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\begin{document}
\large
\noindent G\"{o}teborg ITP 92-??\\
\noindent May 1992\\
\vspace*{10 mm}
\begin{center}
\LARGE \bf SU(1,1) STRINGS\\
\vspace*{10 mm}
\Large \bf Stephen Hwang
\rm and \bf Patrick Roberts\\
\rm Institute of Theoretical Physics\\
S-412 96 G\"{O}TEBORG, SWEDEN\\
\vspace*{10 mm}
\large
\bf abstract \rm \\
\end{center}
\large
We investigate a string moving on an $SU(1,1)$ group manifold.
\newpage
\pagestyle{plain}
\normalsize \noindent
\setcounter{section}{1}
\section{General Structure}
\setcounter{subsection}{3}
\subsection{Partition Functions}
In order to address the issue of modular invariance in $SU(1,1)$ string
theories, we shall construct characters for the various representations of
the $\widehat{SU(1,1)}$ Kac-Moody algebra. As we are interested in unitary
models, we will limit our discussion to the range where the Kac-Moody anomaly
takes on values $k<-2$. Furthermore, for discrete representations the spin $j$
must be within the range $k/2 < j < 0$ \cite{H}. As we have shown above, in this
range there are no null states in the highest weight modules so one expects
that the character formula can be constructed in a straight forward manner. As
we shall see, this is not the whole story.
As in \cite{HHRS}, we begin by writing down preliminary characters for the
representations of the discrete series, $\Dis$, in the range $k/2 < j < 0$,
where in the absence of null states we have simply the $J_{-n}^{a}$ generators
acting on a highest weight state:
\be
\Chi^{\pm}_{j,0}(\tau,\theta) &=& Tr_{\hat{\Dis}}\{
e^{2\pi i[(L_0 -c/24)\tau + J^3_0 \theta]}\} \nonumber \\
&=& e^{2\pi i(\frac{(2j+1)^2}{4(k+2)})\tau\mp 2\pi ij\theta}
R(\tau,\pm\theta)^{-1}
\label{eq:ndchar}
\ee
with
\be
R(\tau,\theta) = (1-e^{2\pi i\theta})e^{\pi i\tau/4}\prod_{n=1}^{\infty}
(1-e^{2\pi in\tau}e^{-2\pi i\theta})(1-e^{2\pi in\tau})
(1-e^{2\pi in\tau}e^{2\pi i\theta}).
\label{eq:denr}
\ee
The continuous representations have two series. The principal series $\Cont$,
with $a=0,1/2$, has a spin which is complex; $j=\frac{1}{2} + i\rho$,
$\rho\in\R^+$.
\be
\Chi^{a}_{j,0}(\tau,\theta) &=& Tr_{\hat{\Cont}}\{
e^{2\pi i[(L_0 -c/24)\tau + J^3_0 \theta]}\} \nonumber \\
&=& e^{2\pi i(\frac{(2j+1)^2}{4(k+2)})\tau\mp 2\pi ij\theta}
\sum_{n=-\infty}^{\infty} e^{2\pi i(n+a)\theta}(1-e^{2\pi i\theta})
R(\tau,\theta)^{-1} \nonumber \\
&=& \eta(\tau)^{-3} e^{-2\pi i(\frac{\rho^2}{k+2})\tau}
\sum_{n=-\infty}^{\infty} e^{2\pi i(n+a)\theta}
\label{eq:ncchar}
\ee
where $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$, and $q=e^{2\pi
i\tau}$.
The supplementary series has real spins in a limited range; $-\frac{1}{2}
3$. We know that in the case of
the $N=2$ superconformal algebra (SCA) that a central charge $c>3$ implies
that we have an extended SCA \cite{L} which reduces to a finite algebra for
rational values of $c$ \cite{L,V}. The same mechanism is at work here in the
$\widehat{SU(1,1)}$ case due to the close relationship with the $N=2$ SCA
\cite{DLP}. It was shown in \cite{ZF} that for $c<3$ both the $N=2$ SCA and
the $\widehat{SU(2)}$ current algebra may be decomposed into a parafermion and a
free boson compactified on a circle. The radius of compactification
distinguishes the two theories. Here the same relation exists between the
extended $N=2$ SCA of \cite{DLP} and the $\widehat{SU(1,1)}$ current algebra, but
now the parafermion is one of those treated in \cite{L}.
In order to make this ``extension" of the algebra explicit we introduce the
following transformation of the generators of $\widehat{SU(1,1)}$ which preserves
the algebra:
\be
J^{\pm}_n &\rightarrow& \tilde{J}_n^{\pm} \equiv J^{\pm}_{n\mp 2s}
\nonumber \\
J^{3}_n &\rightarrow& \tilde{J}_n^{3} \equiv J^{3}_{n}-sk\delta_n.
\label{eq:transj}
\ee
Inserting the transformed generators into the expression for the Sugawara
tensor one obtains the transformed Virasoro generators
\be
L_n \rightarrow \tilde{L}_n -2sJ^3_n + s^2k\delta_n.
\label{eq:transl}
\ee
Here the $L_n$ generators are normal ordered with respect to $J^a_n$, and the
$\tilde{L}_n$ with respect to the transformed $\tilde{J}_n^{a}$. We may now
define a state space of the extended Kac-Moody algebra corresponding to the
generators $\tilde{J}_n^{a}$. In this state space there corresponds an s-vacuum
$|0\rangle_s$ for any transformed Virasoro generator, $(\tilde{L}_n)_s$ Although
the algebra is invariant is under this transformation under any value of $s\in\R$,
we may insure world sheet parity automatically for $s\in\Z$. Otherwise, we expect
that the question of modular invariance to be a more delicate issue in this method
of construction based on the chiral algebra.
In order to gain a deeper understanding of this transformation, we shall now
investigate the symmetries of the weight space of an affine Kac-Moody algebra
$\hat{g}$ \cite{K,GW,GO}. This space is spanned by vectors of the form
$\lambda = (\ov{\lambda},k,-n)$, where $\ov{\lambda}$ is a weight vector of
the horizontal algebra $g$, $n$ is the {\em grade} associated with the
non-zero-mode contribution to the conformal weight, and $k$ is the Kac-Moody
anomaly of the representation. The root space is generated by the simple
roots of the horizontal algebra $\alpha_i=(\ov{\alpha}_i,0,0)$,
$i=1,2,\dots,rank(g)$, and one additional simple root
$\alpha_0=(-\ov{\psi},0,1)$, where $\ov{\psi}$ is the highest horizontal root.
On the weight space we can define a scalar product as follows:
\be
(\lambda,\lambda') &=& (\ov{\lambda},k,-n)\cdot(\ov{\lambda}',k',-n')
\nonumber \\
&=& (\ov{\lambda},\ov{\lambda}')-kn'-k'n
\ee
where $(\ov{\lambda},\ov{\lambda}')$ is the ordinary scalar product on the
horizontal weight space. The Weyl group $W$ is generated by reflections over
the simple roots
\be
\sigma_{\alpha_i}(\lambda = \lambda-2\frac{(\lambda,\alpha_i)}
{(\alpha_i,\alpha_i)}\alpha_i,\;\; i=0,1,\dots,rank(g).
\ee
Although $\widehat{SU(1,1)}$ has the same root lattice as $\widehat{SU(2)}$, not all
of the horizontal unitary representations of $SU(1,1)$ enjoy the reflection
symmetry of their compact cousins. On the discrete representations $\Dis$,
the horizontal reflection acts much like time reversal on the Lorentz group.
On these unitary representations it acts as an anti-unitary operator taking
$D_j^+$ to its complex conjugate, $D_j^-$. The fact that $J^3$ and $-J^3$ are
interchanged means that it actually involves time reversal.
On the {\em affine} $\widehat{SU(1,1)}$ weights we may use the extra simple root
$\alpha_0$ to reflect back the horizontal representation yielding a
non-trivial symmetry of the full affine algebra:
\be
t_{\alpha} &=& \sigma_{\alpha_1}(\sigma_{\alpha_0}(\lambda))
\nonumber \\
&=& (\ov{\lambda}-k\ov{\alpha},k,-(n-(\ov{\lambda},\ov{\alpha})+k))
\label{eq:tr}
\ee
where we have used the normalization, $\ov{\alpha}^2=1$. Comparing with
eq.(\ref{eq:transj}) and eq.(\ref{eq:transl}), we see that the first and
third components of eq.(\ref{eq:tr}) coincide with the algebra preserving
transformation for $s=1$. The affine Weyl group decomposes into a semi-direct
product of of the horizontal Weyl group and the group of translations in the
weight space generated by $t_{\alpha}$, where $t_{\alpha}t_{\beta}
=t_{\alpha+\beta}$ \cite{K,GW,GO}. Thus, our algebra preserving
transformation may be interpreted, for integral values of $s$, as the
translation subgroup of the affine Weyl group.
We are now in the position to write down the set of transformed characters,
keeping in mind that the extended $\widehat{SU(1,1)}$ module contains an infinite
sum over this set. For the unitary discrete representations we have
\be
\Chi^{\pm}_{j,s}(\tau,\theta) = Tr_{\hat{\Dis}}\{
e^{2\pi i[(L_0-2sJ_0^3+s^2k -c/24)\tau + (J^3_0-sk)\theta]}\}
\ee
The factor $q^{-2sJ_0^3}$ may be pinned to the $J_0^3$-phase by the
substitution
\be
\theta\rightarrow \theta -2s\tau.
\ee
Then, using the definition of the theta function
\renewcommand{\arraystretch}{.5}
\be
\Varth\!{\left[\begin{array}{c}
{\sss \alpha}\\ {\sss \beta} \end{array}\right]}
\!(\tau) =
e^{2\pi i\alpha\beta}q^{\frac{1}{2}\alpha^2}
\prod_{n=1}^{\infty}(1-q^n)(1+q^{n+\alpha-\frac{1}{2}}e^{2\pi i\beta})
(1+q^{n-\alpha-\frac{1}{2}}e^{-2\pi i\beta})
\ee
we may write
\be
\Chi^{\pm}_{j,s}(\tau,\theta) =
\Varth\!\!
\left[\begin{array}{c}
{\sss 1/2-2s} \\ {\sss \mp\theta+1/2 }\end{array}\right]
\!\!(\tau)^{-1}
q^{(k+2)(s\pm\frac{2j+1}{2(k+2)})^2}e^{-2\pi i\theta(s(k+2)\pm(j+1/2))}
\label{eq:dchar}
\ee
\renewcommand{\arraystretch}{1}
We have earlier claimed that the transformations represent windings in the
$J^3$ direction. To see this, consider the coset theory $SU(1,1)/U(1)$. $L_0$
is then replaced by $L_0^\prime \equiv L_0 -\frac{1}{k} \sum_n : J^3_{-n}
J^3_n :$, which is invariant under the transformations eq.(\ref{eq:transj}) and
eq.(\ref{eq:transl}). We take this as a first indication that the different
sectors represent windings in the $U(1)$ time direction.
If this interpretation is correct, we should sum over all sectors labeled by
integer $s$ to obtain the full character based on the global representations
$D^\pm_j$.
Comparing (\ref{eq:dchar}) with the zero-mode factor in the character for one
circular time-like dimension
\begin{equation}
e^{4\pi i\tau\left( {nT-{\ell \over {2T}}} \right)^2}e^{-i\pi \theta \left( {{\ell
\over {2T}}-nT} \right)} \to e^{2\pi i\tau 2T^2(n-{\ell \over 2T^2})^2}
e^{2\pi i2T^2(n-{\ell \over 2T^2})}
\end{equation}
we observe that the numerator
representing highest or lowest weight states corresponds to a circle of
``radius" $T=\sqrt {-1-{k \over 2}}$ (here $\theta$ has been rescaled.) Thus,
the sector label $s$ enters as a winding number and $2j+1$ as a momentum number.
The denominator contributes the whole tower of other states
required by the Kac-Moody algebra. Clearly, the presence of this factor will
affect how the left-moving and right-moving representations combine to give
modular invariance.
The transformed continuous characters are
\be
\Chi^{a}_{j,s}(\tau,\theta) &=& Tr_{\hat{\Cont}}\{
e^{2\pi i[(L_0-2sJ_0^3+s^2k -c/24)\tau + (J^3_0-sk)\theta]}\} \nonumber \\
&=& \eta(\tau)^{-3} e^{-2\pi i(\frac{\rho^2}{k+2})\tau}
\sum_{n=-\infty}^{\infty}q^{\frac{(n+a+sk)^2}{k}-\frac{(n+a)^2}{k}}e^{2\pi
i(n+a)\theta} ,
\ee
and
\be
\Chi^{E}_{j,s}(\tau,\theta) &=& Tr_{\hat{E_j^0}}\{
e^{2\pi i[(L_0-2sJ_0^3+s^2k -c/24)\tau + (J^3_0-sk)\theta]}\} \nonumber \\
&=& \eta(\tau)^{-3} e^{-2\pi i(\frac{2j+1}{4(k+2)})\tau}
\sum_{m=-\infty}^{\infty}q^{\frac{(n+a+sk)^2}{k}-\frac{(n+a)^2}{k}} e^{2\pi
im\theta} \ee
We have been unable to construct modular invariant partition functions using the
transformed characters of the supplementary series $\Chi^{E}_{j,s}(\tau,\theta)$.
Furthermore, the Clebsch-Gordon coefficients of the horizontal $SU(1,1)$
representations vanish for the tensoring of any discrete or principal
representations into the supplementary. Thus, we may safely leave the
supplementary series out of the remainder of the discussion.
\subsection{Modular Invariance}
We begin our study of modular invariance by remarking that since our string
model has a compact time component, we will run into divergences due to
windings in the real time direction. In this section we will consider the
divergent functions to be formally defined in the sense that the modular
properties may be extracted following formal manipulations. In the next
section we will show how one may regularize these divergences in a way that
preserves the modular properties discussed in the following.
We first consider the simplest extension of our algebra where we assume
$k\in\Z$, and $k<-2$. In \cite{HHRS} the following extended character was
presented for the discrete representations
\be
\Chi_j^{\pm}(\tau,\theta) &=& \sum_{s\in\Z}\Chi_{j,s}^{\pm}(\tau,\theta)
\nonumber \\
&=& \mp \frac{\Th_{\mp(2j+1),k+2}(\tau,\theta)}{
\Th_{1,2}(\tau,\theta)-\Th_{-1,2}(\tau,\theta)}
\ee
where the formal theta function is defined by
\be
\Th_{n,k}(\tau,\theta) = \sum_{m\in\Z+\frac{n}{2k}}
exp[2\pi ik(m^2\tau-m\theta)]
\label{eq:theta}
\ee
If we wish to describe a theory that enjoys time reversal symmetry, we may write
the partition function as
\be
Z^D(\tau,\theta) = \sum_{k/20$. In the transformation given in
eq.(\ref{eq:transj}), we may express the windings numbers, $s\in \frac{1}{2}\Z$,
as $s=r+s'$, where $s'\in q\Z$, and $r=0,1/2,\dots,(2q-1)/2$. The transformation
on the $J_{0}^{3}$ mode is then written as
\be
\tilde{J}_{0}^{3}=J_{0}^{3}-sk=J_{0}^{3}-r\frac{p}{q}-s'k.
\ee
Given that the eigenvalues of this generator are half-integers,
$\langle J_{0}^{3} \rangle\in\frac{1}{2}\Z$, then the transformed
generators have eigenvalues of the form $\langle\tilde{J}_{0}^{3}
\rangle\in r(p/q)+\frac{1}{2}\Z$, where we will refer to the
different values of $r$ as ``$r$-sectors". It is clear that the
$\tilde{J}_{0}^{3}$ eigenvalues belong to the $q$-th covering
group.
Let us define the $r$-shifted generators as:
\be
J^{\pm(r)}_n &\equiv& J^{\pm}_{n\mp 2r} \nonumber \\
J^{3\,(r)}_n &\equiv& J^{3}_n -rk\delta_n
\ee
which implies that
\be
L_n^{(r)} = L_n -2rJ_n^3 + r^2k\delta_n.
\ee
We may now shift each $r$-sector by the remaining $s'$-transformations with
$s'k\in\Z$ in the same manner as for the integral $k$ case:
\be
J^{\pm(r)}_n \ra \tilde{J}^{\pm(r)}_n &\equiv& J^{\pm(r)}_{n\mp 2s'} \nonumber \\
&=& J^{\pm}_{n\mp 2(r+s')} \nonumber \\
J^{3\,(r)}_n \ra \tilde{J}^{3\,(r)}_n &\equiv& J^{3\,(r)}_n-s'k\delta_n \nonumber
\\ &=& J_n^3 -(r+s')k\delta_n
\ee
Thus, the Virasoro generators transform as
\be
L_n^{(r)} \ra \tilde{L}_n^{(r)} &\equiv& L_n^{(r)}-2s'J^{3\,(r)}_n +s'^2
k\delta_n \nonumber \\
&=& L_n -2(r+s')J_n^3 +(r+s')^2k\delta_n.
\ee
We follow the same procedure as in Chapter 3 to construct characters in each
$r$-sector. The ``naive" discrete characters analogous to eq.(\ref{eq:ndchar})
are given as \be
\Chi_{j,0}^{(r)\pm}(\tau,\theta) =e^{2\pi i(k+2)[(r\pm\frac{2j+1}{2(k+2)})^2\tau
-(r\pm\frac{2j+1}{2(k+2)})\theta]} e^{\pm\pi i\theta}R(\tau,\pm\theta)^{-1}
\ee
where the factor $R(\tau,\theta)$ is defined in eq.(\ref{eq:denr}). Since,
\be
(k+2)(r\pm\frac{2j+1}{2(k+2)})^2 = pq (\frac{2rp\pm(2j+1)q}{2pq})^2
\ee
then we may write the sum over $s'$-sectors as
\be
\Chi_j^{(r)\pm}(\tau,\theta) &=& \sum_{s'\in
q\Z}\Chi_{j,s'}^{(r)\pm}(\tau,\theta) \nonumber \\ &=&
\Th_{2rp\pm(2j+1)q,pq}(\tau,\theta) e^{\pm\pi i\theta} R(\tau,\pm\theta)^{-1}
\ee
where the formal (divergent) theta function is defined as in
eq.(\ref{eq:theta}).
Modular invariant combinations of such theta functions are well known
\cite{GQ,DVV} to be
\be
Z(\tau,\theta) =|\eta(\tau)|^{-2} \sum_{l=p}^{-p-1}\sum_{m=-q}^{q-1}
|\Th_{mp-lq,pq}(\tau,\theta)|^2.
\ee
Due to the existence of null states for $j$ outside the range of unitarity,
$0>j>k/2$, the above characters are only valid if we truncate the theory to the
unitary $SU(1,1)$ string. The time-reflection symmetric combination of characters
given in eq.(\ref{eq:dpart}) allows for the restriction of $j<0$:
\be
Z^D(\tau,\theta) = \sum_{2j+1=p}^{-1}\sum_{2r=-q}^{q-1}
(\Chi_j^{(r)+}+\Chi_j^{(r)-})^*(\Chi_j^{(r)+}+\Chi_j^{(r)-})
\ee
where here we have chosen the diagonal invariant. If the above partition is to
contain only highest modules in the unitary range, $0>2j>k$, then $p>k+1>p/q-1$.
This condition places heavy constraints on the allowed values of $p/q=k+2$ for a
unitary, modular invariant theory, namely:
\be
k+2\in\Z &{\rm or}& \frac{1}{k+2}\in\Z.
\ee
The second series represents the only allowed fractional values for the Kac-Moody
anomaly, $k$. The central charge of this series may be written as
\be
c=3+6q,\;\;q=1,2,\dots
\ee
This is related to the series proposed in \cite{GN,Ger} for the consistent
unitary truncation of the state space in Liouville theory. Here we see that it is
the closure of the characters under the modular group on highest weight modules
in the unitary range $0>j>k/2$ which yield this series. The exact relation to the
Gervais-Neveu series is as follows: In \cite{GN} it was observed that a
non-critical string could be unitary if embedded in a space of $(D+1)$ dimensions
with $D=1,7,13,19,25$, and the extra ``time" dimension is generated by the
Liouville field. In our $SU(1,1)$ string, we have a target space of (2+1)
dimensions, so that a {\em critical} $SU(1,1)$ string satisfies the relation,
$c+(D-2)=26$, and the two series are related by $6q+D=25$ in the range
$1\leq D< 25$. Note that the critical 2-D black hole mentioned at the beginning of
this section is also included in this series ($q=4$), as well as its
supersymmetrized version ($q=2$).
It is interesting to note that there exist a certain mapping between these two
infinite series for modular invariant, unitary $SU(1,1)$ strings related to the
usual $R \ra 1/2R$ symmetry of strings on a torus. We have stated earlier that
the radius of the compact time direction, $R$, is related to the Kac-Moody
anomaly by the relation, $2R^2=k+2=p/q$. The mapping, $R \ra 1/2R$ exchanges
$p$ and $q$ so that we may relate theories with central charges of the form
\be
c=\frac{3q+6}{q} \leftrightarrow c=3+6q,\;\;q=1,2,\dots
\ee
The theories with a partition function of the form given in eq.(\ref{eq:dpart})
will have the same number of $SU(1,1)$ highest weight modules, and the
regularized partition functions will be equivalent in the limit $\theta \ra 0$.
We also comment that the conformal weights of the primary fields in the second
series with $c\geq 9$ are all positive.
{\em (DECOUPLING?)}
In order to relate the above partition functions to the covering groups of
$SU(1,1)$, we define
\be
j^{(r)} \equiv \frac{r}{q} +j,
\ee
so that we may consider $j^{(r)}$ to be the spin of a highest weight
representation on the $q$-th covering of $SU(1,1)$. Viewed in this way, each
``wind" now has length $s'=sq$, as if wrapped around the covering manifold. Thus
we may write
\be
Z^D(\tau,\theta) = \sum_{j^{(r)},\ov{\jmath}^{(r)}} M_{j^{(r)},\ov{\jmath}^{(r)}}
(\Chi^+_{\ov{\jmath}^{(r)}}+\Chi^-_{\ov{\jmath}^{(r)}})^*
(\Chi^+_{j^{(r)}}+\Chi^-_{j^{(r)}}).
\ee
The possible matrices, $M_{j^{(r)},\ov{\jmath}^{(r)}}$, may be found using the
known classification of non-unitary $SU(2)$ modular invariants \cite{KW} and the
relations to $SU(1,1)$ invariants given in Section 3.2.
We can also construct modular invariant partition functions from the characters
of the continuous series representations with $k\in\Q$ as straight forwardly as for
integral $k$. Here we must take the $J^3_0$ eigenvalues from the multiple
covering and sum over all all $s$-sectors such that $sk\in\Z$. Thus, the modular
invariant partition functions take the form
\be
Z^C(\tau,\theta)=\int_0^{\infty}d\rho|\sum_{0\leq a<1}
\Chi_{\rho}^a(\tau,\theta)|^2
\ee
where
\be
a 0 \frac{n}{2q},\; n=0,1,\dots,2q-1\}.
\ee
The characters are given in eq.(\ref{eq:cchar}) and $k+2=p/q$. Note that the
modular invariance of the continuous partition function gives no further
constraints on the Kac-Moody anomaly than already assumed, $k\in\Q$.
\section{The $\widehat{SU(1,1)}/\widehat{U(1)}$ Coset}
Recently the coset model $\widehat{SU(1,1)}/\widehat{U(1)}$ has received a flurry of
interest due to the interpretation of ref.\cite{W} that it is an exact solution of
a two dimensional black hole. Although the question of modular invariance of these
models has so far remained elusive, we are now in the position to construct
modular invariant partition functions using the methods of \cite{GQ,ZF,KP,HNY}.
This construction is based on the fact that the $\widehat{SU(1,1)}$ current
algebra may be represented by a free boson and an extended parafermion of Lykken
\cite{L,DLP}.
The Hilbert space of the $\widehat{SU(1,1)}$ current algebra may be decomposed as
\be
\cH_j^{SU(1,1)} = \bigoplus_{m}\cH_{j,m}
\ee
where
\be
\cH_{j,m} = \{J^{a_1}_{-n_1}\cdots J^{a_k}_{-n_k}|0;j,m\rangle,\;\;n>0\}
\ee
If $\psi(z)$ is a parafermion defined in \cite{L}, then the current generators
may be expressed as in \cite{DLP}:
\be
J^3(z) = i\sqrt{\frac{k}{2}}\partial_z \phi(z) \nonumber \\
J^+(z) = \sqrt{k}\,\psi(z):e^{i\sqrt{\frac{2}{k}}\phi(z)} \\
J^-(z) = \sqrt{k}\,\psi^{\dagger}(z):e^{-i\sqrt{\frac{2}{k}}\phi(z)} \nonumber
\ee
where $\phi(z)$ is a free boson normalized such that $\langle\phi(z)
\phi(w)\rangle =-ln(z-w)$. Using this representation on the highest weight states
of the $\widehat{SU(1,1)}$ algebra, we may further decompose the Hilbert space
into a direct product of the parafermionic and free bosonic hilbert spaces: \be
\cH_{j,m} = \cH^{P\!f}_{j,\tilde{m}}\otimes\cH^b_{\tilde{m}}
\ee
where $\tilde{m}$ is the transformed $\tilde{J}^3_0$ eigenvalue of
eq.(\ref{eq:transj}). Using this relation we may factorize the characters for the
discrete and continuous representations of $\widehat{SU(1,1)}$ in the zero
winding sector, $s=0$. \be
\Chi^{\pm}_{j,0}(\tau,\theta) &=& \sum_{\tilde{m}=-\infty}^j
Tr_{D^{\pm}_{j,\tilde{m}}}
\{e^{2\pi i[(\tilde{L}_0-\frac{c}{24})\tau +\tilde{J}^3_0\theta]}\} \nonumber \\
&=&\sum_{\tilde{m}=-\infty}^{\infty}Tr_{D^{P\!f}_{j,\tilde{m}}}
\{e^{2\pi i\tau(L_0^{P\!f}-\frac{c-1}{24})}\}\times
Tr_{\cH^b_{\tilde{m}}}\{e^{2\pi i\tau(L_0^{b}-\frac{1}{24})+2\pi
i\theta\tilde{J}^3_0}\} \\
\Chi^{a}_{j,0}(\tau,\theta) &=& \sum_{\tilde{m}=-\infty}^{\infty}
Tr_{C^{a}_{j,\tilde{m}}}
\{e^{2\pi i[(\tilde{L}_0-\frac{c}{24})\tau +\tilde{J}^3_0\theta]}\} \nonumber \\
&=&\sum_{\tilde{m}=-\infty}^{\infty}Tr_{C^{P\!f}_{j,\tilde{m}}}
\{e^{2\pi i\tau(L_0^{P\!f}-\frac{c-1}{24})}\}\times
Tr_{\cH^b_{\tilde{m}}}\{e^{2\pi i\tau(L_0^{b}-\frac{1}{24})+2\pi
i\theta\tilde{J}^3_0}\}
\ee
The change in the limits for the discrete characters is due to the action of the
free bosonic operators $:e^{\pm\sqrt{\frac{2}{k}}\phi(z)}:$ which will shift the
$m$-eigenvalue in both the positive and negative directions \cite{ACT}.
We have remarked earlier that the zero-mode energy of the
coset theory,
\be
L_0^{P\!f} = L_0 - \frac{1}{k}\sum_{n\in\Z} :J^3_{-n}J^3_n:
\ee
is invariant under the transformation $L_0\ra\tilde{L}_0$ of eq.(\ref{eq:transl}). This would
lead one to expect that all remnants of the winding would disappear from the
parafermionic character. On the contrary, in the product of Hilbert spaces, we
must project onto the {\em transformed} $\tilde{J}^3_0$ eigenvalues which are
certainly not invariant. This projection is what yields a different character for
each winding sector.
In order to factor out the bosonic character to arrive at an explicit expression
for the parafermionic string function of the discrete representations, we must
expand the denominator of eq.(\ref{eq:dchar}) using the identity \cite{T}
\renewcommand{\arraystretch}{.5}
\be
\Varth\!\left[\begin{array}{c}
{\sss \alpha}\\ {\sss \beta} \end{array}\right]\!\!
(\tau)^{-1} = \frac{1}{\eta(\tau)^3}e^{-\pi\alpha}\sum_{t=-\infty}^{\infty}
e^{2\pi i(t-\alpha)(\beta+\frac{1}{2})}\sum_{r=0}^{\infty}
(-1)^r q^{\frac{1}{2}(r-t+\frac{1}{2})^2 -\frac{1}{2}(t-\alpha)^2}
\ee
\renewcommand{\arraystretch}{1}
The naive discrete characters may now be written as
\be
\Chi^{\pm}_{j,0}(\tau,\theta) &=& \mp\eta(\tau)^{-3}
q^{\frac{(2j+1)^2}{2(k+2)}}\sum_{t=-\infty}^{\infty}\sum_{r=0}^{\infty}(-1)^r
q^{\frac{1}{2}r(r+1-2t)}e^{\mp 2\pi i\theta(j+t)} \nonumber \\
&=& \sum_{\tilde{m}=-\infty}^{\infty} (\eta(\tau)D^{(\pm)}_{j,\tilde{m}}(\tau))
(\frac{1}{\eta(\tau)}q^{\frac{\tilde{m}^2}{k}}e^{\pm 2\pi i\theta\tilde{m}}).
\ee
in the second equality we have defined the {\em string function} which here
takes the form,
\be
D^{(\pm)}_{j,\tilde{m}}(\tau) = \mp\eta(\tau)^{-3}\sum_{r=0}^{\infty}(-1)^r
q^{(k+2)(\frac{r}{2}\mp\frac{2j+1}{2(k+2)})^2-
k(\frac{r}{2}\pm\frac{\tilde{m}}{k})^2}.
\ee
We may express this in a more convenient form by letting $N=-(k+2)>0$, and
including the winding sectors explicitly, $\tilde{m}=m-sk$,
\be
D^{(\pm)}_{j,m+s(N+2)}(\tau) = \mp\eta(\tau)^{-3}\sum_{r=0}^{\infty}(-1)^r
q^{(N+2)(s+\frac{r}{2}\mp\frac{m}{N+2})^2-N(\frac{r}{2}\pm\frac{2j+1}{2N})^2}.
\ee
Summing over the winding sectors, we may construct the following modular
function,
\be
c^{2m-1}_{2j+1}(\tau) = \sum_{s\in\Z}(D^{(+)}_{j,m+s(N+2)}(\tau)
+D^{(-)}_{j,m+s(N+2)}(\tau)).
\label{eq:string}
\ee
It can be shown \cite{HNY,ACT,KP} that this function is equivalent to the
absolutely convergent {\em Hecke indefinite modular form}
\renewcommand{\arraystretch}{.5}
\be
c^L_M(\tau) = \sum_{\begin{array}{c}
{\sss -|x|