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\begin{document}
\large
\noindent G\"{o}teborg ITP 91-20\\
\noindent May 1991\\
\vspace*{10 mm}
\begin{center}
\LARGE \bf MODULAR INVARIANCE OF SU(1,1) STRINGS\\
\vspace*{10 mm}
\Large \bf M\aa ns Henningson, \bf Stephen Hwang\\
\rm and \bf Patrick Roberts\\
\rm Institute of Theoretical Physics\\
S-412 96 G\"{O}TEBORG, SWEDEN\\
\vspace*{5mm}
\Large \bf Bo Sundborg\\
\rm Institute of Theoretical Physics\\
Vanadisv\" agen 9\\
S-113 46 STOCKHOLM, SWEDEN\\
\vspace*{10 mm}
\large
\bf abstract \rm \\
\end{center}
\large
We investigate the question of modular invariance for a string moving on
an $SU(1,1)$ group manifold. By including new sectors corresponding to
windings
around the compact time-direction, modular invariance is achieved for
the discrete representations which are free of physical negative norm states.
The critical spacetime dimension is less than $26$.
\newpage \pagestyle{plain} \normalsize \noindent \bf 1. Introduction. \rm
While
there are many explicit examples of string theories in backgrounds with
purely
spatial curvature, string propagation on manifolds with nontrivial \it
spacetime
\rm curvature are poorly understood. They constitute a large class of new
backgrounds with interesting properties. For example, bosonic strings on
curved
non-compact manifolds may have a total spacetime dimension which is lower
than
the critical value 26, since the conformal anomaly can exceed the geometric
dimension. In this sense such string theories are truly lower-dimensional
string theories without the use of 2-d gravity. It will also be necessary to
understand propagation in curved spacetimes before one can say anything
definite
about string cosmology and discuss scenarios for the very early universe.
For the commonly
studied manifolds of type $\cal{M} \times K$, which are products of
Minkowski
space $\cal M$ and some internal manifold $\cal K$ with Euclidean signature,
string oscillations in $\cal K$ are represented by a unitary conformal field
theory. The absence of physical negative norm states is ensured by a total
conformal anomaly $c=26$ and the Minkowski space no-ghost theorem
\cite{BROWER,GODDARD-THORN}. In contrast, when the curvature is non-trivial
also
in the time-like direction one has to study a non-unitary conformal field
theory
(time-like oscillations lead to negative norm states), and prove absence of
physical negative norm states anew.
%There have been discussions about the consistency of the simplest example
%with
%such properties, namely string propagation on the $SU(1,1)$ groupmanifold,
%described by the non-unitary $SU(1,1)$ Wess-Zumino-Novikov-Witten model
%\cite{R1}.
In this letter we consider the simplest case, namely string propagation on
the
$SU(1,1)$ group manifold. Such string theories have previously been proven to
be
unitary \cite{HWANG}, and we will now demonstrate that modular invariance
can be
achieved while satisfying the conditions of this no-ghost theorem.
Thus, these string theories fulfil the basic requirements for consistency --
unitarity and modular invariance. A priori, there is no guarantee that
absence
of physical negative norm states is compatible with modular invariance, but
$SU(1,1)$ is a first non-trivial example after Minkowski space where both
requirements are met.
%The $SU(1,1)$ group manifold ($d=3$ anti-deSitter space) has a non-trivial
%topology $\bf R \rm ^2 \times S^1$, and we therefore expect winding states
%to
%appear in the string spectrum. Indeed, we will find contributions to the
%partition function that behave in close analogy to winding states on $S^1$.
%However, there are differences probably due to the curvature, and we
%interpret
%these sectors av manifestations of a new phenomenon, "non-abelian winding".
%These states may also be of importance in the Lorentzian version of Witten's
%$SL(2,\bf R \rm ) / \bf R$ "black hole" \cite{WITTEN}.
A new feature of $SU(1,1)$ is that modular invariance
requires the introduction of completely new sectors of states, in addition to
the
naive set of states required by the infinitesimal symmetries. These new
sectors
become less mysterious when we recall that the $SU(1,1)$ group manifold
($d=3$
anti-deSitter space) has a non-trivial topology $\bf R \rm ^2 \times S^1$.
Therefore new sectors due to windings around the compact time-direction are
expected. The loop group of maps from $S^1$ to $SU(1,1)$ with pointwise
multiplication splits into disjoint components labelled by the winding number
and the multiplication table of components is given by the homotopy group
$Z$.
An important step of our analysis will be the identification of unitary maps
between different sectors of the quantum state space, each one corresponding
to a different component. The conformal weights and the ``energies" of the
new
states agree with this interpretation, but the embedding in a non-abelian
group and the separation of leftmovers and rightmovers differ from ordinary
U(1) winding states. Thus we are dealing with a new construction,
``non-abelian winding". These new states disappear in the quotient
$SU(1,1)/U(1)$, i.e. in the Euclidean version of Witten's black hole
\cite{WITTEN}, but they may be important in the Lorentzian version.
The $SU(1,1)$ Wess-Zumino-Novikov-Witten model is characterized by a
$s\hat u(1,1)$ Kac-Moody algebra with central extension $x$ and a
stress-energy
tensor obtained by the Sugawara construction. The primary fields are then
labelled by the quantum numbers of the global $SU(1,1)$ symmetry; i.e. the
centre of mass motion of the string. Harmonic analysis on the group manifold
yields normalizable wavefunctions forming representations of $SU(1,1)$ that
fall
into three series $C_j$, $D^+_j$ and $D^-_j$ \cite{BARGMANN,VILENKIN}. The
``spin" $j$ runs over a continuum $j=-\frac{1}{2}+i \kappa$ in the principal
continous series $C_j$ and over negative integral and half integral values in
the discrete series $D^+_j$ and $D^-_j$. One should notice that the trivial
representation is not square integrable on the $SU(1,1)$ group manifold,
which
means that we will not have an invariant ground state of the theory. However,
we can still work in the Hilbert space of square-integrable functions where
an
identity operator is provided by the completeness relation for a basis in
this
space. Since products of regular square-integrable functions are
square-integrable it is consistent with the operator product to use this
Hilbert space, even though the conventional correspondence between operators
and states fails.
The Clebsch-Gordan coefficients for coupling the above representations
indicate
that all three series $C_j$, $D^+_j$ and $D^-_j$ are needed for the operator
product algebra to close. For the continuous series $C_j$ the no-ghost
theorem
\cite{HWANG} only requires $x<-2$ with no restriction on $j$, so finding a
modular invariant partition function only amounts to the determination of the
correct measure over $j$, which cannot cause a conflict with unitarity. For
the
discrete series $D^+_j$ and $D^-_j$ on the other hand, unitarity imposes the
restriction $j>\frac{x}{2}$ ($j=\frac{x}{2}$ being allowed but exceptional),
and one has to check that no representations outside this range appear in the
partition function. Therefore we have chosen to study the crucial modular
properties of the discrete series here. For integral $x$ one may also prove
selection rules which imply that the unitary range of representations is
closed
under operator products. We plan to return to this result and the question of
modular invariance of the continuous series in a later publication.
In the present study
of modular invariance we concentrate on characters of Kac-Moody
representations
rather than on the actual partition function, thus distinguishing states in
the
same representation of $SU(1,1)$ but with different ``energies"
$m$. In doing so we sidestep a divergence of the partition function arising
from
the simple fact that all (unitary) $SU(1,1)$ representations are infinite
dimensional. The characters we arrive at will also suffer from the same
disease
as the character for strings in flat Minkowski space prior to Wick rotation
and
integration over Euclidean momenta; fields of arbitrary large negative
conformal
dimension contribute. We expect both these problems to be solved by a proper
analogue of Wick rotation, and indeed combinations of our $s \hat u(1,1)$
characters will turn out to be of the same analytic form as $s \hat u(2)$
characters.
\bf 2. Representations of $s \hat u(1,1)$. \rm Our conventions are the same
as in
\cite{HWANG}, and the $s\hat u(1,1)$ algebra takes the form \begin{eqnarray}
&& [
J^+_m , J^-_n ] = 2 J^3_{m+n} + x m \delta_{m+n} \nonumber\\ && [ J^\pm_m ,
J^3_n ] = \mp J^\pm_{m+n} \label{algebra}\\ && [ J^3_m , J^3_n ] =
\frac{1}{2} x
m \delta_{m+n}. \nonumber \end{eqnarray} The difference from $s\hat u(2)$
lies in
the hermiticity property \begin{equation} (J^\pm_m)^\dagger = - J^\mp_{-m}.
\end{equation} Note that $J^3$ acts in the time direction, which is compact.
The Virasoro generators are of the Sugawara type \begin{equation} L_n =
\frac{1}{x+2} \sum_{m \in \bf Z \rm} : (J^3_{n-m} J^3_m + \frac{1}{2}
J^+_{n-m}
J^-_m + \frac{1}{2} J^-_{n-m} J^+_m) : \end{equation} and satisfy the
Virasoro
algebra \begin{eqnarray} && [ L_m , L_n ] = (m-n) L_{m+n} + \frac{c}{12} m
(m^2-1) \delta_{m+n} \\ && [ L_m , J^a_n ] = - n J^a_{m+n} \nonumber
\end{eqnarray} with conformal anomaly
\begin{equation} c =\frac{3x}{x+2}.\end{equation}
For the values of $x$ that we consider, $x<-2$, $c$ is greater than $3$, the
geometric dimension. The normal ordering above means treating $J^a_n$ as
creation
operators for $n<0$ and as annihilation operators for $n>0$. The ground
states
are annihilated by the modes $J^a_n$, $n>0$ and form representations of the
global $SU(1,1)$ generated by the zero-modes $J^a_0$. We label them by the
eigenvalues of the Casimir and the Cartan generator: \begin{eqnarray} && Q
|0;j,m> \equiv -( J^3_0 J^3_0 + \frac{1}{2} J^+_0 J^-_0 + \frac{1}{2} J^-_0
J^+_0
) |0;j,m> = -j(j+1) |0;j,m> \\ && J^3_0 |0;j,m> = m |0;j,m>. \nonumber
\end{eqnarray} The $D^+_j$ and $D^-_j$ series of unitary representations
contain
representations with $j=-\frac{1}{2},-1,-\frac{3}{2},\ldots$, and the $m$
values
form a positive increasing sequence $m=-j,-j+1,\ldots$ or a negative
decreasing
sequence $j=j,j-1,\ldots$ respectively.
Representations $\hat D^{\pm}_j$ of the Kac-Moody algebra are spanned by
vectors
\begin{equation} | \psi > = J^{a_1}_{-n_1} J^{a_2}_{-n_2} \ldots
J^{a_k}_{-n_k}
|0;j,m> \; \; , \; \; n_i > 0. \end{equation} To disentangle this space and
count states we need to know if there are any null vectors, i.e. if any
vectors
except the ground states are annihilated by all $J^a_n$, $n>0$. Using the
commutation relations (\ref{algebra}) one may prove that there are no null
vectors for representations with $j>\frac{x}{2}$ (see e.g. ref.
\cite{DIXON-P-L}). It is of great importance that this range is the range
allowed by the $SU(1,1)$ no ghost theorem \cite{HWANG}.
For integral $x$ and $2j$ such that $2j \leq x$ we may for example construct
null
vectors \begin{equation} | \psi > = (J^-_{-1})^{x-2j +1} |0;j,m=-j>.
\end{equation} For positive $x$ and compact groups Gepner and Witten
\cite{GEPNER-WITTEN} have shown that only integrable representations
(containing null vectors) appear. Modifying their argument to the case at
hand
one reaches the opposite conclusion that $SU(1,1)$ representations containing
null vectors decouple from the amplitudes, thus showing that the no-ghost
sector is closed under fusion. The conclusion does not generalize immediately
to non-integral $x$, or to any covering group of $SU(1,1)$. Thus one may
suspect that the size of $SU(1,1)$ is quantized. In any case only integer $x$
can be regarded as consistent without further analysis. We plan to return to
these issues and to the proof of decoupling in a later publication.
In absence of null vectors we can write down preliminary characters
\begin{eqnarray}
&&\chi^+_{j,0} \equiv Tr_{\hat D^+_j} (e^{2\pi i\left(
{(L_0-{\textstyle{c \over {24}}})\tau +J_0^3\theta } \right)} ) =
e^{2\pi i\left(
{{{(2j+1)^2} \over {4(x+2)}}-{1 \over 8}} \right)\tau -2\pi ij\theta }
R(\theta,\tau)^{-1}\\
&&\chi^-_{j,0} \equiv Tr_{\hat D^-_j} ( e^{2\pi i\left(
{(L_0-{\textstyle{c \over {24}}})\tau +J_0^3\theta } \right)} ) =
e^{2\pi i\left(
{{{(2j+1)^2} \over {4(x+2)}}-{1 \over 8}} \right)\tau +2\pi ij\theta }
R(-\theta,\tau)^{-1}\nonumber
\end{eqnarray}
with
\begin{equation}
R(\theta,\tau) \equiv
(1-e^{2\pi i \theta}) \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}) .
\end{equation}
The term ${\textstyle {c \over 24}}$ is chosen so that modular invariance for
component conformal field theories imply modular invariance for the string
theory if $c_{total}=26$. We have not been able to construct modular
invariant
combinations of left and right moving sectors using only the characters
$\chi^\pm_{j,0}$ with $j$ in the ghost-free range ${\textstyle {x \over
2}}0} {(1-e^{-\alpha
})^{mult\,\alpha }}}}. \label{WK}
\end{equation}
$\Lambda$ is the highest
weight, $\varepsilon (w)$ is $1 (-1)$ if $w$ can be written as an even (odd)
number of Weyl reflections, $\rho$ is a vector which can be calculated from
the
simple roots, the product is over positive roots and mult$\,\alpha$ is the
dimension of the root space of $\alpha$. The exponentials are to be
interpreted
as functions on root space acting as $e^\alpha (\beta) =e^{(\alpha,\beta)}$.
We
are concerned with $s\hat u(1,1)$ representations which are neither highest
weight nor integrable in the conventional sense, but Weyl-Kac formula is
still
useful if reinterpreted as below. For $s\hat u(2)$ and $s\hat u(1,1)$ the
vector $\rho =({\textstyle{1 \over 2}}\overline \alpha ,2,0)$, and
introducing
coordinates in root space $\beta =-2\pi i({\textstyle{\theta \over {\sqrt
2}}},\tau ,u)$ one can derive
\begin{equation}
e^{2\pi i\tau ( {{{(j+{{1 \over2}})^2} \over {x+2}}-{1 \over 8}})}
ch\,L(j{\overline \alpha},2,0)={{\Theta _{2j+1,x+2}-\Theta _{-2j-1,x+2}}
\over
{\Theta _{1,2}-\Theta _{-1,2}}}. \label{symchar}
\end{equation}
Here the theta-functions
\begin{equation} \Theta_{n,k}=e^{-2\pi iku}\sum\limits_{m\in {\rm
Z}+{\textstyle{n \over {2k}}}}{\exp (2\pi ik(m^2\tau -m\theta
))}\label{theta}
\end{equation}
represent contributions from summing over Weyl translations in
the Weyl-Kac formula, and the two terms in (\ref{symchar}) are due to the
finite
Weyl group $\left\{ 1,w_{\overline \alpha } \right\}$. These two terms have
precisely the same form as our characters (\ref{char}), only the values of
$x$
and $j$ differ, and a $u$-dependence has been introduced. Note that all
$2j+1$
dependence is modulo $2(x+2)$. Thus only a finite number of representations
enter. The theta functions have simple transformation properties under the
modular group which ensure that the characters (\ref{symchar}) form a unitary
representation of the modular group. Hence the diagonal combination of
leftmoving and rightmoving characters (\ref{symchar}) is modular invariant,
as
proven by Gepner and Witten \cite{GEPNER-WITTEN}.
If we can interpret (\ref{symchar}) as a character for $s\hat u(1,1)$ the
same
argument provides a modular invariant combination also for strings on
$SU(1,1)$. And all $j$ can then be taken to lie in the unitary range! The
minus
sign in (\ref{symchar}) which subtracts null states looks like an obstacle,
but
is in fact precisely what we need. Consider the characters for the global
$su(2)$ (an example of the Weyl-Kac formula):
\begin{equation}
ch(j{\overline\alpha} )(\theta )={{e^{-2\pi ij\theta }-e^{2\pi i(j+1)\theta
}}
\over {1-e^{2\pi i\theta }}}={{e^{-2\pi ij\theta }} \over {1-e^{2\pi i\theta
}}}+{{e^{2\pi ij\theta }} \over {1-e^{-2\pi i\theta }}}
\end{equation}
For $j<0$ this is precisely the character of $D_j^+\oplus D_j^-$, and in the
same
way $s\hat u(2)$ characters can be interpreted as $s\hat u(1,1)$ characters
$\chi_j^+ + \chi_j^-$ for negative $j$. Thereby the proof of modular
invariance
carries through to the time-reversal symmetric spectrum \begin{equation}
\sum\limits_{j=1+{\textstyle{x \over 2}}}^1 {(\chi _j^++\chi _j^-)_L^*(\chi
_j^++\chi _j^-)_R^{ }}
\end{equation}
Note that representations $j=-{\textstyle {1 \over 2}}$ and $j={\textstyle
{1 \over 2}}+{\textstyle {x \over 2}}$ do not contribute. This is satisfying
since $j=-{\textstyle {1 \over 2}}$ states are not normalizable. The no-ghost
theorem \cite{HWANG} applies to all representations in the sum. In addition
to
the diagonal modular invariants discussed above, there are invariants for
$s\hat u(2)$ \cite{CAPELLI,GEPNER-QIU}, which also work for $s\hat u(1,1)$.
Since there are more characters in the non-compact case there may also be
entirely new solutions.
We would like to thank Lars Brink for discussions and encouragement.
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\end{document}