Schedule for: 18w5195 - Fusion Categories and Subfactors

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The categorical property T is a conjecture saying that 1 is an isolated point in the set of global dimensions of fusion categories. I will talk about recent slightly weaker result saying that a similar statement is true if we restrict ourselves to spherical fusion categories.

I will discuss some specific applications of gauging/anyon condensation. This will include some approaches to classification of various types of braided categories (metaplectic, super-modular) and rank-finiteness for braided fusion categories. I will also present some speculations on other possible applications, for example to verify the property F conjecture, and to find a sensible structure theorem for braided fusion categories. This will be based on several joint projects (some completed, some on-going).

This is a report on a joint work with Alexei Davydov. Let C be a braided fusion category. A braided C-module category is a C-module category with an additional symmetry related to the braiding of C and giving rise to representations of pure braid groups. We show that invertible C-module categories form a braided categorical 2-group Pic_br(C) and apply this to classification of braided extensions of C graded by an Abelian group A. We prove that such extensions correspond to braided monoidal 2-functors from A to Pic_br(C). These can be understood as usual braided monoidal functors such that a certain obstruction in the Eilenberg-MacLane abelian cohomology group H^4_ab(A, k^*) vanishes. We describe this obstruction and compute the group Pic_br(C) for several examples.

Although $G$-extensions of modular tensor categories $\mathcal{C}$ are classified by the work of Etingof, Nikshych, and Ostrik, it remains to formulate explicit constructions of $G$-crossed fusion, associativity, and braiding. One would expect that particularly simple examples of $G$-extensions are permutation extensions, where $G=S_n$ acts on the Deligne product of modular tensor categories $\mathcal{C}^{\boxtimes n}$. However, even in this case it was only recently shown by Gannon and Jones that these extensions exist. This followed work of Edie-Michell, Jones, and Plavnik that computed fusion rules for $\mathbb{Z} / 2\mathbb{Z}$-gaugings for modular tensor categories with no non-trivial invertible objects. I will share some new examples and recent progress on the structure of fusion rules for permutation extensions. This is based on work in progress joint with Eric Samperton, as well as with Julia Plavnik, Corey Jones, Paul Gustafson, Jordyn Harriger, Travis Russell, and Elizabeth Wicks.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss a generalization of both categorical group and categorical Hopf algebra symmetries of modular categories based on linear Hopf monads. The talk is based on the joint work with Shawn Cui and Moji Shokrian Zini.

The Landau-Ginzburg/conformal field theory correspondence is a physics result dating from the late 80s-early 90s which in particular predicts some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. Up to date, we have several examples available yet it lacks a precise mathematical statement. In this talk, we will review the actual state-of-art in this topic and discuss future directions of research.

There is an intimate, elementary, but mostly unspoken relationship between fusion subcategories and Frobenius-Perron dimensions (FPdim): in an arbitrary fusion ring, the Frobenius-Perron dimensions of all summands of a product XY lie in the algebraic number field generated by FPdim(X) and FPdim(Y). This is a powerful observation in the sense that its proof is brief but its implications deep. In joint work with Terry Gannon, we describe the algebraic number fields generated by any object in the unitary modular tensor categories constructed from the representation theory of affine Lie algebras or quantum groups at roots of unity using the existing knowledge of their fusion subcategories. For instance, simple objects of integer dimension in this doubly-infinite family of fusion categories are completely classified by observing when 4cos(pi/n)^2 is an integer for some integer n>2. Implications to de-equivariantizations/simple current extensions/condensations/local module constructions of these categories will be discussed.

In 2014 Evans-Gannon gave an infinite family of potential modular data, and conjectured that the family includes the modular data of the Drinfeld center of a near-group category of an odd abelian group A with multiplicity |A|. Moreover they verified the conjecture for groups with small order. Recently we found 5 infinite families of potential modular data, and generalize the Evans-Gannon conjecture to various classes of quadratic categories (and their de-equivariantization). As it is already the case for the Evans-Gannon's family, there are plenty of examples in the families that are not likely to come from the Drinfeld center of quadratic categories, and whether the corresponding modular tensor categories exist or not would be an interesting problem. This is joint work with Pinhas Grossman.

The notion of weak Morita equivalence between tensor categories was introduced by Müger to conceptualize the standard invariants of subfactors, and became a central ingredient in the larger theory of quantum symmetry. Building on the duality for quantum group actions and module categories over the representation category which goes back to Ostrik, we give a dynamical characterization of categorical Morita equivalence between compact quantum groups. On the algebraic side, this extends Schauenburg's characterization of monoidal equivalence in terms of bi-Hopf-Galois objects. Based on joint work with Sergey Neshveyev.

Assuming a widely-believed conjecture, any action of a finite group $G$ on a holomorphic vertex algebra $A$ determines a "gauge anomaly" $\omega \in H^3(G; U(1))$. The construction is fusion category theoretic: any conformal inclusion $W \subset V$, with $V$ holomorphic, determines a fusion category; $W = V^G$ gives a pointed fusion category. I will explain a subtlety in the construction coming from Galois actions. Specifically, if $\gamma$ is a Galois automorphism, then the anomaly for $\gamma(V)$ is not, as might be assumed naively, $\gamma(\omega)$, but is rather $\gamma^2(\omega)$. The proof relies on a recent construction by Evans and Gannon of vertex algebras with a given gauge anomaly.

We introduce the notion of a reflection fusion category, which is a type of G-crossed category generated by objects of Frobenius-Perron dimension 1 and $\sqrt{p}$, where $p$ is an odd prime. We show that such categories correspond to orthogonal reflection groups over $\mathbb{F}_p$. This allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories. The talk is based on a joint work with Pavel Etingof.

In the ADE quiver theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix, captured by the Coxeter exponents. We formalize related notations and prove such a correspondence for a more general case: this includes the quiver of any module of any semisimple Lie algebra g at any level l. We answer a question posed by Victor Kac in 1994 and a recent comment by Terry Gannon in 2017.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.