﻿ Vertex operator algebras, conformal blocks, and tensor categories-清华大学求真书院

### 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/

## 预备知识 Prerequisites

Complex analysis, differential manifolds, basic notions of Lie algebras

## 参考资料 References

References for vertex operator algebras

Note: I list these books simply because they are often recommended by others. Some topics in these books might be rather advanced or technical for beginners.

1. Frenkel & Ben-Zvi, Vertex algebras and algebraic curves, 2ed, §1-5

2. Kac, Vertex algebras for Beginners

3. Lepowsky & Li, Introduction to vertex operator algebras and their representations

References for conformal blocks

1. Khono, Conformal Field Theory and Topology, §1. (My favorite introductory book on conformal blocks. Brief and concise. Assumes little background knowledge.)

2. Tsuchiya-Kanie, Vertex operators in conformal field theory on P^1 and monodromy representations of braid group. (Very classical paper. Also assumes little background knowledge, although some of the terminology and methods are a bit outdated. Conformal blocks are called “primary fields” in this paper.)

3. My notes “Conformal blocks: vector bundle structures, sewing, and factorization” on my website https://binguimath.github.io/

Note: This monograph is rather comprehensive. We will choose some topics to study and make simplifications when possible. For instance, some results will be proved only for the genus 0 surface so that everything can be worked out using basic complex analysis.)

Don’t read the following references unless you are familiar with algebraic geometry.

4. Frenkel & Ben-Zvi, Vertex algebras and algebraic curves, 2ed, §6,9,10,17,18

5. Tsuchiya-Ueno-Yamada (TUY), Conformal field theory on universal family of stable curves with gauge symmetries. (Very classical paper. Continuation of Tuchiya-Kanie. Notations are complicated.)

6. Ueno, Conformal field theory with gauge symmetry. (Interpretation and clarification of TUY. Notations in TUY are simplified.)

7. Bakalov-Kirillov, Lectures on tensor categories and modular functors, last chapter. (Interpretation of TUY)

• ### Vertex operator algebras and conformal blocks

AbstractConformal blocks are central objects in the study of 2d conformal field theory and vertex operator algebras (VOAs). Indeed, many important problems in VOAs are related to conformal blocks, including modular invariance of VOA characters (the earliest such type of problem is the famous monstrous moonshine conjecture), the construction of tensor categories for VOA representations, the stud...

• ### $\alpha$-induction, tensor categories and operator algebras

AbstractTensor categories play an important role in theory of subfactors in operator algebras in connection to conformal field theory and condensed matter physics. A certain induction procedure called $\alpha$-induction has been studied as a quantum version of the classical induction in group representation theory. I will present this without assuming knowledge on operator algebras