\documentclass[11pt, letterpaper]{amsart}

\usepackage{amsthm, amssymb}
\usepackage{graphicx,tikz}
\usepackage[all]{xy}

\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{arrows}

\setlength{\parindent}{0pt}
\setlength{\parskip}{\baselineskip}

 \theoremstyle{plain}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{thm}{Theorem}[section]
 \newtheorem*{theorem*}{Theorem} %%%%%%%%%%%%%
 \newtheorem*{claim}{Claim}
 \newtheorem*{lemma*}{Lemma}
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{corollary}[theorem]{Corollary}
 \newtheorem{cor}[theorem]{Corollary}
 \newtheorem{proposition}[theorem]{Proposition}
 \newtheorem{prop}[theorem]{Proposition}

 \theoremstyle{definition}
 \newtheorem*{definition}{Definition}
  \newtheorem*{defn}{Definition}
 \newtheorem*{example}{Example}
 \newtheorem*{cexample}{Counter-example}
 \newtheorem*{remark}{Remark}
\newtheorem*{homework}{Homework}
\newtheorem*{question}{Question}
\newtheorem*{answer}{Answer}
\newtheorem*{idea}{Idea}
\newtheorem*{recall}{Recall}
\newtheorem*{note}{Note}
\newtheorem*{fact}{Fact}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cF}{\mathcal{F}} %%%%%%%%%% 
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cQ}{\mathcal{Q}}

\newcommand{\g}{\mathfrak{g}}
\newcommand{\h}{\mathfrak{h}}

\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\id}{\mathbf{1}}

\newcommand{\directSum}{\oplus}
\newcommand{\DirectSum}{\bigoplus}
\newcommand{\tensor}{\otimes}
\newcommand{\Tensor}{\bigotimes}
\newcommand{\acton}{\circlearrowright}


\newcommand{\su}[1]{\mathfrak{su}(#1)} %%%%%%%%%%%%%%%%
\newcommand{\bT}{\mathbb{T}} %%%%%%%%%%%%%%%%
\newcommand{\Trot}{\bT_{rot}} %%%%%%%%%%%%%%%%
\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}


