### Fall 2022

### James Lepowsky

### Subtitle:

Selected topics in vertex operator algebra theory

### Course Description:

The course will focus on selected aspects of vertex operator algebra theory and Monstrous Moonshine, adapted to students' interests,

background, and research needs.

### Text:

Monographs by I. Frenkel-J. Lepowsky-A. Meurman; I. Frenkel-Y.Z. Huang-J. Lepowsky; and J. Lepowsky-H. Li; and other materials available on the internet

### Prerequisites:

Basic algebra

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### Spring 2022

### Yi-Zhi Huang

### Subtitle:

Modular invariance in conformal field theory

### Course Description:

Modular invariance plays an important role in the representation theory of infinite-dimensional Lie algebras, the representation theory of vertex operator algebras, conformal field theory and many other areas. This is an introductory course on the modular invariance in conformal field theory, The following topics will be covered in the course:

1. Modular invariance of graded dimensions of suitable modules for Heisenberg algebras, affine Lie algebras and the VIrasoro algebra.

2. Vertex operator algebras, modules and intertwining operators.

3. q-traces and pseudo-traces.

4. Differential equations.

5. Associative algebras and vertex operator algebras.

6. Modular invariance theorem.

7. Applications (Verlinde formula, rigidity of tensor categories of modules for vertex operator algebras and so on).

### Text:

Lecture notes to be written

### Prerequisites:

Some basic knowledge of first year graduate courses on algebra and complex analysis

### ***********************************

### Fall 2022

### James Lepowsky

### Subtitle:

Intertwining operators and their roles in vertex operator algebra theory

### Course Description:

In general, each semester of Math 557 will cover topics chosen from among the following and related topics:

Vertex operator algebras, basic vertex operator constructions, modules, intertwining operators; relations with q-series theory; convergence, associativity and commutativity for intertwining operators; vertex tensor category of modules; modular invariance, Verlinde formula, modular tensor categories; twisted modules, twisted intertwining operators; Moonshine Module, Monster group and Monstrous Moonshine.

In Fall 2022, the main theme of the course will be the theory of intertwining operators in vertex operator algebra theory. The theory of intertwining operators is central to virtually all aspects of the theory, including vertex tensor categories, conformal field theory in physics, and Monstrous Moonshine, among many other directions. We will use selected, accessible parts of the references listed below, together with selected, accessible parts of work of Y.-Z. Huang, J. Lepowsky and L. Zhang on vertex tensor category theory. We will start from the basics, and relate what we develop to a wide range of the theory.

Students' interests and background will be taken into account. Many research problems will be discussed.

Note: The Lie Group/Quantum Mathematics Seminar, https://sites.math.rutgers.edu/~yzhuang/rci/math/lie-quantum.html will meet at 12:10 on Fridays. Some of the talks will be related to the themes of Math 557.

### Text:

I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs AMS, Vol. 104, Number 494, 1993

I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston,1988

J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2004

Many additional materials to be distributed

### Prerequisites:

Basic algebra

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### Spring 2022

### Yi-Zhi Huang

### Subtitle:

Representation theory of vertex operator algebras and two-dimensional conformal field theory

### Course Description:

The representation theory of vertex operator algebras is in fact equivalent to two-dimensional chiral conformal field theory. This is an introductory course on the representation theory of vertex operator algebras. In this course I will cover the following topics:

1. Vertex operator algebras, modules and intertwining operators. (I will go through this part quickly if the students know some basic material on vertex operator algebras, for example, from Lepowsky's course in Fall 2021.)

2. Associative algebras and intertwining operators.

3. Convergence, associativity and commutativity of intertwining operators.

4. Tensor products of modules for a vertex operator algebra.

5. Introduction to the tensor category of modules for a vertex operator algebra.

### Text:

Yi-Zhi Huang: Lecture notes on vertex algebras and quantum vertex algebras (https://sites.math.rutgers.edu/~yzhuang/rci/math/papers/va-lect-notes.pdf). Lecture notes for this course will be written. Before each lecture, notes will be available for the material to be covered in that lecture.

### Prerequisites:

Basic courses in algebra and analysis. I will start from the very beginning of the theory of vertex operator algebras. It will be very helpful if the students know some basic material on vertex operator algebras and modules, for example, from Lepowsky's course in Fall 2021. But these are not required.

### ***********************************

### Fall 2021

### James Lepowsky

### Subtitle:

Selected topics in vertex operator algebra theory

### Course Description:

In general, each semester of Math 557 will cover topics chosen from among the following and related topics:

Vertex operator algebras, basic vertex operator constructions, modules, intertwining operators. Topics selected from among:

relations with q-series theory; convergence, associativity and commutativity for intertwining operators; vertex tensor category of

modules; modular invariance, Verlinde formula, modular tensor categories; twisted modules, twisted intertwining operators; Moonshine

Module, Monster group and Monstrous Moonshine.

In Fall 2021, Math 557 will focus on the systematic development of vertex operator calculus, based mostly on the early chapters of the listed texts, [FLM] and [LL]. In [FLM], the theory is developed in approximately historical order, including the construction of affine Lie algebras using vertex operators and formal delta-function calculus. This helps motivate the axiomatic notion of vertex operator algebra. In [LL], the algebraic axiom systems for the notion of vertex operator algebra (and of module) are introduced early, based on formal calculus. Each set of axioms highlights crucial aspects of the theory. Their relations and consequences will be developed. All of this is foundational for vertex operator algebra theory and for the mathematics of the closely-related conformal field theory in theoretical physics, and for the many still-expanding applications and consequences in very diverse branches of mathematics and physics.

There will be a lot of discussion of such applications of vertex operator calculus, depending on the interests of the students. Current research directions and problems will be highlighted.

There are no prerequisites for this course beyond basic algebra and a bit of familiarity with Lie algebra theory. Students who already have some experience in vertex operator algebra theory can look forward to gaining new insight.

This course will be online, using Zoom, partly because not all of the participants will be nearby. But I look forward to meeting with students in person, by appointment.

Note: The Lie Group/Quantum Mathematics Seminar,

https://sites.math.rutgers.edu/~yzhuang/rci/math/lie-quantum.html

will meet at 11:00 (not 12:00) on Fridays in Fall 2021. Some of the talks will be related to the themes of Math 557. The seminar will probably be virtual, using Zoom, for at least part of the fall semester. The seminar webpage will be updated as necessary.

### Text:

I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988

J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkh\"auser,

Boston, 2004

and many materials to be distributed

### Prerequisites:

Permission of instructor

### Schedule of Sections:

### Previous Semesters

PLEASE NOTE: The course information changes from semester to semester for this course number. Fall 2021 is the first semester that Math 557 is offered with the course title "Topics in Vertex Operator Algebra Theory." In very many earlier semesters, through Spring 2021,

courses on vertex operator algebra theory and related subjects were given as sections of the courses "16:640:554 - Selected Topics in

Algebra" and "16:640:555 - Selected Topics in Algebra," with subtitles indicating the content.