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\begin{document}
\title{Classification of $LG$-representations and weight polytopes for their characters
}
\author{Speaker: James Tener
\\ Typist: Brent Pym
}
\date{\today}
\thanks{Available online at \texttt{http://math.mit.edu/$\sim$eep/CFTworkshop}. Please email \texttt{eep@math.mit.edu} with corrections and improvements!}
\maketitle
\begin{abstract}
Notes from the ``Conformal Field Theory and Operator Algebras workshop," August 2010, Oregon.
\end{abstract}
The goal is to establish a classification of the irreducible positive-energy representations (PERs). It will mirror the classification of irreducible representations of $SU(N)$. For the purposes of this talk, $N = 3$.
Recall that for $G = SU(3)$, we accept on faith that we may classify representations of $G$ by classifying representations of $\su{3}$. They were described by pictures such as
\includegraphics[scale=.4]{Tuesday10-15amPicture1.jpg}
We think of the highest weight as a \emph{signature} $(f_1,f_2,f_3) \in \Z^3$. For the adjoint representation, the signature is $(1,0,-1)$. Recall the
\begin{theorem*}
The possible highest weights are tuples $(f_1,\ldots,f_N) \in \Z^N$ with $f_1 \ge \cdots \ge f_N \ge 0$. By adding an element of the form $(a,\ldots,a)$ we may take $f_N = 0$.
\end{theorem*}
These signatures lie in the indicated portion of the following diagram:
\includegraphics[scale=.5]{Tuesday10-15amPicture2.jpg}
For $LSU(3)$, we ask the following questions:
\begin{itemize}
\item What is the equivalent of a signature?
\item What are the possible weights?
\end{itemize}
The setup is $(\pi,\cH)$, an irreducible PER of $LG$ at level $\l$. So, $\pi$ is a projective representation of $LG \rtimes \Trot$, and we have an honest representation of $\Trot$. There is a decomposition
$$
\cH = \DirectSum_{n \ge 0} \cH(b)
$$
with $\dim \cH(n) < \infty$ for all $n$, and $\Trot$ acts on $\cH(n)$ by
$$
z \xi = z^n \xi
$$
or, equivalently
$$
r_\theta(\xi) = e^{n\theta} \xi.
$$
The classification is as follows:
\begin{theorem*}
For $(\pi,\cH)$ an irreducible PER, we have
\begin{enumerate}
\item $\cH(0)$ is an irreducible $\su{N}$-module for the embedding $SU(N) \subset LSU(N)$ by constant loops.
\item The signature of $f$ of $\cH(0)$ satisfies
$$
f_1 - f_N \le l
$$
\item Conversely, if $f$ is such a signature, then there is an irreducible $PER$ of $LG$ with $\cH(0) \cong V_f$.
\item This representation is unique up to isomorphism.\footnote{\color{blue}If $\cH$ and $\cH'$ are two irreducible PERs of $LG$ at level $l$ such that $\cH(0) \cong \cH'(0)$ as $SU(N)$-modules, then there is a $U : \cH \to \cH'$ intertwining the projective actions of $LG \rtimes \Trot$.}
\end{enumerate}
\end{theorem*}
For $LSU(n)$ at level $l$, the possible highest weights of $\cH(0)$ are as indicated in the previous picture.
\begin{question}Why is $\cH(0)$ invariant under $SU(N)$?\end{question}
\begin{answer}Recall that $L^{poly}\g$ denotes the trigonometric polynomials with values in $\g$. There exists a representation $\rho$ of $L^{poly} \rtimes \R$ such that
\begin{enumerate}
\item $\pi$ factors through $\rho$:$$\pi(e^x) = e^{\rho(x)}$$
\item If $X(n) = \rho(e^{-int}X)$ for $X \in \g$, we have
$$
[X(n),X(m) ] = [X.Y](n+m) + m l \langle X,Y \rangle
$$
\item The action of $x \in \R$ is given by $$\rho(x) = x D,$$ where $D$ acts on $\cH(n)$ by multiplication by the energy $n$.
\item We have the commutation relation $$[X(n),D] = -nX(n).$$
\end{enumerate}
Then, for $\xi \in \cH(j)$, we simply compute
\begin{align*}
DX(n)\xi &= X(n) D\xi - nX(n)\xi \\
&= (j-n)X(n)\xi.
\end{align*}
The $X(n)$ are analogous to the lowering operators that we employed when studying the representations of $\su{N}$. In particular, taking $n=0$, we see that $\cH(j)$ preserved by constant elements of $L^{poly}\g$, so that $\cH(j)$ is an $SU(N)$-module via the embedding of constant loops.
\hfill$\Box$
\end{answer}
The next picture is the weight diagram for $LSU(2)$ at level $l$. In this picture, the vertical axis is the energy $j$, and the horizontal axis is the weight for $\su{2}$:
\includegraphics[scale=.5]{Tuesday10-15amPicture3.jpg}
The next question is where the third condition comes from. For the sake of Wassermann's paper, irreducible PERs, are sub-representations of $\cF_P^{\otimes l}$, so $\cH(0) \subset \cF_P^{\otimes l}(0)$, and we should look for $SU(N)$-modules in $\cF_P^{\otimes l}(0)$.
We want to find irreducible representations of $SU(N)$ that have signature $(f_1,f_1,\ldots,f_{N-1},0)$. Now,
$$
\cF_P = \bigwedge \cH_P
$$
where $\cH_P = L^2(S^1,V)$ for $V = \C^N$. The space $\cH_P$ contains a copy of $V$, embedded as constant functions, so in $\cF_P^{\otimes l}(0)$, we have an embedded copy of $\left(\bigwedge V\right)^{\otimes l}$.
Recall that the element
$$
e_1^{\otimes(f_1-f_2)} \otimes (e_1 \wedge e_2)^{\otimes(f_2-f_3)} \otimes \cdots \otimes (e_1\wedge \cdots \wedge e_{N-1})^{\otimes(f_{N-1}-f_N)}
$$
generates an irreducible $SU(N)$ module of signature $f$. Since
$$
\left(\bigwedge V\right)^{\otimes l} = (\wedge^1 V \oplus \cdots \oplus \wedge^N V)^{\otimes l},
$$
we can generate modules as before, provided that $f_1 \le l$.
\end{document}