriptstyle L}(w_j)
\prod_k \widetilde T^{\rm \scriptscriptstyle L}(\bar w_k)
\prod\limits_{r=1}^N P(z_r,\bar z_r)\,\right\rangle \ .
\end{equation}
The properties of the longitudinal sector which  are  necessary 
for the Lorentz covariance of tree string amplitudes can be summarized as follows \cite{Hadasz:2002gk}\medskip

\noindent {\bf I}
{\it 
The conformal anomaly 
has its universal form given by the (regularized) Liouville action 
on the punctured sphere and
depends on the central charge $c^{\rm \scriptscriptstyle L}=1+6Q^2$ in the standard way.}
\medskip

\noindent {\bf II}
{\it 
For all correlation functions on $N$-punctured spheres 
the operator--operator product expansion (OOPE) and
the operator--puncture product expansion (OPPE) hold:}
\begin{eqnarray}
\label{TTOPE}
T^{\rm \scriptscriptstyle L}(w)T^{\rm \scriptscriptstyle L}(z)
&=&{1+6Q^2\over 2(w-z)^4 }
+ {2 \over (w-z)^2}T^{\rm \scriptscriptstyle L}(z)
+{ 1\over w-z}\partial T^{\rm \scriptscriptstyle L}(z) + \dots \ , \\
\label{TPOPE}
T^{\rm \scriptscriptstyle L}(w) P(z, \bar z) &=&
{Q^2\over 4(w-z)^2}P(z, \bar z)
+{ 1 \over w-z}\partial P(z, \bar z) + \dots \ .
\end{eqnarray}
 \medskip

\noindent {\bf III}
{\it 
 The conformal  transformations  of the correlation functions
(\ref{Lcorrelators})
are generated by the energy--momentum tensor:}
\begin{eqnarray}
\label{LTtrans}
\delta_{\epsilon}T^{\rm\scriptscriptstyle L}(z)
&=&- {1\over 2\pi i}\oint dw\, \epsilon(w) T^{\rm\scriptscriptstyle L}(w)
T^{\rm\scriptscriptstyle L}(z)   \ ,\\
\label{LPtrans}
\delta_{\epsilon,\tilde\epsilon}P(z,\bar z) &=&
-{1\over 2\pi i}\oint \,d w\,
 \epsilon(w) T^{\scriptscriptstyle L}(w)P(z,\bar z) \\
 \nonumber
&& - {1\over 2\pi i}\oint \,d \bar w\, \tilde\epsilon(\bar w)
\widetilde T^{\scriptscriptstyle L}(\bar w)P(z,\bar z) \ .
\end{eqnarray}
The first property is necessary for the cancelation of non-covariant terms in the expression for the
light-cone tree amplitudes. The relations (\ref{TTOPE}), (\ref{TPOPE}) 
can be easily verified  on the cylinder where the theory is perfectly well defined.
The  requirement that they are also satisfied on arbitrary punctured
sphere  can then be seen as the connection condition 
for the energy--momentum tensor. 
The transformation rules (\ref{LTtrans}), (\ref{LPtrans})  
are essential for the construction of the light-cone string amplitudes 
and for their Lorentz covariance \cite{Hadasz:2002gk}.
Using (\ref{TTOPE}) and (\ref{TPOPE}) they can  be cast in the form
of the conformal Ward identities (CWI):
\begin{eqnarray}
\label{Tward}
\left\langle \,T^{\rm \scriptscriptstyle L}(w)
\prod_r P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L}
&=&\sum_r
\left( {Q^2\over 4(w-z_r)^2} +{1\over w-z_r}{\partial\over \partial z_r}\right)
\left\langle \,
\prod_r P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L} \ ,\\
\label{TTward}
\left\langle \,T^{\rm\scriptscriptstyle L}(u)T^{\rm \scriptscriptstyle L}(w)
\prod_r P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L}
&=&
{1+6Q^2\over 2(u-w)^4 }\left\langle \,
\prod_r P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L} \\
\nonumber
&+& \left({2 \over (u-w)^2} + {1\over u-w}{\partial\over\partial w}\right)
\left\langle \,T^{\rm \scriptscriptstyle L}(w)
\prod_r P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L} \\
\nonumber
&+&\sum_r
\left( {Q^2\over 4(u-z_r)^2} +{1\over u-z_r}{\partial\over \partial z_r}\right)
\left\langle \,T^{\rm \scriptscriptstyle L}(w)
\prod_r P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L}
\ .
\end{eqnarray}
Apart from the complications related with the non-invariant vacuum and the lack 
of operator--state correspondence
the conditions listed above are exact counterparts of almost all
fundamental  properties of standard CFT.  
So are their consequences. Using CWI one can for instance  reduce
an arbitrary correlation function to the correlation function of punctures alone. 
As in the standard CFT the form
of three puncture correlator is determined up to a constant  
\begin{equation}
\label{3_puncture}
 \left\langle \, P(z_1,\bar z_1)P(z_2,\bar
z_2)P(z_3,\bar z_3)\,\right\rangle = {C\over |z_1
-z_2|^{Q^2\over 2}|z_1 -z_3|^{Q^2\over 2}|z_2 -z_3|^{Q^2\over 2} }\ .
\end{equation}
The familiar state--operator
correspondence can be replaced by the state--puncture
correspondence defined by
\begin{eqnarray}
\label{statepuncture}
L^{\rm \scriptscriptstyle L}_{-n_1}\dots
L^{\rm \scriptscriptstyle L}_{-n_N} |\,0\,\rangle^{\rm
\scriptscriptstyle L}
 & \longrightarrow  &
 {\cal L}^{\rm \scriptscriptstyle L}_{-n_1}\dots
 {\cal L}^{\rm \scriptscriptstyle L}_{-n_N}
 \cdot
 P(z,\bar z)
 \\
 & \equiv &
 {1\over (2\pi i)^N} \oint_{C_1} dz_1
 {T^{\rm \scriptscriptstyle L}(z_1)\over (z_1-z)^{n_1-1}}
 \dots
 \oint_{C_N} dz_N{T^{\rm \scriptscriptstyle L}(z_N)\over (z_N-z)^{n_N-1}}
 P(z,\bar z)\nonumber
 \end{eqnarray}
where the contours  of integration are chosen such that $C_i$ surrounds
$C_{i+1}$ for $i=1,\dots,N-1$, and $C_{N}$ surrounds  the point $z$.
Using this prescription one can associate to each state
${\xi\otimes\bar\xi\in{\cal F}^{\rm \scriptscriptstyle L}}\otimes \widetilde{{\cal F}^{\rm \scriptscriptstyle L}}$
a uniquely determined object
$V_0^{\rm \scriptscriptstyle L}(\xi,\bar \xi | z,\bar z)$ which we shall call 
the vertex puncture corresponding to $\xi\otimes \bar \xi$.

The only fundamental property of standard CFT we have not yet required is the 
puncture--puncture product expansion (PPPE). 
With our choice of the space of states  there is only one conformal family of 
vertex punctures and the PPPE would be necessarily of the form
\begin{equation}
\label{PPPE_d}
 P(x,\bar x)P(0,0) = Cx^{-h}\bar x^{-\bar h}
\left[P(0,0) + \beta_1 x\ {\cal L}^{\rm \scriptscriptstyle L}_{-1}\cdot
P(0,0) + \bar\beta_1\bar x\ \bar {\cal L}^{\rm \scriptscriptstyle
L}_{-1}\cdot P(0,0) + \ldots\right]\ ,
\end{equation}
where $C$ is the constant appearing in the three puncture partition
function (\ref{3_puncture}) and the descendant vertex punctures
${\cal L}^{\rm \scriptscriptstyle L}_{-1}\cdot P(0,0),\ $  $\bar {\cal L}^{\rm
\scriptscriptstyle L}_{-1}\cdot P(0,0),\; $ etc. are defined by
(\ref{statepuncture}).
Using PPPE one could in principle reduce all $N$-puncture correlators
to the 3-puncture functions. The consistency conditions of this procedure yields 
in the case of 4-puncture functions
the bootstrap equation \cite{Belavin:1984vu}. One of its consequences in standard CFT is the
restriction on  possible central charge, conformal dimensions, and
fusion rules known as Vafa's condition \cite{Vafa88}. Using
Lewellen's  derivation of this condition \cite{Lewellen89} one can
show that the PPPE (\ref{PPPE_d}) cannot be satisfied.

According to the properties of the longitudinal sector derived so far 
the theory is completely determined once all $N$-puncture correlators
are known
\begin{equation}
\label{Pcorrelators}
\left\langle \,
\prod\limits_{r=1}^N P(z_r,\bar z_r)\,\right\rangle^{\!\!\!\rm \scriptscriptstyle L} \ .
\end{equation}
It should be stressed
that the correlation functions (\ref{Pcorrelators}) can not be expressed as
the vacuum expectation values of some
operators acting in the Hilbert space 
${\cal V}_h\otimes \widetilde{{\cal V}}_h$ ($h={Q^2\over 4}$) of the free theory. Indeed if it were possible
it would imply the PPPE which is excluded by Vafa's condition. 
Thus the only local field in the theory we are looking for is the energy--momentum tensor.
The puncture correlators  (\ref{Pcorrelators})
should be understood as partition functions of such a theory on $N$-punctured spheres
(or in general on higher genus punctured surfaces).


Concluding this section let us briefly comment on the unitarity problem. 
The light-cone approach used in \cite{Hadasz:2002gk} to derive 
the properties  discussed above
incorporates the idea of 
joining--splitting interactions by the assumption that there exists a well defined 
2-dimensional theory on light-cone diagrams. The standard proof of unitarity in this formulation
 is based on the  identification of the light-cone amplitudes as terms of a Dyson
perturbative expansion in the space of multi-string states \cite{kaki,gw}. 
In order to interpret the
integration over moduli of punctured sphere as the integration over interaction times
one uses the Mandelstam map to transform the theory back onto an
appropriate family of light-cone diagrams. 
If the string degrees of freedom are described by standard CFT (as they are in the transverse sector) 
one can use 
the OPE to factorize the light-cone amplitude 
on the free spectrum at any moment between interactions. This in order allows to
interpret the amplitude in terms of interaction vertices and free propagation
yielding the required Dyson expansion structure \cite{Hadasz:2002gk}.
Since in the longitudinal sector the PPPE does not hold  such interpretation is not 
possible. This leaves very little room to maneuver for constructing a consistent theory. 

Although the structure of the perturbative string field theory known form the critical string
breaks down, one can still analyze the light-cone amplitudes in the more general context of the $S$-matrix. 
This brings us back to the early days of string theory when the dual models emerged 
as  perturbative solutions to the basic postulates any $S$-matrix should satisfy.
The assumption  specific to these particular solutions  states that 
the narrow resonance approximation yields a consistent Born term of some perturbative
expansion \cite{Veneziano:dr}. In this framework unitarity on the tree level means that 
the only singularities of tree string amplitudes are simple poles with locations 
determined 
by the free string mass spectrum and with the residues  satisfying the factorization condition.

\section{DOZZ approach}
\setcounter{equation}{0}



Although unitarity provides strong restrictions on possible solutions it can
be hardly used for their practical construction. To proceed one needs 
an explicit model of the longitudinal dynamics. The  only candidate 
is up to now the quantum Liouville theory \cite{sei,gm}. Its  relevance  for the
proper description of the longitudinal 
string excitations has been known  since Polyakov's celebrated papers on conformal anomaly \cite{poly81,poly81b}.
Still, in spite of the great progress achieved in last few years  its application
in non-critical strings in physical dimensions remains a virtually untouched open problem.

The main difficulty is related to the spectrum of the Liouville theory. 
In  commonly adopted approaches it is continuous 
\cite{Curtright:gt}. 
It was conjectured some time ago by  A. and Al. Zamolodchikov \cite{Zamolodchikov:1995aa}
that the continuum spectrum
along with the 3-point functions proposed by Otto and Dorn \cite{Dorn:1994xn}
 satisfy the bootstrap
consistency conditions of standard (in the sense of BPZ \cite{Belavin:1984vu}) CFT. 
This conjecture was recently proved by Ponsot and Teschner
\cite{Ponsot:1999uf,Ponsot:2000mt}
in the weak coupling regime $c>25$. It is believed to hold in the strong coupling
regime $1<c<25$ by the analytic continuation argument \cite{Teschner:2001rv}.
This version of the quantum Liouville theory yields 
 the string field theory interpretation of light-cone amplitudes and unitarity. 
The price one has to pay is the extension of the spectrum of external
string states far beyond the spectrum of the free non-critical Nambu-Goto string we have started with.
This would lead to the continuous family of intercepts and is not acceptable on physical grounds.

A possible way out is to restrict ourselves to  the external states from the  $({Q^2\over 4},{Q^2\over 4})$ 
conformal family. In this case however 
 the tree 4-string amplitude exhibits cut singularities characteristic for the continuous mass spectrum 
of intermediate states. In order to show it let us briefly recall the basic ingredients of the DOZZ
approach to the Liouville theory \cite{Teschner:2001rv}.
The structure constants $C(\alpha_3,\alpha_2,\alpha_1),$ 
which determine the three point correlation function 
\[
G^{\rm\scriptscriptstyle L}_{\alpha_1,\alpha_2,\alpha_3}(z_1,z_2,z_3)=
|z_{12}|^{2\gamma_3}|z_{32}|^{2\gamma_1}|z_{31}|^{2\gamma_2}
C(\alpha_3,\alpha_2,\alpha_1)
\]
where
$\gamma_1=\Delta_{\alpha_1}-\Delta_{\alpha_2}-\Delta_{\alpha_3},
\gamma_2=\Delta_{\alpha_2}-\Delta_{\alpha_3}-\Delta_{\alpha_1}, 
\gamma_3=\Delta_{\alpha_3}-\Delta_{\alpha_1}-\Delta_{\alpha_2},$ 
are given by
\begin{eqnarray}
\label{3pointsZZ}
&&
\hspace*{-1cm}
C(\alpha_3,\alpha_2,\alpha_1)\; = \; 
\left(\left\lbrack \pi^2 \mu\tilde\mu 
\gamma\left(b^{2}\right)\gamma\left(1/b^2\right)\right\rbrack^{Q/2}
\left(b^{2}\right)^{\frac1b - b}\right)^{Q-\alpha_1-\alpha_2-\alpha_3}  \\
&&
\frac{
\Upsilon_0\Upsilon_b(2\alpha_1)\Upsilon_b(2\alpha_2)\Upsilon_b(2\alpha_3)
}
{
\Upsilon_b(\alpha_1+\alpha_2+\alpha_3-Q)
\Upsilon_b(\alpha_1+\alpha_2-\alpha_3)
\Upsilon_b(\alpha_1+\alpha_3-\alpha_2)
\Upsilon_b(\alpha_2+\alpha_3-\alpha_1)}. \nonumber
\end{eqnarray}
The special function $\Upsilon_b(x)$ has an
integral representation convergent in the strip $0<\Re\,x<Q$
\begin{eqnarray}
&&\log\Upsilon_b(x)\; =\; \int_{0}^{\infty}\frac{dt}{t}
\left\lbrack\left(\frac{Q}{2}-x\right)^{2}e^{-t}-
\frac{\sinh^{2}(\frac{Q}{2}-x)\frac{t}{2}}{\sinh\frac{bt}{2}\sinh\frac{t}{2b}}
\right\rbrack, \nonumber
\end{eqnarray}
$\Upsilon_{0}={\rm res}_{x=0}\frac{d\Upsilon_b(x)}{dx},\;$ $\gamma(x)
= \frac{\Gamma(x)}{\Gamma(1-x)}$ and  the ``dual'' cosmological constant
$\tilde \mu$ is related to $\mu$ as
\begin{equation}
\label{mu_dual}
\left\lbrack \pi\tilde\mu \gamma\left(1/b^2\right)
\right\rbrack^b
= 
\left\lbrack \pi\mu \gamma\left(b^2\right)
\right\rbrack^{1/b}.
\end{equation}
Eq. (\ref{3pointsZZ}) is self-dual: it remains unchanged under $b \to
1/b.$ Its form agrees with the three-point coupling constant proposed in
\cite{Zamolodchikov:1995aa} in the weak coupling region of $b \in
\mathbb{R}.$   If one assumes that the
product $\mu\tilde\mu$ remains real and positive in the strong
coupling region, $b = {\rm e}^{i\theta},$ $\theta \in \mathbb{R},$ (it
is straightforward to check that (\ref{mu_dual}) admits such solution),
then (\ref{3pointsZZ}) defines analytic continuation of
the coupling constant to this region
respecting the self-duality condition, which is sufficient for $C$ to
be real there. 


The general four-point function can be written as the $s$-channel
integral
\begin{eqnarray}
\label{four_point}
G^{\rm\scriptscriptstyle L}_{\alpha_4,\alpha_3,\alpha_2,\alpha_1}(z,\bar{z})
\; = \; 
\int_{\mathcal{D}}\!d\alpha
\; C(\alpha_4,\alpha_3,\alpha)C(\bar\alpha,\alpha_{2},\alpha_{1})
\left|\mathcal{F}_{\alpha}
\left[^{\alpha_3\, \alpha_2}_{\alpha_4\, \alpha_1}\right]
(z)\right|^{2}
\end{eqnarray}
where the  conformal block 
$\mathcal{F}_{\alpha}
\left[^{\alpha_3\, \alpha_2}_{\alpha_4\, \alpha_1}\right]
(z)$ is 
represented \cite{Belavin:1984vu} by power series of the form
\begin{eqnarray}
\label{series}
\mathcal{F}_{\alpha}
\left[^{\alpha_3\, \alpha_2}_{\alpha_4\, \alpha_1}\right]
(z)
\; = \; 
z^{\Delta_{\alpha}-\Delta_{\alpha_1}-\Delta_{\alpha_2}}
\sum_{n=0}^{\infty}z^n
\mathcal{F}_{\alpha}
\left[^{\alpha_3\, \alpha_2}_{\alpha_4\, \alpha_1}\right](n).
\end{eqnarray}
As we are interested in the case of all ``external'' $\alpha_i$ equal to
$\frac{Q}{2},$ 
the set $\mathcal{D}$ coincides with the spectrum 
$\mathbb{S} = \frac{Q}{2} + i\mathbb R^{+}$ which we shall parameterize with
$\frac{Q}{2} + iP.$ The coefficients
$\mathcal{F}_{\alpha}
\left[^{\alpha_3\, \alpha_2}_{\alpha_4\, \alpha_1}\right](n)$  
are rational functions of $\Delta_{\alpha} = \frac{Q^2}{4} + P^2$ with
poles located (for $1 < c < 25$) outside $\mathbb{S}.$

From (\ref{3pointsZZ})  it follows that for $b =
{\rm e}^{i\theta}$ 
\begin{equation}
\label{C_square}
C\left({Q}/{2},Q/2,Q/2+iP\right)
C\left(Q/2-iP,Q/2,Q/2\right)
 \; = \; 
q\ {\rm e}^{2H_\theta(P)},
\end{equation}
where $q$ does not depend on $P$ and 
\[
H_\theta(P)
\; = \; 
\int\limits_0^\infty\!\frac{dt}{t}
\left[\cos^2\theta\ {\rm e}^{-t} 
+
\frac{1 - 8\sin^2 \frac{Pt}{2} - \cos (2Pt)\cosh\left(t\cos\theta\right)}
{\cosh\left(t\cos\theta\right) - \cos\left(t\sin\theta\right)}\right]
\; \propto \; -P^2 
\]
for $P^2 \gg 0.$

The 4-string amplitude for the tachionic external states 
can be written as \cite{Hadasz:2002gk}
\begin{equation}
 \label{tachions_1} 
A \; =\; (2\pi)^{d\over 2} [\alpha]^{-{1\over 2}}
\prod\limits_{\mu = 0}^{d-1} \delta\left(\sum\limits_{r=1}^4 p_r^\mu \right)\;
\int\limits_{\mathbb{C}}\!d^2 x\;
|z|^{\frac{p_1\cdot p_2}{2\alpha}} |1-z|^{\frac{p_2\cdot p_3}{2\alpha}}
G^{\scriptscriptstyle L}(z,\bar z)  \ ,
\end{equation}
where 
$G^{\scriptscriptstyle L}(z,\bar z) =
G^{\rm\scriptscriptstyle
L}_{\frac{Q}{2},\frac{Q}{2},\frac{Q}{2},\frac{Q}{2}}(z,\bar{z}) $ and $p_r$ denote external momenta satisfying the on-mass-shell
condition 
${p^2\over 4\alpha}={d-1\over 12}.$ Using (\ref{C_square}) we write
\begin{eqnarray}
\label{integral}
{\cal I} & \equiv & \int\limits_{\mathbb{C}}\!d^2 z\;
|z|^{\frac{p_1\cdot p_2}{2\alpha}} |1-z|^{\frac{p_2\cdot p_3}{2\alpha}}
   G^{\scriptscriptstyle L}(z,\bar z)  \nonumber \\
&= & 
q\int\limits_{\mathbb{C}}\!d^2 z\;
|z|^{\frac{p_1\cdot p_2}{2\alpha}} |1-z|^{\frac{p_2\cdot p_3}{2\alpha}}
\int\limits_{0}^{\infty}\!dP\; {\rm e}^{2H_\theta(P)}
\left|\mathcal{F}_{\frac{Q}{2} + iP}
\left[^{\frac{Q}{2}\, \frac{Q}{2}}_{\frac{Q}{2}\, \frac{Q}{2}}\right]
(z)\right|^{2}.
\end{eqnarray}
If one can change the order of integration, then --- expanding the
integrand in the power series of $z,\,\bar z$ and integrating 
in the vicinity of $z = 0$ one gets
\begin{equation}
\label{poles}
{\cal I} \; \sim \; \sum_n 
\int\limits_0^{\infty}\!dP 
\frac{d_{n,n}(P)}{\frac{s}{4\alpha} + \frac{Q^2}{2} + 2n + 2P^2 - 2}
{\rm e}^{2H_\theta(P)}
\end{equation}
up to terms analytic in the Mandelstam variable $s=(p_1 + p_2)^2$ at 
the finite part of the complex plane.
$d_{n,n}(P)$ are rational functions of $P$ without poles on the real
axis. 

The presence of poles on
the integration contour for $s < -4\alpha\left(\frac{Q^2}{2}  + 2P^2 -
2\right)$ leads to a cuts in ${\cal I}$ in this region. 
Such cuts in the
tree level amplitude indicate a continuous mass
spectrum of the theory.


The cut structure of the scattering
amplitude can be confirmed by choosing a different route in calculating
(\ref{integral}). Due to the convergence 
of the sum defining conformal block and properties of its coefficients 
$\mathcal{F}_{\frac{Q}{2} + iP}
\left[^{\frac{Q}{2}\, \frac{Q}{2}}_{\frac{Q}{2}\, \frac{Q}{2}}\right]\!
(n)$  and $H_\theta(P)$ it is legitimate in
calculating the four-point function 
$G^{\scriptscriptstyle
L}(z,\bar z)$ 
in the vicinity of $z=0$ to change the order of integration over $P$ and
summation over $n$ (see Eq. (\ref{series})). Integrals of the form
\[
\int\limits_0^{\infty}\!dP \; |z|^{2P^2}\ {\rm e}^{2H_{\theta}(P)}\;
\mathcal{F}_{\frac{Q}{2} + iP}
\left[^{\frac{Q}{2}\, \frac{Q}{2}}_{\frac{Q}{2}\, \frac{Q}{2}}\right]\!
(m)\ 
\mathcal{F}_{\frac{Q}{2} - iP}
\left[^{\frac{Q}{2}\, \frac{Q}{2}}_{\frac{Q}{2}\, \frac{Q}{2}}\right]\!
(n)
\]
that arise in this procedure contain terms which for $z \to 0$ behave like
powers of $1/\log|z|.$ When inserted into (\ref{integral}) they will
produce cuts in the complex $s$ plane, as can be concluded from the
equality
\begin{eqnarray*}
\int\limits_0^\epsilon\!dx\; \frac{x^{a-1}}{|\log x|^m}  
& = &
-\frac{1}{\Gamma(m)} \log a + \ldots,
\end{eqnarray*}
where the dots denote terms analytic in $a$ for $a \to 0.$

As the narrow resonance approximation breaks down 
the DOZZ version of the Liouville theory 
does not provide an adequate description of the longitudinal excitations of 
non-critical Nambu-Goto string. Let us mention that 
a possible way out was proposed by Gervais \cite{Gervais:1990be}.
It consists in the truncation of spectrum to some discrete subset.
However due to technicalities involved
\cite{Gervais:1994ec,Gervais:1996vp}
it is still not clear whether this approach may lead to consistent string models in
non-critical dimensions. 

\section{Geometric bootstrap hypothesis}
\setcounter{equation}{0}

As we have seen in Sect.3 if we restrict ourselves to the spectrum of
free non-critical Nambu-Goto string  
the  PPPE has to be lifted.
The only  approach to the Liouville theory
where PPPE is not assumed is the so called geometric formulation of the 2-dim
quantum gravity \cite{Polyakov82,Takhtajan:1993vt,Takhtajan:zi,Takhtajan:1994vt,Takhtajan:1995fd}.
In this approach
the Liouville correlation functions
are defined  in terms of path integral over conformal class of Riemannian metrics
with prescribed  singularities at the punctures.
The results obtained in the case of  parabolic singularities \cite{Takhtajan:1993vt,Takhtajan:zi,Takhtajan:1994vt}
are in perfect agreement with the properties of the
Liouville sector summarized in Sect.3. An additional support for the  geometrical approach
comes from  the fact that some of its geometric
predictions  can be rigorously proved
(\cite{Takhtajan:zi,Takhtajan:1994vt}, and references therein).
In spite of considerable achievements 
the geometric approach  is not yet capable to
produce the puncture correlators. It can however be 
used  to  analyze their factorization
properties. 


%We shall start with some remarks on the spectrum of conformal weights.
In  the weak coupling regime $c>25$ 
we are interested in 
correlators of local operators with conformal weights satisfying the Seiberg bound $\Delta < {Q^2\over 4}$.
If the theory is coupled to conformal matter such operators are necessary for the  gravitational dressing.
They are supposed to correspond to microscopic, non-normalizable states with imaginary Liouville momenta
\cite{sei}. 
In the geometric approach
they are described by conical (elliptic) singularities with the opening angle $\nu$ related to the imaginary Liouville
momentum $P$ by 
$$
\Delta= \frac{Q^2}{4} + P^2 \; = \;  \frac{Q^2}{4}\left(1-\left(\frac{\nu}{2\pi}\right)^2\right)
$$ 
These operators do not appear  when  the Liouville theory is regarded
as the model for the longitudinal  string excitations in physical dimensions $1<c<25$. 
Indeed the derivation of the light-cone amplitudes
shows that  only the parabolic singularities (limiting case of the conical singularity with
the opening angle $\nu=0$) are relevant for the description of external states. 
The states with imaginary Liouville momenta are also not expected in  factorization
of puncture correlators. 



In order to describe the factorization one has to extend  the geometric approach 
to surfaces with finite holes. 
We assume that the Liouville action and the space of metrics in the path integral
are chosen in such a way that 
the classical solution corresponds to the hyperbolic metric with the
curvature $-\mu<0$ and
the geodesic boundary. We also assume that the  conformal weight of the hole 
is 
$$
\Delta_\ell = \frac{Q^2}{4}\left(1 + \frac{\mu \ell^2}{8\pi^2}\right)
$$
where $\ell$ is the length of the hole circumference  measured with
respect to the metric with constant  
negative curvature $R=-\mu$.  It  depends only on the conformal
class of metrics over which the path integral is taken. 
This assumption is motivated by the properties of the energy--momentum 
tensor of the classical hyperbolic solutions on the cylinder and on sphere
with holes (black-hole solutions in 3-dim gravity). The puncture corresponds to the limiting 
case of the hole with zero circumference. 
Since in the case of puncture
the classical conformal dimension does not receive quantum corrections one may expect 
that this is so for the holes as well (up to the renormalization of the cosmological
constant $\mu$). 


Let us now consider the simplest case of 4-puncture correlator.
In the geometric approach it is given by the path integral over the conformal
class of metrics  with parabolic singularities at
puncture locations. Since the conformal weight of the puncture is fixed 
 the space of metric fluctuations   at each puncture is  completely described
by ${\cal L}_n$ operators defined in (\ref{statepuncture}) and coincides
with the space of longitudinal excitations of 
the free string. 
The Liouville action and the space of metrics in the  path integral are
chosen in such a way that for each configuration  
of punctures
one gets a unique  classical solution corresponding to the hyperbolic
metric with scalar curvature $-1.$  
There are three closed geodesics $\Gamma_s,\Gamma_t,\Gamma_u$ in this
geometry separating the punctures into 
pairs (12,34), (13,24), (14,23), respectively.
Let us  cut the
path integral open along the geodesic $\Gamma_s$
 dividing $S$ into the spheres $S_{12},S_{34}$ with one hole and two
punctures ($S=S_{12}\cup S_{34},$ 
$\Gamma_s=S_{12}\cap S_{34}$). 
The classical solution $g$ on $S$ determines classical solutions
$g_{12}$ on $S_{12}$ and $g_{34}$ on $S_{34}.$ 
The initial path integral factorizes into a path integral over the conformal class of $g_{12}$,
 a path integral over the conformal class of $g_{34}$ 
and the integration  over all possible intermediate states.
According to our assumptions the holes on $S_{12}$ and  $S_{34}$
have the same conformal dimension $\Delta_s=
 \frac{Q^2}{4}\left(1 + \frac{\mu \ell_s^2}{8\pi^2}\right)
$ uniquely determined by  the
length $\ell_s$ of the common boundary. It follows that factorization in
each channel involves exactly one conformal family. 
%ground  state.
%The conformal structure of the theory implies that all possible metric  fluctuations along the cut
%are completely described by appropriate ${\cal L}_n$ operators.
%Thus only one conformal family appears in factorization.
Its  conformal weight depends on the channel and the moduli of the surface.
This is in sharp contrast with 
the DOZZ description  of the weak coupling 
regime where the factorization is independent both of the channel and of the moduli and
involves integration over continuous spectrum of conformal families.



Before we formulate the consistency conditions of the geometric factorization 
introduced above 
some comments on the hole-state correspondence are in order. First of
all since the length of the geodesic 
boundary can assume any positive value one needs a continuous spectrum of conformal weights
$\Delta \geqslant {Q^2\over 4}$ of intermediate states. This however does not necessary mean that
the space of states has to be extended. 
The crucial observation is that 
the geometric factorizaton on exactly one conformal family allows to  interpret the conformal 
weight as a characteristic of the energy--momentum tensor behavior
around the hole 
rather than a characteristic of intermediate states attached to its boundary.
Such interpretation is consistent with our choice of the space of free string states.
Indeed for the central charge in the range $1<c<25$ and $\Delta \geqslant {Q^2\over 4}$ 
one can use the oscillator realization of
the Verma module ${\cal V}_{Q^2\over 4}$ to construct 
the local operator $T_{\scriptscriptstyle P}(z)$.
$$
T_{\scriptscriptstyle P}(z)=
\sum\limits_{n}L_n^{\scriptscriptstyle P} z^{-n-2} \;\;\;,
\;\;\;L_n^{\scriptscriptstyle P}\;=\;\left\{
\begin{array}{lll}
L_n^{\scriptscriptstyle L} + 2Pc_n &{\rm for}& n\neq 0\\
L_0^{\scriptscriptstyle L} + {P^2} &{\rm for}& n= 0
\end{array}
\right.
$$ 
where $P$ is a real positive number (the Liouville momentum) \cite{Fairlie}.
One can easyly verify that  $T_{\scriptscriptstyle P}(z)$ satisfies
\begin{eqnarray}
\nonumber
T_{\scriptscriptstyle P}
(w)T_{\scriptscriptstyle P}(z)
&=&{\frac12 (1+6Q^2)\over (w-z)^4 }
+ {2 \over (w-z)^2}T_{\scriptscriptstyle P}(z)
+{ 1\over w-z}\partial T_{\scriptscriptstyle P}(z)
 + \dots \ , \\
\label{Tsingularity}
T_{\scriptscriptstyle P}(z)
\Omega
&=& {\Delta\over z^2}
\Omega
+ {1\over z}L^{\scriptscriptstyle P}_{-1}
\Omega
+ {\rm regular \; terms}\ . 
\end{eqnarray}
where $\Delta = {Q^2\over 4}+{P^2}$.
With an appropriate choice of the coordinates around the hole, the state-hole correspondence 
takes the form
\begin{eqnarray}
\label{statehole}
L^{\rm \scriptscriptstyle P}_{-n_1}\dots
L^{\rm \scriptscriptstyle P}_{-n_N} |\,0\,\rangle^{\rm
\scriptscriptstyle L}
 & \longrightarrow  &
 {\cal L}_{-n_1}\dots
 {\cal L}_{-n_N}
 \cdot
 H_\ell(z,\bar z)
 \\
 & \equiv &
 {1\over (2\pi i)^N} \oint_{C_1} dz_1
 {T(z_1)\over (z_1-z)^{n_1-1}}
 \dots
 \oint_{C_N} dz_N{T(z_N)\over (z_N-z)^{n_N-1}}
 H_\ell(z,\bar z)\ .\nonumber
 \end{eqnarray}
As in the case of punctures  one can associate to each state
${\xi\otimes\bar\xi\in{\cal F}^{\rm \scriptscriptstyle L}\otimes \widetilde{{\cal F}^{\rm \scriptscriptstyle L}}}$
a uniquely determined object
$V_\ell^{\rm \scriptscriptstyle L}(\xi,\bar \xi | z,\bar z)$ which we shall call 
the vertex corresponding to to the state $\xi\otimes \bar \xi$ applied at the 
hole of the circumference $\ell$.
We have assumed  that the location of the hole can be defined 
as a point at which the analytic continuation of the energy--momentum tensor issingular.


Although the correlators of punctures, holes, and in general, vertexes
are not realized as vacuum expectation values 
of operators in some Hilbert space all their conformal properties are 
exactly the same as in the standard CFT. 
In particular  an arbitrary three vertex correlator is determined up to
the structure constant 
$C(\ell_1,\ell_2,\ell_3)$. It takes the form
\begin{eqnarray*}
& &\hspace{-60pt}\left\langle \, 
 V^{\rm \scriptscriptstyle L}_{\ell_1}(\xi_1,\bar \xi_1 |z_1,\bar z_1)
V^{\rm \scriptscriptstyle L}_{\ell_2}(\xi_2,\bar \xi_2 |z_2,\bar z_2)
V^{\rm \scriptscriptstyle L}_{\ell_3}(\xi_3,\bar \xi_3 |z_3,\bar z_3)\,\right\rangle \;= \;C(\ell_1,\ell_2,\ell_3) \\
& \times &{\rho^{\ell_1\,\ell_2\,\ell_3}(\xi_1,\xi_2,\xi_3)\over 
(z_1 -z_2)^{\Delta_1+\Delta_2-\Delta_3}
(z_1 -z_3)^{\Delta_1+\Delta_3-\Delta_2}
(z_2 -z_3)^{\Delta_2+\Delta_3-\Delta_1} }\\
&\times &
{\rho^{\ell_1\,\ell_2\,\ell_3}(\bar\xi_1,\bar\xi_2,\bar\xi_3)\over 
(\bar z_1 -\bar z_2)^{\bar\Delta_1+\bar\Delta_2-\bar\Delta_3}
(\bar z_1 -\bar z_3)^{\bar\Delta_1+\bar\Delta_3-\bar\Delta_2}
(\bar z_2 -\bar z_3)^{\bar\Delta_2+\bar\Delta_3-\bar\Delta_1} }
\end{eqnarray*}
where $\Delta_i = \Delta_{\ell_i} +|\xi_i|$, $\bar\Delta_i = \Delta_{\ell_i} +|\bar\xi_i|$.
The  trilinear forms $\rho^{\ell_1\,\ell_2\,\ell_3}(\xi_1,\xi_2,\xi_3)$, universal for all CFT,
are uniquely determined by CWI \cite{Teschner:2001rv}.
Also the notion of the conformal block 
remains unchanged
\begin{eqnarray*}
{\cal F}_{Q,\,\ell}\left[
^{\ell_3\,\ell_2}_{\ell_4\,\ell_1 }  
\right]\!(z)&=& z^{\Delta_\ell -\Delta_{\ell_2} -\Delta_{\ell_1}}
\sum\limits_{n=0}^{\infty} z^n {\cal F}_{Q,\,\ell}\left[
^{\ell_3\,\ell_2}_{\ell_4\,\ell_1 }  
\right]\!(n)\\
{\cal F}_{Q,\,\ell}\left[
^{\ell_3\,\ell_2}_{\ell_4\,\ell_1 }  
\right]\!(n) &=& \sum\limits_{I,J \in {\cal I}_n} 
\rho^{\ell_4\,\ell_3\,\ell}(\omega,\omega,\xi_I)B_{Q,\ell}^{IJ}(n)
\rho^{\ell\;\;\ell_2\,\ell_1}(\xi_J,\omega,\omega) \ .
\end{eqnarray*}
(In the formulae above $\omega$ denotes the vacuum state in ${\cal
F}^{\scriptscriptstyle L},$
$\left\{ \xi_I \right\}_{I\in {\cal I}_n}$ is a basis on the level $n$ in ${\cal F}^{\scriptscriptstyle L}$,
and $B_{Q,\ell}^{IJ}(n)$ is the inverse to the Gramm matrix on the level $n$ in the Verma module with
the central charge $ c=1 +6Q^2$ and the highest weight 
$\Delta_\ell = \frac{Q^2}{4}\left(1 + \frac{\mu \ell^2}{8\pi^2}\right)$.)


In the simplest case of the four puncture correlator, 
\[
G^{\scriptscriptstyle L}(x,\bar x)
\; = \; \lim\limits_{z,\bar z\to \infty} z^{Q^2\over 2}\bar z{}^{Q^2\over 2}
 \left\langle
  P(z,\bar z)P(1,1)P(x,\bar x)P(0,0)
  \right\rangle\ ,
\]
the geometric factorization in the $s$-channel reads
$$
G^{\scriptscriptstyle L}(x,\bar x)= C(0,0,\ell_s)C(\ell_s,0,0) 
\left|{\cal F}_{Q,\,\ell_s}\!\left[^{\,0\;0\,}_{\,0\;0\,}
\right]\!(x)\right|^2\ ,
$$
where $\ell_s$ is the length of the closed geodesic $\Gamma_s$. 
Since $\ell_s(x,\bar x)$ is a real analytic function of the moduli parameter $x$
of the 4-punctured sphere the amplitude does not factorizes into a holomorphic
and an anti-holomorphic part. 

The basic  consistency condition is that the geometric factorization in each channel yields the
same result. In the slightly more general case of the four hole correlator it leads to the 
geometric bootstrap equations:
\begin{eqnarray*}
&&\hspace{-50pt} C(\ell_4,\ell_3,\ell_s)C(\ell_s,\ell_2,\ell_1)
\left|{\cal F}_{Q,\,\ell_s}\!\left[^{\ell_3\,\ell_2}_{\ell_4\,\ell_1 }  \right]\!(x)\right|^2\\
&=&
C(\ell_4,\ell_1,\ell_t)C(\ell_t,\ell_2,\ell_1)
\left|{\cal F}_{Q,\,\ell_t}\!\left[^{\ell_1\,\ell_2}_{\ell_4\,\ell_3 }  \right]\!(1-x)\right|^2\\
&=&
|x|^{-4\Delta_{\ell_2}} C(\ell_1,\ell_3,\ell_u)C(\ell_u,\ell_2,\ell_4)
\left|{\cal F}_{Q,\,\ell_u}\!\left[^{\ell_3\,\ell_2}_{\ell_1\,\ell_4 }  \right]
\!\left({\textstyle {1\over x}}\right)\right|^2
\end{eqnarray*}
where $\ell_s, \ell_t, \ell_s$ are the lengths of the closed geodesics in corresponding channels.
As in the standard CFT one can promote the geometric bootstrap equations to 
the basic dynamical principle of the theory. 


\section{Discussion and conclusions}
\setcounter{equation}{0}

In this paper we have formulated a new geometric approach to the Liouville theory.
Although the conformal properties and the spectra of conformal weights
are identical to those of the DOZZ approach 
the spaces of states and the factorization properties are essentially different. 

Whether the geometric bootstrap equations provide a plausible alternative to the DOZZ theory 
depends at the first place on 
the existence of their consistent solutions. 
Due to the complicated non-linear nature of the equations a direct analysis of this problem
is prohibitively difficult. It seems that the only strategy available is to construct a
candidate for the structure constants and then to verify whether it satisfies
the geometric bootstrap equations. 

The simplest possibility is to consider the DOZZ structure constant. 
This proposal might be motivated by the analytic continuation of the 3-point 
function from the weak to the strong coupling region and from the elliptic to
the hyperbolic weights. However we are not aware  of any argument in favor of the hypothesis 
that the structure constants solving the geometric bootstrap equations should admit
an analytic continuation to the weak coupling regime. Still verification of this 
proposal is a well posed problem worth further investigations.

The second possible way to find the structure constants is to analyze a path integral
representation of the 3-point function. It requires an appropriate
generalization of the Liouville 
action of the geometric approach from the conical and the puncture singularities to finite
holes. It involves in particular the open problem
of auxiliary parameters
and Polyakov's conjecture in the case of hyperbolic singularities. 
This topic seems to be especially interesting as it would provide a way for 
the semi-classical analysis of the theory by methods analogous to that 
developed by L.\ Takhtajan in the case of the puncture singularities.


Even if the structure constants were known, the verification of the
geometric bootstrap equations would still be a challenging task.
First of all the equations involve the lengths of closed geodesics in each
channel as functions of the locations of punctures. Up to our knowledge
even in the simplest case 
of 4-punctured sphere such functions are not known in a closed
form \cite{Hempel,Wolf}. Secondly the conformal block  
function is also not known in this case and can be studied only by the
numerical methods developed in  
 \cite{Zamolodchikov:ie,Zam}. Both problems are of their own interest
going beyond the present context. 


Let us finally comment on the problem of consistent interactions of non-critical Nambu-Goto 
string. The results of the present paper lead to the no-go theorem of the following
form. {\it The longitudinal dynamics of the non-critical Nambu-Goto string cannot be described by
any standard, in the sense of BPZ, conformal field theory with neither discrete nor continuous spectrum.}
This reduces the problem of non-critical string to the problem of existence of the Liouville
theory based on the geometric bootstrap equation. 
 
\section*{Acknowledgements}

We would like to thank J\"{o}rg Teschner and B{\'e}n{\'e}dicte Ponsot 
for discussion. Z.J. is grateful to Michael Wolf
for the correspondence and bringing to our attention the reference
\cite{Wolf}. The work of L.H. was supported by the EC IHP
network HPRN-CT-1999-000161. Laboratoire de Physique Th{\'e}orique is
Unit{\'e} Mixte du CNRS UMR 8627.

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\end{thebibliography}

\end{document}

