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.
Mikhail Khovanov, Johns Hopkins University, Baltimore, MD, Joshua Sussan, CUNY Medgar Evers, Brooklyn, NY, and Anton Zeitlin, Georgia Institute of Technology, Atlanta, GA
Articles Igor Frenkel and Joshua Sussan, Perspectives on the past, present, and future of representation theory and mathematical physics: An interview of Igor Frenkel by Joshua Sussan Mina Aganagic, Elise LePage and Miroslav Rapcak, Homological link invariants from Floer theory Benjamin Cooper, You Qi and Joshua Sussan, A braid group action on an $A_\infty $-category for zigzag algebras Nora Ganter, Looking for a refined Monster Mee Seong Im and Mikhail Khovanov, From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions Ivan C. H. Ip, Quantum cluster mutations and reduced word graphs Hyun Kyu Kim, A trilogy of mapping class group representations from three-dimensional quantum gravity Andrei Negut, Quantum loop groups for arbitrary quivers Samson L. Shatashvili, On the topics of my conversations with Igor Frenkel Anton M. Zeitlin, Superopers revisited Yongchang Zhu, Variation diminishing operators and unitary representations $SL_2$ over semifields
