Terry Gannon (University of Alberta, APCTP speaker): 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, APCTP speaker): 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, APCTP speaker): 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.
