James Lepowsky
Intertwining operators and their roles in vertex operator algebra theory
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.
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
Basic algebra