Vertex Operator Algebras and Their Automorphisms

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The principal investigator will investigate vertex operator algebras and their automorphism groups. Holomorphic orbifold theory, and rational vertex operator algebras will be studied. Relation among vertex operator algebras, automorphism groups of vertex operator algebras, quantum doubles, dual pairs will be discussed. The automorphism group of a rational vertex operator algebra will be determined. Certain rational vertex operator algebras will be characterized and classified. Several fundamental problems in the theory of vertex operator algebra are expected to be solved. The goal of this project is to lay some further foundations of the theory of vertex operator algebras and their representations. String theory at present is the only candidate for a theory combining all the fundamental interactions. Two dimensional conformal field theory which is an important physical theory describing two-dimensional critical phenomena in condensed matter physics provides a framework for string theory. Vertex operator algebras are symmetry algebras in conformal field theory. Physical theories, perhaps more often than mathematical theories, typically start from particular structure, called the models. On the other hand, mathematical theories abstract from the examples and predict what happens in general. This is a research that starts out in the field of algebra but moves beyond that to touch invariant theory, quantum groups, modular forms, and conformal and topological field theories. A successful study of this project will lead to new mathematical discoveries which will be important both in mathematics and physics.