James Lepowsky
Vertex OP Calculus & Vertex OP Alg Thy
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.
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
Permission of instructor