\documentclass{amsart} \newcommand{\FPdim}{{\rm FPdim}} \usepackage{hyperref} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} \newcommand{\euclid}[1]{\href{http://projecteuclid.org/getRecord?id=#1}{{\tt #1}}} \newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} \newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})} \evensidemargin 0in\oddsidemargin 0in \setlength{\textwidth}{6.3in} \setlength{\textheight}{8.3in} \setlength{\headheight}{.1in} \setlength{\headsep}{.1in} \setlength{\footskip}{.5in} \setlength{\topmargin}{0in} \title{AIM Proposal Description: Classifying Fusion Categories} \begin{document} \maketitle \iffalse \textbf{Organizers:} Scott Morrison (UC Berkeley), Eric Rowell (Texas A\&M), Noah Snyder (Columbia) \textbf{Suggested participants:} \vspace{-0.1cm} \begin{itemize} \item Marta Asaeda (UC Riverside) \item Stephen Bigelow (UC Santa Barbara) \item Dietmar Bisch (Vanderbilt) \item Paul Bruillard (Texas A\&M) \item Richard Burstein (Vanderbilt) \item Pavel Etingof (MIT) \item Shlomo Gelaki (Technion) \item Pinhas Grossman (Cardiff) \item Uffe Haagerup (Odense) \item Seung-Moon Hong (Toledo) \item Masaki Izumi (Kyoto) \item Vaughan Jones (UC Berkeley) \item David Jordan (MIT) \item Michael Mueger (Nejmegen) \item Deepak Naidu (Texas A\&M) \item Sonia Natale (Cordoba) \item Richard Ng (Iowa St.) \item Dmitri Nikshych (New Hampshire) \item Victor Ostrik (Oregon) \item Emily Peters (MIT) \item Zhenghan Wang (Microsoft Station Q) \item Sarah Witherspoon (Texas A\&M) \end{itemize} \textbf{Mathematics Subject Classifications:} 18D10, 46L37, 16T05, 57R56 \fi %\section{Workshop description} The goal of this workshop is to understand and classify small fusion categories. We include fusion categories with additional structure, and we take several interpretations of ``small'' (explicitly described below). We envision that the workshop will establish a program for classification much in the spirit of the meetings on the classification of finite simple groups held in the last half of the 20th century. Fusion categories are a natural generalization of representation categories of finite groups. Further examples include representation categories of finite-dimensional semisimple quasi-Hopf algebras, the subcategory of $N$-$N$ bimodules which are submodules of a tensor power of $M$ for a pair of type $II_1$ subfactors $N \subset M$, and idempotent completions of tangle categories associated with quantum link-invariants (at roots of unity). As the diversity of constructions indicates, fusion categories are of interest in several distinct research areas: representation theory, operator algebras and quantum topology. Researchers in each of these areas have developed different techniques for attacking the classification of fusion categories. This workshop will bring together researchers from these three groups to share these approaches and collaborate on the classification problem. We list a few specific motivations that are relevant to the proposed research groups. \begin{enumerate} \item Search for ``exotic'' fusion categories. Most well-understood examples of fusion categories come from finite groups and quantum groups at roots of unity. Although significant progress has been made in understanding the general structure of fusion categories, a major obstacle has been the lack of good examples from other sources. \item Classify finite-dimensional semisimple quasi-Hopf algebras with (a) few simple objects, (b) of low dimension and (c) with dimension having few prime divisors. \item Classify (2+1)-dimensional topological quantum field theories (TQFTs). Any modular category gives rise to a TQFT, and to certain extent the converse is true. This would have consequences in topological quantum computation as well. \item Classify irreducible finite-depth subfactor pairs $N\subset M$ with small Jones index $[M:N]$. Currently such a classification is known up to $[M:N]\leq 3+\sqrt{3}$. \end{enumerate} \subsection{Small Fusion Categories} There are several statistics on fusion categories that allow us to subdivide them into more manageable subclasses. Some of these statistics are of particular interest in more specialized settings. We will briefly describe these statistics and specialized settings and then explain the various interpretations of ``small'' that we have in mind. A fusion category $\mathcal{C}$ has finitely many (isomorphism classes of) simple objects, including a distinguished \emph{unit object} $\mathbf{1}$. The most basic statistic on $\mathcal{C}$ is the \emph{rank}: the number of such simple objects. For any simple object $X$ we may form its \emph{fusion matrix} $N_X$ with entries the multiplicities $N_{X,Y}^Z=\dim Hom(X\otimes Y,Z)$ where $(Y,Z)$ runs over the (finitely many) pairs of simple objects. The \textit{fusion rules} of $\mathcal{C}$ is the set of these $N_X$ for $X$ simple. The FP-dimension of a simple object $X$ is the largest eigenvalue of $N_X$ which we will denote by $\FPdim(X)$. This coincides with the usual notion of dimension if $\mathcal{C}$ is the representation category of a finite-dimensional semisimple Hopf algebra. Analogously to the Hopf algebra case we define the \textit{global dimension} of $\mathcal{C}$ to be the sum of $\FPdim(X)^2$ over all simple objects $X$. Particularly for topological applications one often considers \emph{braided} fusion categories, that is those for which there is a natural family of isomorphisms $\{c_{X,Y}:X,Y\in\mathcal{C}\}$ where $c_{X,Y}:X\otimes Y\cong Y\otimes X$. In the presence of a braiding an additional structure called a \emph{balancing} permits one to define a categorical trace and the category is called pre-modular. If the braiding is sufficiently non-degenerate (encoded in the invertibility of the $S$-matrix) we say such a category is called \textit{modular}. We first list the specific problems that will serve as a starting place for the workshop, and then give a more detailed description of each. We hope to classify: \begin{itemize} \item all modular categories of rank less than $n$. ($5\leq n\leq 12$, say). \item all fusion categories of global dimension less than $M$. ($M\sim 30$) \item all fusion categories generated by an object $X$ with $\FPdim(X)\leq 5$. \item all fusion categories whose global dimension has at most $k$ prime divisors. ($k$ depending on the number field in which $\FPdim(\mathcal{C})$ resides). \end{itemize} \subsubsection{Low Rank} In the specialized setting of \emph{modular categories} Wang has conjectured that there are only finitely many inequivalent modular categories for any fixed rank $n$. A complete classification of \emph{unitary} modular categories up to rank $4$ is given in \cite{MR2544735} where Galois theory plays a key role. It is known that there are only finitely many inequivalent fusion categories for a fixed set of fusion rules (known as Ocneanu rigidity, see \cite{MR2183279}). So up to finite ambiguity it is enough to classify the Grothendieck semirings of modular categories of rank $n$. Some computational techniques for this have been developed and applied to give a partial classification in rank $5$ \cite{0907.1051}. \subsubsection{Small Global Dimension} Bounding the global dimension $\FPdim(\mathcal{C})$ bounds the eigenvalues of the fusion matrices $N_X$ and hence the (non-negative) integer entries of the $N_X$. Thus by Ocneanu rigidity there are only finitely many fusion categories with global dimension below a given bound. Thus it is feasible to classify all fusion categories with $\FPdim(\mathcal{C})