Lectures on Vertex Operator Algebras and Conformal Blocks

TL;DR

This paper provides comprehensive lecture notes on vertex operator algebras and conformal blocks, emphasizing complex-analytic methods, convergence issues, and motivations from Segal conformal field theory.

Contribution

It develops the foundational theory of VOAs and conformal blocks with detailed proofs of convergence and clarifies subtleties using complex analysis, linking to Segal CFT.

Findings

Proves absolute and locally uniform convergence of key series.

Clarifies formal variable subtleties in VOA theory.

Connects VOA concepts with Segal CFT motivations.

Abstract

These are the lecture notes for a course taught at Tsinghua University in the spring of 2022. In these notes, we develop the basic theory of vertex operator algebras (VOAs) and their conformal blocks using complex-analytic methods. In particular, many well-known subtleties in VOA theory (about formal variables and, e.g., delta-functions) are presented in the form of proving the absolute and locally uniform (a.l.u.) convergence of certain series of complex analytic functions. We also provide many motivations from the perspective of Segal CFT.