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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{Kyoto University, October 29 2010 \\ \url{http://tqft.net/Kyoto-2010}}

\usepackage{multimedia}


\begin{document}

\frame{\titlepage}

\beamertemplatetransparentcovered 

\mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}}

\beamersetuncovermixins 
{\opaqueness<1->{60}} 
{} 

\begin{frame}
\frametitle{Subfactors}
\begin{block}{}
We're interested in inclusions $N \subset M$ of $II_1$ factors. The index $[M:N]$ is the von Neumann dimension of $L^2(M)$ as an $N$-module.
\end{block}

\begin{thm}[Jones, Index for subfactors, '83]
If $[M:N] < 4$, then $[M:N] = 4 \cos^2(\pi/n)$ for some $n$.
\end{thm}

\begin{question}
What indices are possible above 4? What indices are possible when $M$ is the hyperfinite $II_1$ factor?
\end{question}
\end{frame}

\begin{frame}
\frametitle{Principal graphs}

\begin{defn}
The principal graph for a subfactor $N \subset M$ has vertices for the $N-N$, $N-M$, $M-N$ and $M-M$ bimodules, and an edge between $Y$ and $Z$ for each copy of $Z$ appearing inside $Y \tensor X$. \small (Here $X = {}_N M _M$ or ${}_M M_N$ as appropriate.)
\end{defn}

\begin{block}{}
The principal graph has two connected components, the left $N$-modules and the left $M$-modules.
\end{block}

\begin{block}{}
The graph norm is equal to the square root of the index of the subfactor {\tiny(at least when the subfactor is finite depth, or is amenable)}.
\end{block}

\begin{block}{}
Graph norm increases under inclusions.
\end{block}
\end{frame}

\begin{frame}

\begin{thm}[Jones, Ocneanu, Kawahigashi, Izumi, Bion-Nadal]
The subfactors with index less than 4 are exactly

\begin{tabular}{ll}
$A_n  \quad \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (0,0) circle (.5mm) node [above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\node at (3,0) {$\cdots$};
	\filldraw (4,0) circle (.5mm);
	
	\draw (0,0)--(2.5,0);
	\draw (3.5,0)--(4,0);

	\draw[decorate,decoration={brace, mirror}] (0,-.2)--(4,-.2) node[midway, below] {$n$ vertices};
\end{tikzpicture}
$, $n\geq 2$ & index $4 \cos^2(\frac{\pi}{n+1})$ 
\\

$D_{2n} \quad \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (0,0) circle (.5mm) node [above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\node at (2,0) {$\cdots$};
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,.5) circle (.5mm);
	\filldraw (4,-.5) circle (.5mm);

	\draw (0,0)--(1.5,0);
	\draw (2.5,0)--(3,0)--(4,.5);
	\draw (3,0)--(4,-.5);

	\draw[decorate,decoration={brace, mirror}] (0,-.7)--(4,-.7) node[midway, below] {$2n$ vertices};
\end{tikzpicture}
$, $n\geq 2$ & index $4 \cos^2(\frac{\pi}{4n-2})$
\\

$E_6 \text{ and } \overline{E_6} \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (0,0) circle (.5mm) node [above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,0) circle (.5mm);
	\filldraw (2,1) circle (.5mm);

	\draw (0,0)--(4,0);
	\draw (2,0)--(2,1);
\end{tikzpicture}$
&
 index $4 \cos^2(\frac{\pi}{12})$ 
 \\

$E_8 \text{ and } \overline{E_8} \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (-2,0) circle (.5mm) node [above] {$*$};
	\filldraw (-1,0) circle (.5mm);
	\filldraw (0,0) circle (.5mm);
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,0) circle (.5mm);
	\filldraw (2,1) circle (.5mm);

	\draw (-2,0)--(4,0);
	\draw (2,0)--(2,1);
\end{tikzpicture}$
\qquad \qquad
& index $4 \cos^2(\frac{\pi}{30})$
\end{tabular}
\end{thm}
\end{frame}

\begin{frame}
\frametitle{Index exactly 4}
\begin{block}{}
There's a similar classification in terms of extended Dynkin diagrams at index exactly $4$. Here the principal graph is no longer a complete invariant, even up to complex conjugation.
\end{block}

\begin{block}{}
At every index $\geq 4$, there's the `Temperley-Lieb' subfactor, with principal graph $A_\infty$
\end{block}


\end{frame}


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

\begin{frame}
\beamersetuncovermixins 
{\opaqueness<1->{0}} 
{} 

\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 $\mathfig{0.3}{graphs/HA}$, $\mathfig{0.4}{graphs/EHA}$, \ldots
\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 $\mathfig{0.3}{graphs/HA-green}$, $\mathfig{0.4}{graphs/EHA}$, \ldots
\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 $\mathfig{0.3}{graphs/HA-green}$\color{red}{,  $\mathfig{0.4}{graphs/EHA-red}$, \ldots}
\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 $\mathfig{0.3}{graphs/HA-green}$\color{red}{,  $\mathfig{0.4}{graphs/EHA-red}$, \ldots}
\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-> Haagerup (unpublished), \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{Constructing the extended Haagerup subfactor}
\begin{frame}
\frametitle{Can we construct a subfactor with a given principal graph?}

\begin{block}{}
The principal graph determines the dimensions of invariant spaces.
\end{block}

\begin{example}
If $\Gamma = \mathfig{0.4}{graphs/Dn}$, then $$\dim \operatorname{Inv}(X^{\tensor 2k}) = \begin{cases}C_k & \text{for $k \leq n$} \\ C_k + 1 & \text{for $k=n+1$.}\end{cases}$$
\end{example}

\begin{block}{}
This gives clues for \emph{generators and relations} for the representation theory.
\end{block}
\end{frame}

\begin{frame}
\begin{thm}[Bigelow-Morrison-Peters-Snyder, \arxiv{0909.4099}]
If a subfactor with principal graph $\mathfig{0.2}{graphs/EH}$ exists, its planar algebra is generated by an $8$-box $S$, which is a lowest weight vector with eigenvalue $-1$ and relations

\inputtikz{TikzStyles}
\inputtikz{STrains}
\begin{enumerate}
\item $S^2 = \lambda^2 \JW{8}$ {\tiny(here $\lambda = \sqrt{-\frac{1}{5}+2 \operatorname{Re} \sqrt[3]{\frac{117-  65 i \sqrt{3}}{2250}}} $)},
\item \mbox{}\vspace{-0.8cm}
\begin{align*}
	\scalebox{0.4}{\inputtikz{A}}
	        & =  [2] [8]
	\scalebox{0.4}{\inputtikz{B}},
	\end{align*}
\item \mbox{}\vspace{-0.8cm}
	\begin{align*}
	\scalebox{0.4}{\inputtikz{C}} 
	         = \frac{1}{\lambda^2} \frac{1}{[9]} \frac{[20]}{[10]}
        \scalebox{0.4}{\inputtikz{D}}.
	\end{align*}
\end{enumerate}
Otherwise, these relations must be inconsistent.
\end{thm}
\end{frame}

\begin{frame}
\begin{proof}
All these relations must hold (sizes of invariant spaces, traces of projections, a quadratic tangles argument). 

They suffice to evaluate any closed diagram via the ``jellyfish algorithm''.
\begin{align*}
\mathfig{0.19}{jellyfish/network-paths} \rightsquigarrow
\mathfig{0.19}{jellyfish/network-paths-2} \rightsquigarrow
\mathfig{0.19}{jellyfish/network-paths-3} \rightsquigarrow
\mathfig{0.19}{jellyfish/network-paths-4}
\end{align*}

Thus this is the representation theory of \emph{some} subfactor with the right index. The classification results ensure that it has the desired principal graph.
\end{proof}
\end{frame}

\begin{frame}
\begin{block}{}
How do we check that relations are consistent?
Just as every finite group sits inside some $S_n$, we have the following.
\end{block}
\begin{thm}[Jones-Penneys \arxiv{1007.317}, Morrison-Walker]
Every subfactor planar algebra embeds in the graph planar algebra of its principal graph.
\end{thm}
\begin{thm}[BMPS '09]
There is an element of the GPA satisfying the desired relations, so the subfactor $\mathfig{0.2}{graphs/EH}$ exists!
\end{thm}

\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{example}
\vspace{-0.5cm}
$$\mathfig{0.2}{odometer/haagerup} \implies \mathfig{0.6}{odometer/vines}$$
\end{example}

\begin{defn}
A weed represents an infinite family, obtained by either translating or \emph{extending} arbitrarily on the right.
\end{defn}
\begin{example}
\vspace{-0.5cm}
$$\mathfig{0.2}{odometer/haagerup} \implies \mathfig{0.6}{odometer/weeds}$$
\end{example}

\end{frame}

\begin{frame}
\begin{block}{}
The weed $\left(\bigraph{bwd1duals1}, \bigraph{bwd1duals1}\right)$ trivially represents all possible principal graphs. 
\end{block}

\begin{block}{}
We can always convert a weed into a vine, at the expense of finding all possible depth $1$ extensions of the weed {\tiny(which stay below the index limit, and satisfying the associativity condition)} and adding these as new weeds.
\end{block}

\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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\end{frame}

\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, 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}
\end{frame}

\begin{frame}
\frametitle{Triple point obstructions}
\begin{thm}[M-Penneys-Peters-Snyder, part $\mathbb{II}$, \arxiv{1007.2240}]
There are no subfactors in the families
$\cC= \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3},\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right)$ or $\cF=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}\right)$
\small(except possibly for infinite depth subfactors at certain indices above $5$).
\end{thm}
\begin{proof}[Sketch]
If $V$ and $W$ are the objects past the branch point, then 
\vspace{-0.2cm}
\begin{align}
\abs{\dim V - \dim W} & \leq 1,  \tag{connections} \\
\frac{\dim V}{\dim W}+\frac{\dim W}{\dim V}  & = \frac{\lambda+\lambda^{-1}+2}{[m][m+2]} \tag{quadratic tangles}
\end{align}
where $m$ is the supertransitivity, and $\lambda$ is a $m+1$st root of unity.
\end{proof}
\end{frame}

\begin{frame}
\begin{thm}[Calegari-Morrison-Snyder, \arxiv{1004.0665}]
Amongst the infinitely many graphs represented by a vine, there are at most finitely many which are the principal graphs of a subfactor, and there is an effective bound.
\end{thm}

\begin{cor}[Penneys-Tener, part $\mathbb{IV}$, \arxiv{1010.3797}]
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, unpublished]
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}{}
It seems likely that the only subfactor coming from the weeds $\cQ$ or $\cQ'$ is $3311$ $\left(\bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}\right)$.
We're talking to Izumi about this during our visit.
\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, by work of Izumi.
\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 $\frac{1}{2} \left(4+\sqrt{5}+\sqrt{15+6 \sqrt{5}}\right) \sim 5.78339$
%% ruled out by Ostrik's result that formal codegrees lie in the FP field.
%\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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