James Lepowsky
Introduction to Vertex Tensor Categories
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.
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
Basic algebra