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\title{BRST Quantization of String Theory in $AdS_3$}
\author{Ari Pakman \\ Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel \\
E-mail: \email{pakman@phys.huji.ac.il}}

\abstract{We study the BRST quantization of bosonic and NSR
strings propagating in $AdS_3 \times \cn$ backgrounds. The
no-ghost theorem is proved  using the
Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed
representations of affine $\sl$ appear on an equal footing. Possible
generalizations to related curved backgrounds are discussed.}

\preprint{\hepth{0304230}\\RI-04-03}

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\begin{document}

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\section{Introduction}
The BRST quantization of gauge theories was first applied to
string theory in \cite{Kato:1982im}, where a proof of the no-ghost theorem
was given in the BRST context. Once the physical space was identified with the cohomology
of the BRST operator at a particular ghost number, the cohomology itself
is built more or less explicitly,
in order to show that it does not contain states of negative norm.
In particular, in \cite{Kato:1982im} it was shown that, for strings in flat space, physical states
can be isometrically identified with the excitations in
the light-cone gauge.


In \cite{FGZ} Frenkel, Garland and Zuckerman (FGZ) presented a
technically simple but conceptually deep new proof of the no-ghost
theorem in the BRST context, which relies heavily on the fact that the {\it relative}
BRST cohomology\footnote{The relative BRST cohomology is defined imposing the additional constraints
$b_0=L_0=0$ to the BRST cohomolgy (for fermionic strings in the Ramond sector, also $\beta_0=G_0=0$ ). See Appendix A.}
at non-zero momentum
is concentrated at one ghost number\footnote{As shown in \cite{FGZ}
(and in \cite{Figueroa-O'Farrill:1988hu} for the NSR string), the cohomology
of the {\it full} BRST complex at nonzero momentum  consists of two isomorphic copies at consecutive ghost numbers.},
a result known as the ``vanishing theorem''.
Once the latter holds\footnote{It should be noted
that the vanishing theorem
does not hold in less conventional settings.
For example, the BRST analysis of 2D gravity
coupled to $c \leq 1$ matter shows that physical states
are distributed through several ghost numbers \cite{Bouwknegt:1991yg}.}, the proof of the no-ghost theorem is reduced
to showing the equality between the  traces of two operators (the method was discussed in \cite{Spiegelglas}).
The power of the FGZ method lies on its economy: it probes the
unitarity of the physical sector without explicitly
building it.
The method was applied
to bosonic strings in \cite{FGZ,Spiegelglas},
to NRS strings in \cite{Figueroa-O'Farrill:1988hu},
to ${\mathbb{R}}^{1,d-1} \times \cn$~$(d \geq 2)$ backgrounds in \cite{Zuckerman:1987ds}
and to $\mathbb{R}^{1} \times \cn$ backgrounds in \cite{Asano}.

The FGZ proof of the vanishing theorem \cite{FGZ} and its NSR extensions in \cite{Figueroa-O'Farrill:1988hu},
use techniques of homological algebra
and do not depend on the details of the chiral algebra of
the matter conformal
field theory (CFT).
The only assumption needed is that the matter CFT should be an $L_-$-free
Virasoro module\footnote{This means that the Hilbert space of the chiral CFT
can be expressed as a direct sum of spaces, each of which is built by the
action of Virasoro $L_{n <0}$ operators on a Virasoro highest-weight state, such that
states with different $L_{n <0}$ content are linearly independent.
In the superconformal case, the operators $G_{n<0}$ should be included.
The importance of this condition can be seen in a hand-waving way as follows.
Since the  BRST charge $Q$ is built out of the Virasoro modes only (and knows nothing about the
worldsheet chiral algebra),
certain homological statements about $Q$ (such as the vanishing theorem), can  be proved
when a basis for the space upon which $Q$ acts, can itself be obtained from Virasoro modes only.}.
For flat space worldsheet CFT, the latter was shown in \cite{Brower}.

In this work we will apply the FGZ method to bosonic and NSR strings propagating in
$AdS_3 \times \cn$ backgrounds \cite{Malda,Malda2}. Here $\cn$ is a compact unitary CFT,
with the necessary central charge for  the worldsheet CFT to be critical.
These vacua are relevant in the context of the $AdS/CFT$ correspondence \cite{Aharony:1999ti}.
In the $AdS_3/CFT_2$ case, the infinite modes of  the two-dimensional (super)conformal algebras
in the dual boundary CFT can be constructed explicitly
(see \cite{GKS,Giveon:1999jg,Argurio:2000tb,Giveon:2003ku} and references therein).


The vanishing theorem holds in $AdS_3 \times \cn$ vacua,
because the worldsheet CFT is an $L_-$-free Virasoro module,
as shown in \cite{Hwang:1991an,Evans:1998qu} for regular $\sl$ representations and
in \cite{Malda,Pakman:2003cu}
for the spectrally flowed cases\footnote{For fermionic strings, refs. \cite{Evans:1998qu,Pakman:2003cu} consider only the
NS sector. The results can be easily extended to the R sector.
Alternatively, one can consider the $AdS_3 \times \cn$ backgrounds as
a particular case of the backgrounds studied in \cite{Asano},
the timelike $U(1)$ current being the $J^3(z)$ current of  $\sl$.
But since the whole $AdS_3 \times \cn$ CFT is an
$L_-$-free Virasoro module,
the new filtration introduced in \cite{Asano} (which
considers only the timelike $U(1)$, $c=1$ CFT), is not strictly necessary.}.



As in the $\mathbb{R}^{1} \times \cal{N}$ vacua studied in \cite{Asano},
$AdS_3 \times \cn$ is a nontrivial setting to see the
BRST quantization at work since, due to the absence of
a lightcone direction in $AdS_3$,
we cannot identify the action of the ghost degrees of freedom as annihilating
the ``transversal'' string excitations.


The unitarity puzzles posed by string theory in $AdS_3$ backgrounds \cite{Balog:1988jb}
were settled in the old covariant quantization (OCQ) formalism
in \cite{Hwang:1991an,Henningson:1990ua,Evans:1998qu,Pakman:2003cu}. It was shown there that physical states belong
to the $\sl /U(1)$ coset, modulo spurious states. This reduces the problem to the unitarity of the
coset representations, which was proved in \cite{DPL} and \cite{Pakman:2003cu}
for the bosonic and fermionic cases, respectively. Coset unitarity requires a bound for the
the spin of the discrete representations of $\sl$. This bound is rather ubiquitous, and we discuss it in Section 3.1.
As we show in Section 4, one important point in which the OCQ formalism differs from the FGZ approach,
is that for fermionic NRS strings, in the FGZ proof there is no need for the unitarity
of the {\it supersymmetric} $\sl /U(1)$ coset \cite{Pakman:2003cu}, the bosonic results
of  \cite{DPL} being sufficient.
A different approach to string unitarity in $AdS_3$ has been advocated in \cite{Bars:1995mf}.




Both in the OCQ and  FGZ approaches, the general results do not hold when $J^3_0=0$,
since the matter CFT is no longer an $L_-$-free module.
The unitarity of physical states must then be checked by hand\footnote{This is similar to the zero-momentum states in flat space.}.
These exceptional cases have been already studied in \cite{Malda,Evans:1998qu,Pakman:2003cu} for
different representations, and will not be considered here.



The plan of this work is as follows.
In Section 2 we
review the string spectrum in $AdS_3$.
In Section 3 we review the Hodge theory of the string BRST complex,
which is an elegant and clear setting in which to formulate
the FGZ method, which we do in Section 3.1.
In Section 4 we calculate all the traces needed to prove the no-ghost theorem
for the different representations.
Finally, in Section 5 we discuss possible generalizations to other curved backgrounds.
In Appendix A we summarize the essentials of the BRST formalism needed in this work.
Appendices B and C are devoted to technical points
used in the body of the paper. We work in the holomorphic sector.
For closed strings, a similar antiholomorphic copy should be considered.


\section{The affine $\sl$ theory}
\subsection{The bosonic algebra}
The space $AdS_3$ is the universal cover of the group manifold $\sl$, so the action
of a string moving in an $AdS_3$ background is an $\sl$ WZW model. The symmetry
algebra is the affine $\sl$ algebra at level $k$ ($k>2$), generated by three currents $J^{3,\pm}(z)$ satisfying the OPEs
\eqn{opeads}{\eqalign{
J^3(z) J^3(w) \sim & \, -{k/2 \over (z-w)^2} , \cr
J^+(z) J^-(w) \sim & \, {k \over (z-w)^2} + {2 J^3(w) \over z-w},\cr
J^3(z) J^{\pm}(w) \sim & \, \pm { J^{\pm}(w) \over z-w}.
}}
Using the Sugawara construction, the stress tensor is
\eqn{sugawara}{
T^{\ss{\sl}} = {1\over k-2} \frac12   \left(J^+J^- + J^+J^- -2J^3J^3 \right) .
}
and its modes close a Virasoro algebra with central charge $c_{\sl}={3k \over k-2}$.
Expanding the currents in modes as
\eqn{modos}{
J^{3,\pm}(z)= \sum_{n \in \mathbb{Z}} {J^{3,\pm}_n \over z^{n+1}},
}
the current algebra \opeads\ can be written as
\eqn{comm}{
\eqalign{
[J^3_n, J^3_m ]    &= - {k \over 2} n \delta_{n+m,0} \,,\cr
          [J^3_n, J^\pm_m ]  &= \pm J^\pm_{n+m} \,,\cr
          [J^+_n , J^-_m ]   &= -2J^3_{n+m} + kn\delta_{n+m,0}\,,
}}
with all other commutators vanishing.

The highest weight representations of \opeads\ and \comm\,
are built starting from
unitary representations of the $\sl$ Lie algebra  $J^{3,\pm}_0$.
These are characterized by the eigenvalues
of the Casimir operator $\frac12 (J^+_0 \! J^-_0  \! + \! J^-_0 \! J^+_0 ) \! - \! (J^3_0)^2$, which
will be denoted by  $-j(j-1)$, and the states within each representation are labelled by
the eigenvalue of $J^3_0$, which will be denoted by $m$.
These representations are the primary states for the current algebra.
They are annihilated by $J^{3,\pm}_{n>0}$,
and the Fock space of states is
built by acting on them with $J^{3,\pm}_{n<0}$.

The unitary representations of the $\sl$ Lie algebra $J^{3,\pm}_0$
which appear in the spectrum of strings moving in an $AdS_3 \times \cN$ background are \cite{Malda, Malda2}:

\begin{enumerate}
\item {\bf Highest weight discrete representations}
\[ \cD{j}^+ = \left\{ |j;m \rangle : m=j,j+1, j+2, \cdots \right\} \]
where $|j;j \rangle$ is annihilated by $J_0^-$ and $j$ is a  real number such that
$1/2 < j < (k-1)/2$.

\item {\bf Lowest weight discrete representations}
\[ \cD{j}^- = \left\{ |j;m \rangle : m=-j,-j-1, -j-2, \cdots \right\} \]
where $|j;-j \rangle$ is annihilated by $J_0^+$ and $j$ is a  real number such that
$1/2 < j < (k-1)/2$.

\item {\bf Continuous representations}
\[ \cCo{j,\alpha} = \left\{ |j,\alpha;m \rangle : m= \alpha, \alpha \pm 1,\alpha \pm 2, \cdots \right\} \]
where $0 \leq \alpha < 1$ and $j = 1/2 + i s $, where $s$ is a real number.
\end{enumerate}


The  bounds for  $j$ appearing for $\cD{j}^{\pm}$
can be understood in terms of consistency conditions for the primary states.
The lower bound, $ 1/2 < j$,  is necessary for the normalizability
of the primary states when their norm is interpreted as the ${\cal L}^2$
inner product of functions in the $\sl$ group manifold~\cite{DVV}.
Regarding the upper bound, $ j < (k-1)/2$, in \cite{Giveon} it was noted that it is necessary
for the unitarity of the primary states, when their
norm is interpreted as the two-point function
of the vertex operators creating them from the vacuum.
Moreover, adopting either the upper or the lower bound for $j$, the other one
appears when imposing the $w=\pm 1$ spectral flow (see below) to be a symmetry of the spectrum.
Finally, the compelling evidence for the correctness of these bounds on $j$ comes from
the fact that only this range of $j$ appears in the spectrum
of the thermal partition function of the model, computed by
path integral techniques in \cite{Malda2}.

In the computations of Section 4, we will be interested in the characters
of the representations \cite{Malda}, which are very easy to obtain
because  the action of the $J_{n<0}^{3,\pm}$ modes
is free for the representations considered \cite{Kac:fz}.
For $\cD{j}^+$ we have
\eqn{card}{\eqalign{
\chi_j^+(q,z)=\tr_{\cD{j}^+} \left( q^{L_0} z^{J_0^3} \right)
= &\, { q^{- {j(j-1) \over k-2}} z^j \over \prod^{\infty}_{n=1}(1-q^n)(1-q^{n-1}z)(1-q^nz^{-1})} , \cr
= &\, { q^{- {j(j-1) \over k-2}+ \frac18} z^{j - \frac12} \over i\Theta_1(q,z)}.
}}
Eq.\card\ should be understood as a formal series expansion
which converges to \card\ \linebreak when $|q|<|z|<1$. The elliptic
function $\Theta_1(q,z)$ is \cite{mumford}
\eqn{thetauno}
{\eqalign{
\Theta_1(q,z)= & -i z^{-1/2}q^{1/8}\prod_{n=1}^{\infty}(1-q^n)(1-q^{n-1}z)(1-q^nz^{-1})\,, \cr
= & -i \sum_{n \in \mathbb{Z}} (-1)^n z^{n-1/2} q^{{(n-1/2)^2 \over 2}}\,.
}}
For the $\cC_{j,\alpha}$ representations, the character is
\eqn{carc}{
\hat{\chi}_{j=1/2+is,\alpha}(q,z)=\tr_{\cC_{j=1/2+is,\alpha}} \left( q^{L_0} z^{J_0^3} \right)=
\, {q^{s^2 + 1/4 \over k-2} \over \prod^{\infty}_{n=1}(1-q^n)^3}
\sum_{n \in \mathbb{Z}}z^{\alpha+n} \,.
}
The character of $\cD{j}^-$ can be obtained from the spectral flow of $\cD{j}^+$ (see below).



\subsection{$\sl/U(1) \times U(1)$ decomposition and spectral flow}
It will be convenient to decompose both the
currents and the spectrum into the direct product of a
parafermionic $\sl/U(1)$ model and the timelike $U(1)$ model
corresponding to $J^3(z)$.
For the currents $J^{3,\pm}(z)$, this can be performed by representing
them~as
\eqn{decomp}{\eqalign{
J^3(z) = & - \sqrt{\frac{k}2} \d X(z) \,,\cr
J^{\pm}(z) = & \psi^{\pm}(z) e^{\pm \sqrt{\frac2k} X} \,,
}}
with
\eqn{equis}{\eqalign{
X(z)X(w) \sim &  -\log (z-w) \,,\cr
X(z) \psi^{\pm}(w) \sim & \, 0 \,.
}}

\ni
Note that the field $X(z)$ is antihermitian, in order for  $J^3(z)$ to be hermitian
and $J^+(z)^{\dagger}=J^-(z)$.
The fields $\psi^{\pm}$ form an  $\sl/U(1)$ parafermionic theory introduced in \cite{Lykken},
and are a non-minimal generalization of the $SU(2)/U(1)$ parafermions of \cite{Fateev}.
The OPEs of $\psi^{\pm}$ can be read from \opeads\ and \equis.


The stress tensor is similarly decomposed into
\eqn{stressdeco}{
T^{\ss{\sl}} = T^{\ss{\sl/U(1)}} + T^{\ss{U(1)}}\,,
}
with
\eqn{stressuno}{
T^{\ss{U(1)}}= - \frac1k J^3J^3 \,.
}
The decomposition can also be performed on the representations.
Indeed, every representation $\cD{j}^+, \cCo{j}$ gives rise
to an infinite number of parafermionic representations $\lambda_{j,n},\hat{\lambda}_{j,n}$ with
$n \in \hZ$.
The $\cD{j}^+, \cCo{j}$ Hilbert spaces can be decomposed into
\eqn{decompo}{
\eqalign{
\cD{j}^{+} & = \sum_{n \, \in \, \mathbb{Z}} \lambda_{j,n} \otimes \cU{j+n} \,, \cr
\cCo{j,\alpha} & = \sum_{n \, \in \, \mathbb{Z}} \hat{\lambda}_{j,\alpha+n} \otimes \cU{\alpha+n} \,,
}}
where $\cU{m}$ is a highest-weight representation of the timelike current $J^3(z)$ with
$J^3_0$ eigenvalue~$m$. The $\lambda_{j,n},\hat{\lambda}_{j,n}$ representations
are built by acting with the modes of $\psi^{\pm}$ on para\-fer\-mio\-nic highest-weight states.
The reader can find details in \cite{Lykken,Fateev,cha,sfetsos}.
We will be mainly interested in their characters \cite{cha,sfetsos}, which are
\eqn{carpf}{
\eqalign{
\lambda_{j,n}(q) & =\tr_{\lambda_{j,n}} \!\! \lp q^{L_0} \rp =
{q^{- {j(j-1) \over k-2} + \frac{(j+n)^2}k} \over \prod_{n=1}^{\infty}(1-q^n)^2} \sum^{\infty}_{s=0}(-1)^s q^{\frac12 s(s+2n+1) } \,, \cr
\hat{\lambda}_{j,\alpha+n}(q)&= \tr_{\hat{\lambda}_{j,\alpha+n}} \!\! \lp q^{L_0} \rp =
{q^{- {j(j-1) \over k-2} + \frac{(\alpha+n)^2}k} \over \prod_{n=1}^{\infty}(1-q^n)^2} \,.
}}
In Appendix A we show explicitly that the decompositions \decompo\ are, in terms of the characters,
\eqn{cardeco}{\eqalign{
\chi^+_j(q,z) & = \sum_{n \, \in \, \mathbb{Z}} z^{j+n} \, \lambda_{j,n}(q)\, \zeta_{j+n}(q) \,, \cr
\hat{\chi}_{j=1/2+is,\alpha}(q,z) & = \sum_{n \, \in \, \mathbb{Z}} z^{\alpha +n} \, \hat{\lambda}_{j,\alpha+n}(q)\, \zeta_{\alpha+n}(q) \,,
}}
where
\eqn{pfuno}{
\zeta_{m}(q)\equiv \tr_{\cU{m}} \left( q^{L_0} \right) = {q^{-{m^2 \over k}} \over \prod^{\infty}_{n=1}(1-q^n)} \,.
}
%Let's consider the $\cD{j}^+$ case. A vertex  operator
%$\Phi_{j,m}$ which creates a highest weight state $|j;m \rangle$
%can be represented by \eqn{vertex}{ \Phi_{j,m}=\Psi_{j,m} \,\,
%e^{m \sqrt{\frac2{k}}X} } with \eqn{equisphi}{ X(z) \Psi_{j,m} (w)
%\sim 0 } and $\Psi_{j,m}$ being a primary field for the
%$\psi^{\pm}$ currents. The conformal dimensions  of these states
%are \eqn{dimprim}{\eqalign{ \Delta(\Psi_{j,m}) & = -{j(j-1) \over
%k-2} + {m^2 \over k } \cr \Delta(  e^{\ss{m \sqrt{\frac2{k}} X}} )
%& = - {m^2 \over k } }}
%
%All the states in the $\sl$ spectrum with
%$m \geq j$ can be obtained  from $|j;m \rangle$ by acting with the negative modes of
%$J^3(z)$ and $\psi^{\pm}(z)$.
%
%As for the states with $m<j$, they can be obtained from the following
%descendent of $\cD{j}^+$
%\eqn{bajoeme}{
%|j,j-n \rangle \equiv (J^-_{-1})^n |j,j \rangle \,.
%}
%The states \bajoeme\ are such that
%\eqn{propbj}{\eqalign{
%J^3_0 |j,j-n \rangle & = (j-n) |j,j-n \rangle \cr
%J^3_n |j,j-n \rangle & = 0\,, \qquad n>0
%}}
%so they can be created from the vacuum by acting with an operator
%\eqn{vertexdes}{
%\Phi_{j,j-n}= \Psi_{j,j-n} \,\, e^{(j-n) \sqrt{\frac2{k}}X}
%}
%with
%\eqn{equisphides}{
%X(z) \Psi_{j,j-n} (w) \sim 0 \,\,.
%}
%The operators $\Psi_{j,j-n}$ are not parafermionic chiral primaries, but they are $\sl/U(1)$ Virasoro primaries,
%as can be seen by applying the positive modes of $T^{\ss{\sl/U(1)}}$ on \bajoeme.
%Their conformal dimension is
%\eqn{dimprimdes}{
%\Delta(\Psi_{j,j-n})  = -{j(j-1) \over k-2} + {(j-n)^2 \over k } + n
%}
%As in  the $m \geq j$ case, all the states with $m < j$ can be obtained by acting on $|j;j-n \rangle$ with the negative modes of
%$J^3(z)$ and $\psi^{\pm}(z)$.
Now, given any integer $w$, the algebra \comm\ is preserved by the
spectral flow \cite{Malda, Malda2} $J_n^{3,\pm} \rightarrow
\tilde{J}_n^{3,\pm}$ defined by
\eqn{flow}{ \eqalign{
\tilde{J}_n^3 & = J_n^3 - {k \over 2} w \delta_{n,0}  \,, \cr
\tilde{J}_n^\pm & = J_{n \pm w}^\pm \,.
}}
This transformation maps  $L_0$ to
\eqn{flowele}{
\tilde{L}_0= L_0 + wJ^3_0 - \frac{k}4w^2 \,.
}
The spectral flow can be
implemented in  \decomp\ by
\eqn{flowx}{ X(z) \longrightarrow \tilde{X}(z)=X(z) + w \sqrt{\frac{k}{2}} \log z \,,
}
and leaving the
parafermions $\psi^{\pm}(z)$ untouched. For $w=\pm 1$,
the symmetry \flow\ maps the $\cD{j}^{\mp}$ representation
into $\cD{k/2-j}^{\pm}$.
Note that under this mapping the upper and lower bounds of $j$ are interchanged.
Including the $w=\pm1$ spectrally flowed states, we can just consider $\cD{j}^{+}$ representations.

For generic $w$,  it was shown in
\cite{Malda,Malda2} that  the Hilbert space $\cD{j}^+$ is mapped into new
representations which must be included in a consistent description
of a string propagating in $AdS_3 \times \cN$. The way to include these
representations is to take usual representations $\cD{j}^+, \cCo{j,\alpha}$
for flowed operators $\tilde{J}^{3,\pm}_n$,
and measure their quantum numbers with unflowed $J^3_0,L_0$.
Alternatively, instead of this {\it active} procedure, we can use a {\it passive} one,
and obtain the flowed sectors by acting with the unflowed
operators $J^{3,\pm}_n$ on flowed primaries. Starting with an $\sl$ vertex operator
\eqn{vertex}{
\Phi_{j,m}= \Psi_{j,m} \,\, e^{m \sqrt{\frac2{k}}X} \,,
}
where $\Psi_{j,m}$ is a parafermionic primary, the flowed vertex operator
is obtained applying to it the mapping \flowx, yielding
\eqn{vertexflow}{
\Phi_{j,m}^{w}= \Psi_{j,m} \,\,
e^{(m+wk/2) \sqrt{\frac2{k}}X} \,.
}

%The conformal dimension of
%$\Phi_{j,m}^{w}$ can be read from \dimprim\ and \dimprimdes, and
%the decomposition \decompo\ can be generalized to
%\eqn{decompoflow}{ \cD{j}^{w} = \sum_{n \, \in \, \mathbb{Z}}
%\ctD{j,n} \otimes \cU{j+n+kw/2} \,\,. }
\ni
Note that for $\cCo{j, \alpha}$ representations, all the excited physical states
belong to flowed sectors.
This can be seen from the mass-shell
condition,
\eqn{capademasa}{
L_0={\frac14 + s^2 \over k-2} -w(\alpha + n) + \frac{kw^2}{4} + N + h^{\rK} - 1 =0 \,,
}
where $h^{\rK}$ is a highest weight of the inner theory and N is the level. It is
clear that for $w=0$, eq. \capademasa\ can only be satisfied if $N=0$.

The characters of the flowed $\cD{j}^w$ representations are
\eqn{cardf}{\eqalign{
\chi^w_j(q,z)= & \tr_{\cD{j}^w} \left( q^{L_0} z^{J_0^3} \right)
= \tr_{\cD{j}^+} \left( q^{\tilde{L}_0-w\tilde{J}^3_0-kw^2/4} \, z^{\tilde{J}_0^3+kw/2} \right) ,\cr
=& \, q^{-kw^2/4} z^{kw/2} \, \tr_{\cD{j}^+} \left( q^{\tilde{L}_0} \, (zq^{-w})^{\tilde{J}_0^3} \right), \cr
= &\, { (-1)^w q^{\left[- {j(j-1) \over k-2} - (k-2)\frac{\,w^2}{4} -(2j -1)\frac{w}{2} + \frac18 \right]} \,
z^{\left[j + (k-2)\frac{w}{2} - \frac12\right]} \over i\Theta_1(q,z)} \,.
}}
where we have used the property \cite{mumford}
\eqn{pete}{
\Theta_1(q,zq^{-w})=(-1)^w q^{-w^2/2}z^w \Theta_1(q,z) \,.
}
For the $\cCo{j,\alpha}^w$ representations we obtain similarly
\eqn{carcf}{\eqalign{
\hat{\chi}^w_{j=1/2+is,\alpha}(q,z)= & \,\,  \tr_{\cCo{j,\alpha}^w} \left( q^{L_0} z^{J_0^3} \right), \cr
= & \, \, {q^{{s^2 + 1/4 \over k-2}-kw^2/4 -w\alpha} \over \prod^{\infty}_{n=1}(1-q^n)^3}
\sum_{n \in \mathbb{Z}}z^{\alpha+n+ kw/2}q^{-wn} \,.
}}

\ni
As is clear from \vertexflow, the effect of the spectral flow is to
"dislocate" the parafermionic and the $J^3_0$ quantum numbers by an amount $wk/2$.
The decompositions \decompo\ are generalized to
\eqn{decompoflow}{
\eqalign{
\cD{j}^{w} & = \sum_{n \, \in \, \mathbb{Z}} \lambda_{j,n} \otimes \cU{j+n+kw/2} \,,\cr
\cCo{j,\alpha}^{w} & = \sum_{n \, \in \, \mathbb{Z}} \hat{\lambda}_{j,\alpha+n} \otimes \cU{\alpha+n+kw/2} \,.
}}
In Appendix C we show explicitly that, in terms of the characters, the decompositions \decompoflow\ are
\eqn{cardecof}{\eqalign{
\chi^w_j(q,z) & = \sum_{n \, \in \, \mathbb{Z}} z^{j+n+ kw/2} \, \lambda_{j,n}(q)\, \zeta_{j+n+kw/2}(q) \,,\cr
\hat{\chi}^w_{j=1/2+is,\alpha}(q,z) & = \sum_{n \, \in \, \mathbb{Z}} z^{\alpha+n+ kw/2} \, \hat{\lambda}_{j,\alpha+n}(q)\, \zeta_{\alpha+n+kw/2}(q) \,.
}}







\subsection{The supersymmetric algebra}
For fermionic strings in $AdS_3 \times \cN$, we should consider
the current algebra of affine supersymmetric $\sl$ at level k. It is generated by three
dimension-$\frac12$  supercurrents  $\psi^{3,\pm}+ \theta J^{3,\pm}$,
whose modes satisfy
\eqn{commfer}{
\eqalign{
[J^3_n, J^3_m ]    &= - {k \over 2} n \delta_{n+m,0} \,, \cr
          [J^3_n, J^\pm_m ]  &= \pm J^\pm_{n+m} \,,\cr
          [J^+_n , J^-_m ]   &= -2J^3_{n+m} + kn\delta_{n+m,0} \,, \cr
      [J^3_n, \psi^\pm_m]&=\pm \psi^\pm_{n+m} \,,\cr
      [J^\pm_n,\psi^\mp_m]&=\mp 2\psi^3_{n+m} \,,\cr
      [J^\pm_n, \psi^3_m]&=\mp \psi^\pm_{n+m} \,,\cr
      \{ \psi^3_n,\psi^3_m \}&= -{k \over 2} \delta_{n+m,0}\,, \cr
      \{ \psi^+_n,\psi^-_m \}&= k \delta_{n+m,0} \,,
}}
with all other (anti)commutators vanishing.
The modding of $J^a_n$ is integer and that of $\psi^a_n$ half-integer (Neveu-Schwarz) or integer (Ramond).
The worldsheet CFT is supersymmetric,
the Sugawara stress tensor and the supercurrent being
\eqn{tensorT}{
\eqalign{ T^{\sl} &={1 \over 2k}[j^+j^- +j^-j^+] - {1 \over k}j^3j^3 -{1
\over 2k}[\psi^+ \partial \psi^- + \psi^- \partial \psi^+] +{1
\over k}\psi^3 \partial \psi^3  \,,\cr
G^{\sl} &={1 \over k} [\psi^+j^- +
\psi^-j^+] -{2 \over k}\psi^3j^3 - {2 \over k^2}
\psi^+\psi^-\psi^3 \,.
}}
As usual, purely bosonic currents can be defined by
\eqn{bosonicJ}{
\eqalign{ j^a &= J^a - \hat{J}^a \,,  \cr
      \hat{J}^a &= -{i \over k} f^a_{~bc} \psi^b \psi^c \,.
}}
The $j^a$ form an affine $\sl$ bosonic algebra at level $k+2$.
The $\hat{J}^a$ and the $\psi^a$ form a supersymmetric affine $\sl$ algebra
at level $-2$ which commutes with $j^a$. The spectrum
of the theory is the direct product of the Hilbert spaces of both theories.
In the bosonic sector, the reprentations are the ones considered above.
For the $\psi^a$ currents, we have the usual representations for free fermions, for
both Neveu-Schwarz and Ramond sectors. The spectral flow can be also defined
in the $\psi^a,\hat{J}^a$ theory, but it just amounts
to a rearrangement of the usual spectrum, as shown in~\cite{Pakman:2003cu}.










\section{The Hodge theory of the BRST complex}
The formalism reviewed below was developed in \cite{josebrst},
and can be found in \cite{jose}.
For similar results in the context of compact Lie algebras, see \cite{vanHolten:1990xd}.

We will denote by $\cF$ the Hilbert space composed of the ghost plus matter sectors, with the restriction
$b_0=L_0=0$, and also $\beta_0=G_0=0$  for fermionic strings in the Ramond sector.
Although $\cF$ is infinite dimensional
(due to the infinite degeneracy of the base states of the $\sl$ representations), it can be decomposed into
\eqn{efesuma}{
\cF = \bigoplus_{m} \cF(m) \,,
}
where each subspace $\cF (m)$ has $J^3_0=m$ and is finite dimensional.
In the following, we will consider each $\cF(m)$ space separately, although
we will keep the notation $\cF$.


The space $\cF$ has a natural grading under $N_g$ as
\eqn{sumag}{
\cF = \bigoplus_{g \in \mathbb{Z}} \cF_g \,,
}
where $\cF_g$ has ghost number $g$.
The action induced by $Q$ in each $\cF_g$ will de denoted by
\eqn{qun}{
Q_g: \cF_g \rightarrow \cF_{g+1} \,,
}
and the following sequence is a graded complex
\eqn{secuencia}{
\cdots
{\longrightarrow} \cF_{g-1} \stackrel{\ss Q_{g-1}}{\longrightarrow} \cF_{g} \stackrel{Q_{g}}{\longrightarrow}\cF_{g+1}{\longrightarrow}
\cdots \,\,.
}
For each $g$ we define the kernel and the image of $Q$ as
\eqn{kerima}{
\eqalign{
\textrm{Ker}\, Q_g & = \{\xi \in \cF_g \, | \, Q \xi =0 \} \,, \cr
\textrm{Im}\, Q_g & = \{ Q\xi \, | \, \xi \in \cF_g \} \,.
}}
The cohomology of $Q$ in $\cF$ is then defined as
\eqn{defcoho}{
\rH= \bigoplus_g \rH_g \,,
}
with
\eqn{defcohog}{
\rH_g = { \textrm{Ker}\, Q_g \over \textrm{Im}\, Q_{g-1}} \,.
}
Since the indefinite inner product in $\cF$  is non-degenerate,
an orthogonal basis in $\cF$ can be constructed.
Let us define the action of a linear hermitian operator $\cC$ on the states of such a  basis
as $1$ $(-1)$ if the state has positive (negative) norm.
Then for any pair of states $\xi, \eta \in \cF$ the inner product
\eqn{inprodc}{
\inc{\xi}{\eta} \equiv \inn{\xi}{\cC \eta} \,,
}
is clearly positive definite.
The latter implies that $\cC$ maps $\cF_g$ into $\cF_{-g}$
because the antihermitian character of $N_g$ implies that
two states can have nonzero $\inn{}{}$ inner product only if they have opposite ghost number.
Indeed, $\cC$ interchanges the $b_n$ and the $c_n$ modes
of any state (and the $\beta_n$ and $\gamma_n$ for fermionic strings), as we show explicitly
in Appendix B. Moreover, this mapping from $\cF_g$ to $\cF_{-g}$ is a
linear isomorphism because each $\cF_g$ is finite dimensional
and $\cC$ has empty kernel because $\cC^2=1$. Note that the action of $\cC$ is
similar to the action of the Hodge $*$ operator on p-forms.

Under this new inner product $Q$ is no longer hermitian,
but its conjugate $Q^*$ can be obtained from
\eqn{qconj}{
\inc{\xi}{Q\eta} = \inn{\xi}{\cC Q \eta} = \inn{\cC Q\cC \xi}{\cC \eta} = \inc{Q^*\xi}{\eta} \,,
}
where we have used $\cC^2=1$,  so we have $Q^*= \cC Q \cC$. From its definition it is clear that
$\left(Q^* \right)^2 = 0 $ and that $Q^*$ carries ghost number $-1$ and is hermitian under $\inn{}{}$.
As for $Q$, we will denote by $Q^*_g$ the action of $Q^*$ on $\cF_g$.

The positive definiteness of \inprodc\ allows us to decompose $\cF_g$ both as
\eqn{decouno}{
\cF_g = \textrm{Im}\,Q_{g-1} \oplus ( \textrm{Im}\,Q_{g-1} )^{\perp} \,,
}
and as
\eqn{decodos}{
\cF_g = \textrm{Im}\,Q^*_{g+1} \oplus ( \textrm{Im}\,Q^*_{g+1} )^{\perp} \,,
}
where $\lp \cdots \rp ^{\perp}$ means orthogonal with respect to $\inc{}{}$.
But, as can be easily checked, we have $( \textrm{Im}\,Q^*_{g+1} )^{\perp} = \textrm{Ker}\,Q_{g}$
and $( \textrm{Im}\,Q_{g-1} )^{\perp} = \textrm{Ker}\,Q^*_{g}$.
Applying now the decomposition \decouno\ to  $\textrm{Ker}\,Q_{g}$ we have
\eqn{decoker}{
\textrm{Ker}\,Q_{g} = \textrm{Im}\,Q_{g-1} \oplus \cH_g \,,
}
where we have defined
\eqn{hache}{
\cH_g = \textrm{Ker}\,Q_g \cap \textrm{Ker}\,Q^*_g \,.
}
The elements of  $\cH_g$  provide us with a  unique representative
for each cohomology class $\textrm{H}_g$. Indeed, from \hache\  we
see that the elements of $\cH_g$ are annihilated by $Q_g$, and two representatives
of the same class of
$\textrm{H}_g$ always differ by an element belonging entirely to
the first term of \decoker. Moreover, it is easy to check (using that $\inc{}{}$ is positive definite)
that a state belongs to $\cH_g$ iff it is {\bf harmonic}, i.e., it is annihilated by the {\bf Laplacian}
\eqn{laplace}{
\Delta \equiv Q Q^* + Q^* Q \,.
}
Inserting now \decoker\ into \decodos\ we obtain the {\bf Hodge decomposition}
\eqn{hodge}{
\cF_g = \textrm{Im}\,Q_{g-1} \oplus \textrm{Im}\,Q^*_{g+1} \oplus \cH_g \,.
}
It is easy to check that the isomorphism between $\cF_g$ and $\cF_{-g}$ provided by
$\cC$ acts on the terms of \hodge\ as
\eqn{mapeos}{\eqalign{
\cC : \, & \textrm{Im}\,Q_g \longleftrightarrow \textrm{Im}\,Q^*_{-g} \,,\cr
\cC : \, & \cH_g \longleftrightarrow \cH_{-g} \,,
}}
and from the second line of \mapeos\ we find that the
cohomologies $\textrm{H}_g$ and $\textrm{H}_{-g}$ are isomorphic, a result known
as {\bf Poincar\'e duality}.


Now, it is clear that
\eqn{dim}{
\di \cF_g = \di \rker Q_g + \di \rim Q_g \,,
}
and from \decoker\ we have $\di \rker Q_g = \di \rim Q_{g-1} + \di \cH_g$, which implies
\eqn{dimdos}{
\di \cF_g = \di \cH_g + \di \rim Q_g + \di \rim Q_{g-1} \,.
}
Performing an alternating sum of \dimdos\ over all $g$, the last two terms cancel pairwise and we obtain
the {\bf Euler-Poincar\'e identity}:
\eqn{eulerpoincare}{
\sum_g (-1)^g \di \cF_g = \sum_g (-1)^g \di \cH_g \,\,.
}

\subsection{The FGZ proof of the no-ghost theorem}
The physical space of states is defined to be the $Q$ cohomology in $\cF$.
Since the latter is concentrated at zero ghost number,
it can be identified with $\cH_0$, and
the no-ghost theorem asserts that states in $\cH_0$ have positive norm under the
usual inner product $\inn{}{}$.


Note that  from the second line of \mapeos\ we see that $\cC$ maps $\cH_0$ to itself,
so it can be diagonalized
in $\cH_0$ with eigenvalues $+1$ or $-1$ (because $\cC^2=1$), corresponding to states
with positive or negative $\inn{}{}$ norm, respectively.
Calling $N_+$ $(N_-)$ to the number of
$+1$ $(-1)$ eigenvalues, we have
\eqn{des}{
\di \cH_0 \geq N_+ - N_- = \tr_{\cH_0}\cC \,,
}
and it is clear that the no-ghost theorem is proved iff the equality in \des\ holds.


The centrality of the vanishing theorem
comes now to the fore, since
having  $\textrm{H}_{g \neq 0}= 0$ implies, through the Euler-Poincar\'e identity \eulerpoincare,
\eqn{epdual}{
\ind\equiv \sum_{g} (-1)^g \di \cF_g = \sum_{g} (-1)^g \di \cH_g = \di \cH_0 \,,
}
where $\ind$ is the {\bf Euler index} of the BRST complex.
As for $\tr_{\cH_0}\cC$, notice that the isomorphism \mapeos\ under $\cC$ between
$\rim Q_{-1}$ and $\rim Q^*_1$ implies
\eqn{trazanula}{
\tr_{ \ss{ \{ \rim Q_{-1} \oplus \rim Q^*_1 \} } } \, \cC = 0 \,,
}
so from \hodge\ we have
\eqn{trazaefecero}{
\tr_{\cH_0}\cC = \tr_{\cF_0}\cC \,.
}
Similarly, the isomorphism under $\cC$ between $\cF_g$ and $\cF_{-g}$
implies for $g \neq 0$ that
\eqn{trazanulados}{
\tr_{ \ss{ \{ \cF_{g} \oplus \cF_{-g} \} }} \, \cC = 0 \,,
}
and from this we obtain
\eqn{sign}{
\sig \equiv \tr_{\cF}\, \cC = \tr_{\ss{{\bigoplus}_g \cF_g}} \, \cC = \tr_{\cH_0}\cC \,,
}
where $\sig$ is the {\bf signature} of the BRST complex. Putting \des\ -- \sign\ together,
the no-ghost theorem is equivalent to the identity
\eqn{nog}{
\sig = \ind.
}





\section{No-ghost theorem for strings in $AdS_3$}
We will prove the identity \nog\ by comparing
\eqn{see}{\eqalign{
\sig(q,z) =  &  \tr \left(\cC q^{L_0} z^{J_0^3}\right) \,, \cr
\ind(q,t)  =  & \tr \left((-1)^{N_g} q^{L_0} z^{J_0^3}\right) \,,
}}
where the traces will be taken over the different representations.
Note that for our purposes the formal expansions in $z$ and $q$ are of very different nature.
Because of the mass shell condition $L_0=0$, in the expansion in $q$ we are only
interested in the term independent of q.\footnote{Note that the condition $b_0=0$ (and $\beta_0=0$) is already taken into account
by just ignoring the degeneracy of the ghost vacuum in the traces \see.}
On the other hand, we need all the powers of $z$, since the decomposition \efesuma\
was only done  in order to separate $\cF$ into finite-dimensional subspaces.

In the computation of \see, we will use the fact that the trace is multiplicative
under tensor products and additive under direct sums.
We will consider representations with arbitrary flow parameter $w$, which of
course include the regular representations ($w=0$).


\subsection{Bosonic strings}
In \cite{DPL} it was shown that $\sl/U(1)$ representations coming from
$\cCo{j,\alpha}$ representations with $k>2$, and those coming from $\cD{j}$ with $k>2, j < k/2$,
are unitary.
Since these bounds are satisfied in our case, the only
source of negative norms are the modes of $J^3(z)$.
The action of $\cC$ in the $\sl$ sector should clearly be
\eqn{accionc}{\eqalign{
\cC \psi^{\pm}(z) \cC & = \psi^{\pm}(z) \,, \cr
\cC J^3_{-n} \cC & = -J^3_{-n} \,.
}}

\newcommand{\trk}{{\tr_{\rK}\left(q^{L_0}\right)}}
\ni
We decompose $\cF$ into
\eqn{decfb}{
\cF=\cF_{(b,c)} \otimes \cF_{\sl} \otimes \cF_{\cN} \,,
}
and we further decompose the representations of $\cF_{\sl}$ as in \decompoflow-\cardecof.

The signature of the $\cD{j}^w$ representations can now be computed as
\eqn{signatura}{
\sig_{\cD{j}^w} (q,z)
 =   \, \sig_{(b,c)}(q)  \times \sum_{n \, \in \, \mathbb{Z}} z^{j+n+kw/2} \,\, \sig_{\lambda_{j,n}}(q) \, \sig_{\cU{j+n+kw/2}}(q) \times
\sig_{\cN}(q) \,.
}
The traces in \signatura\ can be computed as follows.
In the ghost sector, using the decomposition (\ref{ghostdecom}), we have
\eqn{signabc}{
\sig_{(b,c)}(q) = \tr_{(b,c)} \left(\cC q^{L_0} \right)= q^{-1} \prod_{n=1}^{\infty}(1-q^n)(1+q^n) \,.
}
The signature of the timelike $U(1)$ is easily found from \accionc\ to be
\eqn{signau}{
\sig_{\cU{j+n+kw/2}}(q)=\textrm{Tr}_{{\cal U}_{j+n+kw/2}} \lp {\cal C} q^{L_0}  \rp = {q^{-{(j+n+kw/2)^2 \over k}} \over \prod_{n=1}^{\infty}(1+q^n)} \,.
}
Since the theories $SL(2,R)/U(1)$ and $\cN$ are unitary, their signature is their character.
So from \signatura --\signau\ the total signature is then
\eqn{signais}{
\sig_{\cD{j}^w} (q,z) =
q^{-1}  \prod^{\infty}_{n=1}(1-q^n) \times \sum_{n \, \in \, \mathbb{Z}} q^{{(j+n+kw/2) \over k}^2}
z^{j+n+kw/2} \lambda_{j,n}(q)
\times \trk \,.
}
\vskip 1cm
\ni
The index of the $\cD{j}^w$ representation is
\eqn{index}{
\ind_{{\cD{j}^w}} (q,t)\,
 = \ind_{(b,c)}(q) \, \tr_{\cD{j}^w} \left( q^{L_0} z^{J_0^3} \right) \, \trk \,.
}
In the ghost sector, decomposing $\cF_{(b,c)}$ as
\eqn{ghostdecomdos}{
\cF_{(b,c)} = \bigotimes_{n=1}^{\infty} \left\{ \, |0 \rangle_g, b_{-n}|0 \rangle_g \right\} \otimes \{ \, |0 \rangle_g, c_{-n}|0 \rangle_g \} \,,
}
we have
\eqn{indexbc}{
\ind_{(b,c)}(q)=\tr_{(b,c)} \left((-1)^{N_g} q^{L_0} \right) = q^{-1} \prod^{\infty}_{n=1}(1-q^n)^2 \,,
}
and the index can be written as
\eqn{indexis}{
%\ind_{{\cD{j}^w}} =   \, {  z^{j+kw/2} q^{- {j(j-1) \over k-2} -1}  \prod^{\infty}_{n=1}(1-q^n) \over  \prod^{\infty}_{n=1}(1-q^{n-1}z)(1-q^nz^{-1})}
\ind_{{\cD{j}^w}}= \, q^{-1} \prod^{\infty}_{n=1}(1-q^n)^2  \times \chi^w_j(q,z)\times \trk \,.
}
Finally, the equality between \signais\ and \indexis\ follows immediately from \pfuno\ and \cardecof.

\vskip 1cm
\ni
Proceeding similarly for the $\cC_{j,\alpha}^w$ representations, we obtain
\eqn{sigcon}{\eqalign{
\sig_{\cC_{j,\alpha}^w}& =
q^{-1}  \prod^{\infty}_{n=1}(1-q^n) \times \sum^{\infty}_{n=-\infty} q^{{(\alpha+n+kw/2) \over k}^2}
z^{\alpha+n+kw/2} \hat{\lambda}_{j,\alpha+n}(q)
\times \trk  \,, \cr
\ind_{\cC_{j,\alpha}^w}& = \, q^{-1} \prod^{\infty}_{n=1}(1-q^n)^2  \times \hat{\chi}^w_{j,\alpha}(q,z)\times \trk \,,
}}
and the equality follows again from \pfuno\ and \cardecof.

\subsection{Fermionic strings}
For NSR strings, the computation of the traces is greatly simplified
by decomposing the $AdS_3$ sector at level $k$ into a purely bosonic $\sl$
model at level $k+2$ and three free fermions $\psi^a \, (a=1,2,3)$ (See Section 3.3).
From \commfer, we see that the action of $\cC$ on the free fermions~is
\eqn{accioncfer}{
\cC \psi^{a}_n \cC = (-1)^{\delta_{a,3}} \, \psi^{a}_n(z) \,.
}
The total Hilbert space $\cF$ can then be expressed as
\eqn{decff}{
\cF=\cF_{(b,c)} \otimes \cF_{(\beta,\gamma)} \otimes \cF_{\sl} \otimes \cF_{\psi^a} \otimes \cF_{\cN} \,.
}
Note that the traces on  $\cF_{(b,c)}  \otimes \cF_{\sl} \otimes \cF_{\cN}$ can be taken from the
above bosonic computation, including there the fermionic sectors of $\cN$.
We will compute the traces for $\cD{j}^w$ representations, the $\cCo{j,\alpha}^w$ case being identical.


\vskip 1cm
\ni
\textit{Neveu-Schwarz sector}

\ni
Using the decomposition (\ref{ghdemm}), and \accioncfer, we find
\eqn{sigbg}{
\sig_{(\beta,\gamma)}(q)=\tr_{(\beta,\gamma)} \left(\cC q^{L_0} \right)= \prod_{r=\frac12}^{\infty} {q^{\frac12} \over (1-q^r)(1+q^r)} \,,
}
\eqn{sigpsi}{
\sig_{\psi^a}(q) =\tr_{\psi^a} \left(\cC q^{L_0} \right)=\prod_{r=\frac12}^{\infty} (1-q^r)(1+q^r)^2 \,.
}
The total signature is then,
\eqn{signs}{
\sig^{NS}(q,z)=\sig_{\cD{j}^w}(q,z) \, \sig_{(\beta,\gamma)}(q) \, \sig_{\psi^a}(q)
= q^{\frac12} \times \sig_{\cD{j}^w}(q,z) \times \prod_{r=\frac12}^{\infty} (1+q^r) \,.
}
To find $\ind_{(\beta,\gamma)}$, we decompose
\eqn{bgdecdos}{
\cF_{(\beta,\gamma)} = \bigotimes_{r=1/2}^{\infty} \bigoplus_{N_r=0}^{\infty}
\left\{ \beta^{N_r}_{-r}|0 \rangle_g \right\}
\otimes
\bigotimes_{r=1/2}^{\infty} \bigoplus_{N_r=0}^{\infty}
\left\{  \gamma^{N_r}_{-r}|0 \rangle_g \right\} \,,
}
and then
\eqn{indbg}{
\ind_{(\beta,\gamma)}(q)=\tr_{(\beta,\gamma)} \left((-1)^{N_g} q^{L_0} \right)=\prod_{r=\frac12}^{\infty} {q^{\frac12} \over(1+q^r)^2} \,.
}
We also need
\eqn{indpsi}{
\tr_{\psi^a} \left( q^{L_0}\right)= \prod_{r=\frac12}^{\infty} (1+q^r)^3 \,.
}
So the total index is
\eqn{indns}{
\ind^{NS}(q,z)=\ind_{\cD{j}^w}(q,z) \, \ind_{(\beta,\gamma)}(q) \, \tr_{\psi^a} \left( q^{L_0}\right)
=q^{\frac12} \times \ind_{\cD{j}^w}(q,z) \times \prod_{r=\frac12}^{\infty}(1+q^r) \,.
}
The equality between \signs\ and \indns\ follows then from the bosonic results.


\vskip 2cm
\ni
\textit{Ramond sector}

\ni
The traces are computed similarly, giving for the signatures
\eqn{sigbgr}{
\sig_{(\beta,\gamma)}(q)= \prod_{r=1}^{\infty} {q^{3 \over 8} \over (1-q^r)(1+q^r)} \,,
}
\eqn{sigpsir}{
\sig_{\psi^a}(q)=q^{\frac3{16}} \times \prod_{r=1}^{\infty} (1-q^r)(1+q^r)^2 \,,
}
and
\eqn{sigram}{
\sig^{R}(q,z)=\sig_{\cD{j}^w}(q,z) \, \sig_{(\beta,\gamma)}(q) \, \sig_{\psi^a}(q)
=q^{9 \over 16} \times \sig_{\cD{j}^w}(q,z) \times \prod_{r=1}^{\infty} (1+q^r) \,.
}

\ni
For the computation of the index, we obtain
\eqn{indbgr}{
\ind_{(\beta,\gamma)}(q)= \prod_{r=1}^{\infty} {q^{3 \over 8}  \over(1+q^r)^2} \,,
}
\eqn{indpsir}{
\tr_{\psi^a} \left( q^{L_0}\right)=q^{\frac3{16}} \times \prod_{r=1}^{\infty} (1+q^r)^3 \,,
}
and
\eqn{indram}{
\ind^{R}(q,z)=\ind_{\cD{j}^w}(q,z) \, \ind_{(\beta,\gamma)}(q) \, \tr_{\psi^a} \left( q^{L_0}\right)
=q^{9 \over 16} \times \ind_{\cD{j}^w}(q,z) \times \prod_{r=1}^{\infty}(1+q^r) \,.
}
The equality between \sigram\ and \indram\ follows again from the bosonic results.

Notice that for the computation of $\sig$ in the fermionic case,
we have relied on the unitarity of the
bosonic $\sl/U(1)$ coset, and we did not need the unitarity of the
supersymmetric coset, as in the OCQ \cite{Pakman:2003cu}. This is related  to
the fact that with the FGZ method we bypass the building of the
physical states themselves, which do belong to the {\it supersymmetric} coset,
modulo spurious states.



\section{Discussion}
The applicability of the FGZ method depends on two points.
The first is expressing (a sector of) the matter chiral algebra as a free $L_-$-Virasoro
module. This should be checked in each case, although
the techniques of \cite{Asano,Brower,Malda,Hwang:1991an,Evans:1998qu,Pakman:2003cu}
can be adapted or adopted for a wide family of backgrounds.

The second point is having enough control over the representations of the
worldsheet CFT, so as to compute the traces \see.
Based on the results for $AdS_3 \times \cN$, several related exact CFT backgrounds
could be dealt with successfully. In particular, the BTZ black hole~\cite{Banados:wn} (an orbifold of $AdS_3$) and
pp-waves backgrounds, obtained as Penrose limits of $AdS_3 \times S^3 \times M^4$
(see \cite{Hikida:2003fp} and references therein).

An additional class of related backgrounds are gauged WZW models including an
\linebreak $\sl$ factor,
such as the Lorentzian 2D black hole \cite{Witten:1991yr,DVV} (${\cal M}=\sl/U(1)$),
the cosmological Nappi-Witten model \cite{Nappi:1992kv} (${\cal M}=\sl \times SU(2) /U(1)^2$)
and the  generalization of the latter studied in \cite{Elitzur:2002vw}.
These backgrounds are particularly interesting, because although the matter ${\cal M}$ CFT
(the relative cohomology of the BRST complex in the gauged WZW model)
contains negative-normed states,
it cannot be decomposed as $U(1) \times {\cal M}/U(1)$.
Here we can use to advantage the cohomological analysis of
gauged WZW models in \cite{jose,Hwang:1993nc}, which show that a
vanishing theorem holds in $G/H$ (for certain subgroups $H$).
This allows to compute $\sig$ and $\ind$ on $G/H$ by taking the traces
over $G \otimes  \cF_{ghosts} \otimes H$,
and following a chain of identities similar to those in \epdual\ and \sign.
In this way, as we did in this work, the flowed sectors of the $\sl$ factor in G can be included.
This will be worked out in detail in a future work.







\section*{Acknowledgements}
I thank for discussions G.~d'Appolonio, S.~Elitzur, A.~Giveon, D.~Kazhdan,
B.~Kol, \linebreak A.~Konechny, B.~Pioline, P.A.G.~Pisani, A.~Sever and B.~Zwiebach,
and especially \linebreak J.M.~Figueroa-O'Farrill.
I also thank P.A.G.~Pisani for collaboration at the early stages
of this work.
This work is supported in part by the Israel Academy
of Sciences and Humanities -- Centers of Excellence Program, the
German-Israel Bi-National Science Foundation, and the European RTN
network HPRN-CT-2000-00122.

\vskip 1.5cm
\ni
{\bf Note added:}
After this paper appeared in hep-th,
the work \cite{Asano:2003qb} appeared, where
the methods of \cite{Asano} are applied to $AdS_3 \times \cN$ backgrounds.



\newpage
\appendix
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}

\setcounter{equation}{0}
\section{BRST essentials}
We review here the basics of the $(b,c)$ and $(\beta,\gamma)$ algebra that we need in this work.
More details can be found in \cite{Polchinski:rq}. The zero-mode quantization
has some subtleties, which have been discussed, {\it inter alia}, in \cite{henneaux}.


\subsection{Bosonic strings}
The anticommuting $(b,c)$ reparametrization ghosts, with conformal dimensions $(2,-1)$, have the OPEs
\eqn{opebc}{\eqalign{
b(z)c(w)  & \sim c(z)b(w) \sim  {1 \over z-w} \,,\cr
b(z)b(w)  & \sim  c(z)c(w) \sim \, 0 \,.
}}
The ghost number operator is defined as
\eqn{ng}{\eqalign{
N_g & = - \oint dz \, b(z) c(z) \,, \cr
    & = \sum_{n=1}^{\infty}\left( c_{-n}b_{n} - b_{-n}c_{n}\right) + c_0b_0 - \frac12 \,.
}}
The ordering constant $-\frac12$ can be fixed in several ways \cite{Polchinski:rq},
the most immediate one being the requirement of $N_g$ being antihermitian,
i.e., $N_g^\dagger = -N_g$. For this we use
$b_n^{\dagger}=b_{-n}, c_n^{\dagger}=c_{-n}$.

The charge of the $(b-c)$ modes under $N_g$ is
\eqn{cargasbc}{\eqalign{
[N_g,b_m]= - b_m \,,\cr
[N_g,c_m]= + c_m \,.
}}
The zero modes $b_0, c_0$ satisfy the Clifford algebra $\{b_0,c_0 \}=1$,
so there is a  double-degenerate vacuum $|\uparrow \rangle , |\downarrow \rangle$, which satisfies
\eqn{ghvac}{\eqalign{
c_0 |\downarrow \rangle & = |\uparrow \rangle  \qquad \qquad b_0|\uparrow \rangle  = |\downarrow \rangle \,,\cr
c_0|\uparrow \rangle & = b_0|\downarrow \rangle = c_{n}|\uparrow,\downarrow \rangle  = b_{n}|\uparrow,\downarrow \rangle=0 \,, \qquad n>0.
%c_{n}|\uparrow,\downarrow \rangle & = b_{n}|\uparrow,\downarrow \rangle=0 \qquad n>0.
}}
On the vacua $|\uparrow \rangle , |\downarrow \rangle$ we have $L_0^{(b,c)}=-1$.
The zero-modes algebra implies that the inner product on the ghost space is consistently defined~by
\eqn{ghinp}{\eqalign{
\langle \uparrow | c_0|\uparrow \rangle & = \langle \uparrow | c_0|\uparrow \rangle =1 \,,\cr
\langle \uparrow | c_0|\downarrow \rangle & = 0 \,,
}}
and from this follows the inner product of any exited state.

The BRST charge
\eqn{brstcharge}{
Q = \oint dz \, c(z) \left( T^{M}(z) + \frac12 T^{(b,c)}(z) \right) \,,
}
is an hermitian operator which carries ghost number $+1$, and $Q^2=0$ if $c_M=26$. Here $T^M$ is the matter stress tensor.
In this work we consider the restriction to the space $\cF$ defined by $b_0=L_0=0$,
where $L_0$ is the zero mode of $T=T^M + T^{(b,c)}$.
This chooses the $|\downarrow \rangle$ vacuum,
which will be denoted $|0 \rangle_g$. In $\cF$ we redefine
$N_g$ to be in $\mathbb{Z}$ by assigning ghost number $0$ to $|0 \rangle_g$ and ignoring the last two terms in the expansion \ng.
The action of $Q$ is consistent with the restriction to $\cF$  because $\{ Q,b_0 \} = L_0$.







\subsection{Fermionic strings}
For  NSR strings, we have, on top of the $(b,c)$ system,
the super-reparametrization ghosts $(\beta,\gamma)$, with conformal dimensions $(\frac32,-\frac12)$ and OPEs
\eqn{opebg}{\eqalign{
\gamma(z)\beta(w) \sim \, & {1 \over z-w} \,, \cr
\beta(z)\gamma(w) \sim \, & {-1 \over z-w} \,,\cr
\beta(z)\beta(w) \sim \, & \gamma(z)\gamma(w) \sim \, 0 \,.
}}
The modes of $(\beta,\gamma)$ satisfy
$\beta_r^{\dagger}=-\beta_{-r}, \gamma_r^{\dagger}=\gamma_{-r}$,
where $ r \in \mathbb{Z} \, (r \in \mathbb{Z}+ 1/2)$ for the Ramond (Neveu-Schwarz) sector.
In the Ramond sector, the zero modes again make the vacuum double-degenerate.
We choose $\beta_0=0$ in $\cF$, and denote the selected vacuum also by $|0 \rangle_g$.
We have then,
\eqn{vbg}{
\beta_{r \geq 0}|0 \rangle_{g}  = \gamma_{r>0}|0 \rangle_{g} =0.
}
On $|0 \rangle_g$, we have $L_0^{(\beta,\gamma)}=\frac38 \,\, (L_0^{(\beta,\gamma)}=\frac12)$ for the Ramond (Neveu-Schwarz) sector.
The $|0 \rangle_g$ ghost number is zero, and the operator $N_g$ in \ng\ is extended to satisfy
\eqn{cargasbg}{\eqalign{
[N_g,\beta_r]= - \beta_r \,, \cr
[N_g, \gamma_r]= + \gamma_r \,.
}}
The BRST charge is now
\eqn{brstchfer}{
Q = \oint dz \, \left[ c(z) \left( T^{M}(z) + \frac12 T^{g}(z) \right) +
\gamma(z)\left( G^{M}(z) + \frac12 G^{g}(z) \right) \right] \,,
}
with $Q^2=0$ if $c_M=10$. Here $G^M$ is the matter supercurrent,
$T^{g}=T^{(b,c)} +T^{(\beta,\gamma)}$ and $G^g$ the supercurrent of the superconformal
combined $(b,c)-(\beta,\gamma)$ systems. The $L_0=0$ condition is imposed with $T^M + T^g$.
The space $\cF$ is again defined by $b_0=L_0=0$. In the Ramond sector, also by $\beta_0=G_0=0$, where
$G_0$ is the zero mode of  $G= G^M + G^{g}$. This
restriction is consistent because $\{Q,\beta_0 \} = G_0$.








\setcounter{equation}{0}
\section{The action of $\cC$ in the ghost sector}
In this Appendix we will show that the operator $\cC$, defined in Section 2, interchanges
between $b_n$ and $c_n$ (and $\beta_r$ and $\gamma_r$) modes, thus
mapping $\cF_g$ to $\cF_{-g}$.

We can decompose the ghost sector $\cF_{(b,c)}$ into
\begin{equation}
\cF_{(b,c)} = \bigotimes_{n=1}^{\infty} \left\{ \, |0 \rangle_g, |+,n \rangle \right\} \otimes \{ \, |0 \rangle_g, |-,n \rangle \} \,,
\label{ghostdecom}
\end{equation}
where
\eqn{gemasmenos}{
|\pm,n \rangle \equiv \frac{ ( b_{-n} \pm c_{-n}) }{\sqrt{2}} |0 \rangle_g \,,
}
satisfy
\eqn{gein}{\eqalign{
\langle \pm,n | \pm,m \rangle =&  \pm \delta_{n,m} \,, \cr
\langle +,n | -,m \rangle =&  0 \,,
}}
so from $ \cC | \pm,m \rangle= \pm | \pm,m \rangle$ we have
\eqn{cenghost}{\eqalign{
\cC c_n \cC & = b_n \,, \cr
\cC b_n \cC & = c_n \,.
}}
For fermionic strings, we use the decomposition
\begin{equation}
\cF_{(\beta,\gamma)} = \bigotimes_{r>0} \bigoplus_{N_r=0}^{\infty} \{  |+,N_r \rangle \} \otimes
\bigotimes_{r>0} \bigoplus_{N_r=0}^{\infty} \{ |-,N_r \rangle \} \,,
\label{ghdemm}
\end{equation}
where
\eqn{gemm}{
|\pm,N_r \rangle \equiv \frac{ ( \beta_{-r} \pm \gamma_{-r})^{N_r} }{\sqrt{2N_r!}} |0 \rangle_g \,,
}
satisfy
\eqn{geimm}{\eqalign{
\langle \pm,N_r | \pm,N_s \rangle =&  (\pm 1)^{N_r} \delta_{r,s}\delta_{N_r,N_s} \, \,, \cr
\langle +,N_r | -,N_s \rangle =&  0 \,,
}}
so we have
\eqn{cenghmm}{\eqalign{
\cC \beta_r \cC & = (-1)^{S(r)} \gamma_r \,, \cr
\cC \gamma_r \cC & = (-1)^{S(r)} \beta_r \,,
}}
where $S(r)$ equals $0 \, (1)$ if $r<0 \, (r>0)$.

\setcounter{equation}{0}
\section{$\sl/U(1) \times U(1)$ character decomposition}
In this Appendix we will prove explicitly the character decompositions \cardeco\ and \cardecof.
Let us start with the representation $\cD{j}^+$, whose character is
\eqn{carchi}{
\chi^+_j(q,z)= { q^{- {j(j-1) \over k-2}} z^j \over \prod^{\infty}_{n=1}(1-q^n)(1-q^{n-1}z)(1-q^nz^{-1})} \,.
}
We will use the technique of infinite partial fractions, which has been used in \cite{prod,thorn} in order
to prove similar identities. Define
\eqn{efene}{
f_{N}(q,z)= {1 \over {\prod}_{n=1}^{N} \lp 1-q^{n-1}z\rp \lp 1-q^nz^{-1} \rp } \,,
}
which can be decomposed into partial fractions as
\eqn{partialefe}{
f_{N}(q,z)=\sum^N_{k=1} {a_k(q) \over ( 1 - q^{k-1}z )( 1 - q^kz^{-1} )}\,.
}
The coefficients $a_k(q)$ are
\eqn{aka}{\eqalign{
a_k(q) & = \lim_{z \rightarrow q^k} ( 1 - q^{k-1}z )(  1 - q^kz^{-1} ) f_{N}(q,z) \,, \cr
       & = { 1 - q^{2k-1}  \over \prod_{n=k}^{N+k-1}(1-q^n)\prod^N_{n=1  n \neq k}(1-q^{n-k})} \,, \cr
       & = {(1-q^{2k-1}) (-1)^{k-1} q^{{k(k-1) \over 2}} \over \prod^{N+k-1}_{n=1}(1-q^n) \prod^{N-k}_{n=1}(1-q^n)} \,,
}}
thus for $N \rightarrow \infty$ we have

\eqn{efelimite}{\eqalign{
\lim_{N \rightarrow \infty} f_N(q,z) & = {1 \over {\prod}_{n=1}^{\infty} ( 1-q^{n-1}z)(1-q^nz^{-1} )} \,, \cr
 & = {1 \over \prod^{\infty}_{n=1}(1-q^n)^2 }
\left[ \sum_{k=1}^{\infty} {(1-q^{2k-1})(-1)^{k-1} q^{{k(k-1)
\over 2}} \over (1-q^{k-1}z)(1-q^kz^{-1})} \right] \,.
}}
The
denominators in the sum of \efelimite\ can be expanded and
resummed as
\eqn{sumageo}{\eqalign{
{1 \over (1-q^{k-1}z)(1-q^k z^{-1})} & = \sum^{\infty}_{m,n=0} q^{m(k-1)}z^m q^{nk}z^{-n} \,, \cr
& = {1 \over (1-q^{2k-1})} \left[ \sum^{\infty}_{n=0} q^{n(k-1)}
z^n + \sum^{-1}_{n=-\infty} q^{-nk} z^{n} \right] \,.
}}
Thus
\efelimite\ can be written as
\eqn{efelimitedos}{\eqalign{
{1 \over {\prod}_{n=1}^{\infty} ( 1-q^{n-1}z)(1-q^nz^{-1} )} & =
{1 \over \prod^{\infty}_{n=1}(1-q^n)^2 } \,\,\, \times \cr
 \left[ \sum^{\infty}_{n=0} \sum_{k=1}^{\infty} \right.   z^n (-1)^{k-1} & q^{\frac12 (k - 2n)(k-1)}
  \left. +  \sum_{n=-\infty}^{-1} \sum_{k=1+2n}^{\infty} \! \! \! \! z^n (-1)^{k-1} q^{\frac12 (k - 2n)(k-1)}
\right] \,,
}}
where in the second term we have changed the index $k$ to $k+2n$ in order to make the exponents of $q$ be  the same in both terms.
Now, using that for $n \in \mathbb{Z}$
\eqn{sumacero}{
\sum_{k=1}^{2n}  (-1)^{k-1} q^{\frac12 (k - 2n)(k-1)} = 0 \,,
}
and changing the index $k$ to $s=k-1$ we obtain finally
\eqn{igualdadfinal}{
{1 \over {\prod}_{n=1}^{\infty} ( 1-q^{n-1}z)(1-q^nz^{-1} )} =
{1  \over \prod^{\infty}_{n=1}(1-q^n)^2} \sum^{\infty}_{n=-\infty} \sum^{\infty}_{s=0} z^{n}(-1)^s q^{\frac12 s(s+2n+1)} \,,
}
and
multiplying both sides by
$q^{- {j(j-1) \over k-2}} z^j  \prod^{\infty}_{n=1}(1-q^n)^{-1}$,
the first line of \cardeco\ follows.

From \cardf, we see that the flowed characters can be obtained from $\chi^+_j(q,z)$ by
replacing $z \rightarrow zq^{-w}$, and multiplying the resulting expression by $z^{kw/2}q^{-kw^2/4}$.
So we have,

\eqn{uedt}{\eqalign{
\chi^w_j(q,z)= &
{q^{- {j(j-1) \over k-2}-kw^2/4} \, z^{kw/2} \over
\prod^{\infty}_{n=1}(1-q^n)^3} \sum^{\infty}_{n=-\infty} \sum^{\infty}_{s=0} (zq^{-w})^{n+j}(-1)^s q^{\frac12 s(s+2n+1)} \,, \cr
= &  {q^{- {j(j-1) \over k-2}} z^{j+kw/2}  \over
\prod^{\infty}_{n=1}(1-q^n)^3} \sum^{\infty}_{n=-\infty} \sum^{\infty}_{s=0}
q^{{(j+n)^2\over k}-{(j+n+kw/2)^2\over k}} z^n
 (-1)^s q^{\frac12 s(s+2n+1)} \,,
}}
and this equality is the first line of \cardecof.

The decompositions for $\cCo{j,\alpha}$ representations
are immediate and are left as an exercise.



\newpage




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