Overview of the subject area of the workshop:
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This workshop will build on our 2014 BIRS workshop which brought together researchers in the overlapping fields of subfactor theory and fusion categories. With already so many advances in these fields in the two years since our last workshop, such a reunion conference in 2018 will be long overdue! We would be happy to hold the workshop at either Banff or Oaxaca --- both are fantastic venues.
It is currently a critical moment for research in fusion categories and subfactors.
+ Substantial progress has been made on classification of small examples.
+ Important conjectures have been proposed describing modular tensor categories.
+ Many alternative approaches to the formalisation of enriched quantum symmetries are being pursued.
+ Connections to topological phases of matter are deepening and broadening.
The purpose of this conference is to bring together the key researchers on fusion categories and subfactors, to tackle the research problems arising from these developments, and to develop a coherent view of the way forward.
Fusion categories are generalizations of the representation categories of finite groups and quantum groups at roots of unity. While groups encode symmetry, we say that fusion categories encode quantum symmetry. Unitary fusion categories and closely related unitary modular categories are readily accepted as the correct mathematical formalism to describe topological phases of matter and topological quantum computation.
Subfactors provide a rich source of such unitary tensor categories. Starting with a finite index II_1 subfactor N contained in M, we get two unitary tensor categories of N-N and M-M bimodules generated by M. Moreover, subfactors are universal hosts for quantum symmetries in that every unitary fusion category arises in this way. Examples of unitary modular categories are obtained from the quantum double construction and from conformal nets of von Neumann factors in the algebraic quantum field theory description of CFT.
Both subfactors and fusion categories are of interest to many distinct fields of mathematics and physics, including representation theory, operator algebras, quantum field theory, and statistical mechanics.
We’ve written to 38 experts in the fields of subfactors and fusion categories, with roughly 19 coming from each side. (This makes 42 together with 4 organizers.) We’ve had an overwhelmingly positive response for such a workshop. We will invite many of the participants from the 2014 workshop, including fusion category experts Victor Ostrik ("very happy to come"), Eric Rowell (“thrilled”), Zhenghan Wang ("very happy"), and Sonia Natale ("happy") along with subfactor experts Masaki Izumi ("really like to attend"), David Evans ("enthusiastic"), and Yasuyuki Kawahigashi ("certainly very enthusiastic"). We have also had enthusiastic replies from Pavel Etingof ("interested"), Sorin Popa ("I would love to come"), and Fields Medalist Vaughan Jones ("That sounds like a great conference. I would be very interested in coming.") We also reached out to mathematical physicists studying topological phases including Xiao-Gang Wen (“very enthusiastic”) and Liang Kong (“enthusiastic about attending”) who are very excited about the possibility of this workshop. We have also included several younger mathematicians who are newer to the field.
We have already seen great advances in both subfactor theory and in fusion categories using techniques from the other field. On one hand, subfactors provide a rich source of examples of unitary fusion categories, including the Izumi-Haagerup series and the mysterious extended Haagerup fusion categories. From the other direction, number theoretic techniques afforded by results in fusion categories have led to strong subfactor classification theorems.
As this conference will unite the experts in the fields of subfactors and fusion categories, we expect exciting new discoveries and progress on open questions in both of these fields. This conference will strengthen the collaboration between these two fields, and we will also build new bridges to mathematical physicists studying topological phases of matter.
Statement of the objectives of the workshop:
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This conference will focus on the new objectives which have surfaced since the 2014 Banff conference on subfactors and fusion categories. These topics include classification of `small’ examples, understanding exotic examples, enriched quantum symmetries, and connections to conformal field theory (CFT) via modular tensor categories.
Regarding classification results for 'small' examples, we have new progress by Afzaly-Morrison-Penneys on the classification of small index subfactor standard invariants, bringing the completed classification up to index 5.25. In a parallel vein, there has been significant development of the work initiated by Bisch and Jones on skein theories for a small generator with bounds on dimensions of Hom spaces. In particular, there are now new classifications of categories generated by a trivalent vertex (Morrison-Peters-Snyder), and categories satisfying Yang-Baxter (Liu) or Thurston type relations (C. Jones-Liu-Ren). At this point, significant new ideas are required to extend the classification of small index subfactors. The recent incorporation of number theoretic tools for fusion categories yielded tremendous progress in subfactor classification, and we are eager for new tools with strong applications. For example, the higher Frobenius-Schur indicators of Ng-Schauenburg were useful for Ostrik’s classification of pivotal fusion categories of rank 3, and a better understanding could be very useful for subfactor classification. Morrison has automated the imposition of the known constraints (representation- and number-theoretic, higher Frobenius-Schur indicators, etc) onto possible Grothendieck rings, and is very effective at giving short lists of the possibilities for `small' rank and global dimension. On the other hand, the classification of small skein theories seems ripe for much further progress with current techniques. The methods developed for the classification of categories generated by a trivalent vertex are very general, and we expect to see further progress using these before and during the time of the proposed workshop.
At the 2014 BIRS workshop on subfactors and fusion categories, a breakthrough theorem proving rank finiteness for modular tensor categories was announced. It remains open whether there are finitely many unitary fusion categories of a given rank. This is a question of very significant interest, and many in the community are actively thinking about it. The question is equivalent to rank finiteness for subfactors.
The 'near-group' and 'quadratic' fusion categories are an important class of examples, which appear to be at the edge of tractability at present. In many cases, their classification has been reduced to the description of certain discrete algebraic varieties, and in many small examples, these systems of equations can be solved explicitly. There are tantalising patterns, but presently, we are unable to give a unified, conceptual construction of the entire series. Significant new work by Izumi, building on his earlier work and work of Evans-Gannon, continues to drive this field. Amongst the new examples of quadratic categories is one with group of invertibles Z_2 x Z_4, from which Izumi-Grossman-Snyder gave a new construction of the Asaeda-Haagerup subfactor. This result is significant, because the Asaeda-Haagerup subfactor was formerly one of very few examples which appeared to be utterly unrelated to anything else known in mathematics! We are all excited about the prospect of further systematizing the known examples of subfactors and fusion categories, connecting currently 'exotic looking' examples to well understood constructions.
This new approach to the Asaeda-Haagerup subfactor also allows the computation of its Brauer-Picard groupoid. A number of researchers, including several junior ones, have been developing the tools for computations of Brauer-Picard groupoids. We hope that by the time of the workshop, the Brauer-Picard groupoids for nearly all known examples will be understood. In physical terms, the objects of the Brauer-Picard groupoid correspond to distinct two-dimensional realizations of the same topological phase, together with the morphisms to 'invertible' boundary defects between them. There are other examples of productive connections between the study of fusion categories (in mathematics) and the study of topological phases (in physics), and a major goal of this workshop is to bring together the relevant mathematicians and physicists to develop the theory of fusion categories as the mathematical language for topological phases.
An extremely exciting new direction for the study of tensor categories, at the time of writing this proposal, is that of 'enriched quantum symmetries'. We are now just at the point where a number of competing axiomatizations for the same general idea have been proposed. It remains to understand the connections between them, and determine which approaches best support the available examples. Classical work in category theory provides a language for tensor categories enriched in symmetric tensor categories, and significant recent work develops these ideas for fusion categories or modular tensor categories enriched in super vector spaces, or enriched in pointed (group-like) ribbon categories. (This includes work of Usher, Jaffe-Liu, and others.) There are many new examples, and in particular we are realising that a number of previously mysterious 'holes' in classifications are in fact filled by examples which live in a slightly more general context than previously imagined. For example, there is an ADE classification for small index subfactors, but the even D’s are not actually realized. Instead, they are realized when we extend our view to the super vector space enriched setting. There is also new work being done on quantum symmetries enriched in a braided tensor category, from several alternative viewpoints (e.g. Henriques-Penneys-Tener). Hopefully there are strong connections between these new theories and the description of boundary defects for 3d topological phases. The proposed workshop will make it possible for these various approaches to enriched quantum symmetry to be compared, with the requirement that they describe physical models. Further, at this time, there are only the most preliminary efforts to classify examples of enriched quantum symmetries. By the time of the workshop, there should be quite significant progress in classification of `small’ enriched examples. In a related direction, work of C. Jones and Penneys attempts to lift the entire operator algebraic framework of subfactors and von Neumann algebras to the enriched setting.
Finally, we describe objectives which have recently emerged in the connection between conformal field theory (CFT) and quantum symmetries. Firstly, we point out that recent work of Carpi-Kawahigashi-Longo-Weiner hopefully lead to a translation between two alternate descriptions of CFT, via conformal nets and via vertex operator algebras (VOAs). Starting with a strongly local VOA, they construct a conformal net, after which they can recover the VOA from whence they started. This gives a new construction of known conformal nets, but leaves open the crucial question of comparing the modular tensor category of representations of the VOA and the corresponding conformal net. Second, recent results of Bischoff produce new conformal nets from known examples by looking at fixed points of hypergroups consisting of completely positive maps. Bischoff’s work also focuses on the important conjecture of Kawahigashi that every finite depth subfactor arises from a CFT. If this were the case, one should be able to construct a Hilbert space with a continuous action of the diffeomorphism group of the circle starting with an arbitrary finite depth subfactor. While no such construction is known, Jones was able to produce unitary representations of Thompson’s groups F and T from an arbitrary subfactor planar algebra. In the process, he showed that Thompson’s groups have a rich structure which actually encodes all links and knots!
Related conferences: This 2018 meeting will build on several meetings in subfactors and in fusion categories since our 2014 BIRS workshop. Recent meetings have focused on either the subfactor side, like the 2014 NCGOA at Vanderbilt and the 2015 Oberwolfach meeting on Subfactors and CFT, or they have focused on the fusion category side, like the 2014 AMS MRC workshop at Snowbird for young researchers on Quantum Phases of Matter and Quantum Information and the 2016 CMO workshop on Modular Categories, their representations, classification, and applications. As these two subjects are intimately connected, and very similar ideas are surfacing simultaneously from these two viewpoints, it is crucial for us to reunite now to unify existing results and to collaborate for future breakthroughs.
Workshop press release:
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Classically, the symmetries of a given object are described mathematically by the notion of a group. Over the past few decades, we have seen the emergence of new `quantum’ mathematical objects whose symmetries are best captured by the notion of a fusion category. Thus we say that fusion categories encode quantum symmetry. In turn, each of these fusion categories gives us a new topological field theory and associated quantum invariants for links and 3-manifolds. In recent years, we have seen that unitary fusion categories and closely related unitary modular categories describe the physics of topological phases of matter and topological quantum computation. Subfactors give a rich source of examples of unitary fusion categories. Indeed, all unitary fusion categories arise from subfactors, so subfactors are universal hosts for quantum symmetries. On the other hand, techniques from fusion categories have led to strong classification results for subfactors of small index. The purpose of this workshop is to build on the 2014 BIRS workshop on Subfactors and fusion categories. This conference will bring together the world’s experts in these related fields to unify existing results and to collaborate for future breakthroughs.