Fall 2025
Yi-Zhi Huang
Subtitle:
Associative algebras and vertex operator algebras
Course Description:
The representation theory of Lie algebras is equivalent to the representation theory of universal enveloping algebras of Lie algebras. In particular, the universal enveloping algebras of Lie algebras and their quotients play an important role in the representation theory of Lie algebras.
For vertex operator algebras, though we also have universal enveloping algebras constructed by Frenkel and Zhu, they are not very useful since we consider only lower-bounded generalized modules. It is therefore important to find an associative algebra associated to a vertex operator algebra such that the category of modules for the associative algebra is equivalent to the category of lower-bounded generalized modules for the vertex operator algebra. In the case all lower-bounded generalized modules are completely reducible, such an algebra was found by Zhu. In the general case, such an algebra was found by Huang. In this course, we will study these algebras and their applications in the representation theory of vertex operator algebras. We will also study other related associative algebras.
Here is a list of topics to be covered in the course:
1. Vertex operator algebras and modules.
2. Universal enveloping algebras of vertex operator algebras.
3. Zhu algebra and zero-mode algebra.
4. The higher level generalizations of Zhu algebra by Dong-Li-Mason.
5. Huang's associative algebras and lower-bounded generalized modules.
6. Twisted generalizations of the associative algebras above.
7. Applications of the associative algebras above.
Text:
No textbook. Papers will be discussed in the classes.
Prerequisites:
Some knowledge of first year algebra and complex analysis for graduate students
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Fall 2024
Yi-Zhi Huang
Subtitle:
Vertex operator algebras and orbifold conformal field theory
Course Description:
Orbifold conformal field theories play an particularly important role in conformal field theory and in the applications of conformal field theory. The moonshine module vertex operator algebra constructed by Frenkel-Lepowsky-Meurman is the first example of orbifold conformal field theories. Many results on the construction and classification of self-dual (or holomorphic) vertex operator algebras of central charge 24 and on the generalized moonshine depend heavily on abelian orbifold conformal field theory. The nonabelian orbifold conformal field theory still needs to be developed.
This is an introductory course on the representation theory of vertex operator algebras and orbifold conformal field theory. The topics covered in the course include:
1. Vertex operator algebras and examples.
2. Modules for vertex operator algebras and examples.
3. Twisted modules for vertex operator algebras and examples.
4. Construction of modules and twisted modules.
5. Intertwining operators and twisted intertwining operators.
5. The orbifold theory conjectures.
6. The moonshine module as an orbifold theory.
7. G-crossed tensor categories.
Text:
No textbook. Papers and chapters in books will be discussed in the classes.
Prerequisites:
Basic courses in algebra and analysis. I will start from the very beginning of the theory of vertex operator algebras.
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Spring 2024
Yi-Zhi Huang
Subtitle:
Vertex operator algebras and tensor categories
Course Description:
Two-dimensional conformal field theory can be studied using the representation theory of vertex operator algebras. Suitable module categories for vertex operator algebras have structures of vertex tensor categories. These vertex tensor categories give braided tensor categories and, when the vertex operator algebras satisfy strong conditions, give modular tensor categories.
In this course, I will discuss the construction of these vertex tensor categories and braided tensor categories. Below are the detailed topics to be covered in this course:
1. Vertex operator algebras, modules and intertwining operators.
2. Tensor product modules and their construction.
3. Associativity of intertwining operators and associativity isomorphisms.
4. Skew-symmetry and commutativity of intertwining operators and braiding isomorphisms.
5. Vertex tensor categories and braided tensor categories.
6. Rigidity, twists and modularity.
Text:
No text book. Lectures are based on papers and lecture notes that are available online.
Prerequisites:
First year graduate algebra and analysis courses.
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Fall 2023
James Lepowsky
Subtitle:
Introduction to Vertex Tensor Categories
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.
In Fall, 2023, I will review basic foundations of the theory, for the benefit of less-experienced and also more-experienced students. I will include some discussion of Yi-Zhi Huang's geometric recasting of the notion of vertex operator algebra. Then I will develop properties of intertwining operators among modules for a vertex operator algebra, which leads to the concept of the vertex tensor category of modules for a suitable vertex operator algebra. This in turn leads to the vertex-operator-algebraic approach to the construction of topological invariants, via work of Huang and others. I will relate all of this to conformal field theory and string theory as developed in physics.
When there is not enough time to carry out details in class, I will provide extensive references and invite discussion outside of class time for interested students, including discussion of the basic foundations as necessary. In the later part of the course, students will be invited, on a purely voluntary basis, to give brief talks to the class on topics that they are interested in.
Text:
Basic references:
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, Boston, 1988
I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs AMS, Vol. 104, Number 494, 1993
J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2004
Many other materials, available online
Prerequisites:
Basic algebra
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Spring 2023
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
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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.
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Fall 2021
James Lepowsky
Subtitle:
Vertex OP Calculus & Vertex OP Alg Thy
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.