2022 Spring YMSC Course: Vertex Operator Algebras, Conformal Blocks, and Tensor Categories

Time and Venue:

2.21~5.13 (Mon./Thurs.) 19:20-20:55
Online: Zoom Meeting ID: 361 038 6975 Passcode: BIMSA Offline:近春西楼报告厅 (Jinchunyuan West Building) Lecture Hall

Lecture Notes and Videos

Lectures on Vertex Operator Algebras and Conformal Blocks

Videos (Chinese starting from the second lecture) are available on OneDrive and Bilibili

Schedule

2/21 Segal’s picture of 2d CFT; motivations of VOAs and conformal blocks

2/24 Virasoro algebras

2/28 Change of boundary parametrizations

3/3 Definition of VOAs

3/7 Definition of VOAs

3/10 Jacobi identity and its consequences

3/14 Consequences of Jacobi identity

3/17 Constructing examples of VOAs

3/21 Constructing examples of VOAs

3/24 Local fields

3/28 Local fields; n-point functions for vertex operators

3/31 Proof of reconstuction theorem; VOA modules

4/2 Contragredient modules

4/7 Change of coordinate theorems

4/11 Definitions of conformal blocks and sheaves of VOAs

4/14 Definitions of conformal blocks and sheaves of VOAs; pushforward in sheaves of VOAs

4/18 Lie derivatives in sheaves of VOAs; families of compact Riemann surfaces

4/21 Families of compact Riemann surfaces and parellel sections of conformal blocks

4/25 Sheaves of coinvariants and conformal blocks

4/28 Connections and local-freeness of sheaves of coinvariants and conformal blocks

5/9 Projective structures and uniqueness of connections; sewing and propagation of conformal blocks

5/12 Sewing and factorization of conformal blocks; tensor categories of VOA modules

Course Description:

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 spaces of conformal blocks form a vector bundle with (projectively flat) connections. (2) Sewing conformal blocks is convergent (3) Factorization property.

Prerequisites

• Complex analysis
• You should be familiar with the materials taught in any course or textbook called “Differential Manifolds”. You should also know the definition of its complex analog: complex manifolds.
• You should have some rough idea of how the representations of Lie algebras and Lie groups are related.

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.

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

• I.Frenkel, Huang, Lepowsky, On axiomatic approaches to vertex operator algebras and modules

• Kac, Vertex algebras for Beginners

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

References for conformal blocks

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

• 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.)

• My notes: Conformal Blocks: Vector Bundle Structures, Sewing, and Factorization

Our course could be viewed as a friendly introduction to the topics in this monograph. When studying these topics, we will 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.

The following references require a solid background in algebraic geometry, which we do NOT require in our course.

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

• 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.)

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

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