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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}
































