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\author[Scott Morrison]{Scott Morrison \\ \url{http://tqft.net/} \\ joint work with Vaughan Jones, Emily Peters and Noah Snyder}
\institute{UC Berkeley / Miller Institute}
\title{Small fusion categories and subfactors, \II}
\date{Fusion categories and Applications, Baylor, October 18 2009 \\ \url{http://tqft.net/waco2}}



\begin{document}

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\begin{frame}
       \frametitle{Outline}
       \tableofcontents
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\section{Small objects in fusion categories}

\AtBeginSection[]
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       \frametitle{Outline}
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\begin{frame}

%\begin{thm}[Haagerup et al. $+ \varepsilon$]
%Any object in a fusion category with dimension at least $2$ has dimension at least $\sqrt{3+\sqrt{3}} \sim 2.17533$.
%\end{thm}

\begin{question}
What is the spectrum of possible dimensions for fusion objects?
\end{question}

\begin{block}{}
Dimensions less than $2$ are of the form $2 \cos\left(\frac{\pi}{k+3}\right)$.
\end{block}

{
\mode<beamer>{\setbeamercolor{block title}{bg=blue!80!black}}
\begin{block}{Theorem {\tiny (modulo some corners)} (Jones-Morrison-Peters-Snyder)}
\it Any object in a fusion category with dimension at least $2$ has dimension at least $\sqrt{\frac{5+\sqrt{21}}{2}} \sim 2.1889$.
\end{block}
}

\begin{conj}
This lower bound is realised by a fusion category with fusion graph $\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$.
\end{conj}
\end{frame}

\begin{frame}
\frametitle{Principal graphs}
\begin{block}{Fusion graphs}
The fusion graph for a fusion category $\cC$ with self-dual object $X$ encodes $- \tensor X$ in the fusion ring.
\end{block}

\begin{block}{Principal graphs for subfactors}
The principal graph pair for a subfactor $N \subset M$ has vertices for the $N-N$, $N-M$, $M-N$ and $M-M$ bimodules, and encodes the tensor products $- \tensor_M {}_M M_N$ and $- \tensor_N {}_N M_M$.
\end{block}

\begin{block}{}
The graph norm is the quantum dimension of the generating object. In the subfactor case, this is $\sqrt{[N:M]}$.
\end{block}

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

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

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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-> Earlier this year with Stephen Bigelow and Emily Peters, we constructed the last missing case. \arxiv{0909.4099}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Could these be alternating parts?}
\begin{thm}[Etingof-Nikshych-Ostrik \arxiv{math.QA/0203060}] % Corollary 8.54
Dimensions of objects in fusion categories are cyclotomic integers.
\end{thm}
\begin{block}{}
The index of a subfactor is cyclotomic, since it is the sum of the dimensions of objects at depths $0$ and $2$, all in the even part.
\end{block}
\begin{block}{}
For any subfactor $\dim_q \left({}_N M_M\right) = \sqrt{[N \subset M]}$. Thus the square root of the index of an alternating part must also be cyclotomic.
\end{block}
\end{frame}

\begin{frame}
\begin{cor}
The three subfactors below index $3+\sqrt{3}$ can not be the alternating parts of a fusion category.
\end{cor}
\vspace{-7mm}
\begin{align*}
\operatorname{index}\left(\mathfig{0.15}{graphs/Haagerup-green}\right) & = \frac{5+\sqrt{13}}{2} \\
\operatorname{index}\left(\mathfig{0.225}{graphs/EH-green}\right) & = \frac{8}{3}+\frac{2}{3} \operatorname{Re} \sqrt[3]{\frac{13}{2} \left(-5-3 i \sqrt{3}\right)} \\
\operatorname{index}\left(\mathfig{0.3}{graphs/HA-green}\right) & = \frac{5+\sqrt{17}}{2} 
\end{align*}
\begin{thm}[Haagerup's classification + Corollary]
There are no fusion objects with dimensions in $(2, \sqrt{3+\sqrt{3}})$.
\end{thm}
\end{frame}

\section{Classifying to $\frac{5+\sqrt{21}}{2}$ and beyond!}

\AtBeginSection[]
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\begin{frame}{Classification to a given index}

Begin with $\cF = \eset$ and $\cW = \{ \bigraph{gbg1}\}$.
\begin{itemize}
\item[$\smash{\mathfig{0.07}{trees}}$] $\cF$: a finite set of graphs, any \emph{translate} of which might be a principal graph
\item[$\smash{\mathfig{0.07}{dandelion}}$] $\cW$: a finite set of graphs, any \emph{extension} of a \emph{translate} of which might be a principal graph.
\end{itemize}


\begin{block}{Let the weeds grow}
Take a graph from $\cW$, move it to $\cF$, and add to $\cW$ all its ``depth 1 extensions'' that remain below the index limit.
\end{block}

$\implies$ we can increase the minimal depth in $\cW$.

\begin{block}{Do some gardening}
\begin{itemize}
\item Graphs in $\cF$ ($\cW$) must satisfy the `(weak) Ocneanu test'.
\item `Quadratic tangles' tests can be applied to both $\cF$ and $\cW$, by proving certain inequalities.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Subfactors with index at most $\frac{5+\sqrt{21}}{2}$}

So far, we can't empty $\cW$. Nevertheless, we have a result which is almost strong enough for our purpose!

\begin{thm}[Jones-Morrison-Peters-Snyder]
Below index $\frac{5+\sqrt{21}}{2}$, the principal graph must be one of
\begin{enumerate}
\item [\I]
$\bigraph{gbg1v1v1v1p1v1x0p0x1v1x0p0x1} $
$\bigraph{gbg1v1v1v1v1v1v1v1p1v1x0p0x1v1x0p0x1}$
$\bigraph{gbg1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1} $
$\bigraph{gbg1v1v1v1p1p1v0x0x1v1}$ and 
$\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$, or
\item[\II] a very high translate of a certain list of 7 graphs, or
\item[\III] a very high rank graph beginning $\mathfig{0.7}{2221-badseed}$.
\end{enumerate}
\end{thm}
\end{frame}

\begin{frame}
\frametitle{What about case \II?}
We need to worry about (even-)translates of the graphs
$\bigraph{gbg1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v0x1x0x0v1}$, $\bigraph{gbg1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1}$, $\bigraph{gbg1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x1}$, $\bigraph{gbg1v1v1v1p1v1x0p1x0p0x1v0x1x0p0x0x1v0x1}$, $\bigraph{gbg1v1v1p1v1x0p1x0p0x1v1x0x0p0x0x1v0x1}$, $\bigraph{gbg1v1v1v1p1p1v1x0x0}$ and  $\bigraph{gbg1v1v1v1v1p1p1v1x0x0v1}$.

\begin{block}{Cyclotomicity}
Any given translate can be easily ruled out (we've checked $\sim 45$). It's hard to check cyclotomicity for families (c.f. Asaeda-Yasuda).
\end{block}

\begin{block}{Quadratic tangles}
We've already ruled out some other families using \emph{quadratic tangles}. We expect to eventually rule out the rest too.
\end{block}
\end{frame}

\begin{frame}
\frametitle{What about case \III?}
\begin{block}{Case \III{} makes us sad}
At this point, we can't make $\cW$ empty:  $\cW = \left\{ \mathfig{0.7}{2221-badseed}\right\}$.
\end{block}
\begin{block}{}
Quadratic tangles techniques have already ruled out the other parity, and intermediate subfactor techniques have ruled out $$\bigraph{gbg1v1p1p1v1x0x0}, \bigraph{gbg1v1p1p1v0x1x0p0x0x1v1x0p0x1}\quad \text{and} \quad \bigraph{gbg1v1p1p1v0x1x0p0x0x1v1x0p0x1p0x1}.$$
\end{block}
\begin{block}{}
We can show that any such example has large rank ($>20$), and more computer time increases this (but non-linearly).
\end{block}
\end{frame}

\begin{frame}
\begin{block}{}
All 5 graphs in case \I{} have been realized as principal graphs of subfactors:
\begin{itemize}
\item [GHJ] $\bigraph{gbg1v1v1v1p1p1v0x0x1v1}$ is a GHJ subfactor, with index $3+\sqrt{3}$.
\item[Xu] $\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$ can be constructed indirectly from the conformal inclusion $(G_2)_3 \subset (E_6)_1$, with index $\frac{5+\sqrt{21}}{2}$.
\end{itemize}
\end{block}

\begin{block}{}
Only $\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$ can be the alternating part of a fusion category.
\end{block}

\begin{cor}
Either something exists in Cases \II{} and \III, or the smallest dimension above $2$ in a fusion category is $\sqrt{\frac{5+\sqrt{21}}{2}}$.
\end{cor}
\end{frame}

\begin{frame}
\begin{block}{}
Is $\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$ the alternating part of a fusion category?
\end{block}
\begin{block}{}
We think it is the alternating part arising from a self-dual object: the fusion graph would be the same graph.
\end{block}

\begin{block}{Clues}
\begin{itemize}
\item Xu constructed it via a conformal inclusion.
\item Izumi constructed it using the  Cuntz algebra.
\item The planar generator is at an \emph{odd} depth.
\item The fusion ring passes all of Ostrik's $d$-number tests.
\end{itemize}
\end{block}

\begin{block}{Open question}
Can you find a generators mod relations planar algebra presentation of $\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$. Does it require a shading?
\end{block}
\end{frame}

\begin{frame}
\frametitle{Beyond $\frac{5+\sqrt{21}}{2}$}
\begin{block}{Sketch of classification to index $5$}
\begin{itemize}
\item[\I] The same $5$ subfactors, $H$, $EH$, $AH$, $3311$ and $2221$.
\item[\II] 17 infinite families of translates, with no small super-transitivity examples.
\item[\III] One new `bad seed'
$$\bigraph{gbg1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1},$$
with no small rank examples.
\end{itemize}
\end{block}

Improving Cases \II{} and \III{} will require new `quadratic tangles' style techniques.
\end{frame}
\end{document}
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