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\author[Scott Morrison]{Scott Morrison \\ \url{http://tqft.net} \\ joint work with Jones, Penneys, Peters, Snyder, Tener}
\title{Classifying subfactors up to index 5}
\date{DARPA kickoff, UCLA, October 8 2010 \\ \url{http://tqft.net/UCLA-2010}}

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\begin{document}

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\section{Haagerup's classification to index $3+\sqrt{3}$}

\begin{frame}
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{} 

\frametitle{Haagerup's list}
\begin{itemize}
\item<1-> In 1993 Haagerup classified possible principal graphs for subfactors with index less than $3+\sqrt{3}$:
\only<1|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/EEH}, \ldots$,
\item $\mathfig{0.3}{graphs/HA}$,
\item \vspace{0.25cm} $\mathfig{0.15}{graphs/hexagon}, \mathfig{0.225}{graphs/Ehexagon}, \ldots.$
\end{itemize}}
\only<2|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/EEH}, \ldots$,
\item $\mathfig{0.3}{graphs/HA-green}$,
\item \vspace{0.25cm} $\mathfig{0.15}{graphs/hexagon}, \mathfig{0.225}{graphs/Ehexagon}, \ldots.$
\end{itemize}}
\only<3|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH}\color{red}{, \mathfig{0.3}{graphs/EEH-red}, \ldots}$,
\item $\mathfig{0.3}{graphs/HA-green}$,
\item \vspace{0.25cm} \color{red}{$\mathfig{0.15}{graphs/hexagon-red}, \mathfig{0.225}{graphs/Ehexagon-red}, \ldots$}.
\end{itemize}}
\only<4>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH-green}\color{red}{, \mathfig{0.3}{graphs/EEH-red}, \ldots}$,
\item $\mathfig{0.3}{graphs/HA-green}$,
\item \vspace{0.25cm} \color{red}{$\mathfig{0.15}{graphs/hexagon-red}, \mathfig{0.225}{graphs/Ehexagon-red}, \ldots$}.
\end{itemize}}
\item<2-> Haagerup and \href{http://dx.doi.org/10.1007/s002200050574}{Asaeda \& Haagerup (1999)} constructed two of these possibilities.
\item<3-> \href{http://www.springerlink.com/content/c4885q02dflfttwm/}{Bisch (1998)} and \href{http://arxiv.org/abs/0711.4144}{Asaeda \& Yasuda (2007)} ruled out infinite families.
\item<4-> Last year we (Bigelow-Morrison-Peters-Snyder) constructed the last missing case. \arxiv{0909.4099}
\end{itemize}
\end{frame}

\section{The classification up to index 5}
\begin{frame}
\frametitle{Classification statements}
We work with \emph{principal graph pairs}, which describe the simple bimodules for the subfactor, along with their tensor products with the generating bimodule, and which bimodules are dual.
\begin{example}[The Haagerup subfactor's principal graph pair]
$$\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$$
\end{example}
The pair must satisfy an associativity test: $$(X \tensor Y) \tensor X \iso X \tensor (Y \tensor X)$$

\begin{block}{}
We can efficiently enumerate such pairs with index below some number $L$ up to any rank or depth, obtaining a collection of allowed \emph{vines} and \emph{weeds}.
\end{block}
\end{frame}

\begin{frame}
\begin{defn}
A vine represents an integer family of principal graphs, obtained by \emph{translating} the vine.
\end{defn}

\begin{defn}
A weed represents an infinite family, obtained by either translating or \emph{extending} arbitrarily on the right.
\end{defn}

\begin{block}{}
If the weeds run out, the enumeration is complete.
This happens in favourable cases (e.g. Haagerup's theorem up to index $3+\sqrt{3}$), but generally we stop with some surviving weeds, and have to rule these out `by hand`.
\end{block}
\end{frame}

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\begin{frame}
\frametitle{The classification up to index 5}
\begin{thm}[Morrison-Snyder, part $\mathbb{I}$, \arxiv{1007.1730}]
Every (finite depth) $II_1$ subfactor with index less than $5$ sits inside one of 54 families of vines (see below), or 5 families of weeds:
\begin{align*}
\cC &= \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3},\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right), \\
	\cF &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}\right), \\
	\cB &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right), \\
	\cQ  &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3} \right), \\
	\cQ' &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3} \right).
\end{align*}
\end{thm}

\begin{thm}[M-Penneys-Peters-Snyder, part $\mathbb{II}$, \arxiv{1007.2240}]
Using quadratic tangles techniques, there are no subfactors in the families
$\cC$ or $\cF$.
\end{thm}
\end{frame}

\begin{frame}
\begin{thm}[Calegari-Morrison-Snyder, \arxiv{1004.0665}]
In any family of vines, there are at most finitely many subfactors, and there is an effective bound.
\end{thm}

\begin{cor}[Penneys-Tener, part $\mathbb{IV}$, conjecture/work in progress]
There are only four possible principal graphs of subfactors coming from the 54 families
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1v1v1x2v2x1}, \bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p1x0duals1v1v1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right)$
\item $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}\right).$
\end{itemize}
\end{cor}
\end{frame}

\begin{frame}
\frametitle{Recent results}
\begin{thm}[Morrison-Penneys-Peters-Snyder, part $\mathbb{V}$, Tuesday]
There are no subfactors coming from the weed $\cB =\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right )$
\end{thm}
\begin{proof}
A connection on the principal graph only exists at a certain index (one for each supertransitivity), but the only graphs with exactly that index are certain infinite graphs which are easily ruled out.
\end{proof}

\begin{block}{Work in progress, Wednesday}
Also by a connection argument (inspired by Izumi), it seems likely that the only subfactor coming from the weeds $\cQ$ or $\cQ'$ is $3311$. $\left(\bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}\right)$
\end{block}
\end{frame}

\begin{frame}
We're thus very close to completing the classification up to index 5:
\begin{conj}
There are exactly ten subfactors other than Temperley-Lieb with index between $4$ and $5$.
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$,
\item $\left(\bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1v1v1x2v2x1}, \bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p1x0duals1v1v1v1v1x2}\right)$,
\item $\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right)$,
\item The 3311 GHJ planar algebra (MR999799), with index $3+\sqrt{3}$   $\left(\bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}\right)$,
\item Izumi's self-dual 2221 planar algebra (MR1832764), with index $\frac{5+\sqrt{21}}{2}$ $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}\right)$
\end{itemize}
along with the non-isomorphic duals of the first four, and the non-isomorphic complex conjugate of the last.
\end{conj}
\end{frame}

\section{Beyond 5: examples and prospects}
\begin{frame}
\frametitle{Index exactly $5$}
There are 5 principal graphs that come from group-subgroup subfactors, and these are known to be unique.
\begin{itemize}
\item  $\left(\bigraph{bwd1v1p1p1p1duals1v4x3x2x1}, \bigraph{bwd1v1p1p1p1duals1v4x3x2x1}\right)$ \tiny $1 \subset \Integer/5\Integer$
\item $\left(\bigraph{bwd1v1p1v1x1v1duals1v1x2v1}, \bigraph{bwd1v1p1v1x1v1duals1v1x2v1}\right)$ \tiny $\Integer/2\Integer \subset D_{10}$
\item $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v2x1x3}\right)$ \tiny $\Integer/4\Integer \subset \Integer/5\Integer \rtimes \operatorname{Aut}(\Integer/5\Integer)$ 
\item $\left(\bigraph{bwd1v1v1v1p1p1v0x0x1p0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x1x0p0x0x1v1x0p0x1duals1v1v1x2x3v2x1}\right)$  \tiny $A_4 \subset A_5$
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x1v1x0v1duals1v1v1x2v1x2v1}, \bigraph{bwd1v1v1v1p1v0x1p0x1v1x0p1x0p0x1v0x1x0v1duals1v1v1x2v1x2x3v1}\right)$  \tiny $S_4\subset S_5$
\end{itemize}
We still have a few other possibilities to rule out
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1p1v0x1x0p0x0x1v1x0p0x1duals1v1v1x3x2v2x1}, \bigraph{bwd1v1v1v1p1p1v0x1x0p0x0x1v1x0p0x1duals1v1v1x3x2v2x1}\right)$
\item $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v1x2x3}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1p1p1v1x0x0p0x1x0v1x0v1v1duals1v1v1v1x2x3v1v1}, \bigraph{bwd1v1v1v1v1v1p1p1v1x0x0p0x1x0v1x0v1v1duals1v1v1v1x2x3v1v1}\right)$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{To index $2\tau^2 \sim 5.23607$ and beyond}
Beyond index 5, complete classification is still daunting. We can still fish for examples (only supertransitivity $> 1$)! Some are already known, but most appear to be new. There aren't yet guarantees that any of these exist, however.
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3},
\bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}\right)$ \\ (from $SU_q(3)$ at a root of unity, index $\sim 5.04892$)
\end{itemize}
At index $2\tau^2 \sim 5.23607$
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1v1p1v1x1p0x1duals1v1v1v1x2}, \bigraph{bwd1v1v1v1v1p1v1x1p0x1duals1v1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1p1p1v0x1x0p0x1x0p0x0x1v0x0x1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0p0x1x0p0x0x1v1x0x0p0x1x0p0x0x1duals1v1v1x2x3v2x1x3}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1p1p1v1x0x0p0x1x0p0x0x1v1x0x0p0x1x0v1x0p0x1duals1v1v1v2x1x3v2x1}, \bigraph{bwd1v1v1v1v1p1p1v1x0x0p0x1x0p0x0x1v1x0x0p0x1x0v1x0p0x1duals1v1v1v2x1x3v2x1}\right)$
\item $\left(\bigraph{bwd1v1v1p1v1x0p1x0p0x1p0x1v0x1x0x1duals1v1v1x4x3x2}, \bigraph{bwd1v1v1p1v1x1v1v1duals1v1v1v1}\right)$
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1p1v1x1p0x1v0x1duals1v1v1x2}, \bigraph{bwd1v1v1p1v1x1p0x1v0x1duals1v1v1x2}\right)$ \\ (``Haagerup $+1$'' at index $\frac{7+\sqrt{13}}{2} \sim 5.30278$)
\item $\left(\bigraph{bwd1v1v1p1p1v0x1x1p0x0x1duals1v1v1x2}, \bigraph{bwd1v1v1p1p1v0x1x1p0x0x1duals1v1v1x2}\right)$ at $\frac{1}{2} \left(4+\sqrt{5}+\sqrt{15+6 \sqrt{5}}\right) \sim 5.78339$
\item $\left(\bigraph{bwd1v1v1v1v1p1p1v0x1x1p0x0x1duals1v1v1v1x2}, \bigraph{bwd1v1v1v1v1p1p1v0x1x1p0x0x1duals1v1v1v1x2}\right)$ at $3+2\sqrt{2} \sim 5.82843$
\end{itemize}
And at index $6$
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1p1v1x0p1x1p0x1v1x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1p1v1x0p1x1p0x1v0x1x0v1duals1v1v1x2x3v1}\right)$
\item $\left(\bigraph{bwd1v1v1p1v1x0p1x1p0x1v0x1x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1p1v1x0p1x1p0x1v0x1x0v1duals1v1v1x2x3v1}\right)$
\end{itemize}
and several more!
\end{frame}

\begin{frame}
\frametitle{Summary and prospects}

\begin{block}{}
The classification of subfactors up to index 5 is almost finished.
\end{block}
\begin{block}{}
We can look further out; there are several new examples, but it's sparser than anyone expected. New methods using connections may allow complete classifications to higher indices.
\end{block}
\begin{block}{}
Our techniques also apply to fusion categories. Fusion categories with objects of dimension $2 \cos(\pi/n)$ have been used in topological quantum computing. We expect to obtain strong new classification results for dimension slightly above $2$.
\end{block}
\end{frame}

\end{document}
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