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On the multiplet W-algebras

Hold Date
2022-05-20 16:00〜2022-05-20 17:00
C-513 and Zoom meeting
Object person
Shoma Sugimoto (Kyushu University)

Vertex operator algebra (VOA) was introduced in the 1980s as a mathematical formulation of two-dimensional conformal field theories and is an interesting subject that relates to various areas of mathematics. Until now, semisimple VOAs have been mainly studied, but recently, from the viewpoint of higher-dimensional quantum field theory and quantum topology, the study of non-semisimple VOAs (logVOAs) has attracted much attention. One of the main (and almost the only) known examples of logVOA is the multiple W-algebra (associated to a simple Lie algebra g) which has been well studied when g is of type A_1. However, the multiple W-algebra associated to a general g has a complicated structure and conventional algebraic methods do not work, so there have been no results for many years. In this talk, the speaker will prove various basic properties of multiple W-algebras using a geometric approach. First, we give the algebraic and geometric definitions of multiple W-algebras and show that they coincide (Feigin-Tipunin conjecture, 2010). This allows us to use various powerful theorems in geometry for the study of multiple W-algebras. In particular, by using a duality theorem, we construct irreducible modules, determine the G×W_k(g)-module structures, and compute the q-characters. Finally, we discuss the relationship between the aforementioned results and quantum topology, as well as future prospects.