Some results and conjectures in the theory of vertex operator algebras

by Dr Shashank Kanade (University of Denver)

Thursday 9 Dec 2021, 11:00 → 12:00 Asia/Kolkata

Via Zoom

Abstract: Individual vertex operators arose in the mathematical literature nearly four decades ago in Lepowsky-Wilson's Lie algebraic proof of the Rogers-Ramanujan identities. Vertex operator algebras (VOAs) were also central to Borcherds' proof of the moonshine conjecture -- the moonshine module constructed by Frenkel-Lepowsky-Meurman and used in Borcherds' proof is a VOA. Since their inception, the study of VOAs has seen a rapid growth guided by various conjectures in mathematics and physics. Most well-known VOAs are in some way connected to affine Lie algebras and their study is naturally related to representation theory, tensor categories, algebraic combinatorics and number theory. In this talk, I will survey a selection of results and conjectures pertaining to these topics. I will focus on (a subset of) -- 1. Rogers-Ramanujan-type identities related to affine Lie algebras, 2. Tensor categorical aspects related to conformal embeddings of VOAs, 3. Some problems in the representation theory of twisted affine Lie algebras at non-integrable levels. Parts of the talk will be based on joint works with my collaborators.