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\begin{document}

\title{The symmetric monoidal 3-category of conformal nets 
}
 \author{Speaker:  Braxton Collier
  \\ Typist: Brent Pym
   }
 \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}

The goal of this talk is to give a definition of symmetric monoidal 3-categories, show that conformal nets form such a category and relate the subject to Chern-Simons theory.  (This is work with Bartles, Douglas and Henriques.)

\section{Motivation: Topological QFT}

Let $Bord^n_{n-1}$ be the category of bordisms of $n$-manifolds.  This category is a symmetric monoidal category with operation given by disjoint unions and identity object the empty set $\varnothing$.

\begin{definition}
An $n$-dimensional topological quantum field theory or TQFT is a symmetric monoidal functor
$$
Z : Bord^n_{n-1} \to (Hilbert, \otimes)
$$
\end{definition}

$\Omega Bord^n_{n-1} = End(\varnothing) = \mbox{closed }$n$\mbox{-manifolds}/\mbox{diffeomorphisms}$

$\Omega Hilb = End(\C) = \C$

So $Z$ assigns $\C$-valued diffeomorphism invariant to closed $n$-manifolds.

$Bord^n_k$ is the symmetric $(n-k)$-category which we think of as manifolds with bordisms of bordisms of ..., We have
$$
\Omega Bord^n_k \cong Bord^n_{k+1}
$$

\begin{definition}
Let $\cC$ be a symmetric monoidal $n$-category with $\Omega^{n-1} \cC \cong Hilb_\C$.  A \emph{$\cC$-valued local TQFT} is a symmetric monoidal functor $Bord^n_0 \to \cC$
\end{definition}

\begin{theorem}[BDH]
There exists a symmetric monoidal 3-category $CN$ whose objects are conformal nets and $\Omega^2 CN \cong Hilb$.
\end{theorem}

\begin{theorem}[Cobordism hypotheis of Baez-Dolan, 99\% certainty proof by Hopkins-Lurie]
Framed local $\cC$-valued $n$-dimensional TQFTs are in one-to-one correspondence with dualizable objects in $\cC$.
\end{theorem}

\begin{theorem}
$A \in CN$ is dualizable if and only if it is the direct sum of irreducible conformal nets with finite $\mu$-index.
\end{theorem}

\section{Warm-up: algebras and bimodules}

We assign to a $0$-dimensional manifold an algebra $A$.  Given a 1-morphism between points with algebras $A$ and $B$, we assign an $(A,B)$-bimodule $V$.  We compose these bimodules using the tensor product.  To a 2-morphism, we assign a bi-module homomorphism between the corresponding bi-modules.  There are two ways to compose these morphisms (horizontal and vertical) and in this case we ask that they agree.

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\begin{remark}
We can thing of a symmetric monoidal category as a bi-category with a 1-object, so a symmetric monoidal category is something at least 4-categorical in nature.
\end{remark}

There is a symmetric monoidal category of algebras with $\otimes_\C$.  There is also a symmetric monoidal category of bimodules with $\otimes$:
$$
\left({}_AV_B\right) \otimes \left({}_{A'}V'_{B'}\right) = {}_{A\otimes A'}(V\otimes V')_{B \otimes B'}.
$$
In this case the functors $s$ and $t$ with take an arrow to its source and target are symmetric monoidal functors $Bimod \to Alg$ with
$$
\boxtimes : Bimod\ {}^s\negthickspace\times^t Bimod \to Bimod
$$

The upshot is that $(Alg,Bimod)$ is a category object in the $2$-category $SMC$ of symmetric monoidal categories.  Let us, then, think of conformal nets as a bicategory object in $SMC$.


\section{Conformal nets revisited}

\begin{definition}
We call $Int$ the (topological) category whose objects are oriented intervals and whose morphisms are smooth embeddings, which are not necessarily orientation preserving.  The topology on the hom-sets is given by point-wise convergence.
\end{definition}

Notice that we are allowing more information than just the transformation given by Mobi\"us transformations.
\begin{definition}
A conformal net is a continuous functor
$$
\cA : Int \to vN\mbox{-}alg
$$
from intervals to von Neumann-algebras satisfying the usual axioms, as well as
\begin{itemize}
\item For $\phi : I \to I$, such that $\phi$ is the identity in a neighbourhood of $\partial I$, then $\cA(\phi) : \cA(I) \to \cA(I)$ is inner
\item If $\phi : I \to J$ is orientation preserving (resp. reversing) then $\cA(\phi)$ is a homomorphism (resp. anti-homomorphism).
\end{itemize}
\end{definition}

\section{Conformal nets and 2-algebras}
Given a conformal net $\cA$, let $A = \cA([0,1])$.  Define the standard inclusions
$$
i,j : [0,1] \to [0,2]
$$
where $i$ is the inclusion and $j$ is inclusion plus translation one unit to the right.  Also pick an isomorphism
$$
s : [0,2] \to [0,1]
$$
which has derivative 1 in a neighbourhood of $\partial[0,2]$.  We define
$$
\mu : A \times A \to A
$$
by 
$$
\mu(x,y) = s_*(i_*(x)j_*(y)) = (si)_*(x)(sj)_*(y) =\colvect{x \\ y}
$$

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\begin{claim}
There exists $v \in A$ such that
\begin{enumerate}
\item $$ v \colvect{\colvect{x \\ y} \\ z} v^{-1} = \colvect{x \\ \colvect{y \\ z}} $$
\item $$ v^2 = \colvect{1 \\ v} v \colvect{v \\ 1}$$
\end{enumerate}
\end{claim}

We get identities like

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The morphisms in the category are:
\begin{itemize}
\item 1-morphisms: defects
\item 2-morphisms: sectors
\item 3-morphisms: homomorphisms of sectors
\end{itemize}

\begin{definition}
A \emph{bicolored interval} is an interval $I$ with two subintervals $I_w$ and $I_b$ (the \emph{white} and \emph{black} intervals) with $I = I_w \cup I_b$ and such that either
\begin{enumerate}
\item $I_w = \varnothing$,
\item $I_b = \varnothing$, or
\item $I = I_w \cup I_b$
\end{enumerate}
together with a coordinate function $c : nbd(I_w \cap I_b) \to \R$.
\end{definition}

We can define a category $Int_{bc}$ of bicolored intervals.

\begin{definition}
A \emph{defect $D : A \to B$ for $A,B \in CN$} is a cosheaf $D : Int_{bc} \to vN\mbox{-}alg$ such that
$$
D|_{\mbox{white int.}} = A
$$
and 
$$
D|_{\mbox{black int.}} = B
$$
\end{definition}

\section{Sectors}
Consider intervals $I$ in $S^1$ such that either $i \notin I$ or $-i \notin I$.  We can bicolor such intervals: call the pieces to the left of $\pm i$ black and those to the right of $\pm i$ white.

We have bimodules with defects and a composition using Connes fusion:

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There is a natural isomorphism between the different ways of fusing, which uses the machinery we've been discussing this week.

\end{document}