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\begin{document}

\title{Sugawara's formula and the action of $\operatorname{Diff}$($S^1$) on positive energy representation 
}
 \author{Speaker: Nick Rozenblyum  (MIT)
 \\ Typist: Mike Hartglass}
 \date{\today}
 \thanks{Available online at \texttt{http://math.mit.edu/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!
$G$ simply connected Lie Group, and $\tilde LG$ cecntral extension, and we're looking at positive energy representations of this.  These representations are nice, and in particular one feature is that there is a projective action of $\operatorname{Diff}(S^1)$ on positive energy reps.\\

We will be looking at the Lie algebra $\operatorname{Lie}(\operatorname{Diff}(S^1))$ of vector fields on $S^1$.  We can take a Fourier expansion of such things, i.e. 
\[
\sum_n a_n e^{in\theta} \frac{d}{d\theta}.
\]
We'll restrict to polynomials.  Let $L_n = -ie^{in\theta} \frac{d}{d\theta} = t^{n+1} \frac{d}{dt}$.  These have commutation relations
\[
[L_n,L_m] = (m-n)L_{n+m}.
\]
Algebraic Geometry version: formal power series on the disk, $\mathcal{O} := \operatorname{Spec}\C(t)$.\\

circle $\leadsto$ $\mathcal{K}$ formal functor disk \

Consider vector fields on these  $\operatorname{Der} \mathcal{K} = \C(t) \frac{d}{dt}$.

To understand the projective action of $\DiffS$, we have to understand the central extensions of $\DiffS$.  There exists a universal central extension of $\operatorname{Lie}(\DiffS)$, called the {\bf Virasoro algebra}. 
\[
0 \to \C k \to \operatorname{Vir} \to \operatorname{Der} \mathcal{K} \to 0
\]
 It is generated by the $L_n$ (for $n \in \Z$), and a cecntral element $k$.  The bracket now looks like
\[
[L_m,L_n] = (m-n)L_{m+n} + \frac{m^3 - m}{12} \delta_{m,-n} k.
\]
A module $M$ over $\operatorname{Vir}$ has central charge $c \in \C$ if $k$ acts by $c$.\\

let $g$ be a simple finite dimensionl Lie algebra.  We have a cetral extension $$0 \rightarrow \mathbb{C}I \rightarrow \widehat{g} \rightarrow g \rightarrow 0$$ and an acton of Diff$(S^{1})$ hich gives an action of $V_{in}$ on $\widehat{g}$  $$[L_{m}, X[n]] \equiv -nX[m + n]$$ for $X \in g$ $X[n] = X \otimes t^{n}$.\\

PERs:  $\widehat{LG} \rtimes \mathbb{T}_{rot}$ $V = \oplus_{n} V(n)$ $\frac{d}{d\theta} \in$ Lie$(\mathbb{T}_{rot})$ $$[\frac{d}{d\theta}, X[m]] = -mX[m]$$ which implies $v \in V(n)$ implies $$X[m]v \in V(n - m)$$ 

Casimir:  Fix an inner product on $g$ $\langle , \rangle$.  If $\theta$ is the lowest root then $\langle \theta, \theta \rangle = 2$ and if $X_{j}$ is an orthonormal basis of $g$ then $$\Omega = \sum_{j}X_{j}^{2}$$ fix $\Omega$ in the center of $U(g)$ the universal enveloping algebra.\\

In the adjoint rep, $\Omega = 2h' \cdot I$ where $h'$ is the dual coxeter number.  In the case $G = SU(N)$ then $h' = N$.  In general, if $V$ is a highest weight representation of weight $\lambda$ then $$\Omega = \langle \lambda, \lambda + 2\rho \rangle \cdot I$$ where $\rho = \frac{1}{2}($sum of positive roots).  \\

Consider $\sum_{j, n \in \mathbb{Z}} X_{j}[n]X_{j}[-n]$ a natural question is if this is central and what the sum even means.  Thinking like a physicist, we proceed and play with the power series to get $$\sum_{j, n > 0} (X_{j}[n]X_{j}[-n] + X_{j}[-n]X_{j}[n]) + \sum_{i}X_{i}^{2}$$ $$=\sum_{j, n > 0} 2(X_{j}[n]X_{j}[-n]) + [X_{j}[-n], X_{j}[n]] + \sum_{j}X_{j}^{2}$$ $$=\sum_{j, n \in \mathbb{Z}} X_{j}[n] X_{j}[-n] = \sum_{j, n > 0} [X_{j}[-n], X_{j}[n]$$  

$\sum_{j, n > 0} [ , ] = \sum_{n > 0} l\cdot n \cdot dim(g) = (: X[m]Y[n]:)$ is by definition $X[m]Y[n]$ if $n \geq m$ and $y[n]X[m]$ otherwise.\\

 Consider $\Delta_{0} = \sum_{j, n \in \mathbb{Z}} : X_{j}[-n]X_{j}[n]:$.  If $V$ is a PER of $LG$ then $\Delta_{0}$ acts on $V$.  $\Delta_{0}$ is \emph{not} in the center.  $$[Y[m], \sum_{j} : X_{j}[-n]X_{j}[n]:] = Z_{n+m} - Z_{n}$$ for $m \neq \pm n$ and is equal to $Z_{n+m} - Z_{n} + mY[m]I$ if $m = \pm n$ where $Z_{n} = \sum_{j, k} \alpha_{jk}X_{k}[n]X_{j}[m - n]:$ and $[Y, X_{j}] = \sum_{k}\alpha_{jk}X_{k}$ (in $g$).\\

Add all of these up.  We get $$[Y[m], \Delta_{0}] = m(2YI + \Omega Y)[m] = 2M(l + h')Y[m]$$ $$=2(l + h')\frac{d}{d\theta}(Y[m]).$$  We can learn two thigs:\\

1.) We can get a central element $\Delta = \Delta_{0} + 2(l + h')\frac{d}{d\theta}$\\

2.) $L_{n} = \frac{d}{d\theta} = -\frac{1}{2(l + h')}\Delta_{0}$\\    

We can define the higher $\Delta_{m}$ by $$\Delta_{m} = \sum_{j, n \in \mathbb{Z}}  :X_{j}[m-n]X_{j}[n]:$$ Then $L_{m} = -\frac{1}{2(l+h')}\Delta_{m}$\\

Theorem:  These satisfy the Virasaro relations with central charge $$K = \frac{l \cdot dim(g)}{l + h'}$$  




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