Vertex Operator Algebras, Elliptic Genus and Conformal Nets

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The PI plans to study the structure and representation theory for vertex operator algebras, the connection between vertex operator algebras and geometry, and the connection between algebraic and analytic approaches to 2 dimensional conformal field theory. In particular, the following topics will be studied: (1) The Frenkel-Lepowsky-Meurman's uniqueness conjecture of the moonshine vertex operator algebra. The goal is to prove the conjecture completely and to obtain a characterization of the Griess algebra (independent of the monster simple group). This is a part of program of classification of holomorphic vertex operator algebras of central charge 24. (2) The chiral ring (which was defined by physicists in the study of super conformal field theory and super string theory) of a N=2 unitary vertex operator superalgebra and elliptic genus of a Calabi-Yau manifold. The purpose is to understand the role that the chiral ring plays in the structure and representation theory of vertex operator superalgebra, and to investigate to what extend the chiral ring of the vertex operator superalgebra associated to a Calabi-Yau manifold determines the elliptic genus of the manifold. (3) The connection between vertex operator algebra (an algebraic approach to conformal field theory) and conformal nets (an analytic approach to conformal field theory). This is a long term program. The goal is to construct vertex operator algebras and conformal nets from each other. This will lead to a unification of the algebraic and analytic approaches to 2 dimensional conformal field theory. The theory of vertex operator algebra provides an algebraic foundation for the 2 dimensional quantum conformal field theory in physics and is also deeply related to many important areas of mathematics such as representation theory, group theory, modular forms, topology invariants, and C*-algebras. The planned research links vertex algebra operator algebras with topology, C*-algebras and quantum field theory, and has important applications in both mathematics and physics.