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.
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
Mikhail Khovanov, Johns Hopkins University, Baltimore, MD, Joshua Sussan, CUNY Medgar Evers, Brooklyn, NY, and Anton Zeitlin, Georgia Institute of Technology, Atlanta, GA
