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\begin{document}
\title{Morning session: Overview
}
\author{Speaker: Andr\'{e} Henriques
\\ Typist: Emily Peters
}
\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}
%%% Start typing here!
{\bf First, some algebra:}
Main point in the beginning of Wasserman's paper: There is a strong analogy
\begin{tabular}{l | l}
Lie groups & Conformal field theory \\
\hline
(compact simple connected & ie, conformal nets: strongly additive, split \\
and simply-connected) & finite $\mu$-index \\
\hline
$G=SU(N)$ classification of irreps & $\tilde{LG}_\ell$, $\ell>0 $ classification of irreps\\
by tableau of height at most $N$. (where & by tableau of height at most $N$ and \\
deleting a column of height $N$ & width at most $\ell$ \\
doesn't change the rep) & \\
\end{tabular}
and we've been studying how to move from the left column to the right.
\begin{defn}
{\em split:} Given disjoint intervals $I$ and $J$, consider the algebraic tensor product $\cA(I) \tensor \cA(J)$; the two different completions are equal.
\end{defn}
Some examples of tableau and the corresponding representations:
\includegraphics[scale=.3]{Friday9amPicture1.jpg}
{\em Piere rule}: $- \tensor V_{[k]}$ = add $k$ boxes, no two in same row; sum over the corresponding irreducibles $V_\lambda$, with mutliplicity one.
The goal is to show the very same statement, with addition of an admissibility condition $f_1-f_N \leq \ell$ on tableau:
$- \boxtimes H_{[k]}$= add $k$ boxes, no two in same row; sum over the corresponding irreducibles $H_\lambda$, with mutliplicity one.
\begin{example}
\includegraphics[scale=.3]{Friday9amPicture2.jpg}
\end{example}
Here's one difference between the two pictures: Free fermion rep has only one irrep while loop group has lots of them.
We have an action of $\tilde{LG}$ on Fock space.
Notice that $LG$ reps correspond $L\g reps$, which correspond to $L^{pol}\g$ reps, where we take the finite energy parts of the vector spaces and no longer have to worry about unbounded operators. This is very useful for classification results. For example, an $L^{pol}\g$ rep gives us a rep of a dense subgroup of $\tilde{LG}$; so classification of $L^{pol}\g$ reps gives us an upper bound on the number of $LG$ reps; then we use the action of $\tilde{LG}$ on Fock space to actually construct these reps.
{\color{blue}Virasoro algebra/ $Diff(S^1)$.}
{\bf There is also an analytic side to this story: }
The main technical tool for studying (type III) von Neumann algebras is Tomita-Takesaki theory. One can build noncommutative $L^p$ spaces such that $L^\infty (M)=M$, $L^1(M)=M_*$ (the predual), and $L^2(M)$ is some hilbert space -- all of these are equalities as bimodules of $M$. Further if $M=\cA(\tikz \draw (0,0) arc (0:180:2mm);)$ then $L^2(M)$ is the vacuum rep.
The *-operator on $L^2(M)$, typically called $J$, acts via
\begin{tikzpicture}
\draw (0,0) circle (1cm);
\draw (-2,0)--(2,0);
\draw[<->] (1.2, .5)--(1.2, -.5);
\end{tikzpicture}
Is this breaking symmetry? No because we chose a special upper half circle already $M=\cA(\tikz \draw (0,0) arc (0:180:2mm);)$.
We also have a modular flow
$\Delta_\phi^{it}(\xi)=\phi^{it} \xi \phi^{-it}$
with $\phi=\langle \cdot \Omega, \Omega \rangle$; this acts via
\begin{tikzpicture}
\draw (0,0) circle (1cm);
\draw (-2,0)--(2,0);
\draw[->] (170:1.2cm) arc (170:10:1.2cm);
\draw[->] (-170:1.2cm) arc (-170:-10:1.2cm);
\end{tikzpicture}
--this is a geometric implementation of the modular flow (along the lines of the Bisognano-Wichman theorem).
We get factoriality because the modular group is ergodic, and that these factors are type $III_1$.
Thus: free fermions are a factor representation. Being a factor representation tells us that the vNa we get by completing $\tilde{L_I(G)_\ell}$ acting on $H_0$ or $\tilde{L_I(G)_\ell}$ acting on $H_\lambda$; a priori it's not clean at all why the two different completions $(\tilde{L_I (G)}_\ell)''$ should be the same; however, they both sit inside Fock space, and are both factor reps, so are canonically isomorphic. Thus we can talk about ``the'' level $\ell$ representation
At this point we can talk about "the" conformal net associated to the loop group.
We can view these as bimodules for the algebra $M$, where we secretly identify the top and the bottom intervals via $J$;
{\color{blue} (Finite index condition means)} We get a braided rigid ribbon category. Many of these properties follow from there being only finitely many $L^{pol}\g$ reps.
\begin{question}
Can you say something about fusion of conformal nets coming from loop groups?
\end{question}
\begin{answer}
Ah, so the question is, we
have a rep of this style
\includegraphics[scale=.3]{Friday9amPicture3.jpg}
but also need to know, say, that the left interval acts.
Solution: use the formalism Yoh used; $\rho, \sigma \in \operatorname{End}(\cA(\tikz \draw (0,0) arc (0:180:2mm);))$ that act trivially near the boundary. "Localized endomorphisms."
Use that $H_\rho \simeq H_0$ equivariantly w.r.t.~\tikz \draw (0,0) arc (60:-240:2mm);
\end{answer}
{\bf We also had talks about primary fields:}
Defined the beasts (Arturo classified them using algebraic approach, constructed them using fermions), Anatoly defined some function (four-point function) via power series, and we'll need to know that the value of the function at a certain point in non-zero. This is why we calculated the transport coefficients.
\end{document}