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\begin{document}

\title{Boundary CFTs and their classification via Frobenius algebras}
 \author{Speaker:  Emily Peters \\ Typist: Corbett Redden
 }
 \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!

\newcommand\cO{\mathcal O}
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Idea/Motivation: We're really doing this because we'd like to do all this in 4-dimensions, but its so intractable that we do it in 1 or 2 dimensions.  Boundary CFTs are a nice intermediate point.

We always assume locality in 1-dimension: if $I\cap J =\emptyset$ then $[\cA(I), \cA(J)]=0$ (they commute).  In 2-dimensions, we want them to be in their causal complements.

Let $\cO_1, \cO_2 \in M^2$, $\cO_1$ in the causal complement of $\cO_2$. 

\begin{tikzpicture}
	\draw [thick, <->] (-1,0)--(5,0);
	\draw [thick, <->] (0,-1)--(0,5);
	\draw (0,-.5)--(5,4.5);
	\draw (0,1.5)--(3.5,5);
	\draw (0,3)--(4,-1);
	\draw (0,4.5)--(5, -.5);
	
	\node at (1.625,2.125) {$\cO_1$};
	
	\draw (3,1.8)--(4,2.8)--(5,1.8)--(4,0.8)--(3,1.8);
	
	\node at (4,1.8) {$\cO_2$};
\end{tikzpicture}

\reminder{Finish picture}

\begin{defn}Complete Rationality (of a net on $\R$) means you have
\begin{enumerate}
\item Split property: $I \cap J = \emptyset \Rightarrow \cA(I) \vee \cA(J) = \cA(I)\otimes \cA(J)$
\item Strong additivity: $\cA((a,b)) \vee \cA((b,c)) = \cA((a,c))$
\item Finite index: $\mu_2 < \infty$
\end{enumerate}
\end{defn}

Let $M_+=$ positive Minkowski space $= \{(t,x) \> | \> x>0\}$.  Consider double cones $\cO$

\begin{tikzpicture}
	\draw [thick, <->] (-1,0)--(5,0);
	\draw [thick, <->] (0,-1)--(0,5);
	\draw (0,-.5)--(5,4.5);
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	\draw (0,3)--(4,-1);
	\draw (0,4.5)--(5, -.5);
	
	\draw[decorate, decoration=brace] (-.1,-.5) -- node [left] {$I$} (-.1,1.5);
	\draw[decorate, decoration=brace] (-.1,1.5)--node [left] {$K$}(-.1,3);
	\draw[decorate, decoration=brace] (-.1,3)-- node [left] {$J$}(-.1,4.5);
	\draw[decorate, decoration=brace] (-.6,-.5)-- node [left] {$L$} (-.6,4.5);
	
	\node at (1.625,2.125) {$\cO$};
\end{tikzpicture}
.

\subsection{Boundary CFTs}
\begin{defn}A boundary CFT over a given 1-d conformal net $\cA$ is an assignment $\cO \mapsto \cB_+(\cO) \subset B(H)$ satisfying locality, isotony, equivariance with respect to an action of $PSL_2(\R)$ on $M_+$ and $H$, and 
\begin{itemize}
\item existence and uniqueness of a vacuum vector $\Omega \in B(H)$,
\item covariance: $U(g) \cB_+(\cO) U(g)^* = \cB_+(g\cO)$ when $g\cO$ is a double cone,
\item an action $\pi$ of the net $\cA$ on $H$, covariant under $PSL_2(\R)$; i.e.
\[ U(g) \pi(\cA(I)) U(g^*) = \cA(g I), \]
\item Joint irreduciblity: Where $\pi(\cA)'' = $ the vNA generated by all of $\pi(\cA(I))$, then
\[ \pi(\cA)'' \vee \cB(\cO) = B(H) .\]
\end{itemize}
\end{defn}

\begin{example} Trivial BCFT over $\cA$ is
\[ \cO \mapsto \cA_+(\cO) := \cA(I) \vee \cA(J) \]
Dual to trivial
\[ \cO \mapsto \cA_+^{dual} (\cO) := \cA(K)' \cap \cA(L).\]
\end{example}

\subsection{Relations between different mathematical objects}
Fix a particular 1d CN $\cA(I)$ and examine BCFT over $\cA$.  Focus on Haag dual BCFTs over $\cA$.  These are in 1-1 correspondence with chiral extensions of $\cA$, i.e. 1d CN which extend $\cA$.  These in turn are classified by superfactors of $\cA$ (of index $\mu_2$).  These are in bijection with Frobenius algebras (in some category coming from $\cA$).

\begin{tikzpicture}
\node [text width=3cm] (BCFT)  at (0,2)  {BCFT over $\cA$};
\node [text width=3cm] (Haag dual BCFT) at (0,0) {Haag dual BCFT over $\cA$};
\node [text width=2.5cm] (extensions) at (3.5,0) {Chiral extensions of $\cA$};
\node [text width=2cm] (super) at (7.5,0) {superfactors of $\cA$ of index $\mu_2$};
\node [text width=4cm] (Frobenius) at (12,0) {Frobenius algebras in (a suitable category coming from) $\cA$};

\draw [->] (BCFT) -- (Haag dual BCFT) node [midway, right] {};
\draw [->] (Haag dual BCFT) to [in=150, out=30] (extensions) node [midway, above] {$(\cdot)^{gen}$};
\draw [->] (extensions) to [in=-30, out=-150] (Haag dual BCFT) node [midway, below] {$(\cdot)^{ind}$};
\draw [<->] (extensions) -- (super) node [midway, above, blue] {};
\draw [->] (super) to [in=150, out=30] (Frobenius) node [midway, above] {Q-systems};
\draw [->] (Frobenius) to [in=-30, out=-150] (super) node [midway, below] {conditional expectation and basic construction};

\end{tikzpicture}

 The construction going from a Haag-dual BCFT over $\cA$ to a chiral extension is $gen$, and going the other way is $incl$.

\begin{defn}By locality $\cB_+(\cO) \subset \cB_+(\cO')'$.  The BCFT $\cB$ is {\it Haag dual} if this inclusion is an equality.
\end{defn}


\begin{defn}Given a BCFT $\cB_+$ over $\cA$, its boundary net $\cB^{gen}$ is given by 
\[ \cB^{gen}(I) := \cB_+(W_I), \]
where $W_I$ is the finite wedge determined by $I$, and $\cB_+(W_I)$ is the algebra generated by $\cB_+(O)$ for all $O \subset W_I$.  This is possibly non-local, though relatively local with respect to $\cA$; i.e.
\[ [ \cA(I), \cB(J) ] = 0.\]
\end{defn}

\begin{thm}[or definition] Given an irreducible (non-local) chiral extension $\cB$ of $\cA$, the induced BCFT 
\[ B_+^{ind}(\cO) := \cB(L) \cap \cB(K)'.\]
Then, we check that
\begin{itemize}
\item $(\cB_+^{ind})^{gen} = \cB$
\item $(\cB^{gen})^{ind}_+ = \cB^{dual}$
\end{itemize}
\end{thm}
\ \\
{\bf Facts:}
\begin{itemize}
\item Given chiral extensions $I \mapsto \cB(I) \supset \cA(I)$, we have a consistent family of conditional expectations
\[ \varepsilon_I: \cB(I) \to \cA(I) \]
such that $I \subset J \Rightarrow \varepsilon_I |_J = \varepsilon_J$. 
\item If irreducible and finite index, then $\varepsilon_I$ is implemented by $\varepsilon: \cB \to\cA$.
\end{itemize}

\begin{thm}[Reeh--Schlieder] The vaccum vector $\Omega$ is cyclic and separating for any $\cB(I)$.
 \end{thm}


\begin{thm}Classifying chiral extensions $\cB$ of $\cA$ is equivalent to classifying ``extensions" of $\cA(I)$.
\end{thm}

\subsection{Superfactors to Frobenius algebras}  What is the ``some category coming from $\cA$?"  Objects are elements in $\End(A)$, and morphisms are intertwiners $a\in \cA$ such that for $a\in (\rho, \sigma)$, then $a\rho(x) = \sigma(x) a$.

{\bf Frobenius algebra} in a category $\cC$ consists of an object $Q$, multiplication $m:Q\otimes Q \to Q, \> \eta :1 \to Q$ such that $(Q, m, \eta)$ is a monoid.  $\Delta: Q\to Q\otimes Q,  \epsilon: Q\to 1$ such that $(Q, \Delta, \epsilon)$ is a co-monoid.  And, with $I=H$ relation: $(m\otimes 1) \circ (1 \otimes \Delta) = \Delta \circ m$.  \

\includegraphics[scale=.3]{Friday4pmPicture1.jpg}

\begin{example}$G$ finite group with group ring $\C[G] = \C\{ g \in G\}$.  Then 
\begin{align*}
m(g,h) &= gh \\
\epsilon (\sum a_g g) &= a_e \\
\Delta(g) &= \sum_{ab = g} a \otimes b\\
\eta (1) &= 1
\end{align*}
\end{example}

\begin{example}Subfactors and the canonical endomorphism.  Let $N\subset M$ be type III$_1$ subfactors, $J_N, J_M$ the modular conjugations of $N, M$ (with respect to a cyclic and separating vacuum vector $\Omega \in H$).
\begin{align*}
\gamma: M&\longrightarrow N \\
x &\longmapsto J_N J_M x J_M^* J_N
\end{align*}
\end{example}

Given $N\subset M$, we define a Frobenius algebra in $\End(M)$.  Let $\gamma$ be the canonical endomorphism $\gamma = \iota \overline \iota$, where $\iota: N\into M$, $\overline \iota: M\to N.$\footnote{$\overline \iota = \iota^{-1} \circ \gamma$, where $\gamma$ is from the example above.}

\includegraphics[scale=.3]{Friday4pmPicture2.jpg}

Given a Frobenius algebra in $\End(M)$, it gives a subfactor $E:M\to N$ by defining $E(x)= m \rho(x) \Delta$.

\includegraphics[scale=.3]{Friday4pmPicture3.jpg}


Proof: This is a bimodule map whose image is an algebra!  Want to show
\[ E(xE(y)) = E(x)E(y) \]

\includegraphics[scale=.3]{Friday4pmPicture4.jpg}

End proof.
\end{document}

