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\begin{document}

\title{Questions and commentss
}
 \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!

Projective $SL_2(\R)$ representations on various spaces.  Some facts (they might seem contradictory).

$SL_2(\R)$ acts on Fock space $\cF_P^{\tensor \ell}$.

Every level $\ell$ $LG$ representation appears in $\cF_P^{\tensor \ell}$ (this is Wasserman's definition of level).

\includegraphics[scale=.4]{Thursday9amPicture1.jpg}

--this picture explains something about the appearance of $N$ (of $SU(N)$) in this setting.  

$SL_2(\R)$ acts only {\em projectively} on $H_\lambda$.  

Why isn't it an honest action?  $H_{\lambda}$ appears with huge multiplicity in $\cF_P^{\tensor \ell}$ -- the $\lambda$-isotypical component of $\cF_P^{\tensor \ell}$ can be written as $H_\lambda \tensor (\text{Multiplicity space})$ -- both components are projective reps of $SL_2(\R)$, and when you tensor them together the cocycles cancel.  






\begin{question} What's up with the braiding on the category of conformal nets?
\end{question}

Universal central extension of $SL_2(\R)$ (or $PSL_2(\R)$) is $\tilde{SL_2(\R)}$, extension of $SL_2(\R)$ with fiber $\Z$.

\includegraphics[scale=.4]{Thursday9amPicture2.jpg}

Given a representation $H_\lambda$ of a net (drawn in a circle), rotation by $\pi$ introduces an isomorphism between $\cA(I)$ and $\cA(-I)$.  Define $H_\lambda^\pi$ to be $H_\lambda$ with actions twisted by rotation by $\pi$.  This is a new representation of the conformal net, which is isomorphic to the previous one, but not canonically.  The half-twist  $\theta$-- the element lifting $\pi$ -- is an $LG$-equivariant map $H_\lambda \rightarrow H_\lambda^pi$.

The braiding
$\beta:  H_\lambda \boxtimes H_\nu \rightarrow H_\nu \tensor H_ \lambda $ is given by $\theta_{H_\lambda \boxtimes H_\nu}^{-1} (\theta_{H_\lambda } \circ \theta_{ H_\nu})$.

Noah:  is there any easy way to check this is actually a braiding, other that just satisfying the braid relations?

\includegraphics[scale=.3]{Thursday9amPicture3.jpg},
\includegraphics[scale=.3]{Thursday9amPicture4.jpg}

\begin{question} Can you say something about how the primary fields appear in the Segal picture?
\end{question}

$V$ is a rep of $G$, $H$  are reps of $LG$

A primary field is an $\tilde{LG}_\ell \rtimes S^1$-equivariant map 
\begin{equation}C^\infty(S^1,V_k)\tensor H_j \rightarrow H_i,
\end{equation} 
which can be unbounded.

This encodes the same information as an $\tilde{LG}_\ell$-equivariant map 
\begin{equation}
H_k \boxtimes H_J \rightarrow H_i ,
\end{equation}
even though the second is positive energy and the first isn't.  

(2) is equivalent to $\Hom_{\tilde{L_IG}}(H_0,H_k)\otimes H_j \rightarrow H_i$.

I'm given $f \in C^\infty(S^1,V_k)$.  What do I get from this function?  From (1) I get $H_j \rightarrow H_i$.  
I also get (ah, now comes the circular reasoning) -- 
if you already believe the equivalence between the $j=0$, $i=k$ case of the correspondence, 
then $f \in C^\infty(S^1,V_k)$ induces a map $H_0 \rightarrow H_k$.  
Here we've used the obvious map $H_k \boxtimes H_0 \rightarrow H_k$.

So, a primary field takes a function $f$, produces a map $H_0 \rightarrow H_k$.

Okay, end circular reasoning.  I also get, from $f$, using (2), an element in $\Hom(H_0,H_k).$

Hmm ... can anyone say anything about the Segal picture?
\end{document}