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\begin{document}
\newpage\pagenumbering{arabic}
90 - 12\\
March 1990
\vspace{15mm}
\begin{center}
{\huge Modular Invariance of Minimal Models from Self-Dual Lattices}
\\[1cm]{\large Patrick Roberts}\\
\\[4mm]{\sl Institute of Theoretical Physics\\
CTH/GU\\S-412 96 G\"{o}teborg\\SWEDEN\\Bitnet\\ TFRPR@SECTHF51}\\
\end{center}
\vspace{1cm}
\begin{abstract}
The A-D-E sequence of modular invariant partition functions for Virosoro minimal
models is constructed in the Coulomb gas representation. We show that these partition
functions can be written as sums and differences of even self-dual lattice partition
functions.
\end{abstract}
\newpage
\section{Introduction}
The study of conformal field theories has proceeded at a considerable pace
over the past few years. Not only are conformal field theories of interest in two
dimensional statistical systems, but they also describe the classical vacuum of string
theories. These considerations have given motivation to a program to classify
all rational conformal field theories. Since constraints of modular invariance severely
limits the set of such theories, the classification of modular invariant partition
functions has received much attention recently \cite{CIZ}-\cite{SY}. The invariants of the
SU(2) Kac-Moody algebras, as well as their related coset models, have been shown by Cappelli,
et al.\cite{CIZ} to fall into an A-D-E classification scheme, but other extended algebras have
resisted attempts at such a classification. What is unfortunate is that there is no general
method for constructing the modular invariant partition functions, the ÒexceptionalsÓ being
the notorious case.
In the study of string theories, the covariant lattice approach \cite{LSW} was
developed to ensure modular invariance from the outset. By using a bosonic construction
of strings it was found that if the set of possible winding numbers and momenta of the
bosonic fields formed an even self-dual lattice, then the model would be automatically
modular invariant \cite{N1}. One of the short-comings of this approach has been that it was
limited to level-1 Kac-Moody and related algebras. With the developement of the Coulomb
gas representation \cite{FF}-\cite{FL} this situation has changed dramatically. In this
construction one introduces a background charge into a free bosonic field theory that affects
the central charge. With such a free field representation of a conformal field theory
one would expect to find similar constraints on the winding momentum as in covariant
lattice theories. It has been proposed \cite{CN} that all rational conformal field
theories may be represented by Coulomb gas models, but in the following we restrict ourselves
to the simple case of the Virosoro minimal models with $c<1$.
\section{Coulomb Gas Representation and Characters of Minimal Models}
The free field representation of minimal models has been presented in detail by many
authors \cite{FF}-\cite{F}, so we shall only sketch the main points leaning heavily
on the work of Caselle and Narain \cite{CN}. One begins by introducing a free bosonic
theory with a stress tensor of the form;
\begin{equation}
T(z) = -\frac{1}{2}(\partial_{z}\varphi(z))^{2} + i\alpha_{o}\partial_{z}^{2}\varphi(z)
\end{equation}
where we have choosen the normalization such that
\begin{equation}
<\varphi(z)\varphi(w)> = -\ln(z-w).
\end{equation}
The central charge is given by
\begin{equation}
c = 1 - 12\alpha_{o}^{2}
\end{equation}
and we recover the unitary series if we set
\begin{equation}
\alpha_{o} = \frac{1}{\sqrt{2m(m+1)}}\; .
\end{equation}
One represents highest weight states by vertex operators of the form
\begin{equation}
V_{\beta}(z) = :e^{i\beta\varphi(z)}:\; .
\end{equation}
The conformal dimension of $V_{\beta}(z)$ is given by
\begin{equation}
\Delta_{\beta} = \frac{1}{2}(\beta^{2}-2\alpha_{o}\beta) \; .
\end{equation}
In order to describe the possible values of $\beta$ it is convenient to define an abelian Lie
algebra ${\cal U}(\sqrt{2m(m+1)})$ with ``root''-lattice
\begin{equation}
\Gamma_{R}({\cal U}(\sqrt{2m(m+1)})) =
\{ k\rho \mid\rho = \sqrt{2m(m+1)}\;,\:k\in{\bf Z}\}.
\end{equation}
The dual of this lattice is the ``weight''-lattice defined by
\begin{equation}
\Gamma_{W}({\cal U}(\sqrt{2m(m+1)})) =
\{ k\alpha_{o} \mid\alpha_{o} = \frac{1}{\sqrt{2m(m+1)}}\;,\:k\in{\bf Z}\}
\end{equation}
which contains the root-lattice as a sublattice. One may speak of conjugacy classes as cosets
of the weights with respect to the root lattice:
\begin{equation}
(\ell) = \{\alpha_{o}\ell + \rho k\mid\ell,\:k\in{\bf Z}\}\; .
\end{equation}
When we say that $\beta$ belongs to the one dimensional lattice
\begin{equation}
\beta\in\Gamma_{1} = {\cal U}(\sqrt{2m(m+1)})
\end{equation}
we mean that $\beta$ belongs to the union of some set of conjugacy classes of which the
root lattice (7) must be one.
Now we introduce screening charges with vanishing conformal dimension:
\begin{equation}
Q^{\pm} = \int V_{\alpha_{\pm}}(z)dz\; ,\; \Delta_{\alpha_{\pm}} = 1.
\end{equation}
It has been shown \cite{FF}-\cite{F} that one may use these screening charges to construct null
states of the theory if the operator
\begin{equation}
V_{\beta}^{r\pm}(z) =
\int\cdots \int V_{\alpha_{\pm}}(z_{1}) \cdots V_{\alpha_{\pm}}(z_{r})V_{\beta-r\alpha_{\pm}}(z)
\prod_{i=1}^{r}dz_{i}
\end{equation}
is well defined. This is found to be the case for $V_{\beta}^{p+}(z)$ and
$V_{\beta}^{q-}(z)$ if
\begin{equation}
\beta (p,q) = \alpha_{o}(mp-(m+1)q)\; ;\; p,q\in{\bf Z}\; .
\end{equation}
We may identify the null states with points on the lattice $\Gamma_{1}$ by introducing
the operation\cite{CN}
\begin{equation}
g:\beta(p,q)\mapsto\beta(p,-q).
\end{equation}
Given a highest weight state $V_{\beta}(z)$, we know that we have a null state
$V_{g\beta}^{q-}(z)$ at level $N = pq$ above $V_{\beta}(z)$.
Using this knowledge about null states we may write down the character \cite{FF}-\cite{CN}
corresponding to the highest weight state $V_{\beta}(z)$; \begin{equation}
\chi_{\beta}(\tau) = \frac{1}{\eta (\tau )}\sum_{k=- \infty}^{\infty}
(q^{\frac{1}{2}(\beta -\alpha_{o}+k\rho )^{2}}-q^{\frac{1}{2}(g\beta -\alpha_{o}+k\rho )^{2}})
\end{equation}
where $\rho = \sqrt{2m(m+1)}$ is a root of ${\cal U}(\sqrt{2m(m+1)})$ , and
\begin{equation}
q = e^{2\pi i\tau},\;\;\eta (\tau ) = q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n}).
\end{equation}
We see that the sum is over the ${\cal U}(\sqrt{2m(m+1)})$ conjugacy classes generated by the
weights $\beta$ and $g\beta$. With the addition of more bosons to the theory, this character
formula generalizes to extended Virasoro algebras.
\section{Partition Functions}
Given the form of the character we may construct partition functions:
\begin{equation}
Z(\bar{\tau},\tau ) = \sum_{i,j} N_{ij} \bar{\chi}_{i}(\bar{\tau})\chi_{j}(\tau )
\end{equation}
for which we will seek modular invariant solutions. Furthermore, for a sensible theory we
require that the vacuum is unique, $ie. \; N_{00} = 1$, and all $N_{ij}$ are non-negative
integers.
In the spirit of the covariant lattice approach we introduce a {\em Lorenzian} lattice
\begin{equation}
\Gamma_{1,1} = {\cal U}(\sqrt{2m(m+1)})\times{\cal U}(\sqrt{2m(m+1)})
\end{equation}
with signature $\; (-,+)$, and consisting of the vectors
\begin{equation}
\vec{\beta} = (\beta_{L} ;\beta_{R}) \in\Gamma_{1,1}
\end{equation}
belonging to some subset of all pairs of ${\cal U}(\sqrt{2m(m+1)})$ conjugacy classes which
will be specified later. Here we apply the convention of ref. \cite{CN} where
$\beta_{L,R}=\beta +\alpha_{o}$, so that the conformal dimension of a highest weight state is
given by \begin{equation}
\Delta_{\beta_{L,R}} = \frac{1}{2}(\beta^{2}_{L,R}-\alpha_{o}^{2}) \; .
\end{equation}
The partition function may be written as
\begin{equation}
Z(\bar{\tau},\tau ) = \frac{1}{4} \sum_{(\vec{\beta})\in\Gamma_{1,1}}
\bar{\chi}_{\beta_{L}}(\bar{\tau})\chi_{\beta_{R}}(\tau )
\end{equation}
or,
\begin{equation}
Z(\bar{\tau},\tau )
= \frac{1}{4}\frac{1}{\bar{\eta}\eta}\sum_{\vec{\beta}\in\Gamma_{1,1}}
(\bar{q}^{\frac{1}{2}\beta_{L}^{2}}q^{\frac{1}{2}\beta_{R}^{2}}+
\bar{q}^{\frac{1}{2}(g\beta_{L})^{2}}q^{\frac{1}{2}(g\beta_{R})^{2}}-
\bar{q}^{\frac{1}{2}(g\beta_{L})^{2}}q^{\frac{1}{2}\beta_{R}^{2}}-
\bar{q}^{\frac{1}{2}\beta_{L}^{2}}q^{\frac{1}{2}(g\beta_{R})^{2}})
\end{equation}
where the sum in (21) is over {\em conjugacy classes} of $\Gamma_{1,1}$,
and in (22) we sum over all points on the lattice. One may be worried that the lattice sum
contains unphysical states. In terms of the operation $g$, these states are characterized
by the property that they are mapped onto themselves. Thus their characters vanish and
they decouple from the theory.
We assume that $g$ is an automorphism of $\Gamma_{1,1}$. Thus we may simplify (22) using
\begin{equation}
\sum_{\vec{\beta}\in\Gamma_{1,1}}(\bar{q}^{\frac{1}{2}\beta_{L}^{2}}q^{\frac{1}{2}\beta_{R}^{2}})=
\sum_{\vec{\beta}\in\Gamma_{1,1}}(\bar{q}^{\frac{1}{2}(g\beta_{L})^{2}}q^{\frac{1}{2}(g\beta_{R})^{2}}),
\end{equation}
and since $g^{2} = 1$,
\begin{equation}
Z(\bar{\tau},\tau ) = \frac{1}{2}\frac{1}{\bar{\eta}\eta}\sum_{\vec{\beta}\in\Gamma_{1,1}}
(\bar{q}^{\frac{1}{2}\beta_{L}^{2}}q^{\frac{1}{2}\beta_{R}^{2}}-
\bar{q}^{\frac{1}{2}(g\beta_{L})^{2}}q^{\frac{1}{2}\beta_{R}^{2}}).
\end{equation}
Defining the lattice $\Lambda_{1,1}$ containing the vectors
\begin{equation}
(g(\beta_{L}) ;\beta_{R}) \in\Lambda_{1,1}
\end{equation}
we may write the partition function as
\begin{equation}
Z(\bar{\tau},\tau ) = \frac{1}{2}\frac{1}{\bar{\eta}\eta}(\sum_{\vec{\beta}\in\Gamma_{1,1}}-
\sum_{\vec{\beta}\in\Lambda_{1,1}})
\bar{q}^{\frac{1}{2}\beta_{L}^{2}}q^{\frac{1}{2}\beta_{R}^{2}}.
\end{equation}
It is well known \cite{N1} that if $\Gamma_{1,1}$ is even, {\em ie.} if for all
$\vec{\beta}\in\Gamma_{1,1}\: ,\; \vec{\beta}^{2}\in 2{\bf Z}$, then the lattice sum
over $\Gamma_{1,1}$ is invariant under $\tau\rightarrow\tau + 1$. Further, if
$\Gamma_{1,1}$ is self-dual, that is if the volume of the unit cell is one and all scalar
products among lattice vectors are integers, then the lattice sum is invariant under
$\tau\rightarrow -\frac{1}{\tau}$. Thus, the lattice partition function of an even
self-dual lattice is modular invariant.
Since the operation of $g$ on a lattice point changes the conformal dimension of
the corresponding vertex operator by an integer, one may show that if $\Gamma_{1,1}$
is even and self-dual then $\Lambda_{1,1}$ is also even self-dual. This implies that
if $\Gamma_{1,1}$ is even self-dual, then $Z(\bar{\tau},\tau )$ is modular invariant.
The simplest two dimensional Lorentzian lattice one may construct which is even
self-dual is the one where the conjugacy classes of the two components are equivalent. This
corresponds to the {\em diagonal}, or A-invariant in the classification of ref \cite{CIZ}, and
in the following we shall call this lattice $\Gamma_{A}$. It is worth noting here that the
vacuum is represented by the two points \begin{equation}
\vec{\beta} = (\alpha_{o} ;\alpha_{o}) ,(-\alpha_{o} ;-\alpha_{o})
\end{equation}
{\em ie.} belonges to the conjugacy classes ((1);(1)) and ((-1);(-1)) of $\Gamma_{A}$.
Due to the factor of $\frac{1}{2}$ in eq. (26) we have ensured that the vacuum state
appears once in the partition function.
\section{Shifted Lattices}
Given an even self-dual lattice, one may construct other even self-dual lattices by
using the ``shift method'' presented in detail in \cite{LSW}. To shift a self-dual lattice
$\Gamma_{1,1}$ one considers a vector $\vec{\delta}$ which is not an element of
$\Gamma_{1,1}$, but for some smallest non-zero integer $n$,
\begin{equation}
n\vec{\delta} \in \Gamma_{1,1}
\end{equation}
We then define the sublattice
\begin{equation}
\Gamma_{o} = \{\vec{\beta}_{o}\mid\vec{\beta}_{o}\in\Gamma_{1,1}\;
and\; \vec{\beta}_{o}\cdot\vec{\delta}\in{\bf Z}\}.
\end{equation}
The new even self-dual lattice is the the union of lattices
\begin{equation}
\Gamma_{\delta} = \biguplus_{k=0}^{n-1}(\Gamma_{o} + k\vec{\delta}).
\end{equation}
We now shift the diagonal lattice $\Gamma_{A}$ by vectors belonging to the following conjugacy
classes: \begin{eqnarray}
\vec{\delta}_{a} & \in & ((-a(m+1)) ; (a(m+1))) \\
\vec{\delta}_{a}' & \in & ((-am) ; (am))
\end{eqnarray}
where $a\in {\bf Z}$.
Clearly these do not belong to $\Gamma_{A}$, but
\begin{eqnarray}
m\vec{\delta}_{a} & \in \Gamma_{A} \\
(m+1)\vec{\delta}_{a}' & \in \Gamma_{A}
\end{eqnarray}
if $m$ and $a$ are coprimes, otherwise the order of the shift is reduced by their common factor.
It is convenient here to introduce the standard parametrization of highest weight states
for minimal models:
\begin{equation}
\vec{\beta}(p,q;p,q) \in ((mp-(m+1)q);(mp-(m+1)q)) \subset \Gamma_{A}
\end{equation}
In this notation
\begin{eqnarray}
\vec{\delta}_{a}\cdot\vec{\beta}(p,q;p,q) = a(p-(\frac{m+1}{m})q) \;\; mod \: {\bf Z} \\
\vec{\delta}_{a}'\cdot\vec{\beta}(p,q;p,q) = a((\frac{m}{m+1})p-q) \;\; mod \: {\bf Z}
\end{eqnarray}
Thus, a shift by $\vec{\delta}_{a}$ leaves the $p$-parameter unaffected, and the
$\vec{\delta}_{a}'$-shift ignores $q$. In the following we concentrate on $\vec{\delta}_{a}$
and keep in mind that the same analysis follows for $\vec{\delta}_{a}'$ where the roles of
$p$ and $q$ are reversed.
We now work out an interesting example with $m=12$. This case has three modular
invariants\cite{CIZ}: the {\em diagonal} $(A_{11},A_{12})$ which we have already constructed, the
{\em complementary} $(D_{7},A_{12})$, and the {\em exceptional} $(E_{6},A_{12})$. Here we
introduce the notation \begin{equation}
Z_{A} = \frac{1}{2}\frac{1}{\bar{\eta}\eta}\sum_{p=0}^{m}
\sum_{q=-m}^{m-1}[ (p,q;p,q)-(p,-q;p,q)].
\end{equation}
By shifting $\Gamma_{A}$ we can construct a new modular invariant partition function
\begin{equation}
Z_{\delta_{a}} = \frac{1}{2}\frac{1}{\bar{\eta}\eta}(\sum_{\vec{\beta}\in\Gamma_{\delta_{a}}}-
\sum_{\vec{\beta}\in\Lambda_{\delta_{a}}})
\bar{q}^{\frac{1}{2}\beta_{L}^{2}}q^{\frac{1}{2}\beta_{R}^{2}}
\end{equation}
If $a=1$ then the unshifted sector contains the conjugacy classes $((mp);(mp))$ and
$((mp+(m+1)m);(mp+(m+1)m))$ so that the $\vec{\delta_{1}}$ shifted lattice
$\Gamma_{\delta_{1}}$ may be described by
\begin{equation}
\Gamma_{\delta_{1}} = \{((mp+(m+1)q);(mp-(m+1)q)) \mid \;p= 0,...,m\; ;\: q=-m,...,m-1\}
\end{equation}
This means that $\Gamma_{\delta_{1}} = \Lambda_{1,1}$ so that $Z_{\delta_{1}} =
-Z_{A}$ which alone yields negative coefficients $N_{ij}$ in the partition function when
expressed as in eq. (17). Furthermore, the sum $Z_{A}+Z_{\delta_{1}}$ vanishes.
If $a=2$ then the lattice $\Gamma_{\delta_{2}}$ contains the conjugacy classes
\begin{equation}
\Gamma_{\delta_{2}}=\left\{ \begin{array}{c}
((mp+(m+1)2r);(mp-(m+1)2r)) \\
((mp+(m+1)(6-2r));(mp+(m+1)(6+2r))) \\
p = 0,1,...,12 \; ; r = -6,-5,...,5
\end{array}
\right\}
\end{equation}
Thus,
\begin{equation}
Z_{\delta_{2}} = \frac{1}{2}\frac{1}{\bar{\eta}\eta}\sum_{p=0}^{12}
\sum_{r=-6}^{5}[ (p,-2r;p,2r)-(p,2r;p,2r)+(p,6-2r;p,6+2r)-(p,-6+2r;p,6+2r)].
\end{equation}
This contains no vacuum state, so as it stands it does not constitute a partition function
for a conformal field theory. On the other hand, the sum $Z_{A}+Z_{\delta_{2}}$ does
contain the vacuum state, and the first two terms in the brackets of (42) exactly
cancel the $q = even$ terms in $Z_{A}$. The last two terms may be rewritten such that
\begin{equation}
Z_{A}+Z_{\delta_{2}} = \frac{1}{2}\frac{1}{\bar{\eta}\eta}\sum_{p=0}^{12}\{
\sum_{q=odd}^{11}[ (p,q;p,q)-(p,-q;p,q)]+\sum_{q=even}^{10}[(p,q;p,12-q)-(p,-q;p,12-q)]\}.
\end{equation}
In terms of non-vanishing characters
\begin{equation}
Z_{A}+Z_{\delta_{2}} = \frac{1}{2}\sum_{p=0}^{12}\{
\sum_{q=odd}^{11}\mid\chi_{p,q}\mid^{2}+
\sum_{q=even}^{10}\bar{\chi}_{p,q}\chi_{p,12-q}\}.
\end{equation}
which we recognise as the complementary invariant $(D_{7},A_{12})$.
If we let $a=3$, then
\begin{equation}
\Gamma_{\delta_{3}}=\left\{ \begin{array}{c}
((mp+(m+1)3s);(mp-(m+1)3s)) \\
((mp-(m+1)(4-3s));(mp-(m+1)(4+3s))) \\
((mp-(m+1)(8-3s));(mp-(m+1)(8+3s))) \\
p = 0,1,...,12 \; ; s = -4,-3,...,3
\end{array}
\right\}
\end{equation}
As above, we find here that $Z_{\delta_{3}}$ contains no vacuum state, and the only combination
which can be expressed in terms of characters as in eq. (17) with non-negative
$N_{ij}$-coefficients and a non-degenerate vacume is
\begin{equation}
Z_{A}+Z_{\delta_{2}}+Z_{\delta_{3}} = \frac{1}{2}\sum_{p=0}^{12}\{
\mid\chi_{p,1}+\chi_{p,7}\mid^{2}+\mid\chi_{p,4}+\chi_{p,8}\mid^{2}+
\mid\chi_{p,5}+\chi_{p,11}\mid^{2}\}
\end{equation}
which is the exceptional invariant $(E_{6},A_{12})$.
All other values of $a$ in $\vec{\delta_{a}}$ yield partition functions equivalent to one of
the above so we have exhausted all possible invariants for $m=12$ using shifts of this form.
This method may be applied to all other values of $m$ and we summerize the results in relation
to the classification of \cite{CIZ} as follows: \begin{equation}
\begin{array}{lll}
{\bf Complementary\: Invariants} \\
\; m=2j\; ,\; j\in {\bf Z}^{+} \\
\; \; Z_{D}=Z_{A}+Z_{\delta_{2}} \\
{\bf Exceptional\: Invariants} \\
\; m=2\cdot 2\cdot 3 \\
\; \; Z_{E_{6}}=Z_{A}+Z_{\delta_{2}}+Z_{\delta_{3}} \\
\; m=2\cdot 3\cdot 3 \\
\; \; Z_{E_{7}}=Z_{A}+Z_{\delta_{2}}+Z_{\delta_{3}} \\
\; m=2\cdot 3\cdot 5 \\
\; \; Z_{E_{8}}=Z_{A}+Z_{\delta_{2}}+Z_{\delta_{3}}+Z_{\delta_{5}} \\
\end{array}
\end{equation}
By replacing the values of $m$ with $m+1$, one finds the same invariants where
$\vec{\delta_{a}}'$ replaces $\vec{\delta_{a}}$.
\section{Conclusion}
We have shown how to write modular invariant partition functions of the Virasoro minimal
models in terms of sums and differences of even self-dual lattice partition functions.
As in the covariant lattice approach to string theory, this places the study of modular
invariances into the study of self-dual lattices. A major advantage of this approach is that
{\em exceptional} invariants may be studied on the same footing as the {\em diagonal}
and {\em complementary} invariants.
It would be interesting to know the relation between these shift vectors and the action
of the Weyl group\cite{SW} on the lattice $\Gamma_{1,1} = {\cal U}(\sqrt{2m(m+1)}\: )^{2}$. One
would then expect to find some group theoretic reason for this particular set of modular
invariants, and perhaps shed some light on the mysterious A-D-E classification.
One may apply these methods to any other Coulomb gas representation of conformal field
theories with more bosons as long as the screening charges can be expressed as
\begin{equation}
Q_{a} = \int :e^{i\vec{\alpha_{a}}\cdot\vec{\varphi}(z)}:dz\; ,\; \Delta_{\vec{\alpha_{a}}} = 1.
\end{equation}
In the important case of the Kac-Moody algebras, the holomorphic winding momenta live on a
Lorentzian lattice\cite{N2}-\cite{Ne}. This can be inconvenient for the construction of lattice
partition functions because one must limit the sum to ensure that all states have non-negative
conformal dimension. A possible remedy to this difficulty would be to {\em Euclideanize}
the lattice \cite{LSW} and study its modular properties as in the even lattice formulation of
string theories. One may hope that the study of these and other
lattices will lead to a general classification of the modular invariant partition
functions for Kac-Moody algebras.
\section*{Acknowledgements}
I wish to thank Per Salomonson and Bengt Nilsson for discussions and many helpful suggestions on
the manuscript.
\include{refMM}
\end{document}