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\begin{document}
\newpage\pagenumbering{arabic}
ITP 90 - 26\\
August 1990\\
revised: January 1991
\vspace{15mm}
\begin{center}
{\Large\bf Modular Invariants of
Ka\v{c}-Moody Algebras\\[3mm]
from\\[4mm]
Self-Dual Lattices}
\\[1cm]{\large Patrick Roberts\\[4mm] Haruhiko Terao\\
[4mm]}{\sl Institute of Theoretical Physics\\
CTH/GU\\S-412 96 G\"{o}teborg\\SWEDEN\\Bitnet\\
TFRPR@SECTHF51}
\end{center} \vspace{1cm}
\begin{abstract}
A general method for the construction of
modular invariant partition functions of
Ka\v{c}-Moody algebras using a covariant lattice approach
is presented. The A-D-E series for $A_1^{(1)}$ conformal
invariant theories and partition functions of $A_2^{(1)}$
are constructed explicitly to demonstrate the technique.
\end{abstract}
\newpage
\section{Introduction}
Two dimensional conformal field theories have attracted a great deal of interest
due to their relation to string theories, statistical systems, and recently,
two dimensional gravity. Since modular invariance of the partition
function severely restricts the variety of possible conformal invariant
field theories, the classification of modular invariant theories has received
much attention \cite{r}-\cite{w}. A well known example of this
classification is the A-D-E series for $A_1^{(1)}$ and $c<1$ minimal conformal
theories \cite{ciz1,gq,ciz2,k}. However, it has not yet been known how to
construct modular invariant partition functions generally, especially the so
called exceptional invariants.
Recently, one of us \cite{r} has reconstructed modular invariant partition
functions for minimal conformal field theories by means of the covariant lattice
approach. This approach, developed in the context of string theory,
automatically ensures the modular invariance of the model if the corresponding
lattice is even and self-dual. Moreover, various even self-dual lattices can be
generated by the so called ``shift method" in a rather general way. Indeed it
has been shown that the A-D-E sequence of partition function for the minimal
models may be constructed using this method \cite{r}.
In the following we show that this lattice approach is applicable to the
$A_1^{(1)}$ systems as well. We find all possible shifts acting non-trivially
and discuss the classification problem. We also find that the lattice structure
for the minimal conformal field theories are identical to this $A_1^{(1)}$
case. We conclude with a discussion of the extention of this lattice approach
to the affine partition functions in terms of $A_2^{(1)}$ characters. Although
these characters are much more complicated than in the $A_1^{(1)}$ case, one
finds that the symmetries of the corresponding lattices make this approach
quite managable, and we give examples of known modular invariant partition
functions.
\section{$A_1^{(1)}$ Characters}
The partition function of a conformal system is known to be expressable as a
sesquilinear combination of characters $\chi_l$ ;
\begin{equation}
Z=\sum_{l,\bar{l}}N_{l,\bar{l}}\chi_l\chi_{\bar{l}}^*.
\end{equation}
Here we concentrate on the conformal systems with Ka\v{c}-Moody $\hat{g}$
algebra which is the affine extention of the group $g$. The characters for
$\hat{g}$ are defined by
\begin{equation}
\chi_{\lambda}^{k}(\tau,\theta^{i}) =
tr_{{\cal H}_{\lambda}}[q^{L_0-\frac{c}{24}}e^{i\theta^iT^i}],\hspace{.5in}
q=e^{2\pi i\tau},
\end{equation}
where $k$ is the level of the algebra $\hat{g}$, c is the central charge,
$T^i$ are the generators of the Cartan subalgebra of $g$, and $\lambda$ denotes
an irreducible unitary highest weight representation of $g$. These characters
may be expressed compactly in terms of theta functions, yielding the Weyl-Ka\v{c}
formula which is the affine extention of the Weyl formula for the characters of
the group $g$:
\begin{equation}
\chi_{\lambda}^{k}(\tau,\theta^{i}) =
\frac
{\sum_{\sigma \in W(g)}\epsilon(\sigma)\Theta_{\sigma(\lambda+\rho),k+\tilde{h}}
(\tau,\theta^{i})}
{\sum_{\sigma \in W(g)}\epsilon(\sigma)\Theta_{\sigma(\rho),\tilde{h}}
(\tau,\theta^{i})},
\end{equation}
where $W(g)$ denotes the Weyl group of $g$, $\epsilon(\sigma)=det(\sigma)$, $\rho$
denotes half the sum of the positive roots of $g$, and $\tilde{h}$ is the dual
Coxeter number. The function $\Theta_{\lambda,k}(\tau,\theta^{i})$ is the
classical $g$ theta function defined by
\begin{equation}
\Theta_{\lambda,k}(\tau,\theta^{i})=
\sum_{n\in M+\frac{\lambda}{k}}q^{\frac{1}{2}kn^{2}}e^{ikn^i\theta^i},
\end{equation}
where M denotes the root lattice of $g$. For further details on these formulas
we refer to references \cite{gw,gko}.
In the SU(2) case this formula becomes quite simple because the root lattice is
one dimensional and the Weyl group is simply composed of $\{\pm 1\}$. The
character for the spin j representation is found to be
\begin{eqnarray}
\chi_{2j+1}^{k}(\tau,\theta) & = & \frac{1}{\Pi(\tau,\theta)}
\{ \Theta_{2j+1,k+2}-\Theta_{-(2j+1),k+2}\}
(\tau,\theta) \nonumber\\
& = & \frac{1}{\Pi(\tau,\theta)}
\sum_{l\in {\bf Z}}\{ q^{p(l+\frac{n}{2p})^2}e^{i\theta 2p(l+\frac{n}{2p})} -
q^{p(l-\frac{n}{2p})^2}e^{i\theta 2p(l-\frac{n}{2p})} \},
\end{eqnarray}
where we introduced
$\Pi(\tau,\theta)=\Theta_{1,2}-\Theta_{-1,2}$
and we let $n=2j+1$ and
$p=k+2$. It should be noted here that $n$ must satisfy
$0