Categorification of integral group rings extended by one dimension | BIMSA-Tsinghua Quantum Symmetry Seminar

Abstract

Fusion categories have established themselves in the past two decades as indispensable objects of study across representation theory, low-dimensional topological quantum field theory, conformal field theory, and quantum computation. But this field of study is at a cross-roads since the production of high-level machinery and abstraction is far out-pacing the set of known examples. At the core of this crisis is the "categorification" problem of determining when there exists a fusion category realizing a given fusion ring, i.e. the combinatorial skeleton of a fusion category. The most elementary fusion rings are the integral character rings of finite groups; it is still a vast open problem to determine which fusion rings with exactly one noninvertible element have corresponding fusion categories. In this talk we will discuss the categorification problem for fusion rings whose basis elements take one of two dimensions. This is the setting where almost all known "exotic" examples of fusion categories live. At least the first half of the talk will be understandable by a general audience. ([2208.07319] Categorification of integral group rings extended by one dimension (arxiv.org)).