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\begin{document}

\title{Conformal nets
}
 \author{Speaker:  Corbett Redden
 \\ Typist:  Pavel Safronov
 }
 \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

\section{More M\"{o}bius group}
It is the group of conformal automorphisms of $D\subset\C$. It is denoted by $PSU(1,1)\cong PSL_2(\R)$.

\begin{align*}
S^1-\text{picture}&\quad\R-\text{picture} \\
z &\leftrightarrow x \\
SU(1,1)&\leftrightarrow SL_2(\R)
\end{align*}

It consists of translations $T_tx=x+t$, dilations $D_tx=e^{-2\pi t}x$ and rotations $R_tz=e^{-2\pi it}z$.

\underline{Fact}: we have an isomorphism of manifolds $SU(1,1)\cong D\times T\times R$.

\underline{Corollary}: $SU(1,1)$ acts transitively on $\{I\}_{I\subset S^1}$, and
\begin{itemize}
\item isotropy group of $z\in S^1$ is $D\times T$.
\item isotropy group of $\{z_1,z_2\}\in S^1$ is $D$.
\end{itemize}

\section{Definitions of conformal nets and examples}

\begin{defn}
A (vacuum) conformal net, denoted by CN (VCN), is a collection of von Neumann algebras $\{\cA(I)\}_{I\subset S^1}$, parametrized by open, connected, non-dense intervals, that satisfy the following axioms

\begin{enumerate}
\item (Isotony) $I\subset J\Rightarrow \cA(I)\subset\cA(J)$.

\item (Locality) $I\subset J'=S^1\backslash\overline{J}\Rightarrow\cA(I)\subset\cA(J)'$.

\item (M\"{o}bius covariance) There exists a representation $PSU(1,1)\rightarrow U(H)$, such that $\pi(g)\cA(I)\pi(g)^*=\cA(gI)$.

\item (Positive energy) $R\subset PSU(1,1)$ should be positive energy.

Vacuum nets also satisfy:

\item (Vacuum) There exists a unique (up to a factor) vacuum vector $\Omega\in H$, $\Omega$ invariant under the M\"{o}bius action and $\{\bigcup_{I\subset S^1}\cA(I)\}'"\Omega$ is dense in $H$.
\end{enumerate}
\end{defn}

Remark: although $PSU(1,1)$ acts projectively, $U(1)\subset PSU(1,1)$ acts honestly.

\underline{Example}: $\pi: \tilde{LG}_\ell\rightarrow U(H)$ be an IPER $\cA(I)=\pi(\tilde{L_IG})''$.

\begin{defn}
An irreducible representation is called a vacuum representation if it has a vacuum $\Omega$ invariant under $PSU(1,1)$.
\end{defn}

\begin{thm}
$\cA$ is a conformal net. If $\pi$ is a vacuum representation, then $\cA$ is a vacuum conformal net.
\end{thm}
\begin{proof}
\begin{enumerate}
\item $I\subset J\Rightarrow \tilde{L_IG}\subset \tilde{L_JG}\Rightarrow \pi(\tilde{L_IG})''\subset\pi(\tilde{L_IG})''$.

\item $I\cap J=\emptyset$, then $[\tilde{L_IG},\tilde{ L_JG}]=1\Rightarrow\cA(I)$ commutes with $\cA(J)$.

\item $PSU(1,1)$ acts conformally on $D\subset S^1$, therefore canonically implemented.

\item True by assumption.

\item True by definition.
\end{enumerate}
\end{proof}

\section{Properties}

\begin{thm}[Reeh-Schlieder]
If $\cA$ is a vacuum conformal net, then $\Omega$ is cyclic for each $\cA(I)$: $\overline{\cA(I)\Omega}=H$.
\end{thm}
\underline{Corollary}: $\Omega$ is cyclic and separating for each $\cA(I)$.
\begin{proof}
$\Omega$ is separating for $\cA(I)\Leftrightarrow\Omega$ is cyclic for $\cA(I)'$. But $\Omega$ is cyclic for $\cA(I')\subset\cA(I)'$.
\end{proof}

\underline{Summary}: $I\rightarrow\cA(I)$, we have a cyclic and separating $\Omega$, so can use Tomita-Takesaki theory. $S_I(A\Omega)=A^*\Omega$. From this we get the modular operators:
\begin{align*}
J_I\cA(I)J_I&=\cA(I)' \\
\Delta_I^{it}\cA(I)\Delta_I^{-it}=\cA(I)
\end{align*}

\begin{thm}

$\cA$ is a vacuum conformal net.

\begin{itemize}
\item All $\cA(I)$ are type III$_1$ factors.

\item If inequality (almost always satisfied), then $\cA(I)$ is hyperfinite and there is a unique (up to an isomorphism) type III$_1$-factor.
\end{itemize}
\end{thm}

Question: why is it a factor?

Answer: it follows from the axioms of a conformal net, not true for higher dimensions.

Let $j_{S_+}\in SU_-(1,1)$ be the flip.

\begin{thm}[Geometric modular operators]
$\cA$ is a vacuum conformal net.
\begin{enumerate}
\item $\pi$ extends to $PSU_{\pm}(1,1)\stackrel{\pi}{\rightarrow}U_{\pm}(H)$ such that $J_{S_+}=\pi(j_{S_+})$.

\item $\Delta^{it}_{S_+}=\pi(D_t)$.
\end{enumerate}
\end{thm}
\begin{proof}
\begin{enumerate}
\item Check homomorphism for $D, R, T$.
\item Work with equivariance properties.
\end{enumerate}
\end{proof}

\begin{thm}[Haag duality]
If $\cA$ is a vacuum conformal net, then $\cA(I')=\cA(I)'$.
\end{thm}
\begin{proof}
Because of the M\"{o}bius covariance, it suffices to show only for $S_+$.

$J_{S_+}\cA(S_+)J_{S_+}=\cA(S_+)'=\pi(j_{S_+}\cA(S_+)\pi(j_{S_+})=\cA(S_+')$.
\end{proof}

\section{Representations}

\begin{defn}
A representation of a conformal net $\cA$ on a Hilbert space $H_\pi$ is a collection of representations $\{\pi_I\}_{I\subset S^1}$, $\pi_I:\cA(I)\rightarrow B(H_\pi)$, such that
\begin{enumerate}
\item (Consistency) $I\subset J\Rightarrow \pi_I=\pi_J\left|_{\cA(I)}\right.$.

\item There exists a representation $\pi^m:PSU(1,1)\rightarrow PU(H_\pi)$. $\pi^m(g)\pi_I(-)\pi^m(g)^*=\pi_{gI}(\alpha_{g-})$. Here $\alpha$ is a conjugation using the M\"{o}bius representation on $\cA$.

\item Rotations in $\pi^m$ are generated by a positive operator.
\end{enumerate}
\end{defn}

Question: is it true that $\cA(I)$ and $\cA(J)$ commute in the representation if $I$ and $J$ are disjoint?

Examples:
\begin{itemize}
\item Identity representation: $\pi(\cA(I))=\cA(I)$.

\item Let $\cA_0$ be a vacuum conformal net of level $\ell$ IPER of $LG$. $\pi:\tilde{LG}_\ell\rightarrow U(H_\pi)$ an IPER. Obtain a representation of $\cA_0$.
\end{itemize}

Since $\pi_0,\pi$ are subrepresentations of $\pi^{\otimes\ell}=$ factor representation, then local equivariance property from Min's talk to guarantee the map $\pi_0(L_IG)''\rightarrow\pi(L_IG)''$.
\end{document}