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\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 $\FPdim(\mathcal{C})<M$ for some prescribed threshold $M$.  This would generalize known results for semisimple Hopf algebras \cite{MR2294999} (up to global dimension 60).


\subsubsection{Small-dimensional Object}
Small-dimensional objects in fusion categories are closely related to the classification of subfactors of small index.  In particular, finite-depth finite index subfactors of the hyperfinite $II_1$ factor correspond precisely to algebra objects in unitary fusion categories and the index of the subfactor is the FP-dimension of the algebra object.  Since $X \otimes X^*$ is always an algebra object, the classification of subfactors of small index gives a partial classification of small objects in a unitary fusion category.   In order to get better classifications of small objects in fusion categories we would like to relax the unitarity condition, and improve the understanding of when an algebra object in a fusion category can be realized as $X \otimes X^*$ for some object $X$ in a larger fusion category.

The current status of the classification of subfactors of small index is as follows.  If the index is less than $4$ then it must be of the form $4\cos(2\pi/\ell)^2$, and such subfactors have an ADE classification (see \cite{MR996454} for the outline of this result, and \cite{MR1193933, MR1145672, MR1313457, MR1308617} for more details).  Extending this classification to $[M:N] < 3+\sqrt{3}$ was initiated by Haagerup \cite{MR1317352} and has been completed recently \cite{MR1686551,MR1625762, MR2472028,0909.4099}.  Recent progress has been made extending Haagerup's techniques to $[M:N] < 5$.

An independent arithmetic approach \cite{1004.0665} gives a classification of all possible FP-dimensions less than $76/33$ of any object in any (not necessarily unitary) fusion category.  However, this approach does not give a classification of all fusion categories with an object of one of the allowed dimensions.

\subsubsection{Global Dimension with Few Prime Divisors}

The global dimension $\FPdim(\mathcal{C})$ is an algebraic integer, and under additional assumptions even $\FPdim(\mathcal{C})/\FPdim(X)^2$ is an algebraic integer for any simple object $X$.  A result of Etingof, Nikshych and Ostrik \cite{MR2183279} states that $\FPdim(X)$ is a rational integer for all objects $X$ if, and only if $\mathcal{C}$ is equivalent to the category of representations of some semisimple finite-dimensional quasi-Hopf algebra $A$.  A well-understood subclass of such \emph{integral} fusion categories are those that are \emph{group-theoretical}.  For example it is known that any integral fusion category of global dimension $p^n$ with $p$ prime is group-theoretical.  If $\FPdim(\mathcal{C})$ has relatively few prime divisors (in an appropriate number field) or is a rational integer then number-theoretic techniques (diophantine analysis etc.) can be brought to bear.







\subsection{Participants}  Each of the potential participants listed above has expertise in computational or theoretical aspects of the classification problems described above.
By bringing together people who have been working on different aspects of the above problems we will be able to take a hybrid approach to the classification problems.  As an example of this, the classification of fusion categories of small global dimension naturally splits into three regimes:
categories which are built from $U_q(su_2)$ via extension theory,
categories with relatively few simple objects, and categories which are related to subfactors of small index.
Each regime requires expertise from a different field.
\subsection{Format}  We will adhere to the usual format of having talks in the mornings and discussions in the afternoons.  Roughly, we expect the workshop to proceed as follows.  First, we will have talks which explain how to translate between the language of subfactors and fusion categories,
thereby allowing people working in different fields to work together.  Discussions on the first afternoon would be on some explicit classification problems, such as those mentioned above.  The results of these discussions would be presented the second morning, brief descriptions of each problem and its current status.  The second afternoon we will break into groups to discuss these problems in groups.
The third morning we will discuss available computational tools, which would lead to  discussions in which people can suggest improvements and new functionalities in the afternoon.   On the morning of the last day we will have talks on the updated status of the problems presented on the second morning and a list of feasible goals going forward.

\subsection{Expected outcomes}
As progress on classification problems are often incremental and not widely known, at a minimum the participants will leave the workshop fully up-to-date on the current status, technology and interest on the problems.  The organizers hope to write a survey paper soon after the workshop that would include a list of ``small'' fusion categories.  In the long-term we expect that the cross-germination of ideas will lead to significant progress on the problems and collaboration among the participants.  Potentially this will be the first of many periodically held workshops on the subject.
\subsection{Timeliness}
A major recent impetus for attacking the classification of fusion categories is the application to topological phases and quantum computation.  This was the original motivation for \cite{MR2544735}, for example.  Recently developed computational techniques and new theoretical results are scattered across the literature of several research communities and the time is ripe to unify these approaches.  Although several recent meetings with classification of fusion categories as a sub-topic have been held (egs. at Indiana University,  2008; Baylor University, 2009; Vanderbilt University, 2010) no conference dedicated solely to the classification problem has been held.
\subsection{About the organizers}
Scott Morrison is a Miller Fellow in the Mathematics department at the University of California, Berkeley. He received his Ph.D. in 2007 from Berkeley, working with Vaughan Jones, and was subsequently a postdoc at Microsoft Station Q in Santa Barbara. His research interests include subfactors, link homology, and topological quantum field theories. 

Eric Rowell is an assistant professor in the Mathematics department at Texas A\&M University.  He received his Ph.D. in 2003 under Hans Wenzl and was a postdoc at Indiana University from 2003-2006 under Zhenghan Wang.  His primary research interests are in braided fusion categories particularly the associated braid group representations and low-rank classification.  He co-organized an international conference ``Modular Categories and Applications'' at Indiana University in 2009, as well as two recent special sessions of the AMS.

Noah Snyder is an NSF Postdoctoral Fellow at Columbia University working with Dylan Thurston.  He received his Ph.D. in 2009 from U.C. Berkeley where his advisor was Nicolai Reshetikhin.  His main research interests are the topology and arithmetic of fusion categories and subfactors.  He organized two graduate seminars, an RTG workshop, and was academic coordinator for one summer of Canada/USA Mathcamp.

%\bibliographystyle{alpha}
%\bibliography{../../bibliography/bibliography}

\newcommand{\noopsort}[1]{}\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{BMPS09}

\bibitem[AH99]{MR1686551}
Marta Asaeda and Uffe Haagerup.
\newblock Exotic subfactors of finite depth with {J}ones indices
  {$(5+\sqrt{13})/2$} and {$(5+\sqrt{17})/2$}.
\newblock {\em Comm. Math. Phys.}, 202(1):1--63, 1999.
\newblock \mathscinet{MR1686551} \doi{10.1007/s002200050574}
  \arxiv{math.OA/9803044}.

\bibitem[AY09]{MR2472028}
Marta Asaeda and Seidai Yasuda.
\newblock On {H}aagerup's list of potential principal graphs of subfactors.
\newblock {\em Comm. Math. Phys.}, 286(3):1141--1157, 2009.
\newblock \mathscinet{MR2472028} \doi{10.1007/s00220-008-0588-0}
  \arxiv{0711.4144}.

\bibitem[Bis98]{MR1625762}
Dietmar Bisch.
\newblock Principal graphs of subfactors with small {J}ones index.
\newblock {\em Math. Ann.}, 311(2):223--231, 1998.
\newblock \mathscinet{MR1625762} \doi{http://dx.doi.org/10.1007/s002080050185}.

\bibitem[BMPS09]{0909.4099}
Stephen Bigelow, Scott Morrison, Emily Peters, and Noah Snyder.
\newblock Constructing the extended {H}aagerup planar algebra, 2009.
\newblock \arxiv{0909.4099}, to appear \emph{Acta Mathematica}.

\bibitem[BN91]{MR1193933}
Jocelyne Bion-Nadal.
\newblock An example of a subfactor of the hyperfinite {${\rm II}\sb 1$} factor
  whose principal graph invariant is the {C}oxeter graph {$E\sb 6$}.
\newblock In {\em Current topics in operator algebras ({N}ara, 1990)}, pages
  104--113. World Sci. Publ., River Edge, NJ, 1991.
\newblock \mathscinet{MR1193933}.

\bibitem[CMS10]{1004.0665}
Frank Calegari, Scott Morrison, and Noah Snyder.
\newblock Cyclotomic integers, fusion categories, and subfactors, 2010.
\newblock With an appendix by Victor Ostrik. To appear in Communications in
  Mathematical Physics. \arxiv{1004.0665}.

\bibitem[ENO05]{MR2183279}
Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik.
\newblock On fusion categories.
\newblock {\em Ann. of Math. (2)}, 162(2):581--642, 2005.
\newblock \mathscinet{MR2183279} \doi{10.4007/annals.2005.162.581}
  \arxiv{math.QA/0203060}.

\bibitem[Haa94]{MR1317352}
Uffe Haagerup.
\newblock Principal graphs of subfactors in the index range
  {$4<[M:N]<3+\sqrt2$}.
\newblock In {\em Subfactors ({K}yuzeso, 1993)}, pages 1--38. World Sci. Publ.,
  River Edge, NJ, 1994.
\newblock \mathscinet{MR1317352} available at
  \url{http://tqft.net/other-papers/subfactors/haagerup.pdf}.

\bibitem[HR09]{0907.1051}
Seung-Moon Hong and Eric~C. Rowell.
\newblock On the classification of the {G}rothendieck rings of non-self-dual
  modular categories, 2009.
\newblock With an appendix by Victor Ostrik. To appear in Journal of Algebra
  \arxiv{0907.1051}.

\bibitem[Izu91]{MR1145672}
Masaki Izumi.
\newblock Application of fusion rules to classification of subfactors.
\newblock {\em Publ. Res. Inst. Math. Sci.}, 27(6):953--994, 1991.
\newblock \mathscinet{MR1145672} \doi{10.2977/prims/1195169007}.

\bibitem[Izu94]{MR1313457}
Masaki Izumi.
\newblock On flatness of the {C}oxeter graph {$E\sb 8$}.
\newblock {\em Pacific J. Math.}, 166(2):305--327, 1994.
\newblock \mathscinet{MR1313457} \euclid{euclid.pjm/1102621140}.

\bibitem[Kaw95]{MR1308617}
Yasuyuki Kawahigashi.
\newblock On flatness of {O}cneanu's connections on the {D}ynkin diagrams and
  classification of subfactors.
\newblock {\em J. Funct. Anal.}, 127(1):63--107, 1995.
\newblock \mathscinet{MR1308617} \doi{10.1006/jfan.1995.1003}.

\bibitem[Nat07]{MR2294999}
Sonia Natale.
\newblock Semisolvability of semisimple {H}opf algebras of low dimension.
\newblock {\em Mem. Amer. Math. Soc.}, 186(874):viii+123, 2007.
\newblock \mathscinet{MR2294999}, \arxiv{math/0305038}.

\bibitem[Ocn88]{MR996454}
Adrian Ocneanu.
\newblock Quantized groups, string algebras and {G}alois theory for algebras.
\newblock In {\em Operator algebras and applications, Vol.\ 2}, volume 136 of
  {\em London Math. Soc. Lecture Note Ser.}, pages 119--172. Cambridge Univ.
  Press, Cambridge, 1988.
\newblock \mathscinet{MR996454}.

\bibitem[RSW09]{MR2544735}
Eric Rowell, Richard Stong, and Zhenghan Wang.
\newblock On classification of modular tensor categories.
\newblock {\em Comm. Math. Phys.}, 292(2):343--389, 2009.
\newblock \mathscinet{MR2544735} \arxiv{0712.1377}
  \doi{10.1007/s00220-009-0908-z}.

\end{thebibliography}


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