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\title{Classifying fusion categories}
\author[Morrison]{Scott Morrison}
\institute{Miller Institute / Mathematics}
\date{June 4, 2010}

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	\begin{block}{Background}
		$$\mathfig{0.9}{kyoto}$$
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		\begin{columns}
			\begin{column}{.4\linewidth}
				\begin{block}{Bio}
					\begin{itemize}
						\item I'm from $\mathfig{0.15}{australia}$.
						\item Berkeley Ph.D 2007.
						\item Microsoft Station Q 2007-2009.
						\item Miller Fellow since mid-2009.
						\begin{itemize}
							\item Miller host  Vaughan Jones.
						\end{itemize}
					\end{itemize}
				\end{block}
			\end{column}
			\begin{column}{.4\linewidth}
				\begin{block}{Research interests}
					\begin{itemize}
					\item Higher-dimensional algebra (fusion categories, $n$-categories)
					\item Quantum field theory and homotopy theory
					\item Subfactors and von Neumann algebras
					\end{itemize}
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      \begin{block}{What is a fusion category?}
      	In the \textbf{Standard Model} of particle physics, we have%
	%\begin{wrapfigure}{r}{0.25\textwidth}\begin{center}$$\mathfig{0.25}{standard-model}$$\end{center}\end{wrapfigure}\vspace{-0.7cm}
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	\begin{description}
		\item[fundamental particles]
			leptons and quarks in $3$ generations, and $4$ bosons
			$$\mathfig{0.25}{standard-model}\hspace{9cm}$$
		\item[interactions]
			described by trivalent graphs ($\sim$ Feynman diagrams)
			\vspace{-1cm}
			\begin{increaseindent}{4cm}
			\begin{columns}
			\begin{column}{\linewidth}
			\begin{example}[beta decay]
				$\mathfig{0.2}{beta_decay}$  \parbox{18cm}{Beta decay transmutes one chemical element into another, e.g. $$\leftidx{_{55}^{137}}{Cs}{} \to \leftidx{_{56}^{137}}{Ba}{} + e^- + \nu_e$$}
			\end{example}
			\begin{example}[Higgs boson at the LHC?]
				At high energies, Higgs bosons might be produced.
				$$\mathfig{0.2}{particle_tracks} \overset{???}{\scalebox{3.0}{$\rightsquigarrow$}} \mathfig{0.2}{higgs_production}$$
			\end{example}
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		\item[amplitudes for histories]\mbox{}\\
			\begin{itemize}
				\item	quantum mechanical `probabiliities'
				\item described by a complicated \emph{Lagrangian}
					$\tikz{\path[use as bounding box] (0,0) -- (0,0); \node at (9.5,0) {$\mathfig{0.31}{sm-lagrangian}$};}$
				\item depend on position, momentum, energy, etc.
			\end{itemize}
	\end{description}
	\end{increaseindent}
	
	\bigskip
	
	A \textbf{fusion category} is a combinatorial abstraction of this setting.
	\begin{increaseindent}{2cm}
	\begin{itemize}
		\item throw out all geometry and dynamics!  (no position, no momentum)
	\end{itemize}
	\end{increaseindent}
	
	A fusion category has
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	\begin{itemize}
		\item finitely many \emph{particle types}
		\item combinatorial rules describing \emph{particle interactions}
		\item an \emph{amplitude} for each history
	\end{itemize}
	\end{increaseindent}
	satisfying a \emph{locality condition}: the amplitude for a large history can be computed from the amplitudes for constituent parts.
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	\begin{block}{Example: the golden category}
		\begin{description}
		\item[\textbf{Particles}]\mbox{}\\
		Just one type of particle, called \scalebox{1.4}{$\tau$}.
		
		\item[\textbf{Interactions}]\mbox{}\\
		When two $\tau$ particles interact, they can either
			\begin{increaseindent}{2cm}
			\begin{itemize}
				\item annihilate, producing the vacuum
				$$\tikz{\draw (0,0) node[below] {$\tau$} .. controls (0,2) and (3,2) .. (3,0) node[below] {$\tau$};}$$ 
				\item or combine to form a single $\tau$ particle.
				$$\tikz{\foreach \q in {90,210,-30} {\node at (\q:2) {$\tau$}; \draw (\q:1.5) -- (0,0);}}$$
			\end{itemize}
			\end{increaseindent}
		%(Note there are no conserved quantities in this universe!)
		
		We write this symbolically as $\tau \otimes \tau \iso 1 \directSum \tau$, %(This explains the name of the category: the Golden Ratio $\frac{1+\sqrt{5}}{2}$ satisfies  $\tau^2 = 1 + \tau$.)
		or graphically as 
		$\tikz[baseline=-0.5cm]{\path[use as bounding box] (-2,-1) rectangle (5,1); \draw[fill] (0,0) circle (0.2) node[below=5pt] {vacuum} -- (3,0) circle (0.2) node[below=5pt] {$\tau$}; \draw (3,0) .. controls (5,-2) and (5,2) .. (3,0);}$
		
		\item[\textbf{Amplitudes}] can be calculated by local rules:
		\begin{align*}
			\tikz[baseline]{\draw (0,0) circle (1cm);} & = \frac{1+\sqrt{5}}{2} \\
			\tikz[baseline=1.3cm]{\draw (-1,0) -- (0,1) -- (1,0) (0,1) -- (0,2) -- (-1,3) (0,2) -- (1,3);} & =  \frac{3-\sqrt{5}}{2} \;\tikz[baseline=1.3cm]{\draw (-1,0) .. controls (-0.5,1.5) .. (-1,3) (1,0) .. controls (0.5,1.5) .. (1,3);} + (2-\sqrt{5}) \tikz[baseline=1.3cm]{\draw (-1,0) .. controls (0,1) .. (1,0) (-1,3) .. controls (0,2) .. (1,3);}
		\end{align*}
		
		\textbf{Amazing fact:} any way you use these rules, you get a \emph{consistent answer}!
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	\begin{block}{Quantum computing}
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		\emph{``Fusion categories are just mathematical toys!''} \hfill ... or are they?
		
		\medskip
	
		
		The \textbf{Fractional Quantum Hall Effect} (low temperature, high magnetic field, 2d electron gases) seems to be described by certain fusion categories!
		
		$$\mathfig{0.5}{PanPRL}$$
		
		The ``quasi-particle excitations'' of the system satisfy the rules of a fusion category. In the low temperature limit, the combinatorial abstraction reflects real lab benchtop physics!
		
		\bigskip
		
		It may be possible to build a computer using the FQHE: the golden category is \textbf{universal for quantum computing}. For now, people are trying to build simpler devices to characterise the systems.
		$$\mathfig{0.3}{interferometer}\qquad\qquad\mathfig{0.235}{junction}$$
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	\begin{block}{Classification}
		Classifying all fusion categories is probably too hard.
		
		We'd like to understand all the `small' ones.
		
		\vspace{-0.5cm}
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		\begin{theorem}[many people, 2009-2010, \url{http://tqft.net/ncgoa2010}]
			We can classify all the possible combinatorial interactions for (subfactor) fusion categories with ``index less than $5$''.
			\begin{increaseindent}{2cm}
			\begin{itemize}
				\item several well-understood families (including the golden category)
				\item five `exotic' examples
				 $$ \hspace{-2cm} \evenH \qquad \evenEH \qquad \evenAH$$ $$\evenGHJ \qquad \evenO $$
			\end{itemize}
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		\end{theorem}
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	\begin{block}{Constructing exotic examples}
		That result restricts what is possible. Just recently, we discovered the last missing \textbf{exotic fusion category} from this classification!
		
		\vspace{-0.6cm}
		
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		\begin{theorem}[Bigelow-Morrison-Peters-Snyder, 2009, \href{http://arxiv.org/abs/0909.4099}{arXiv:0909.4099}]
			There really is a fusion category with particle types: $$\evenEH$$
		\end{theorem}
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		\begin{proof}
			\begin{description}
			\item[\textbf{Interactions}] are described by graphs which are $3$- or $8$-valent,
			\item[\textbf{Amplitudes}] are defined via the ``jellyfish algorithm'':
				$$\mathfig{0.4}{network-paths} \qquad %\mathfig{0.4}{echeng-jellyfish-lake-palau}
					\mathfig{0.4}{jellyfish}$$
				 using certain rules, e.g.:
				$$\mathfig{0.6}{relation2}$$
			Proving that these rules give consistent answers is \textbf{hard}! \qedhere
			\end{description}
		\end{proof}
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