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\author[Emily Peters, Noah Snyder]{Emily Peters \\ \url{http://euclid.unh.edu/~eep} \\ Noah Snyder \\ \url{http://math.columbia.edu/~nsnyder} \\ joint work with F. Calegari, Jones, Morrison, Penneys}
\title{Classifying subfactors up to index 5}
\date{Shanks Workshop on Subfactors and Tensor Categories \\ \url{http://euclid.uhn.edu/~eep/Shanks2010.pdf}}

\usepackage{multimedia}


\begin{document}

\begin{thm}
There are 10 subfactors with \\ index in $(4,5]$, and probably no more!
\end{thm}

\begin{frame}
\textbf{$2221$}, Izumi/Xu/Ostrik, 
\end{frame}

\begin{frame}
\begin{itemize}
\item index $\frac{5+\sqrt{17}}{2} \simeq 4.56155$
\item constructed in [Asaeda-Haagerup, math.OA/9803044]
\end{itemize}


\begin{itemize}
\item index $\frac{8}{3}+\frac{2}{3} \operatorname{Re} \sqrt[3]{\frac{13}{2} \left(-5-3 i \sqrt{3}\right)} \simeq 4.3772$
\item constructed in [Bigelow-Morrison-Peters-Snyder, math/0909.4099]
\end{itemize}
\end{frame}

\begin{frame}
\begin{thm}[Etingof-Nikshych-Ostrik, math.QA/0203060]\hspace{0cm}\\
The dimensions of objects in fusion categories are cyclotomic integers.
\end{thm}

\begin{cor}
The index of a subfactor is a cyclotomic integer.
\end{cor}

\begin{thm}[Calegari-Morrison-Snyder, math/1004.0665]\hspace{0cm}\\
For a given graph, the square of the graph norm is cyclotomic for only finitely many supertransitivities.
\end{thm}

\begin{thm}[Calegari-Morrison-Snyder, math/1004.0665]\hspace{0cm}\\
The spectrum of dimensions for objects in a fusion category is discrete in $(2,73/33)$.
\end{thm}
\end{frame}

Any exception must be a translated extension of

with rank $\geq 45$ or a translated extension of

with rank $\geq 20$v

How do we prove classification results?
\begin{itemize}
\item Use the \emph{odometer} to generate families of principal graphs: \emph{vines} and \emph{weeds}.
\item Find \emph{obstructions} to eliminate or truncate families.
\item \emph{Construct} subfactors for the surviving principal graphs.
\end{itemize}

Classification of subfactors up to index 5.

Classification theorem.

The odometer.

Combinatorial obstructions.

Pruning vines.

Pulling out weeds.

Open questions.

Acknowledgements.

\textbf{How does the odometer work?}

The odometer takes a graph pair, and finds all (finitely many) \\
ways to extend the depth by 1, such that:
\begin{itemize}
\item The index stays below 5.
\item Certain combinatorial tests are satisfied.
\end{itemize}

\textbf{Why finitely many?}

\begin{itemize}
\item If there are many vertices at the next depth, some vertex at the previous depth has high valence.
\item No edge multiplicity can be higher than $\sqrt{5}$.
\end{itemize}

\parbox{9cm}
{\textbf{Efficient enumeration}

Index is an increasing function on graphs. We treat the new piece of the adjacency matrix as an `odometer'.

\begin{itemize}
\item Start at the top left of the matrix.
\item Can we increase the current entry?
\begin{description}
\item[yes] Record the new graph, and return the top left.
\item[no] Set the current entry to zero, and move to the next entry.
\end{description}
\item Terminate when you've exhausted the matrix entries.
\end{itemize}
}

\parbox{5cm}{This approach only enumerates graph pairs where both graphs have the same depth!}


\begin{thm}[Haagerup, 1994, MR1317352]\hspace{0pt}\\
All possible principal graphs up to index $3+\sqrt{3}$ \\are represented by the vines:
\end{thm}

\begin{center}
Move a graph $\Gamma$ from $\mathcal{W}$ to $\mathcal{V}$,\\
run the odometer on $\Gamma$, \\
and add the results to $\mathcal{W}$.
\end{center}

$\mathcal{W} = \Big\{,,, \Big\}$

\parbox{5cm}{
There's strong evidence that $$5/2 \simeq SU(2)_2,$$ and
some evidence that $$12/5 \simeq SU(2)_3.$$
}

rings : invertible bimodules :: fusion categories : subfactors

Studying subfactors is studying \textbf{symmetries of fusion categories}.


The principal graph encodes multiplicities for $- \otimes M$.

This does not necessarily determine the full tensor product multiplicities.

Indeed there might not be any!

\begin{thm}[Bisch]\hspace{0pt}\\
There are no associative fusion rules extending those encoded by:
\end{thm}

\textbf{Sketch.} Work mod $2$. Compute $\operatorname{dim} \operatorname{Hom}(2Y \otimes Y, Y) = 0$.\\But as a polynomial in $X$, $2Y = 1 + X(\cdots)$, and \\$\operatorname{dim} \operatorname{Hom}(X^k \otimes Y, Y) = 0$ for $k>1$, by the symmetry of the graph.

\begin{minipage}{7.5cm}
Asaeda noticed that the $n$-supertransitive Haagerup graph has cyclotomic index only for $n = 3$, $n = 7$, and possibly for $n > 55$.

\begin{thm}[Asaeda-Yasuda]\hspace{0pt}\\
For all $n>7$, the index of the \mbox{$n$-supertransitive} Haagerup graph is not cyclotomic.
\end{thm}

\begin{cor}
Thus there are no subfactors with this principal graph.
\end{cor}
\end{minipage}


The main theorem relies on recent contributions from many people:

\textbf{Frank Calegari/Noah Snyder} (bounds from cyclotomicity), \textbf{Dave Penneys/James Tener} (computing these bounds algorithmically), \textbf{Vaughan Jones} (obstructions from quadratic tangles, graph planar algebra techniques), \textbf{Dave Penneys/Emily Peters/Noah Snyder} (applying these obstructions to weeds), \textbf{Noah Snyder} (the odometer, applications of intermediate subfactors), \textbf{Feng Xu/Victor Ostrik} (identifying 2221), \textbf{Stephen Bigelow/Emily Peters/Noah Snyder} (skein theory for the Haagerup family).


Forthcoming papers
\begin{itemize}
\item Towards the classification of subfactors with index at most 5, Morrison-Snyder.
\item Applications of quadratic tangles to subfactors with magic numbers 10, Morrison-Penneys-Peters-Snyder.
\item Quadratic tangles for quadruple points, Jones.
\item ... and probably two or three more!
\end{itemize}

\begin{minipage}{8cm}
For today:

A \emph{subfactor} is a finite depth, finite index, irreducible, extremal $II_1$ subfactor $N \subset M$.

It's better to think about the representation theory (standard invariant): the $2$-category of bimodules between $N$ and $M$.
\end{minipage}


\begin{itemize}
\item unitary,
\item pivotal,
\item two 0-morphisms ($N$ and $M$),
\item semisimple (finitely many simple $1$-morphisms),
\item generated by a fixed simple $N \xrightarrow{X} M$.
\end{itemize}


For any given vine $\Gamma$, we compute a constant $N(\Gamma)$. Increasing the supertransitivity above $N(\Gamma)$ ensures that the index is \emph{not} cyclotomic.

Often this constant is in the range $50$ to $200$. Checking cyclotomicity seperately for every value below this is possible with a modern computer, but sometimes takes a few minutes.


\begin{minipage}{10cm}
\textbf{Sketch.}

Let $\Gamma_n$ be the translate of $\Gamma$ with $n$ vertices.

We first compute a constant $K(\Gamma)$, so $$\sum_{\lambda} \lambda^2 -2 = 2n + K(\Gamma),$$
where $\lambda$ ranges over the eigenvalues of $\Gamma_n$.

With a bounded number of exceptions, $\lambda$ is a real cyclotomic integer. With finitely many exceptions, which occur boundedly often, $$\mathcal{M}(\lambda) = \frac{1}{Gal(\Gamma)} \sum_\sigma (\sigma \lambda)^2 - 2 > \frac{9}{4}.$$

Let $R$ be the bound on the exceptional cases above.
$$\frac{9}{4} (n-R) < 2n + K(\Gamma)$$
\end{minipage}

\begin{frame}
\begin{thm}[Jones, recent proofs Jones-Penneys \& Morrison-Walker]
Given a subfactor $\cP$ with principal graph $\Gamma$, there is an inclusion $$\cP \hookrightarrow GPA(\Gamma)$$
of $\cP$ into the graph planar algebra of $\Gamma$.
\end{thm}

\begin{thm}[Morrison]
Given a subfactor $\cP$ with principal graph starting as\\

and $\operatorname{dim}(P)=\operatorname{dim}(Q)$, either
\begin{enumerate}
\item $n \geq 1$, or
\item the vertices $P'$ and $Q'$ connect at the next depth, or
\item $\cP$ is Haagerup.
\end{enumerate}
\end{thm}

\begin{cor}
The `bad seed' is at least $7$-supertransitive.
\end{cor}
\end{frame}


\begin{minipage}{10cm}
\textbf{Sketch.}

If $n=0$, there is an low weight vector $S \in \cP_4$ satisfying $$S^2 = f^{(4)}.$$

If $P'$ and $Q'$ are not connected by a path of length two, there are no such elements of the `supported at depths $\leq 5$' subalgebra of $GPA(\Gamma)_4$, unless the index is the index of Haagerup.
\end{minipage}


\textbf{Example}

\textbf{Results}

\textbf{Obstructions}

\textbf{How does the odometer work?}

\textbf{Families of principal graphs}

\textbf{Running the odometer}

\parbox{8cm}{The odometer (appears to) run forever on the weeds. We could learn more about the weeds, by running the odometer further. We'd know the graphs to higher depths, but there'd also be many more of them!}

\textbf{What are we going to do about all these vines and weeds!?}


\textbf{Haagerup's triple point obstruction}

\textbf{Ocneanu's square test}

\textbf{Magic numbers}

\begin{minipage}{9cm}
\begin{thm}[Haagerup, MR1317352]\hspace{0pt}\\
Given two dual triple points, for each dimension preserving bijection between the sets of neighbours, there must be a pair of vertices which are not in bijection which have an `unexpected' path of length two between them.
\end{thm}

In practice:
\begin{cor}
Often, the principal graph and dual principal graph must be different!
\end{cor}

\begin{example}
If the principal graph starts as

then the dual principal graph must start as
\end{example}
\end{minipage}

\begin{lem}[Ocneanu]
The multiplicity of $Z$ inside $(X \tensor Y) \tensor X$ is the same as its multiplicity inside $X \tensor (Y \tensor X)$. Both can be computed using the principal graphs and dual data.

\begin{align*}
(X \tensor Y) \tensor X & = (Y^* \tensor X^*)^* \tensor X \\
X \tensor (Y \tensor X) & = ((Y \tensor X)^* \tensor X^*)^*
\end{align*}
\end{lem}

This relates the two principal graphs. The only `easy' way to satisfy this condition is for the graphs to be the same.v


\begin{minipage}{9.5cm}
We don't just use the index limit while running the odometer.

There are certain \emph{combinatorial} obstructions, even without fixing the supertransitivity or the end of the graph.
\end{minipage}


\begin{minipage}{9cm}
Every planar algebra is a representation of the \textbf{annular Temperley-Lieb category}.

Irreducible representations are cyclic, generated by a `lowest weight vector'.

The multiplicities of lowest weight vectors can be read off from either the principal graph or the dual principal graph. \emph{These sequences of numbers must coincide.}
\end{minipage}


\begin{minipage}{7.5cm}The tension between these obstructions (``same same, but different'') is extremely effective! Very few graphs survive.

\hspace{0pt}\\

(For graphs with quadruple points, life is harder, but there aren't many below index $5$.)
\end{minipage}

\begin{minipage}{13cm}
Studying the representations of annular TL, we discover that certain \emph{quadratic tangles} are actually linear combinations of \emph{annular consequences} of a generator. We can compute components of projections: these must sum up to the entire quadratic tangle.

\begin{thm}[Jones]\hspace{0pt}\\
For a graph beginning

with $r=\operatorname{dim}P/\operatorname{dim}Q$ and $\omega$ the rotational eigenvalue of the generator in $\cP_{n+1}$,
$$r+\frac{1}{r}-2 = \frac{\omega + \omega^{-1} + 2}{[n+1][n+3]}$$
\end{thm}

\begin{cor}[Morrison-Penneys-Peters-Snyder]
For two of the weeds,

we can compute $r$ as a function of $n$ and the index, and prove an inequality that contradicts this identity.
\end{cor}
\end{minipage}

\begin{minipage}{9cm}
\textbf{Putting it all together:}

\begin{itemize}
\item We ran the odometer, and found $38$ vines and $6$ weeds.
\item All the vines can be eliminated, using cyclotomicity.
\begin{itemize}
\item Except for finitely many cases: $8$ in fact.
\end{itemize}
\item Some of the weeds we can deal with, using quadratic tangles.
\item Two survive.
\begin{itemize}
\item Using the graph planar algebra, we can increase the minimum supertransitivity.
\item We can run the odometer overnight, and get a lower bound on the number of vertices.
\end{itemize}
\end{itemize}

What's left to do? \textbf{Rule out the exceptions.}
\end{minipage}


An important invariant is the \textbf{principal graph}.
\begin{itemize}
\item A vertex for each simple bimodule, and
\item an edge from $X_N$ to $Y_M$ for each copy of $Y$ inside $X \tensor_N {} M$.
\end{itemize}

\begin{minipage}{9.5cm}
In fact

\begin{thm}[Popa, Jones/Shlyakhtenko/Walker, M\"uger. et al.]\hspace{0pt}\\
Such a $2$-category is always the representation theory of a subfactor!
\end{thm}

From this point of view, the category is central, but sometimes it's convenient to realise it as the representation theory of something.
\end{minipage}

The main theorem relies on recent contributions from many people:\\
\textbf{Frank Calegari/Noah Snyder} (bounds from cyclotomicity), \textbf{Dave Penneys/James Tener} (computing these bounds algorithmically), \textbf{Vaughan Jones} (obstructions from quadratic tangles, graph planar algebra techniques), \textbf{Dave Penneys/Emily Peters/Noah Snyder} (applying this obstructions to weeds), \textbf{Dave Penneys/Kevin Walker} (graph planar algebra embedding theorem), \textbf{Noah Snyder} (the odometer, applications of intermediate subfactors), \textbf{Feng Xu/Victor Ostrik} (identifying 2221), \textbf{Stephen Bigelow/Emily Peters/Noah Snyder} (skein theory for the Haagerup family), \textbf{Anton Geraschenko} (solving large systems of quadratics), \textbf{mathoverflow} (putting the right people in contact).


\begin{center}
Take the `even parts'\\ (a.k.a source and target).

$(X^* \tensor X)^{\tensor k}$ and $(X \tensor X^*)^{\tensor k}$ \\are Morita equivalent.
\end{center}

\end{document}
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