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\begin{document}
\title{Notes from the Operator Algebras and CFT Workshop}
\author{Scott Carnahan}
\begin{abstract}
Notes August 16-21 2010. Other people's notes appear at:
\texttt{http://math.mit.edu/\~{}eep/CFTworkshop}
\end{abstract}
\maketitle
%\tableofcontents
\section{Monday, 8/16/10}
%\subsection{9am Andre Henriques, Overview and introduction}
%
%Main question: What is Conformal Field Therory?
%
%There are many possible answers to that question that live in very different areas of mathematics. One of these answers has to do with operator algebras. Other key words are "vertex algebras" and the formalism of Graeme Segal. There is an axiomatic setup that uses the formalism of operator algebras, more specifically von Neumann algebras, and these objects are called conformal nets.
%
%A conformal field theory is some vague notion that some will call a conformal net, and others will define differently. There are parallel constructions, but no known functors.
%
%Most prominent examples come from loop groups of compact Lie groups (
%``loop group nets''), and Dirac free fermions. (Loop groups are only conjectured to satisfy Segal's CFT axioms.)
%
%Start with $G$ a compact Lie group, (simply connected). $LG = Map_{C^\infty} (S^1,G)$. $\widetilde{LG}$ is a central extension of $LG$ by $S^1$. The set of such extensions is isomorphic to $\mathbb{Z}$, so this depends on a choice of element $\ell \in \mathbb{Z}$, called the level.
%
%One has a representation of $\widetilde{LG}$ on a separable Hilbert space $H_0$. If $\ell < 0$, then such a representation does not exist (rather, negative level reps are not unitarizable).
%
%Given $I \subset S^1$, consider $L_I G = \{ \gamma: S^1 \to G \mid \gamma(z) = e \, \forall z \not\in I \}$. Pull back the central extension to a central extension $\widetilde{L_I G}$. This group acts on $H_0$. We can then consider the von Neumann algebra $A(I)$ generated by all operators in $\widetilde{L_I G}$. (this includes taking the weak closure)
%
%The assignment $I \mapsto A(I)$ is a conformal net. A conformal net is a Hilbert space, together with a collection of algebras of operators acting on the Hilbert space - satisfying some axioms.
%
%$SL_2(\mathbb{R})$ also acts on $H_0$. There is an $SL_2(\mathbb{R})$-invariant vector $\Omega \in H_0$, called the vacuum. $SL_2(\mathbb{R})$ appears as the group of algebraic automorphisms of $S^1$ as a real projective line. It is a subgroup of all diffeomorphisms.
%
%In summary, for every compact Lie group $G$ and every integer $\ell$, we have a conformal net. This is true in all setups of CFT (conjecturally in Segal's setup).
%
%Wasserman doesn't talk about conformal nets in general. He works with $SU(N)$. Fortunately, most of his constructions generalize to conformal nets, i.e., there isn't much that is specific to $SU(N)$. When he describes representations of the loop group, we can replace it with representations of a conformal net.
%
%What is a representation of a conformal net? Recall that for each interval $I$, we have an algebra $A(I)$, and for each inclusion $I \subset J$, we have an algebra map $A(I) \to A(J)$. A representation is a Hilbert space $H_\lambda$ together with a system of maps $A(I) \to B(H_\lambda)$, such that $A(I) \to A(J) \to B(H_\lambda)$ agrees with $A(I) \to B(H_\lambda)$. It is important to exclude $S^1$ from the possible intervals. For loop groups, the algebra of all operators is just $B(H)$, and it might not act on $H_\lambda$.
%
%The local loop groups sit inside the algebras $A(I)$, so we get representations of all of these groups, and whatever they generate. It turns out the groups $\widetilde{L_I G}$ together generate a representation of $\widetilde{LG}$.
%
%We get a 1-1 correspondence between representations of the loop group net and projective representations of $LG$ that are positive energy. This restricts correspondences between net reps of a specific level and projective reps of a given central charge.
%
%What is Wasserman's big innovative idea? He defines a tensor product operation on these loop group representations using techniques from von Neumann algebras. Suppose we have a bimodule ${}_A H_B$ and another bimodule ${}_B K_C$. If these were just algebras, we could just do a tensor product of $H$ with $K$. In von Neumann algebras, there is an operation that behaves formally like tensor product, called Connes fusion. It is written ${}_A H \underset{B}{\boxtimes} K_C$.
%
%Warning: It is not true that given $h \in H$ and $k \in K$, one obtains an element of $H \boxtimes K$.
%
%More nets: Subintervals yield subalgebras, and disjoint intervals yields commuting algebras. Given left and right half-circles, we get two left actions by the algebras corresponding to intervals. We want a right action. Fortunately, if $I \cong J$ is orientation-reversing, we get $A(I) \cong A(J)^{op}$. We use reflection in the vertical line as our orientation-reversing isomorphism. We get $A(I)-A(I)$-bimodules, and given two of them, we can use Connes fusion to get a third bimodule.
%
%We have a concrete question: $H_\lambda \boxtimes H_\mu \cong \bigoplus_\nu N_{\lambda\mu}^\nu H_\nu$, so what are the integers $N_{\lambda\mu}^\nu$?
%
%The answer is quite pretty: $(Rep(LG_\ell),\boxtimes)$ is a quotient of $(Rep(G),\otimes)$. This is all very explicit, and we can compute the numbers explicitly. This is true in general for groups $G$ , done for vertex algebras. It is not known for general groups in conformal net language. It is difficult to do - not just because people are lazy. Conjecturally, we get the semisimplified representation category of a quantum group at a root of unity. The representation category is known to be modular, but the identification has not been done.
%
%\subsection{10:15am Hiro - Representations of $SU(N)$, Pieri rule, fusion rules of $LSU(N)$}
%
%Say we have two irreducible representations of a group $G$ ($=SU(N)$). Representations of a group $G$ are a semisimple category, so we can ask how to their tensor product decomposes into irreducible representations.
%
%$V_f \otimes V_g = \bigoplus_h N^H_{fg} V_h$. This formula is in general kind of complicated, but we can say what it is for special $f$ and $g$.
%
%We'd like to consider reps of a central extension of the loop group. We can label them by $f$ and $g$ as well. Consider $H_f \boxtimes H_g = \bigotimes_{h'} N^{h'}_{fg} sgn(\sigma) H_{h'}$. Here, $\sigma$ is an element of the affine Weyl group $\Lambda_0 \rtimes S_N$.
%
%Crash course in the representation theory of groups, in particular, $SU(N)$:
%
%Two principles in representation theory:
%\begin{enumerate}
%\item Complex representations of $G$ are in correspondence with complex representations of $\mathfrak{g}$.
%\item Irreducible representations of $\mathfrak{g}$ correspond to complex representations of $\mathfrak{g} \otimes \mathbb{C}$.
%\end{enumerate}
%
%We will work with the representations of $SU(3)$. Hopefully, at the end of this, you will understand highest weight vectors and signatures.
%
%\textbf{Example 1:} Take the Lie algebra $\mathfrak{su}(3)$. Its complexification is $\mathfrak{sl}_3(\mathfrak{C})$. Consider the action of this Lie algebra on itself, called the adjoint representation. $X(v) = [X,v] = Xv-vX$. This is a representation because of the Jacobi identity. There is a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{sl}_3$, given by diagonal matrices. We can decompose $\mathfrak{sl}_3$ into eigenspaces under the action of $\mathfrak{h}$. The zero eigenspace is $\mathfrak{h}$ itself, of dimension 2. Other eigenspaces are spanned by $E_{ij}$ for $i \neq j$, that is one at the $(i,j)$ entry and zero elsewhere. These together with $\mathfrak{h}$, span $\mathfrak{sl}_3$. If $X = diag(\alpha_1, \alpha_2, \alpha_3)$, then $X(E_{ij}) = (\alpha_i-\alpha_j)E_{ij}$. We get 6 one-dimensional eigenspaces, and we can place them in a symmetrical arrangement in the plane $\mathfrak{h}^\vee$. This plane is called the space of weights, and we have a weight decomposition $\mathfrak{g} = \bigoplus_{\alpha \in \mathfrak{h}^\vee} \mathfrak{g}_\alpha$.
%
%Off-diagonal matrices will shift weights around.
%
%\textbf{Claim:} If $\beta$ is the weight corresponding to $E_{ij}$, then $v_\beta$ takes $\mathfrak{g}_\alpha$ to $\mathfrak{g}_{\alpha + \beta}$.
%
%\begin{proof} Take $X \in \mathfrak{h}$.
%\[ \begin{aligned}
%X(v_\beta(v_\alpha)) &= v_\beta(X(v_\alpha)) + [X,v_\beta](v_\alpha) \\
%&= v_\beta(\alpha(X)v_\alpha + \beta(X)v_\beta(v_\alpha) \\
%&= (\alpha(X) + \beta(X))v_\beta(v_\alpha)
%\end{aligned} \]
%\end{proof}
%
%Raising operators are those that take things up and to the right (by convention). They kill things that are up and to the right, because we run out of nonzero weight spaces. $E_{ij}$ for $i_k f} V_g$. Here, $g >_k f$ means $g$ is given by adding $k$ boxes to $f$, but no two in the same row. For example, $V_\square \otimes V_\square = V_(2,0,\dots,0) \oplus V_{1,1,0,\dots,0}$. When $N=2$, the latter summand is trivial.
%
%Jump to section 34:
%
%Note: when looking at representations of $\widetilde{LSU(N)}$ at level $\ell$, signatures need to be permissible, i.e., $f_1-f_N \leq \ell$. (This barrier is the far wall of the alcove.)
%
%\textbf{Theorem:} (Verlinde formula)
%
%If $V_f \otimes V_g = \bigoplus_h N^h_{fg} V_h$, then $H_f \boxtimes H_g = \bigoplus_h N_{fg}^h sign(\sigma_h) H_{h'}$.
%
%Action of affine Weyl group: $\Lambda_0 = \{ (N+\ell)(m_i) \mid m_i \in \mathbb{Z}^n, \sum m_i = 0 \}$. $W_{aff} = \Lambda_0 \rtimes S_N$, where the action of the symmetric group permutes coordinates.
\subsection{11:30am Min Ro, Hilbert spaces, polar decomposition, spectral theorem, von Neumann algebras}
Let $H$ be a Hilbert space. $S \subset B(H)$ a set of bounded linear operators.
The commutant of $S$ is $S' = \{ a \in B(H) \mid ab=ba \, \forall b\in S \}$.
The adjoint of an operator $a$ is the unique operator $a^*$ that satisfies $(a\xi,\eta) = (\xi, a^* \eta)$ for all $xi,\eta \in H$. $a$ is self-adjoint if $a^* = a$. $a$ is normal if $a^*a = aa^*$. $a$ is unitary if $aa^* = a^*a = 1$. $a$ is a projection if $a^* = a = a^2$.
There is a rough dictionary, where self-adjoint operators correspond to real-valued functions, and projections correspond to characteristic functions.
Convergence: $a_\lambda \to a$ in strong operator topology if and only if we have pointwise convergence in norm: $\lim_\lambda \Vert a_\lambda \xi \Vert = \Vert a \xi \Vert$. $a_\lambda \to a$ in the weak operator topology if we have pointwise convergence in inner products: $\lim_\lambda |(a_\lambda \xi,eta) | = |(a\xi,\eta)|$ and $|(a\xi,\eta)| \leq \Vert a \xi \vert \cdot \Vert \eta \Vert$.
\textbf{Theorem} (DCT) If $M$ is a unital $*$-subalgebra of $B(H)$, then TFAE
\begin{enumerate}
\item $M'' = M$
\item $M$ is weak operator closed.
\item $M$ is strong operator closed.
\end{enumerate}
\textbf{Definition} A von Neumann algebra $M \subset B(H)$ is a unital $*$-subalgebra that satisfies $M'' = M$.
The center of $M$ is $Z(M) = M \cap M'$.
$M$ is a factor if $Z(M) = \mathbb{C} \cdot 1$.
Examples: $B(H)$ is a factor, and if $H$ is finite dimensional, this is just $M_n(\mathbb{C})$. We can also take direct limits of matrix algebras, e.g., for $F_n$ the Fibonacci sequence, take a limit of inclusions $(M_{F_{n-1}} \oplus M_{F_n} \to M_{F_n} \oplus M_{F_{n+1}}$ by $(a,b) \mapsto (b,\binom{a0}{0b})$.
If $S \subset B(H)$ is a self-adjoint subset, then $S'$ is a von Neumann algebra, and $S''$ is the smallest von Neumann algebra containing $S$. In particular, for any $a \in B(H)$, we can construct $W^*(a) = \{a,a^* \}''$.
Let $X$ be a $\sigma$-finite measure space $(X = \bigcup_{i=1}^\infty E_i)$, then $L^\infty(X)$ includes into $B(L^2(X))$, and this is a von Neumann algebra. Conversely, for any commutative von Neumann algebra $A$, there is a measure space $X$ such that $A \cong L^\infty(X)$.
Let $X$ be compact, $T^2$. $B_b(X)$ is the space of complex Borel bounded functions. $Sp(a) = \{ \lambda \in \mathbb{C} \mid \lambda \cdot 1 -a \text{ not invertible} \}$. This is a nonempty compact set.
\textbf{Theorem:} (Borel functional calculus)
Let $a \in B(H)$ be normal. There is a $*$-homomorphism $B_b(sp(a)) \to W^*(a)$ taking $f \mapsto f(a)$. If $(f_n) \subset B_b(sp(a))_{sa}$, and $f_n \to f$, then $\lim f_n(a) = f(a)$.
\textbf{Definition:} Let $X$ be compact Hausdorff. A spectral measure relative to $(X,H)$ is a map $E$ from the Borel sets of $X$ to the projections of $H$ such that
\begin{enumerate}
\item $E(\emptyset) = 0$, $E(X) = 1$
\item $E(\bigcup_{n=1}^\infty S_i)$ converges to $\sum_{i=1}^\infty E(S_i)$ in the strong operator topology, if $S_i$ are pairwise disjoint.
\item $E(S_1 \cap S_2) = E(S_1) E(S_2)$.
\end{enumerate}
\textbf{Definition:} $\int_X f(\lambda) d(E,\lambda)$ can be defined.
\textbf{Theorem:} (Spectral theorem) Let $a \in B(H)$ be normal. There exists a unique spectral measure relative to $(sp(a),H)$ such that $a = \int_{sp(a)} \lambda d(E,\lambda)$.
\textbf{Theorem:} (Polar decomposition) Let $a \in B(H)$ with $a^*a$ positive (i.e., its spectrum is contained in the nonnegative reals). Then there is a square root $(a^*a)^{1/2}$. There is a unique $u \in B(H)$ such that $u$ is a partial isometry (i.e., $u^*u$ and $uu^*$ are projections), $a=u(a^*a)^{1/2}$, and $\ker(u) = \ker(a^*a)^{1/2}$.
If $a$ is densely defined and closed (i.e., the graph is closed in $H \times H$), then this also works. (NB: there is also a notion of closable operator, whose graph is not closed, but has the property that the closure of the graph is the graph of something else.)
\textbf{Definition:} Let $S \subset B(H)$ be a self-adjoint subset. We consider two types of representation of $S$. ($K$ is some other Hilbert space.)
\begin{enumerate}
\item $\pi: S \to U(K)$ a group.
\item $\pi: S \to B(K)$ a $*$-homomorphism.
\end{enumerate}
We can associate a von Neumann algebra by $S'$ or $S''$. We call either a factor representation if $S'$ is a factor. This is an analogue of a piece of an isotypical decomposition - canonical, not necessarily irreducible.
If $M = S'$, then the projections in $M$ correspond one-to-one to subrepresentations of $(\pi,K)$.
\textbf{Proposition:} If $(\pi,H)$ is a factor representation of $S$, and $\pi_1, K_1)$, $(\pi_2,K_2)$ are two subrepresentations, then
\begin{enumerate}
\item there is a unique $*$-isomorphism $\Theta: \pi_1(S)'' \to \pi_2(S)''$ such that $\Theta(\pi_1(x)) = \pi_2(x)$ for all $x\in S$.
\item For $X = Hom_S(K_1,K_2)$, then $\overline{XK_1} = K_2$.
\item $\Theta(a)T = Ta$ for all $a \in \pi_1(S)'', T \in X$.
\item If $X_0 \subset X$ such that $\overline{X_0K_1} = K_2$, then $\Theta(a)$ is the unique $b \in \pi_2(S)''$ such that $bT = Ta$ for all $T \in X_0$.
\end{enumerate}
%\subsection{2:45pm Ryan Grady, Segal's quantization criterion, actions of $LSU(N)$ and of $Diff(S^1)$ on the fermionic Fock space}
%
%
%We'll construct a representation of $LU(N)$ by considering sections of the vector bundle $S^1 \times \mathbb{C}^N \to S^1$, and studying Clifford algebras and Fock representations.
%
%Let $(H, \langle, \rangle )$ be a Hilbert space, and let $Cliff(H)$ be the corresponding Clifford algebra. It is the unital $*$-algebra generated by $c(f)$ for all $f \in H$, subject to the relations $c(f) c(g) + c(g)c(f) = 0$, $c(f)c(g)^* + c(g)^*c(f) = \langle f, g \rangle$. There is an explicit realization as a quotient of the tensor algebra: $Cliff(H) = T(H)/(f \otimes f - \langle f,f \rangle)$.
%
%$Cliff(H)$ acts on $\bigwedge H$ by the ``wave representation''. $\pi(c(f)) := f\wedge$. There is a distinguished vector $\Omega = 1 \in \bigwedge^0 H \cong \mathbb{C}$, called ``vacuum''. It is cyclic.
%
%$a(f) = c(f)^*$, the annihilation operator. $a(f)(w_0\wedge \dots \wedge w_n) := \sum_{j=0}^n (-1)^j \langle f, w_j \rangle (w_0 \wedge \dots \wedge \widehat{w_j} \wedge \dots \wedge w_n)$.
%
%\textbf{Fact:} $a(f)$ and $c(f)$ are adjoint with respect to the inner product $\langle w_0 \wedge \dots \wedge w_n , \eta_0 \wedge \dots \wedge eta_n \rangle = \det( \langle w_i, \eta_j \rangle )$.
%
%\textbf{Proposition:} $\bigwedge H$ is irreducible as a $Cliff(H)$ representation.
%
%\begin{proof}
%Let $T$ be an endomorphism of $\bigwedge H$ that commutes with all $a(f)$s. Then $T\Omega = \lambda \Omega$ for some scalar $\lambda$, since $Omega$ spans the intersection of all kernels of annihilation operators. If $T$ also commutes with all creation operators, then $T = \lambda I$.
%\end{proof}
%
%\textbf{Unitary structures}
%
%Let $(V, (,))$ be a real Hilbert space. A unitary structure is an element $J \in O(V)$ such that $J^2 = -I$. $V_J$ is then a complex Hilbert space, with Hermitian inner product $\langle v, w \rangle := (v,w) + i(v,Jw)$.
%
%If $V$ already has a complex structure, we can define a projection operator $P_J = \frac12 (I - iJ) \in End(V_J)$. A choice of $P$ is equivalent to a choice of new complex structure. We will use this to define a representation much in the spirit of the wave representation.
%
%\textbf{Definition:} The Fermionic Fock space is $\mathcal{F}_P = \bigwedge(PH) \hat{\otimes} \bigwedge(P^\perp H)^*$.
%
%This is an irreducible representation of $Cliff(V_J)$ via $\pi_P(c(f)) = c(Pf) \otimes 1 + 1 \otimes c((P^\perp f)^*)^*$.
%
%\textbf{Theorem:} (I. Segal-Shale equivalence criterion)
%The Fock representation $\pi_P$ and $\pi_Q$ are unitarily equivalent if and only if $P-Q$ is Hilbert-Schmidt, i.e., for $\{ e_i \}$ a basis for $H$, $\sum_i \Vert (P-Q)e_i \Vert^2 < \infty$.
%
%For $u \in U(H)$, we get an automorphism of $Cliff(H)$ by $c(f) \mapsto c(uf)$. $u$ is said to be implemented in $\pi_P$ (or $\mathcal{F}_P$) if $\pi_P(c(uf)) = U \pi_P(c(F))U^*$ for some unitary $U$ (in $U(\mathcal{F}_P)$, unique up to a scalar).
%
%\textbf{Proposition:} $u \in U(H)$ is implemented if and only if $[u,P]$ is Hilbert-Schmidt.
%
%\textbf{Definition:} The restricted unitary group $U_{res}(H) = \{ u \in U(H) \mid u \text{ is implemented in } \mathcal{F}_P \}$.
%
%We have a representation $U_{res}(H) \to PU(\mathcal{F}_P)$, called the basic representation.
%
%If $u \in U(H)$ and $[u,P] = 0$, then $u$ is implemented in $\mathcal{F}_P$, and is ``canonically quantized''.
%
%Now we construct a representation of $LU(N)$. Let $H = L^2(S^1) \otimes \mathbb{C}^N$, and let $P: H \to H_{\geq 0}$. The target is the Hardy space of boundary values of functions that are holomorphic on the unit disc. For $f \in C^\infty(S^1, End(\mathbf{C}^n)$, we define an operator $m(f)$ on $H$ via multiplication.
%
%\textbf{Fact:} $\Vert [P,m(f)] \Vert_2 \leq \Vert f' \Vert_2$. In particular, $f \in LU(N)$ is implemented in $\mathcal{F}_P$, so we get a projective representation of $LU(N)$ on $\mathcal{F}_P$, called the fundamental representation.
%
%Let $SU_\pm(1,1) = \left\{ \begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha} \end{pmatrix} \mid |\alpha|^2 - |\beta|^2 = \pm 1 \right\}$. This is a semidirect product of $SU(1,1)$ with an orientation reversing transformation on the circle. Fractional linear transformations $z \mapsto \frac{\alpha z + \beta}{\overline{\beta}x + \overline{\alpha}}$ yield a unitary representation $H = L^2(S^1) \otimes \mathbb{C}^N$ via $(V_g \cdot f)(z) = \frac{f(g^{-1}(z))}{\alpha - \overline{\beta}z}$. Note that if $|z| < 1$ and $|\alpha| > |\beta|$, then $(\alpha - \overline{\beta}z)^{-1}$ is holomorphic. This implies $V_g$ commutes with $P$, and the action is canonically quantized.
%
%For $F = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$, $(V_F \cdot f)(z) = \frac{f(z^{-1})}{z}$, so $V_F P V_F = I - P$.
%
%There is an action of $U(1)$ on $H$, via multiplication by the constant function $z$. This is canonically quantized, so we let $U_Z$ be the action on $\mathcal{F}_P$. The $U(1)$ action is called the ``charge operator''. The inclusion of $PH$ into the Fock space has charge 1, and the inclusion of $P^\perp H$ has charge $-1$. Higher wedge powers have suitable charges.
%
%\textbf{Proposition:} If $\pi$ is the representation of $LSU(N)$ on fermionic Fock space $\mathcal{F}_P$, and $U_z$ is the $U(1)$ action, then $\pi(g) U_z \pi(g)^* = U_z$.
%
%\subsection{4:00 Owen Gwilliam, The central extension of $LG$, positive energy representations, Lie algebra cocycles}
%
%
%My goal today is to tell you what a positive energy representation is, and how to get a handle on one, and tomorrow, James will talk about positive energy reps of $LSU(N)$.
%
%\textbf{Outline:}
%\begin{enumerate}
%\item Define and motivate positive energy representations
%\item projective representations and central extensions
%\item reducing questions to the loop algebra
%\end{enumerate}
%
%Let $G$ be a compact connected Lie group, and let $T$ be the circle as a group.
%
%\textbf{Definition:} A positive energy representation of $LG$ is a topological vector space $E$ with
%\begin{enumerate}
%\item a projective representation of $LG$, i.e., for $\gamma \in LG$, get $U_\gamma \in GL(E)$, such that $U_\gamma U_{\gamma'} = c_{\gamma \gamma'} U_{\gamma \gamma'}$ for $c_{\gamma \gamma'} \in \mathbb{C}^\times$.
%\item an intertwining action of $T$, i.e., an operator $R_\theta$ on $E$ for each $\theta \in T$, such that $R\theta U_\gamma R_\theta^{-1} = U_{R_\theta \gamma}$.
%\item under the weight decomposition of $E$ by $T$, $E = \bigoplus_{n \in \mathbb{Z}} E(n)$ then $\dim E(n) < \infty$ and $E(n) = 0$ for $n < 0$.
%\end{enumerate}
%
%Concrete examples: $G = SU(N)$, $V = \mathbb{C}^N$, $H = L^2(S^1,V)$, and $P: H \to H$ is projection onto nonnegative Fourier modes. Then $\mathcal{F}_P = \bigwedge(PH) \hat{\otimes} \bigwedge(P^\perp H)^*$, the fermionic Fock space, is positive energy and level 1. The $\ell$th tensor power is level $\ell$.
%
%\textbf{Theorem:} (Pressley-Segal 9.3.1) A positive energy representation of $GL$ is
%\begin{enumerate}
%\item completely reducible into irreducible positive energy representations
%\item unitary
%\item extend to holomorphic representations of $LG_\mathbb{C}$.
%\item admit a projective intertwining action of $Diff_+(S^1)$.
%\end{enumerate}
%The first three are directly analogous to the theory of compact Lie groups.
%
%Segal says you can think of positive energy representations as boundary conditions of holomorphic representations of the semigroup $\mathbb{C}^times_{\leq 1} \times \widetilde{LG_\mathbb{C}}$. Something about CFT on a cylinder.
%
%More about projective representations:
%
%\textbf{Definition:}
%A projective unitary representation of a group $G$ on a Hilbert space $V$ is a continuous homomorphism $G \to PU(V) = U(V)/T$ (with strong operator topology on $U(V)$ - most of the other topologies on $B(V)$ yield the same topology on $U(V)$, but the norm topology doesn't admit enough representations).
%
%We have $U(V)$ a central extension of $PU(V)$, and a map from $G$ to $PU(V)$, so we can pull back the central extension to get a group $\tilde{G}$ and an honest unitary representation. A central extension of $LG$ yields a circle bundle, and it has a first Chern class $c_1 \in H^2(LG,\mathbb{Z})$.
%
%\textbf{Exercise:} Calculate $H^2(LSU(N),\mathbb{Z})$.
%
% An alternative idea is to consider a central extension of the Lie algebra $L\mathfrak{g}$ by $\mathbb{R}$. Elements are pairs $(X,a)$ for $X \in L\mathfrak{g}$ and $a \in \mathbb{R}$. The bracket has the form $[(X,a),(Y,b)] = ([X,Y], \omega(X,Y))$ for $\omega: L\mathfrak{g} \times \mathfrak{g} \to \mathbb{R}$. The combination of skew-symmetry and Jacobi identity implies $\omega([X,Y],Z) + \omega([Y,Z],Z) + \omega([Z,X],Y) = 0$.
%
%\textbf{Proposition:} (Pressley-Segal 4.2.4) For $\mathfrak{g}$ semisimple, every continuous $G$-invariant 2-cocycle $\omega$ for $L\mathfrak{g}$ has the form $\omega(X,Y) = \frac{1}{2\pi} \int_0^{2\pi} \langle X(\theta), Y'(\theta) \rangle d\theta$ with $\rangle,\langle$ a $G$-invariant symmetric bilinear form on $\mathfrak{g}$.
%
%Note that $H^3(\mathfrak{g})$ is the space of $\mathfrak{g}$-invariant symmetric bilinear forms on $\mathfrak{g}$. Given $\langle,\rangle$, we get the cocycle that takes $X,Y,Z$ to $\langle X, [Y,Z] \rangle$. Given a 2-cocycle on $L\mathfrak{g}$, we get a 2-form on $LG$.
%
%\textbf{Theorem:} (Pressley-Segal 4.4.1) If $G$ is simply connected, then
%\begin{enumerate}
%\item A 2-cocycle $\omega$ on $L\mathfrak{g}$ deines a group extension if and only if $[\omega/2\pi] \in H^2(LG,\mathbb{Z})$.
%\item If we have a group extension, then it is unique (up to isom?)
%\item Such an $\omega$ is an integral multiple of $\omega_{basic}$ (when $G$ is simple).
%\end{enumerate}
%$\omega_{basic}$ corresponds to the inner product on $\mathfrak{g}$ such that $\langle \theta,\theta \rangle = 2$, where $\theta$ is the highest weight.
%
%\textbf{Example:} $\mathfrak{g} = \mathfrak{su}(N)$. $\langle X,Y \rangle = -tr(XY)$. For $N=2$, $\theta = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$, so $-tr(\theta^2) = 2$.
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%\textbf{Definition:} The level of a central extension $\widetilde{LG}$ of $LG$ is the integer $\ell$ such that $\omega_{\widetilde{LG}} = \ell \omega_{basic}$.
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%Return to positive energy:
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%\textbf{Definition:} A positive energy representation of $LG$ is an honest representation of $\widetilde{LG} \rtimes T$, satisfying a positive energy condition. The $T$ on the right is the ``rotation circle'' (or ``energy circle'') $T_{rot}$, and the central circle is labelled $T_{CE}$.
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%We have a Cartan subgroup $T_{rot} \times T_G \times T_{CE} \subset \widetilde{LG} \rtimes T_{rot}$, where $T_G$ is a maximal torus for $G$. This gives us a decomposition $E = \bigoplus_{n,\lambda,\ell} E(n, \lambda, \ell)$
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%Try passing to the Lie algebra. $L^{poly}\mathfrak{g} = \{ \sum_{finite} X_n \}$, and $L^{poly}\mathfrak{g} \otimes \mathbb{C} \cong \mathfrak{g}_\mathbb{C}[z,z^{-1}]$. We want a correspondence between positive energy representations of $LG$ and positive energy representations of $L^{poly}\mathfrak{g}$. Energy is given by $\mathbb{R} \to PU(E)$ via $t \mapsto \pi(exp(tx))$. Abstractly, for $E$ a positive energy representation of $LG$, $L^{poly}\mathfrak{g} \to LG \to PU(E)$, where the first arrow is exponsntiation, and the second is the representation. For each $x \in L\mathfrak{g}$, we get a 1-parameter subgroup, and by Stone's theorem, we get $X \in End(E)$ such that $e^{tX} = \pi(e^{tx})$. This timples we get a projective representation $\rho: L^{poly}\mathfrak{g} \to End(E)$.
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%Let $d$ denote an infinitesimal generator of $T_{rot}$ on $E$. $d|_{E(n)} = n$ and $R_\theta = e^{i \theta d}$.
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%\textbf{Theorem:} (Wasserman) Let $E$ be a level $\ell$ representation of $LG$. Then
%\begin{enumerate}
%\item $E^{fin} := \bigoplus^{alg}_{n \geq 0} E(n) \subset E$ is preserved by $\rho$.
%\item We can choose lifts such that $[d,\rho(X)] = i\rho(x')$ on $E(n)$.
%\item $[\rho(x),\rho(y)] = \rho([x,y]) + i\ell \omega_{basic}(x,y)$.
%\end{enumerate}
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%Concretely, $H = L^2(S^1,V)$, $V = \mathbb{C}^N$. $Cliff(H)$ acts on $\mathcal{F}_P$, the fermionic Fock space. $L^{poly}V = \{$ polynomials in $e^{in\theta} \} \otimes V \to Cliff(H)$ by $\pi$. $e^{in\theta} \otimes v$ is taken to $c(e^{in\theta}v)$. $L^{poly}\mathfrak{su}(N) \to Cliff(H)$. The complexification $\mathfrak{sl}_N$ of $\mathfrak{su}(N)$ is generated by $E_{ij}$. We just write a formula $E_{ij}(n) = \sum_{m >0} e_i(n-m) e_j(-m)^* - \sum_{m \geq 0} e_j(m)^* e_i(m+n)$.
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%\textbf{Theorem:} (Wasserman)
%\begin{enumerate}
%\item $[X(m), a(f)] = a(Xe^{im\theta} f)$ for $f \in L^{poly}V$.
%\item $[d,X(m)] = -mX(m)$.
%\item $[X(n),Y(m)] = [X,Y](n+m) + m \langle X,Y \rangle \delta_{n+m,0}$.
%\end{enumerate}
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%\section{Tuesday, 8/17/10}
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