Topics in Vertex Operator Algebras

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Principal Investigator: Chongying Dong Proposal Number: DMS - 0245548 Institution: University of California-Santa Cruz Abstract: The principal investigator proposes to study the structure and representation theory of vertex operator algebras, connections between vertex operator algebras and geometry, number theory in mathematics and conformal field theory in physics. He plans to investigate the following: (1) Connections among the monster simple group, the monstrous moonshine, permutation orbifolds and elliptic genera. The goal is to understand a new mysterious appearance of the permutation orbifolds in the monstrous moonshine and to understand the moonshine geometrically. (2) Coset construction. The coset construction is an important way to construct new conformal field theory from a given one. The goal is to prove a duality theory of Schur-Weyl type and determine the module category of a coset vertex operator algebra. (3) Subalgebras of vertex operator algebras. The goal is to understand the rationality of vertex operator algebras. This research has fundamental applications to orbifold theory. (4) Characterizations of certain finitely generated vertex operator algebras. This research leads to the classification of rational vertex operator algebras with small central charges. The theory of vertex operator algebra is a new and very rapidly developing area of mathematics. The vertex operator algebra is essentially the chiral algebra in conformal field theory and string theory (which is the leading candidate for the "theory of everything" in mathematical physics) in physics. The proposed research lays some algebraic foundations of conformal field theory. The proposed research also has a lot of applications in many areas of mathematics, such as group theory, topology and geometr