Schedule

The program is divided into invited overview talks (90min), invited research talks (60min) and contributed talks (30min).

Overview talks

Iván Angiono (Universidad Nacional de Córdoba): Quantum groups nowadays: classification and examples

Since the appearance of quantized enveloping algebras in the 1980s, Hopf algebras have been considered in different branches of mathematics. Different generalisations have led for example to consider their analogues in the context of super algebras, or, thinking of the classification of pointed tensor categories, co-quasi-Hopf algebras.

In this talk we will recall different invariants that are fundamental in classification in the finite-dimensional context, together with recent classification results for pointed (co-)quasi-Hopf algebras, and as many examples as possible.

Johannes Flake (University of Bonn): Lie theory in tensor categories

(Symmetric) tensor categories provide a natural framework for representation theory, commutative algebra, and algebraic geometry. A deep theorem by Deligne classifies a large collection of symmetric tensor categories in terms of (super)groups, namely, the symmetric tensor categories of moderate growth in characteristic zero. In the past couple of years, our understanding of tensor categories beyond Deligne's theorem has evolved significantly, both in positive characteristics and in the case of tensor categories without a growth condition. We will discuss which symmetric tensor categories are currently known, and what is the state of a new kind of Lie theory that is being developed for some of them.

Pinhas Grossman (UNSW): A tour of subfactor theory (from an algebraic point of view)

In the early 1980s Vaughan Jones discovered a surprising connection between operator alagebras and low-dimensional topology. Since then, subfactor theory has been closely entwined with the fields of quantum topology, representation theory of quantum groups, and conformal field theory. In this talk, we will explain what a subfactor is and describe some algebraic ideas and constructions in subfactor theory, focusing on examples rather than technicalities. In particular, we will discuss the relationship between subfactors and several algebraic structures, including tensor categories, (bi)module categories, 2-categories, and planar algebras. If time permits, we will also discuss some ongoing classification programs for subfactors and tensor categories.

Shashank Kanade (University of Denver): Tensor categories for vertex operator algebras: a brief introduction

I'll provide an overview of the main constructions regarding tensor structure on representation categories of vertex operator algebras given by Huang--Lepowsky(--Zhang). The exposition will focus on highlights, omitting precise statements. Then, I'll provide a brief overview of some developments that happened until about a decade ago, leaving the remarkably exciting developments of the recent times to the luminaries.

Julia Plavnik (Indiana University Bloomington): Classifying and constructing fusion categories

The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories also have applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others.

In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories. We will give some concrete examples and introduce some important invariants. Then we will give an overview of the current situation of the classification program for modular categories. We will also present some constructions of fusion and modular categories such as gauging, bicrossed product, and, we will focus on zesting. If time allows, we will present some open questions and conjectures in the area.

Abstracts

Terry Gannon (University of Alberta): The orbifolds of lattice VOAs

There is an ever-deepening relationship between VOAs and tensor categories. In particular, VOA orbifold theory corresponds to G-crossed braided categories and the extension theory of Etingof-Nikshych-Ostrik. In my talk I'll explain the practical implications of the category theory to VOA orbifold theory, in particular in extending beyond the G cyclic (or abelian) case. I'll prove conjectures of Mason-Ng describing the orbifolds $V^G$ of lattice theories where the group G fixes all V-modules. The other extreme, where $G$ acts fixed-point-freely on the V-modules, will also be discussed (when G=Z/2 this reduces to the familiar Tambara-Yamagami categories, but there is no need to limit to abelian $G$). This is joint work with Andrew Riesen.

Mee Seong Im (Johns Hopkins University):  Entropy, cocycles, algebraic K-theory and diagrammatics

I will discuss how cocycles appear in a graphical network. Furthermore, the Shannon entropy of a finite probability distribution has a natural interpretation in terms of diagrammatics. I will explain the diagrammatics and their connections to infinitesimal dilogarithms and entropy. If I have time, I will talk about how algebraic K-theory appears in diagrammatics.

Kenichi Shimizu (Shibaura Institute of Technology): Properties of the category of local modules

This talk is based on my recent joint works with Thomas Creutzig, Robert McRae, and Harshit Yadav. Given a commutative algebra A in a braided monoidal category C, the category C_A^loc of local A-modules in C is defined as a certain full subcategory of the category of A-bimodules in C. As has been pointed out by Pareigis, the category C_A^loc has the natural structure of a braided monoidal category inherited from that of C. The category of local modules plays an important role in the study of conformal field theory as it closely relates to extensions of vertex operator algebras and their representation theory. I will introduce several results guaranteeing C_A^loc to have nice properties. Suppose that C is a closed monoidal category. A central observation is that C_A^loc is a closed monoidal category with the same internal Hom functor as the category of A-bimodules in C. This yields handy criteria for C_A^loc to be rigid or Grothendieck-Verdier. As an application, one can show that C_A^loc is a braided finite tensor category provided that C is a braided finite tensor category and the category of A-bimodules is a finite tensor category (or, equivalently, A is an indecomposable exact commutative algebra in C). If, furthermore, A is symmetric Frobenius, then C_A^loc is actually a ribbon finite tensor category. I will also discuss relevant results and questions on commutative algebras and symmetric Frobenius algebras in braided finite tensor categories and the Witt equivalence of non-degenerate braided finite tensor categories.

Simon Lentner (Hamburg University): Categorical structures appearing in conformal field theory

The logarithmic Kazhdan Lusztig correspondence is a conjectural equivalence between the representation theory of quantum groups to the representation theory of certain vertex algebras. I will explain the categorical language and recent results with T. Creutzig and M. Rupert, that have allowed us to settle this conjecture in several rank 1 cases. At the same time the approach produces a rather general type of (non semisimple) modular tensor category, extending a given one, that is a good candidate for the representation theory of a kernel-of-screening vertex subalgebra. This has applications much beyond the original intention of the conjecture.

Robert McRae (Tsinghua University): Rigidity of tensor categories for vertex operator extensions and subalgebras

In this talk, I will discuss recent joint works with Thomas Creutzig, Kenichi Shimizu, Harshit Yadav, and Jinwei Yang. Let A be a commutative algebra in a braided monoidal category C. For example, A could be a vertex operator algebra (VOA) extension of a VOA V in a category C of V -modules. First, assuming that C is a finite braided tensor category, I will discuss conditions under which the category C_A of A-modules in C and its subcategory C_A^loc of local modules inherit rigidity from C. These conditions are based on criteria of Etingof and Ostrik for A to be an exact algebra in C. As an application, we show that if a simple non-negative integer-graded vertex operator algebra A contains a strongly rational vertex operator subalgebra V, then A is also strongly rational, without requiring the dimension of A in the modular tensor category of V-modules to be non-zero. Second, assuming that C is a Grothendieck-Verdier category (which means that C admits a weaker duality structure than rigidity), I will discuss conditions under which C inherits rigidity from C_A^loc. These conditions are motivated by free field-like VOA extensions A of a vertex operator subalgebra V where A is often an indecomposable V -module. As an application, we show that the category of weight modules for the simple affine VOA of sl_2 at any admissible level is rigid, using Adamović's inverse quantum Hamiltonian reduction of the simple affine VOA of sl_2.

Cris Negrón (University of Southern California): Quantum Frobenius kernels at arbitrary roots of 1

I will discuss constructions of quantum Frobenius kernels at (completely) arbitrary roots of 1.  As one output, we associate a finite-dimensional (non-semisimple) modular tensor category to any pairing of a simply-connected reductive algebraic group with an even order root of 1.  I will explain the field theoretic motivations for this work, where quantum groups at "bad" roots of unity appear naturally.

Yoshiko Ogata (RIMS, Kyoto University): Operator algebraic approach to topological orders

Topological phases of matter are hot topics in mathematical physics. In quantum statistical mechanics, dynamics are given by linear operators called Hamiltonians with certain localities. Hamiltonians with a spectral gap between the lowest spectrum and the rest are said to be gapped. Such gapped Hamiltonians can be considered to be in a physically "normal" phase. We say two Hamiltonians are equivalent if they can be connected smoothly without closing the gap. Finding invariants of this classification is an exciting mathematical problem. In this talk, I would like to explain the operator algebraic approach to this problem.

Florencia Orosz Hunziker (University of Colorado): Relaxed fusion rules and rigidity of weight modules for sl2 and the N=2 superconformal algebra

The N=2 universal superconformal algebra N(3-6/t) has a coset realization that relates its representations to those of the universal affine sl_2 vertex operator algebra V^{t-2}(sl_2). At admissible levels t=u/v  for coprime u ∈ \mathbb{Z}_{\geq 2},  v ∈ \mathbb{Z}_{\geq 1}, the coset realization factors to the respective simple quotients V_{u/v-2}(sl_2) and the minimal model superconformal algebras N(u,v). At fractional admissible levels t=u/v for coprime  u,v ∈ \mathbb{Z}_{\geq 2}, both V_{u/v}-2}(sl_2)  and  N(u,v) are irrational vertex algebras and only recently, the vertex tensor structure developed in generality for vertex operator algebra modules by Lepowsky, Huang and Zhang, was established on the category of V_{u/v-2}(sl_2) weight -modules and that of N(u,v) weight -modules for these fractional admissible levels by Creutzig. Using the coset realization and the vertex tensor category structure, we prove that the fusion rules between  V_{u/v-2}(sl_2)  relaxed highest weight modules conjectured by Creutzig and Ridout in 2013 hold. We also establish the rigidity of the categories of weight modules for  V_{u/v-2}(sl_2) and N(u,v) at admissible levels and central charges respectively. This talk is based on joint work with H. Nakano, A. Ros Camacho and S. Wood.

Bregje Pauwels (Macquarie University): Approximation in triangulated categories

Given a module over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful.  In other words, every object in the derived category of a ring has a sequence of ‘simpler’ objects converging to it.

In general, a triangulated category is called approximable if this type of ‘approximation by simpler objects’ is possible. The notion of approximability in triangulated categories (due to Amnon Neeman) is fairly new and not very well understood yet. But the early evidence is that the new techniques, while at first sight purely categorical, are very powerful.

In this talk, I will discuss some applications of this new technique and explain when the derived category of an algebraic stack is approximable. 

Eric Rowell (Texas A&M University): Two possibly new monoidal categories 

In two unrelated projects, my collaborators and I happened upon monoidal categories that we had never seen before.  One is related to a fresh look at the problem of classifying solutions to the Yang-Baxter equation, while the other arose from an attempt to make our zesting construction more palatable to physicists.  These are joint works with Martin and Torzewska, and with Delaney, Galindo, Plavnik and Zhang. 

Christoph Schweigert (Hamburg University): Skein theoretic methods for CFT correlators

We present a skein theoretical construction based on a graphical calculus for pivotal bicategories. The graphical calculus for a pivotal monoidal category and the bicategory of Frobenius algebras internal to it are related by a Frobenius monoidal functor. It induces a relation between the skein modules that encodes information on CFT correlators and their mapping class groups.

Daniel Tubbenhauer (University of Sydney): Big data approaches to representation theory

This talk offers a friendly introduction to how topological data analysis and other big data methods can be applied in representation theory. The focus is on quantum knot invariants and Kazhdan-Lusztig theory as the main examples.

Simon Wood (Cardiff University): Duality beyond rigidity

Many of the best studied types of tensor categories enjoy a notion of duality called rigidity (which among other nice properties implies that the tensor product is exact, if the underlying category is abelian). However, not all sources of interesting tensor categories obligingly produce rigid ones. In particular, the natural notion of duality for a category of modules over a vertex operator algebra is Grothendieck-Verdier duality, which will be the focus of this talk. 

I will discuss some recent results on tensor categories with Grothendieck-Verdier duality structures, module categories over these and why these results are encouraging for conformal field theory.

Contributed talks

Tyler Franke (The University of Melbourne): Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model

We revisit the phenomenon of hidden symmetry breaking in a spin chain with quantum group symmetry: the q-deformed Affleck-Kennedy-Lieb-Tasaki (qAKLT) model. We explain why the non-local Kennedy-Tasaki duality transformation that was previously proposed to relate the string order to a local order parameter does not provide an adequate description of hidden symmetry breaking in the qAKLT model. We then present a modified non-local transformation, based on a generalization of Witten's conjugation to frustration-free lattice models, that is capable of resolving this issue.

Jared Heymann (The University of Melbourne): Categorical symmetry resolved entanglement entropy and boundary conformal field theory

In recent years, the notion of a global symmetry in quantum field theory has been generalised. Symmetries go beyond those of groups and can be non-invertible. Similarly to ordinary symmetries, it is interesting to understand the interplay between non-invertible symmetries and physical quantities, such as the entanglement entropy. In this talk, I discuss the symmetry resolution of the entanglement entropy with respect to non-invertible symmetries in 1+1d conformal field theories, with an emphasis on the difference between the representation theory of boundary condition changing operators and bulk operators.

Ana Kontrec (RIMS, Kyoto University): On the structure of some subregular W-algebras

One of the most important families of vertex algebras are affine vertex algebras and their associated W-algebras, which are connected to various aspects of geometry and physics. Recently, the affine W-algebras associated to the subregular nilpotent element W^k(\g, f_{sub}) have attracted a lot of interest. In this talk I will present some recent results about structure and representation theory of certain subregular W-algebras at integer levels. This is joint work with D. Adamovic.

Peter McNamara (The University of Melbourne): The Spin Brauer Category

The Brauer category is a tool that controls the representation theory of (special) orthogonal Lie groups and Lie algebras. A drawback is it doesn't see the spin representations. We introduce and study a spin version, the Spin Brauer category, which sees the entire representation theory of type B/D Lie algebras. This is joint work with Alistair Savage.

Monique Müller Lopes Rocha (Universidade Federal de São João del-Rei): On exact factorization of fusion categories

The concept of exact factorization of fusion categories was introduced by Gelaki in 2017 and is a categorical generalization of the concept of exact factorization of finite groups. We will show some properties of exact factorization of fusion categories and present a result on how to construct an exact factorization from two fusion categories and some data. This is a joint work with Héctor Martín Peña Pollastri and Julia Plavnik.

Shigenori Nakatsuka (FAU Erlangen-Nurnberg): On the Structure of W-algebras

The W-algebras are vertex algebras obtained from affine Lie algebras through quantized Drinfeld-Sokolov reductions parametrized by nilpotent orbits. They are fundamental objects for studying Whittaker models for affine Lie algebras. Recently, it has been conjectured that the W-algebras are organized along the closure relations of nilpotent orbits: more precisely, two W-algebras should be reconstructed from each other if the corresponding two nilpotent orbits are in closure relations. In this talk, I will discuss the motivation, the current status of this conjecture, and various applications to the representation theory. The talk is based on several joint works with Creutzig-Fasquel-Linshaw, Fasquel-Fehily-Fursman, and Fasquel-Kovalchuk.

Joe Newton (The University of Sydney): Finite symmetric and exterior algebras in tensor categories

In characteristic zero, every symmetric tensor category of moderate growth fibres over super vector spaces. The parity of an object in such a category can be determined by its symmetric and exterior powers, with even and odd objects having non-vanishing symmetric and exterior powers respectively. However, in characteristic p>0 there are examples of objects whose symmetric and exterior powers both vanish. I will discuss a method to classify symmetric tensor categories generated by an object whose maximal non-zero powers sum to p. These are described in terms of Verlinde categories of algebraic groups, and along the way I will describe a useful decomposition of such Verlinde categories. This gives evidence towards a conjecture that all finitely generated semisimple symmetric tensor categories of moderate growth can be obtained from Verlinde categories. This talk is based on joint work with Kevin Coulembier and Pavel Etingof.

Thomas Quella (The University of Melbourne): Symmetry-protected topological phases with quantum group invariance

In recent years there were tremendous efforts to understand phases of quantum matter that involve the preservation or breaking of various types of generalized symmetries in the framework of higher categorical symmetries, including so-called topological or symmetry-protected topological phases.

In this talk I will review recent efforts to understand topological phases of 1D spin chains that are protected by quantum group symmetry. Key example will be the q-deformed AKLT model for which an exact Matrix Product State representation of the ground state can be found. We will then highlight non-trivial entanglement features and dualities in this model.

Alex Sherman (The University of Sydney): Homogeneous spaces in Verlinde p

Homogeneous spaces play an important role in the geometry and representation theory of algebraic groups (think Peter-Weyl, Borel-Weil-Bott, symmetric spaces, toric varieties, and much more).  The existence of homogeneous superspaces was proven by Masuoka and Zubkov, and they have similary found abundant use in super-representation theory.  In this vein, I will discuss the existence of homogeneous spaces in the symmetric tensor category Ver_p, along with some nice applications in representation theory.  Part of a joint work with Kevin Coulembier.

William Stewart (TU Munich): Domain walls and oplax natural transformations

In the Atiyah-Segal framework for topological quantum field theory, a domain wall is a functor out of a decorated bordism category.  In this talk I will illustrate an equivalence between topological domain walls and oplax natural transformations. I’ll show how this equivalence provides an extension of the usual cobordism hypothesis to the setting of domain walls.

Organisers

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