From Representation Theory to Mathematical Physics and Back

This volume is a proceedings of a workshop at the Simons Center for Geometry and Physics from May 31- June 4, 2022. The workshop highlighted progress in the areas of vertex operator algebras, conformal field theory, categorification, low dimensional topology and representation theory of affine Lie algebras, loop groups, and quantum groups. In the past 40 years, string theory gave rise to the mathematical theory of vertex operator algebras, which led to the construction of representations of affine Lie algebras and the Moonshine module of the Monster group. These mathematical constructions have in turn led to ideas about 3-dimensional quantum gravity. In another direction, the discovery of the Jones polynomial led to a physical construction of 3-dimensional topological quantum field theories (TQFTs), which in turn advanced many mathematical developments in quantum groups and low dimensional topology. Louis Crane and Igor Frenkel introduced the categorification program with the goal of upgrading 3-dimensional TQFTs coming from representation theory of quantum groups to 4-dimensional TQFTs. This idea gave rise to the development of link homologies constructed from representation-theoretic, algebraic-geometric, combinatorial, and physical structures. Articles in this volume present both classical and new results related to these topics. They will be interesting to researchers and graduate students working in mathematical aspects of modern quantum field theory.