UQSL- Fall 2020

8/27/2020- David Reutter, Max Planck Institute:

Semisimple topological field theories and exotic smooth structures.

A major open problem in quantum topology is the construction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4-manifolds. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of vector-space-valued oriented 4-dimensional TQFTs, including `unitary field theories' and `once-extended field theories' which assign algebras or linear categories to 2-manifolds. If time permits, I will give a concrete expression for the value of a semisimple TQFT on a simply connected 4-manifold and explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a non-simply connected 4-manifold. Throughout, I will use the Crane-Yetter field theory associated to a ribbon fusion category as a guiding example. This is based on arXiv:2001.02288.

9/3/2020- Andrew Schopieray, Pacific Institute for the Mathematical Sciences:

The importance of norm and trace for fusion categories.

An incredible amount of literature on fusion categories focuses on the weakly integral, i.e. fusion categories whose Frobenius-Perron dimension is an integer. This includes representation categories of finite groups and quasi-Hopf algebras, (weakly) group-theoretical fusion categories, Tambara-Yamagami categories, etc. We will demonstrate how two rudimentary concepts from number theory (norm and trace) dictate the structure of fusion categories which lie far from the cozy world of rational integers. In particular, we will discuss the ongoing classification of fusion categories whose global dimension is a prime rational integer.

9/10/2020- Pavel Etingof, Massachusetts Institute of Technology: TBA

New incompressible symmetric tensor categories in positive characteristic.

Let $k$ be an algebraically closed field of characteristic $p>0$. The category of tilting modules for $SL_2(k)$ has a tensor ideal $I_n$ generated by the $n$-th Steinberg module. I will explain that the quotient of the tilting category by $I_n$ admits an abelian envelope, a finite symmetric tensor category ${\rm Ver}_{p^n}$, which is not semisimple for $n>1$. This is a reduction to characteristic $p$ of the semisimplification of the category of tilting modules for the quantum group at a root of unity of order $p^n$. These categories are incompressible, i.e. do not admit fiber functors to smaller categories. For $p=1$, these categories were defined by S. Gelfand and D. Kazhdan and by G. Georgiev and O. Mathieu in early 1990s, but for $n>1$ they are new. I will describe these categories in detail and explain a conjectural formulation of Deligne's theorem in characteristic $p$ in which they appear. This is joint work with D. Benson and V. Ostrik.

9/17/2020- Florencia Orosz Hunziker, University of Colorado, Boulder:

Tensor categories arising from the Virasoro algebra.

In this talk we will discuss the tensor structure associated to certain representations of the Virasoro algebra. In particular, we will show that there is a braided tensor category structure on the category of C1-cofinite modules for the Virasoro vertex operator algebras of arbitrary central charge. This talk is based on joint work with Jinwei Yang, Thomas Creutzig, Cuibo Jiang and David Ridout.

9/24/2020- Cain Edie-Michell, Vanderbilt University:

Symmetries of modular categories and quantum subgroups

Since the problem was introduced by Ocneanu in the late 2000's, it has been a long-standing open problem to completely classify the quantum subgroups of the simple Lie algebras. This classification problem has received considerable attention, due to the correspondence between these quantum subgroups, and the extensions of WZW models in physics. A rich source of quantum subgroups can be constructed via symmetries of certain modular tensor categories constructed from Lie algebras. In this talk I will describe the construction of a large class of these symmetries. Many exceptional examples are found, which give rise to infinite families of new exceptional quantum subgroups.

10/1/2020- André Henriques, University of Oxford:

Extended chiral CFT and a new kind of anomaly

I will describe ongoing work with James Tener, in which we construct unitary chiral CFTs à la Segal, and I'll explain why it is in fact easier to directly construct extended chiral CFT as opposed to non-extended ones. On our way, we have found a new kind of anomaly which is related to unitarity: Hilbert spaces whose norm is only well-defined up to a positive scalar.

10/8/2020- Stuart White, University of Oxford:

Simple amenable operator algebras

The last decade has seen dramatic advances in the structure and classification of simple amenable C*-algebras, driven by strong parallels with results for amenable von Neumann algebras in the 70's and 80's. In this survey talk, I'll try and explain where we are right now, how this parallels the von Neumann algebraic situation, and also why attendees of a quantum symmetries seminar might want to know about this. I won't assume any prior knowledge of C*-algebras beyond their formal definition.

10/15/2020- Chelsea Walton, Rice University:

Universal Quantum Semigroupoids

In a recent paper (https://arxiv.org/abs/2008.00606), Hongdi Huang, Elizabeth Wicks, Robert Won, and I introduce the concept of a universal quantum linear semigroupoid (UQSGd). This is a weak bialgebra that coacts on a (not necessarily connected) graded algebra A. Our main result is that when A is the path algebra kQ of a finite quiver Q each of the various UQSGds introduced in our work is isomorphic to the face algebra attached to Q (an important weak bialgebra due to Hayashi). Most of the talk will be dedicated to setting up context and terminology towards the main result. So even if you’re new to weak bialgebras (as I was last year), you’ll be able to follow along.

10/22/2020- Fiona Burnell, University of Minnesota:

An introduction to fracton order

In recent years, a new type of order -- reminiscent of, but qualitatively distinct from, the topological order that is described mathematically by modular tensor categories and topological quantum field theory -- has emerged in condensed matter physics. This "fracton" order is decidedly geometrical in nature, but also describes physical phenomena, such as statistical interactions between particles and robust ground state degeneracies, that are usually associated with topologically ordered systems. Using a series of examples, I will describe a set of characteristics common to all fracton orders, and discuss some different variations of the concept of fracton order from the physics literature. I will then review some ideas for describing how fracton orders can be obtained by coupling many "layers" of topological orders.

10/29/2020- Dmitri Nikshych, University of New Hampshire:

Braided module categories

Let M be a module category over a braided fusion category C. A C-module braiding on M is an additional symmetry related

to the braiding of C and giving rise to representations of Artin braid groups of type B. I will show that braided C-module categories form a braided monoidal 2-category equivalent to Z(Mod(C)), the 2-center of the monoidal 2-category Mod(C) of C-module categories. As an application, I will explain that braided extensions of C graded by a group A correspond to braided monoidal 2-functors from A to the braided 2-categorical Picard group of C, consisting of invertible braided C-module categories. Such functors can be explicitly described using the Eilenberg-MacLane abelian cohomology. I will also discuss a (conjectural) characterization of minimal modular embeddings of C in terms of Z(Mod(C)). This is a joint work with Alexei Davydov, based on arXiv:2006.08022.

11/5/2020- Guillermo Sanmarco, Universidad Nacional de Córdoba:

Pre-Nichols algebras with finite Gelfand-Kirillov dimension

Finite dimensional Nichols algebras of diagonal type are generalizations of (positive parts) of small quantum groups that play a crucial role in the classification of finite dimensional pointed Hopf algebras with abelian group of group-likes. Some combinatorial tools of quantum groups were extended to this setting, which led to the classification of such Nichols algebras.

When one attempts to classify certain Hopf algebras with finite Gelfand-Kirillov dimension Nichols algebras are not enough and one is naturally led to study pre-Nichols algebras with the same restriction on the growth. In this talk we report our progress in the problem of determining all pre-Nichols algebras of diagonal type with finite GKdim. We reduce the original problem to the one of constructing a pre-Nichols algebra with finite GKdim that projects onto all other such algebras. Our main tools are the so-called distinguished pre-Nichols algebras (which generalize De Concini-Kac-Procesi big quantum groups) and normal extensions of braided Hopf algebras.

11/12/2020- Christoph Schweigert, Universität Hamburg:

Topological field theories with boundaries: from tensor networks to Frobenius-Schur indicators

State sum models of Turaev-Viro type have various applications. Here, we explain their application to projected entangled pair states (PEPS) in tensor network models and to a deeper understanding of equivariant Frobenius-Schur indicators.

11/19/2020- César Galindo, Universidad de los Andes:

Braided Zesting and its applications.

In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. I will present a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

The talk is based on the manuscript https://arxiv.org/abs/2005.05544 a joint work with Colleen Delaney, Julia Plavnik, Eric C. Rowell, and Qing Zhang.

12/3/2020- Colleen Delaney, Indiana University, Bloomington:

Ribbon zesting can produce modular isotopes.

We observe that the modular categories of Mignard and Schauenburg can be related through the ribbon zesting construction. Thus zesting is capable of producing different modular categories with the same modular data. We then use zesting to explain the phenomenology of topological invariants such as the W-matrix and Borromean tensor that go beyond the modular data, and offer a perspective for applying quantum knot theory to the study of (2+1)D topological order.

This talk in based on joint work in progress with Sung Kim and Julia Plavnik.

12/10/2020- Cris Negron, University of North Carolina, Chapel Hill:

Geometries for finite tensor categories.

I will describe three appearances of geometry in studies of finite tensor categories. Or, more accurately, studies of areas of mathematics which take as inputs finite tensor categories. Here we consider cohomology, support theory, and field theories. I will describe how the relevant geometries provide an interesting point of intersection between these fields. This talk will essentially be a survey, and I will try to focus on open questions.