Vertex operator algebras, conformal blocks, and tensor categories

Vertex operator algebras (VOAs) are mathematical objects describing 2d chiral conformal field theory. The representation category of a “strongly rational” VOA is a modular tensor category (which yields a 3d topological quantum field theory), and conjecturally, all modular tensor categories arise from such VOA representations. Conformal blocks are the crucial ingredients in the representation theory of VOAs.

This course is an introduction to the basic theory of VOAs, their representations, and conformal blocks from the complex analytic point of view. Our goal in the first half of this course is to get familiar with the computations in VOA theory and some basic examples. The second half is devoted to the study of conformal blocks. The goal is to understand the following three crucial properties of conformal blocks and the roles they play in the representation categories of VOAs. (1) The space of conformal blocks forms a vector bundle with (projectively flat) connections. (2) Sewing conformal blocks is convergent (3) Factorization property.

I will type lecture notes and post them on my website https://binguimath.github.io/

Prerequisites

Complex analysis, differential manifolds, basic notions of Lie algebras