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\begin{document}
\title{The central extension of $LG$, positive energy representations, Lie algebra cocycles}
\author{Speaker: Owen Gwilliam \\ Typist: Corbett Redden
}
\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!
\newcommand\Lpg{L^{\operatorname{poly}}\fg}
\newcommand\omb{\omega_{\text{basic}} }
\newcommand\T{\mathbb T}
\newcommand\cF{\mathcal F}
\newcommand\Diff{\operatorname{Diff}}
%\newcommand\to{\rightarrow}
\newcommand\wt{\widetilde}
\newcommand\fg{\mathfrak g}
The talk will have three main parts:
\begin{enumerate}
\item define \& motivate positive energy representations;
\item projective representations \& central extensions;
\item reducing questions to the loop algebra.
\end{enumerate}
Throughout the talk, $G$ denotes a compact connected Lie group and $\T$ the circle as a group.
\section{Positive energy representations}
At first acquaintance, the definition of positive energy representation probably seems a little weird and unmotivated. We'll get the awkward introduction to them out the way before explaining why we'll be spending so much time with PERs this week.
\begin{defn}A {\em positive energy representation} (PER) of $LG$ is a topological vector space $E$ with:
\begin{enumerate}
\item a projective representation $LG \to PU(E)$, meaning that for $\gamma \in LG$ you can choose $U_\gamma \in GL(E)$ (not continuously) and then $U_\gamma U_{\gamma '} = c_{\gamma \gamma'} U_{\gamma \gamma'}$, where $c_{\gamma \gamma'} \in \C^*$;
\item an intertwining action of $\T$, so that $R_\theta$, the operator for $\theta \in \T$ on $E$, satisfies $R_\theta U_\gamma R_\theta ^{-1} = U_{R_\theta \gamma}$;
\item under the weight decomposition of $E$ by $\T$,\footnote{Note that this action is not the $U_z$ action in Wassermann, but is the rotation subgroup of the M\"obius group.}
\[
E=\bigoplus_{n \in \Z} E(n),
\]
with $\T$ acting by $e^{in\theta}$ on $E(n)$, we have
\[
\begin{cases} E(n)=0, &n<0, \\ \dim E(n) <\infty, & n \geq 0. \end{cases}
\]
This is known as the {\em positive energy condition}.
\end{enumerate}
\end{defn}
\subsection{Concrete examples} Ryan's talk provides the crucial example of a PER for this week and he's already verified the relevant properties. Let $G=SU(n)$, $V=\C^n$ the standard representation, $\cH=L^2(S^1,V)$, and $P:\cH \to \cH$ projection onto non-negative Fourier modes. Then the fermionic Fock space
\[ \cF_P = \Lambda(P\cH) \hat\otimes \Lambda (P^\perp \cH)^*\]
is a positive-energy representation. In fact, this is a {\it level one} representation, and $(\cF_P)^{\otimes \ell}$ is a positive-energy representation of level $\ell$.
\subsection{Why study PERs?}
There are two types of reasons for focusing our attention on PERs: mathematical and ``physical." From the point of view of math, the most convincing reason is that PERs are well-behaved.
\begin{thm}[9.3.1 in \cite{PS86}]
A positive energy representation of $LG$ is
\begin{itemize}
\item completely reducible into irreducible positive energy representations,
\item unitary,
\item and extends to holomorphic representations of its complexification $LG_\C := Maps(S^1, G_\C)$.
\end{itemize}
Furthermore, the representation admits a projective intertwining action of $\Diff_+ (S^1)$.\footnote{This is the group of diffeomorphisms of the circle that preserve orientation.}
\end{thm}
This theorem tells us that a PER acts a lot like a representation of a compact Lie group, since the first three properties are what make life so nice when studying compact Lie groups. The last property insures that reparametrizing the circle does not affect a PER: the PERs are preserved under a huge space of $LG$ automorphisms. In a sense, being a PER is a kind of finiteness condition that insures we can get complete control over the representation theory.
Although I'm know next to nothing about physics, I'll explain why I guess that the positive energy condition is natural from a physics point of view, and I hope that this week will provide better explanations. Here are three thoughts:
\begin{itemize}
\item We focus on projective representations because the space of states for a quantum mechanical system is really $\mathbb{P}(H)$, the projective space of the Hilbert space $H$, so it's only reasonable to expect that a group act projectively on $H$.
\item When studying quantum mechanics, the ``energy" of a particle corresponds to the infinitesimal generator of translation along its worldline (this is what Schr\"odinger's equation says). When studying QM along a circle, the analog of translation is rotation. The positive energy condition is then the physically reasonable assertion that the accessible energy states are bounded below.
\item Segal says that a PER is the ``boundary condition" of a holomorphic representation of the semigroup
$\C_{\leq 1}^\times \rtimes \wt{LG}_\C.$
That is, PERs extend to the study of maps of the annulus into $G_\C$. Thus PERs ought to capture at least the annular part of a CFT (at least in the Segal style).
\end{itemize}
\section{Projective representations and central extensions}
We now explore what a projective representation is and how to construct invariants that classify such representations.
\begin{defn}A {\em projective unitary representation}\footnote{The unitary group uses the strong operator topology (or any equivalent topology) and then $PU$ has the quotient topology. You would get in trouble if you used the norm topology to define $U(V)$.} of a group $G$ on a Hilbert space $V$ is a group homomorphism
\[ \rho: G \to PU(V) := U(V) / \T,\]
where $PU$ denotes the projective unitary group obtained by quotienting out by the subgroup of scalar multiples of the identity.
\end{defn}
We would like to lift this to an actual representation, which will require taking a central extension of the group. That is, we pull back the short exact sequence defining $PU$:
\[ \xymatrix{
1 \ar[r] &\T \ar[r] &U(V) \ar[r] &PU(V) \ar[r] &1 \\
1 \ar[r] & \T \ar[r] \ar@{=}[u] & \wt G \ar[r] \ar@{-->}[u]^{\rho} &G \ar[r] \ar[u]^{\rho}& 1
}
\]
Thus, given a projective representation $\rho: G \rightarrow PU$, we obtain an extension $\wt G$ and an honest representation $\rho: \wt G \rightarrow U$. Conversely, given an honest representation of a central extension of $G$, we get a projective representation of $G$. Thus we can just study extensions of $G$ and honest representations to get a handle on projective representations.
Another consequence is that we obtain an invariant for a projective representation: the associated central extension. We can give an invariant for an extension in two ways.
{\bf Idea 1:} Thinking like topologists, we note that any extension
\begin{align*}
1 \to \T \to \wt{LG} \to LG \to 1
\end{align*}
defines a circle bundle and hence an element
\[ c_1 \Big( \wt{LG} \to LG \Big) \in H^2(LG;\Z).\]
We can try to use this cohomological information to help construct group extensions or narrow down possible ones. (There's a really great story if you head down this road of mixing topology and representation theory, leading to twisted K-theory, Chern-Simons, and so on, but we'll not pursue it.)
\begin{homework}
Check that $H^2(LSU(N);\Z)\cong \Z$. How does the level of $\cF_P$ compare to the first Chern class of the associated extension?
\end{homework}
{\bf Idea 2:} Thinking like algebraists, we note that at the Lie algebra level we get a central extension
\begin{align*}
0 \to \R \to \wt{L \fg} \to LG \to 0 .
\end{align*}
The extended Lie bracket has the form
\[ \Big[ (X,a), (Y,b) \Big] = \Big( [X,Y], \omega(X,Y) \Big) \]
for $(X,a), (Y,b) \in L\fg \oplus \R \cong \wt {L\fg}$ (as vector spaces). Consequently, the functional $\omega: L\fg \times L\fg \to \R$ must be skew-symmetric and satisfy a Jacobi identity, i.e.
\[ \omega([X,Y],Z) + \omega([Y,Z],X) + \omega([Z,X],Y) = 0.\]
This implies $\omega$ will be a 2-cocycle and live in $H^2(L\fg, \R)$.
There is a beautiful characterization of such cocycles.
\begin{prop}[4.2.4 of \cite{PS86} ]For $\fg$ semisimple, every continuous $G$-invariant 2-cocycle $\omega$ for $L\fg$ has the form
\[ \omega(X,Y) = \frac 1 {2\pi} \int_0^{2\pi} \langle X(\theta), Y'(\theta) \rangle \> d\theta \]
where $\langle \cdot, \cdot\rangle$ is a $\fg$-invariant symmetric bilinear form on $\fg$.
\end{prop}
The proof is quite simple: you simply observe that such a cocycle extends to the complexification $L\fg_\C$ and apply Fourier series. It's important to observe that the $G$-invariant condition is irrelevant. Since $LG$ acts on $L\fg$ by the adjoint action, it acts on cocycles, and two cocycles are cohomologous if they live on the same orbit. Now notice that the constant loops $G \subset LG$ act on cocycles as well, but since $G$ is compact, we can do the averaging trick to replace any cocycle by a cohomologous $G$-invariant cocycle.
Thanks to this theorem, it's enough to know about symmetric $\fg$-invariant bilinear forms on the finite-dimensional Lie algebra $\fg$. This is a well-known (to others, probably!) piece of mathematics. In particular, $H^3(\fg, \R)$ is isomorphic to the space of symmetric bilinear $\fg$-invariant forms on $\fg$, via the map
\begin{align*}
\operatorname{Sym}^2(\fg)^G &\longrightarrow H^3(\fg) \\
\langle \cdot, \cdot \rangle &\longmapsto \langle \cdot, [ \cdot, \cdot] \rangle.
\end{align*}
(There's something interesting here in relation to the Chern-Simons action, but I'm not competent enough to comment \dots)
It's natural to ask now whether given these invariants (either topological or Lie-theoretic), we can reconstruct a central extension of $LG$. For instance, every 2-cocycle on $L\fg$ yields a 2-form on $LG$ by left translation. This would be the first Chern class of the extension if it existed, so we know this 2-form has to be integral. If you follow this train of thought and have familiarity with the Chern-Weil story of bundles, curvature, and connections, then the following theorem is unsurprising.
\begin{thm}[4.4.1 of \cite{PS86}] Let $G$ be compact and simply-connected. Then
\begin{itemize}
\item A 2-cocycle $\omega$ on $L\fg$ defines a group extension if and only if $[\omega/2\pi] \in H^2(LG;\Z)$.
\item If it defines a group extension, the group extension is unique.
\item If $G$ is simple, such an $\omega$ is an integral multiple of $\omega_{\text{basic}}$, where $\omega_{\text{basic}}$ is a rescaling of the Killing form so that the longest coroot has length 2.
\end{itemize}
\end{thm}
\begin{example}
For $G=SU(N)$, the Killing form is $\omega_{basic}$:
\[ \langle X,Y \rangle = -\operatorname{Tr}(XY).\]
Note that for $SU(2)$, there is only one coroot and we verify
\[ \langle \theta, \theta \rangle = -\operatorname{Tr} \Big( \begin{pmatrix} i&0 \\ 0 & -i \end{pmatrix}^2 \Big) = 2 .\]
\end{example}
\begin{defn}If $G$ is simple, the {\it level} of a central extension $\wt{LG}$ of $LG$ is the integer $\ell \in \Z$ such that $\omega_{\wt{LG}} = \ell \omega_{\text{basic}}.$ For simply-connected semisimple $G$, the level $\ell$ lives in $H^3(G;\Z) \cong H^4(BG;\Z)$.
\end{defn}
\subsection{Redefining PERs}
With the language of central extensions now available to us, we can give a more succinct definition of a positive energy representation. We denote any central extension of $LG$ by $\wt{LG}$, and if we want to specify the level $\ell$ extension, we use $\wt{LG}_\ell$. Thus in the definition of PER, we replace ``projective representation of $LG$" by ``representation of $\wt{LG}$." Likewise, saying ``there is an intertwining action of $\T$" means that the rotation action of $\T$ on $LG$ plays nicely with the representation. This property can be rephrased by saying the PER is a representation of the semidirect product
\[
1 \rightarrow \wt{LG} \rightarrow \wt{LG} \rtimes \T \rightarrow \T \rightarrow 1,
\]
where $\T$ acts by rotation on $\wt{LG}$. In sum, we obtain the following.
\begin{defn} A {\em positive energy representation} of $LG$ at level $\ell$ is an honest representation of $\wt{LG}_\ell \rtimes \T$ satisfying the positive-energy condition for the action of $\T$ by rotation.
\end{defn}
Once you pick a maximal torus for $G$ (and a splitting of the semidirect product), you have the abelian subgroups
\[ \T_{\text{rot}} \times T_G \times \T_{CE} \subset \wt{LG} \rtimes \T_\text{rot}.\]
There are many circle actions appearing this week, so I've tried to distinguish the circle from the central extension, $\T_{CE}$, from the rotation circle, $T_{\text{rot}}$. They play different roles so watch out!
A positive energy representation decomposes under the action of this big torus into a finer sum of weight spaces
\[ E = \bigoplus_{(\text{energy } n,\, \lambda, \text{ level } l)} E(n, \lambda, \ell),\]
where $\lambda$ is the weight for the action of $T_G$.
Notice that for an irreducible PER, only one level can appear by Schur's lemma: the action of $\T_{CE}$ is central, and hence commutes with everything in sight, so the decomposition of $E$ by level is preserved under the full action of $\wt{LG}$. In consequence, when thinking about irreducible PERs, we can restrict our attention to the weight space for $T_G \times \T_{\text{rot}}$, and this is quite useful in understanding the affine Weyl group.
\section{Reducing to the loop algebra}
The loop group is a big, rather unwieldy thing, and we'd like a smaller, more algebraic object to work with. A first step, familiar from Lie theory, is to work with its Lie algebra, but even $L\fg$ is too unwieldy. Instead, we introduce $\Lpg$, the polynomial loops into $\fg$, for which explicit computations are easy. This is the space of smooth maps into $\fg$ that are given by trigonometric polynomials:
\[
\Lpg := \left\{ \sum_{\text{finite}} X_n \cos(n\theta) + Y_n \sin(n\theta) \, :\, X_n, Y_n \in \fg\right\}
\]
Clearly it is dense in $L\fg$. Its complexification is simply
\[ L^{poly}\fg \otimes \C \cong \fg_\C [z, z^{-1}],\]
where $z \leftrightarrow e^{in\theta}$.
We want a correspondence between representations of $LG$ and representations of $\Lpg$. That is, we want a bijection:
\begin{align*}
\{ \text{PERs of }LG \} \longleftrightarrow \{ \text{PERs of } \Lpg \}
\end{align*}
It is not so hard to construct a map going to the right, but there is some work involved in showing it's a bijection. Once we have the bijection, we can often reduce questions about $LG$ to purely algebraic computations with $\Lpg$ (for instance, see Nick's talk on the action of the Virasoro).
Abstractly, a PER $E$ of $LG$ induces a representation of $\Lpg$ in the obvious way:
\[ \Lpg \overset{\exp}\longrightarrow LG \overset{\pi}\longrightarrow PU(E).\]
For each $x \in \Lpg$ you get a 1-parameter subgroup
\begin{align*}
\R &\longrightarrow PU(E)\\
t &\mapsto \pi( \exp(tx)).
\end{align*}
By Stone's theorem, we get a skew-adjoint operator $X\in \operatorname{End} (E)$ such that
\[ e^{tX} = \pi( e^{tx}).\]
(This depends on picking a lift to $U(E)$, so $X$ is defined up to a character of $\R$.) Thus we have a projective representation
\[ \rho: \Lpg \to \operatorname{End} E.\]
Wassermann showed this representation is well-behaved.
Let $d$ denote the infinitesimal generator of $\T_{\text{rot}}$ on $E$, so that
\[ d\Big|_{E(n)} = n \quad \& \quad R_\theta = e^{i\theta d}.\]
\begin{thm}[\cite{Was98}]
Let $E$ be a level $\ell$ representation of $LG$. Then
\begin{itemize}
\item The space of {\em finite energy vectors} $\displaystyle E^{\operatorname{fin}} \subset E$ is preserved by $\rho$;\footnote{This is the dense subspace of $E$ given by taking the algebraic direct sum of the weight space $E^{\operatorname{fin}} = \bigoplus_{n\geq 0}^{alg} E(n)$.}
\item We can choose lifts (of the 1-parameter subgroups) such that
\[[d, \rho(x)] = i \rho(x')\] on $E^{\operatorname{fin}}$;
\item $[\rho(x), \rho(y)] = \rho([x,y]) + i \ell \omb (x,y)$.
\end{itemize}
\end{thm}
This theorem says that we get a level $\ell$ PER for $\Lpg$. Notice that the second property says the Lie algebra of rotations intertwines correctly with our PER for $\Lpg$, and the third property says the level is respected.
In the paper \cite{Was98}, Wassermann proves this theorem in a very concrete way. The proofs usually boil down to a simple combination of functional analysis and explicit computation. Recall our concrete example of PERs at the beginning of the talk.
Let $\cH = L^2(S^1,V) = L^2(S^1, \C^N)$, and recall that $\operatorname{Cliff}(\cH)$ acts on the fermionic Fock space $\cF_P$.\footnote{The action uses Wassermann's notation. It would be $c(e^{in\theta}v)$ in Ryan's notation.} We thus obtain an action of polynomial loops in $V$ on $\cF_P$:
\begin{align*}
L^{\text{poly}}V = \{ \text{polynomials in }e^{in\theta} \} \otimes V &\overset{\pi}\longrightarrow \operatorname{Cliff}(\cH), \\
e^{in\theta} \otimes v &\longmapsto a(e^{in\theta}v).
\end{align*}
We want a map
\[ L^\text{poly}su(N) \overset{\pi}{\longrightarrow} \operatorname{Cliff}(\cH)\]
which would give us a PER for $L^\text{poly} su(N)$.
Observe that $su(N)_\C = sl(N,\C)$ is generated by the elementary matrices $E_{ij}$. It is enough to say where $\pi$ would send these generators. Let $\{ e_1, \ldots, e_n\}$ be the standard basis for $V=\C^N$. Set $e_k(n) = \pi( e^{-in\theta} \otimes e_k)$, and
\[ E_{ij}(n) = \sum_{m > 0} e_i(n-m) e_j(-m)^* - \sum_{m\geq 0} e_j(m)^* e_i(m+n).\]
\begin{thm}[\cite{Was98}] \
\begin{itemize}
\item $[X(m), a(f)] = a(X e^{im\theta}f)$ for $f\in L^{\operatorname{poly}} V$.
\item $[d, X(m)] = -mX(m)$
\item $[X(n), Y(m)] = [X,Y] (n+m) + m\langle x,Y\rangle \delta_{n+m, 0}$
\item $ \displaystyle \langle Xe^{-in\theta}, Ye^{-in\theta} \rangle = \int \langle X,Y \rangle_{\operatorname{basic}} e^{-in\theta} (-im) e^{-im\theta} \> d\theta$, and therefore you have a level 1 representation.
\end{itemize}
\end{thm}
The proof consists primarily of straightforward computation. You can then extend it to $\cF_p^{\otimes \ell}$ to obtain the level $\ell$ representations of $LSU(N)$.
\begin{thebibliography}{Was}
\bibitem[PS]{PS86}
Andrew Pressley and Graeme Segal.
\newblock {\em Loop groups}.
\newblock Oxford Mathematical Monographs. The Clarendon Press Oxford University
Press, New York, 1986.
\bibitem[Was]{Was98}
Antony Wassermann.
\newblock Operator algebras and conformal field theory. {III}. {F}usion of
positive energy representations of {${\rm LSU}(N)$} using bounded operators.
\newblock {\em Invent. Math.}, 133(3):467--538, 1998.
\end{thebibliography}
\end{document}