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\begin{document}

\title{Tomita-Takesaki theory for fermions
}
 \author{Speaker:  Dmitri Pavlov
 \\ 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!
Outline:
\begin{enumerate}
\item Review of Tomita-Takesaki theory
\item Examples
\begin{enumerate}
\item Modular Theory for fermions
\item Segal's CFT
\end{enumerate}
\end{enumerate}

\section{Review}

Notation:  $L_p:=L^{1/p}$.  We change notation because these L's form a graded algebra, and lower indices indicate covariance (in topology).

\begin{defn}[Definition/Theorem]
If $M$ is a von Neumann algebra, $L_*(M)$ is a $\C_{Re\geq 0}$-graded complex unital *-algebra, with maps
$L_p(M) \times L_q(M) \rightarrow L_{p+q}(M)$ and $*:L_p(M) \rightarrow L_{\bar{p}}(M)$.  

$L_0(M) \simeq M$ as *-algebras, and $L_1(M) \simeq M_*$ -- canonically isomorphic to the predual.  What's more, these are isomorphic as bimodules. (Analog to the Riesz lemma from functional analysis, $(L_1)^* \simeq L_0$.)  (Bimodule structure on predual:  $f \in M_*$, $m,x \in M$:  $(mf)(x):=f(xm)$.)  

There is also a trace $tr:L_1 \rightarrow \C$ such that $tr(xy-yx)=0$.

$z \in L_p^+$, $p \in \R \iff$ there exists $y$, $y^*y=z$.
If $p \in \C_{Re \geq 0}, q \in \R_{\geq 0}$.

\begin{align*}
L_q^+(M) &\rightarrow L_{qp}(M) \\
z &\mapsto z^p
\end{align*}
and if $p\in\R_{>0}$ then $L_q^+ \rightarrow L_{qp}^+$ is a bijection.

If the real part of~$p$ is zero,
then the last map can be extended
to unbounded measures and their powers:
$\hat{L_q^+}\to L_{pq}$; elements $\phi \in \hat{L_1^+}$ are called {\em weights}

\end{defn}

\begin{question}
How do you define these $L_p$ for complex $p$?
\end{question}

\begin{answer}
If $M$ is commutative,
choose some measure $\mu$.  $L_p(\mu):=\{ f | \int \vert f \vert^{1/Re(p)} < \infty \}$ if $Re(p)>0$, or equal to the set of bounded functions if $Re(p)=0$.
\end{answer}

\begin{defn}
The {\em modular automorphism group:}
$M$ a von Neumann algebra, $\phi \in \hat{L_1^+}(M)$, $t \in \mathbb{I}:=\{ x \in C | Re(x)=0\}$.  Then $\sigma_t^\phi(x)=\phi^t x \phi^{-t}$, $\sigma_t^\phi \in Aut(L_p(M))$
\end{defn}

$\sigma_s^\phi(xy)=\phi^sxy \phi^{-s}=\phi^x x \phi^{-s} \phi^s y \phi^{-s}=\sigma_s^\phi(x) \sigma_s^\phi(y)$ so it's a homomorphism.  

Also easy to show $\sigma_s^\phi (\sigma_ t^\phi(x)) = \sigma_{s+t}^\phi(x)$.

\begin{defn}
{\em Radon-Nikodym derivative}
$\phi, \psi \in L_1^+$ and $t \in \mathbb{I}$.  $(D\phi:D\psi)_t=\phi^t\psi^t \in L_0$.  Note that the imaginary power makes unbounded things, bounded.
\end{defn}

\begin{theorem}(KMS condition) Kubo-Martin-Schwinger

For any $M$,
there is a bijection between weights $\phi \in \hat{L_1^+}(M)$, and continuous one-parameter groups of elements in $L_t$, $t \in \mathbb{I} \mapsto U(t) \in L_t(M)$ such that $U(s+t)=U(s)U(t)$ and $U(s)^*=U(-s)$.  The isomorphism is $U(t) = \phi^t$.
\end{theorem}

\section{Examples}

1.  Suppose $H$ is a $\C$-hilbert space and $K$ is a closed real subspace such that $K\cap iK = 0$ and $K+iK$ is dense in $H$.  $Cl^{alg}(K)$ acts on $\Lambda H$ (a Hilbert space) by the creation and annihilation operators.  $Cl(K)$ is the von Neumann algebra generated by $Cl^{alg}(K)$.  $Cl(K^\perp)$ acts on $\Lambda H$; This makes $\Lambda H$ a $Cl(K)$, $Cl(K^\perp)$ bimodule.  Each of these is actually the commutant of the other on $\Lambda H$, whence $\Lambda H \simeq L_{1/2}(Cl(K))$.  The vacuum vector $\Omega \in L_{1/2}(Cl(K))$ gives a finite weight by letting $\Omega = \phi^{1/2}$, $\phi \in L_1^+$.

For example, if $H=L_{1/2}(M)$ and $\phi \in \hat{L_1^+}$, let $K=\overline{M_{sa}\phi^{1/2}\cap L_{1/2}(M)}$.  $\Delta^t\in Aut(L_{1/2}(M))=Aut(H)$, and $* \in \tilde{Aut}(L_{1/2}(M))=\tilde{Aut}(H)$.

\begin{theorem}[Jones-Wasserman]
\begin{enumerate}
\item $\Lambda H$ is an invertible bimodule in the category of bimodules; the left and right actions are commutants of each other.
\item  $*:L_{1/2}(Cl(K))=\Lambda H \acton$.  For 
$\psi \in \Lambda H$, $\psi= a \wedge b \wedge c \wedge \cdots \wedge z$,
$\psi^*=z^* \wedge \cdots \wedge a^*$.
\item $\sigma_t^\phi:  \Lambda H \acton$.  $\psi \in \Lambda H$, $\psi = a \wedge b \wedge \cdots \wedge z$:  
$\sigma_t^\phi(\psi) = \sigma_t(a) \wedge \cdots \sigma_t(z)$.
\end{enumerate}
\end{theorem}





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