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\author[Scott Morrison]{Scott Morrison \\ \url{http://tqft.net}}
\title{Fusion categories and subfactors}
\date{Caltech colloquium, November 10 2010 \\ \url{http://tqft.net/Caltech-2010}}

\usepackage{multimedia}

\begin{document}

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\section{What is a ...?}
\begin{frame}
\frametitle{What is a fusion category?}
\begin{defn}
A fusion category is a linear semisimple $\tensor$-category with duals, with finitely many simple objects.
\end{defn}
\begin{example}
The representation theory of any finite group is a fusion category.
\end{example}
\begin{example}
The representation category of $U_q(\mathfrak{g})$ at a root of unity is not semisimple, but after quotient by the `negligible ideal' it becomes a fusion category.
\end{example}

These are great examples to keep in mind --- but are there fusion categories that don't `essentially' come from one of these two sources?
\end{frame}

\begin{frame}
`With duals' means
\begin{itemize}
\item There's a functor $* :  \cC \to \cC$ that reverses tensor products, and $** = \id_\cC$.
\item There are maps
\begin{align*}
\tikz{\draw (0,0) arc (180:0:0.4);}: & V^* \tensor V \to \id &  \text{(pairing)} \\
\tikz{\draw (0,0) arc (-180:0:0.4);}: & \id \to V \tensor V^* &  \text{(copairing)}
\end{align*}
 for each object $V$, such that
\begin{itemize}
\item $\tikz[baseline]{\draw (0,0.5) -- +(0,-0.5) arc (-180:0:0.4) arc (180:0:0.4) -- + (0,-0.5);} = \tikz[baseline]{\draw (0,0.5) -- +(0,-1);} = \tikz[baseline]{\draw (0,0.5) -- +(0,-0.5) arc (0:-180:0.4) arc (0:180:0.4) -- + (0,-0.5);}$\;, \quad and
\item $\tikz[baseline]{\node[draw,rectangle](f) at (0,0) {$f$}; \draw (f.south) -- +(0,-0.3); \draw (f.north) arc (180:0:0.4) -- +(0,-0.8);} = \tikz[baseline]{\node[draw,rectangle](f) at (0,0) {$f^*$}; \draw (f.south) -- +(0,-0.3); \draw (f.north) arc (0:180:0.4) -- +(0,-0.8);} $
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{More examples}
\begin{example}[Quantum $SU(2)$ at a root of unity]
The irreps of $SU(2)$ (or of $U_q(\mathfrak{sl}_2)$) are indexed by natural numbers (the `highest weight', which is one less than the dimension), and the tensor product rule is
\vspace{-0.3cm}
$$V_a \tensor V_b = \DirectSum_{c=\abs{a-b}}^{a+b} V_c.$$
\vspace{-0.3cm}

When $q$ is a root of unity, the tensor category has a `negligible ideal'. We need to quotient by this to get a semisimple category. Only finitely many objects survive; when $q$ is a $4n-4$th root of unity, it's the first $n$ objects. The `truncated tensor product' is
\vspace{-0.3cm}
$$V_a \tensor V_b = \DirectSum_{c= \abs{a-b}}^{\min\{a+b,2n-1-a-b\}} V_c.$$
\vspace{-0.3cm}
\end{example}
\end{frame}

\begin{frame}
\begin{block}{}
Representations of finite groups always form a \emph{symmetric} tensor category; representations of quantum groups form a \emph{braided} tensor category. Both of these are special situations, however.
\end{block}

\begin{example}
The category of $G$-graded vector spaces, with $G$ a finite group, is neither symmetric nor braided when $G$ is nonabelian.
\end{example}
\end{frame}

\begin{frame}
\frametitle{Dimensions}
\begin{defn}
A \emph{dimension function} on a fusion category is a homomorphism from the Grothendieck ring to $\mathbb{C}$.
\end{defn}

\begin{defn}
The \emph{principal graph} for an object $X \in \cC$ has vertices $\text{Obj}(\cC)$, and an edge from $Y$ to $Z$ for each summand of $Z$ in $Y \tensor X$.
\end{defn}

\begin{defn}
The \emph{Frobenius-Perron dimension} of $X$ is the largest eigenvalue of the principal graph for $X$.
\end{defn}

The Frobenius-Perron dimension is always an algebraic integer.
\end{frame}

\begin{frame}
In a $\tensor$-category with duals, endomorphisms have a trace:
\begin{align*}
\tr{f} & = \tikz[baseline]{\node[draw,circle](f) {f}; \draw (f.north) arc(180:0:0.25) -- +(0,-0.65) arc(0:-180:0.25);}  \\
        & = p_{X^*} \compose (f \tensor \id_{X^*}) \compose c_X
\end{align*}
where $p_{X^*} : X \tensor X^* \to \id$ is the duality pairing, and $c_X : \id \to X \tensor X^*$ is the copairing.

\begin{defn}
The \emph{categorical dimension} of an object is $\tr{\id_X}$.
\end{defn}

If the fusion category is \emph{unitary} (there's a $*$-structure on $\operatorname{Hom}$ spaces, so $\langle x, y\rangle = \tr{y^* x}$ is positive definite), then the Frobenius-Perron and categorical dimensions agree.

\end{frame}

\begin{frame}
\frametitle{What is a subfactor?}
\begin{block}{}
\begin{itemize}
\item A subfactor is an inclusion of von Neumann algebras $N \subset M$ each with trivial centre.
\item We're interested in  $II_1$ factors {\tiny(no minimal projections, the identity is finite)}.
\item The \emph{index} of $N \subset M$ is the von Neumann dimension of $M$ as an $N$ module.
\end{itemize}
\end{block}

\begin{defn}
The ``even part of $N \subset M$'' is the collection of $N{-}N$ bimodules generated by ${}_N M_N$.
It is a unitary semisimple $\tensor$-category. If the index is finite, it has duals.
\end{defn}
The dimension of the object $M$ is the index of $N \subset M$.
\begin{block}{}
When there are finitely many simple objects (so the even part is a fusion category), we say $N \subset M$ is finite depth.
\end{block}

\end{frame}

\begin{frame}
\frametitle{Algebra objects in fusion categories}
In the even part of $N \subset M$, there is a special object ${}_N M_N$, which has an algebra structure. From this, we can recover the subfactor.

\begin{thm}
For every algebra object $\cA$ in a \emph{unitary} fusion category $\cC$, there is a finite depth $II_1$ subfactor $N \subset M$ so $$(\cC, \cA) \iso ({}_N\text{mod}_N, M).$$
\end{thm}
\end{frame}

%% begin noop
\noop{
\begin{frame}
\begin{block}{}
The category of $\cA-\cA$-bimodule objects in $\cC$ form another unitary fusion category $\cC'$. The category of $1-\cA$-bimodule objects in $\cC$ provides a (categorical) Morita equivalence between $\cC$ and $\cC'$.
\end{block}

More generally
\begin{thm}[Ostrik]
If $\cX$ is a bimodule category between unitary fusion categories $\cC$ and $\cD$, the `internal endomorphisms' of any object $X \in \cX$ is an algebra object in $\cC$, and
\begin{align*}
\cX &\iso {}_{1}\text{mod}_{\underline{\text{End}}(X)} \\
\cD &\iso {}_{\underline{\text{End}}(X)}\text{mod}_{\underline{\text{End}}(X)}.
\end{align*}
\end{thm}
\end{frame}

\begin{frame}
\begin{block}{}
We can think of a finite depth subfactor as a pair of unitary fusion categories, a Morita equivalence between them, and a chosen object in the bimodule category.
\end{block}

Because of this close relationship between unitary fusion categories and subfactors, we often take advantage of both settings to prove theorems.
\end{frame}
}
%% end noop

\begin{frame}
\begin{block}{}
In fact, a finite depth subfactor is equivalent to either
\begin{enumerate}
\item a unitary $\tensor$-category $\cC$, an algebra object $A \in \cC$, and a chosen object $X$ in the category of $A$-module objects, or
\item a pair of unitary $\tensor$-categories $\cC$ and $\cD$, a (categorical) Morita equivalence $\cX$ between them, and a chosen object $X \in \cX$.
\end{enumerate}
\end{block}

Because of this close relationship between unitary fusion categories and subfactors, we often take advantage of both settings to prove theorems.
\end{frame}

\section{Classification}
\subsection{Dimensions}
\begin{frame}
\frametitle{What's out there?}
The dimensions of fusion objects are highly constrained. Jones proved the first result in this direction.
\begin{thm}[Jones, Index for subfactors, '83]
If $1 < \dim V < 2$, then $\dim V = 2 \cos(\pi/n)$.
\end{thm}
\begin{proof}
These are the only real algebraic integers less than 2 which are maximal amongst their conjugates.
\end{proof}
\end{frame}

\begin{frame}
\begin{thm}[Coste-Gannon, '94]
Every dimension is a cyclotomic integer.
\end{thm}
\begin{proof}
Entries of the $S$-matrix of the Drinfeld center are cyclotomic.
\end{proof}

\begin{thm}[Calegari-Morrison-Snyder, CMP '10]
If $2 < \dim V < 76/33$, then $\dim V$ is one of
$$\frac{\sqrt{7} + \sqrt{3}}{2},
\sqrt{5}, 1 + 2 \cos(2 \pi/7), \frac{1 + \sqrt{5}}{\sqrt{2}}, \frac{1 + \sqrt{13}}{2}$$
\end{thm}
\begin{proof}
These are the only real cyclotomic integers less than $76/33$ which are maximal amongst their conjugates.
\end{proof}
\end{frame}

\subsection{Enumerating graphs}
\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}
\frametitle{Classification statements}
\begin{block}{}
We also remember which bimodules are dual to each other.
\end{block}
\begin{example}[The Haagerup subfactor's principal graph]
$$\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$$
\end{example}
The principal graph must satisfy an associativity test: $$(X \tensor Y) \tensor X \iso X \tensor (Y \tensor X)$$

\begin{block}{}
We can efficiently enumerate such pairs of graphs 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, but generally we stop with some surviving weeds, and have to rule these out `by hand`.
\end{block}
\end{frame}


\subsection{Results}
\begin{frame}
\frametitle{Classification results for index less than $3+\sqrt{3}$}
\begin{thm}[Haagerup, '93]
All subfactors other than $A_\infty$ with index in the interval $(4, 3+\sqrt{3})$ are represented by the following vines:
\begin{enumerate}
\item $ \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1}, \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$ \\
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1x2v1}\right)$
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x1duals1v1v1x2v1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1duals1v1v1x2v1x2}\right)$
\end{enumerate}

\end{thm}

\begin{block}{}
This was a favourable case, where the enumeration ran out of weeds.
\end{block}
\end{frame}

\begin{frame}
\begin{thm}
There are exactly three subfactors in this range, with principal graphs
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1v1v1x2v2x1} \bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p1x0duals1v1v1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1} \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right)$
\end{itemize}
\end{thm}
\begin{proof}
\begin{itemize}
\item Asaeda-Haagerup '98 constructed the first two examples.
\item An unpublished result of Haagerup's, and results of Bisch '98, Asaeda-Yasuda '07 showed that there are no others except possibly the third example.
\item Bigelow-Morrison-Peters-Snyder constructed the `extended Haagerup' subfactor last year. 
\end{itemize}
\vspace{-0.5cm}
\end{proof}
\end{frame}

\begin{frame}
\frametitle{Classification results for index less than $5$}
\begin{thm}[Morrison-Snyder \arxiv{1007.1730}]
All subfactors other than $A_\infty$ with index between $4$ and $5$ are represented by 43 families of vines, or the following $5$ weeds.
\begin{align*}
\cC &=  \left(\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0v1p1duals1v1v1x2v1}, \bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1v0x1x0p0x0x1duals1v1v1x2v3x2x1}\right), \displaybreak[1]\\
     \cF &= \left(\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v1x0p1x0p0x1v0x1x0p0x0x1v1x0p0x1p0x1duals1v1v1x2v1x2v2x1}, \bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x0x1p1x0x0x0v1x0x0p0x1x0p0x1x0p0x0x1v0x0x0x1p1x0x0x0p0x1x0x0duals1v1v1x2v4x2x3x1v3x2x1x4}\right), \displaybreak[1]\\
      \cB &= \left(\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v1x0p0x1v1x0p0x1v1x0p0x1duals1v1v1x2v1x2v2x1}, \bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x0x0x1p1x0x0x0v1x0p0x1v0x1p1x0duals1v1v1x2v4x2x3x1v1x2}\right), \displaybreak[1]\\
      \cQ &=\left(\bigraph{bwd1v1v1v1p1p1v1x0x0duals1v1v1x3x2}, \bigraph{bwd1v1v1v1p1p1v1x0x0duals1v1v1x3x2}\right)\\
    \cQ' &= \left(\bigraph{bwd1v1v1v1p1p1v1x0x0duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v1x0x0duals1v1v1x2x3}\right)
\end{align*}
\end{thm}
\end{frame}

\begin{frame}
\frametitle{Eliminating vines}
\begin{thm}[Calegari-Morrison-Snyder, CMP '10]
In any vine, only finitely many graphs have a cyclotomic index.
With much better bounds, all but finitely many graphs have a multiplicity free eigenvalue which is not cyclotomic.
\end{thm}

\begin{block}{}
Either condition is sufficient to eliminate a possible subfactor.
\end{block}

\begin{itemize}
\item Penneys-Tenner \arxiv{1010.3797} have recently developed algorithms for efficiently computing these bounds,
\item and computed them for the 43 vines in our enumeration.
\item They looked at the finitely many cases remaining from the vines, and found obstructions for all but one graph.
\end{itemize}
\end{frame}

\begin{frame}
All of the weeds have been killed off:
\begin{itemize}
\item $\cC$, $\cF$ and $\cB$ by Morrison-Penneys-Peters-Snyder \\($\cC$, $\cF$ in \arxiv{1007.2240}, $\cB$ unpublished),
\item$\cQ'$ and $\cQ$ by Izumi-Jones-Morrison-Snyder (last week!).
\end{itemize}

\begin{thm}
The only subfactors with index in the interval $[3+\sqrt{3}, 5)$ are
\begin{description}
\item[$A_\infty$] at every index,
\item[`3311'] $\left(\bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}\right)$ \\with index $3+\sqrt{3}$ (Goodman-de la Harpe-Jones, '89),
\item[`2221'] $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}\right)$ \\ with index $\frac{5+\sqrt{21}}{2}$ (Izumi, '01).
\end{description}
\end{thm}

\begin{block}{}
These results complete the classification of subfactors with index less than $5$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Indices of hyperfinite subfactors}
\begin{question}
Every index is realised by a subfactor with principal graph $A_\infty$. Which indices are possible for hyperfinite subfactors?
\end{question}
\begin{itemize}
\item Popa has shown that the smallest index above $4$ that can be realised by a hyperfinite $II_1$ subfactor is the square of the graph norm of $E_{10}$, approximately $4.02642$.

\item The waters here are deep, analytic techniques are required, and little is known.
\end{itemize}
\end{frame}

\section{Constructions}
\begin{frame}
\frametitle{Can we construct a subfactor with a given principal graph?}

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

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

\begin{block}{}
These data strongly constrain \emph{generators and relations} for the representation theory.
\end{block}
\end{frame}

\begin{frame}
\begin{block}{}
If a subfactor with principal graph $\mathfig{0.2}{graphs/EH}$ exists, its representation theory 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^2 = -\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}
\end{block}
\end{frame}

\begin{frame}
\begin{thm}[Bigelow-Morrison-Peters-Snyder, Acta Math. '09]
There is at most one subfactor with these generators and relations.
\end{thm}

\begin{proof}
These relations 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*}
\end{proof}
\end{frame}

\begin{frame}
Two things could go wrong:
\begin{itemize}
\item The relations are inconsistent (i.e. this is a presentation of the `zero subfactor').
\item The resulting representation theory is not unitary.
\end{itemize}

\end{frame}

\begin{frame}
How do we check that relations are consistent?
Every finite group sits inside some $S_n$. Analogously, we have

\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. Unitarity is inherited from the GPA,  so the subfactor $\mathfig{0.2}{graphs/EH}$ exists!
\end{thm}

\end{frame}

\section{Consequences}

\begin{frame}
\begin{block}{}
Just as the exceptional Lie groups were discovered via the Killing-Cartan program of classification, the classification of small index subfactors is producing examples of \emph{exotic subfactors}.
\end{block}

By contrast, fusion categories with integer dimension objects are conjectured to be `weakly group-theoretical', essentially definable in terms of group-theoretic data.

\begin{block}{}
The new exotic subfactors and fusion categories we are finding
\begin{itemize}
\item constrain possible structural theorems,
\item give counterexamples to conjectures, and
\item give interesting examples of new modular data and thus exotic $3$-manifold invariants.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Exotic fusion categories}

A classical theorem of Brauer shows that the representation theory of any finite group can be defined over a cyclotomic field. (The same holds for quantum groups at roots of unity.) Etingof, Nikshych and Ostrik asked if this is true of every fusion category.

\begin{thm}[Morrison-Snyder, Transactions of the AMS '10]
The even parts of the Haagerup and extend Haagerup subfactors cannot be defined over any cyclotomic field.
\end{thm}
\begin{proof}
Using the skein theory, we produce a canonical element of the ground field which is not cyclotomic.
\end{proof}
\end{frame}

\begin{frame}
\frametitle{Summary}
\begin{block}{Recent progress}
\begin{itemize}
\item Constructed the missing case, extended Haagerup, from earlier classifications.
\item Developed uniform new techniques from number theory to reduce classifications to finite problems.
\item The classification of subfactors with index less than $5$.
\end{itemize}
\end{block}

There's still plenty more to do:

\begin{block}{}
\begin{itemize}
\item The classification beyond $5$ looks difficult; perhaps we'll need new techniques.
\item Computer searches show that the classification remains sparse, but also find new candidate examples.
\item It will be interesting to try to construct these!
\end{itemize}
\end{block}
\end{frame}

\begin{frame}[plain]
\begin{tikzpicture}[remember picture,overlay]
	\node[at=(current page.center)] {
		\includegraphics[width=0.95 \paperwidth]{../diagrams/MapOfSubfactors}
        };
\end{tikzpicture}
\end{frame}


\end{document}
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Graphs between $3+\sqrt{5}$ and $6$ for the map of subfactors:
$\bigraph{bwd1v1v1p1p1v0x1x1p0x0x1duals1v1v1x2}$
$\bigraph{bwd1v1v1p1p1p1v0x1x0x0p0x0x1x0p0x0x0x1duals1v1v1x2x3}$
$\bigraph{bwd1v1v1p1p1p1v0x1x0x0p0x0x1x0p0x0x0x1duals1v1v2x1x3}$
$\bigraph{bwd1v1v1v1p1p1p1v0x0x0x1p0x0x0x1duals1v1v1x2x3x4}$
$\bigraph{bwd1v1v1p1v1x1p0x1v0x1duals1v1v1x2}$
$\bigraph{bwd1v1v1v1p1p1v0x0x1p0x0x1v1x1duals1v1v2x1x3v1}$
$\bigraph{bwd1v1v1v1v1v1p1p1v0x0x1p0x0x1v1x0p0x1v1x1duals1v1v1v2x1x3v2x1}$

{5.78339, 5.82843, 5.82843, 5.79129, 5.30278, 5.56155, 5.41421}