Vertex Operator Algebras and Mathematical Conformal Field Theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Huang and Lepowsky intend to solve a range of problems related to vertex (operator) algebra theory and conformal field theory, and their relations with and applications to a variety of areas of mathematics and physics. They intend to use new and evolving algebraic, geometric and analytic ideas and methods. Huang proposes to continue to develop an algebraic, analytic and geometric theory underlying conformal field theory and to apply the results obtained to the solution of problems in algebra, geometry and topology. Lepowsky intends to continue to develop vertex operator algebra theory and its relations with conformal field theory, infinite-dimensional algebraic structures, number theory and the theory of combinatorial identities. Huang and Lepowsky also propose to solve problems related to the remarkable circle of ideas called ``monstrous moonshine,'' to geometric uniformizations and to logarithmic tensor product theory. The varied topics discussed in this proposal are in fact deeply connected with one another, and the solutions of certain of these problems are expected to be useful in the analysis and solution of others. A long-term goal of Huang and Lepowsky is to contribute to the deeper development of a mathematical theory that will enhance the conceptual understanding of many known and still-unknown connections among many branches of mathematics and theoretical physics. Huang and Lepowsky will continue to be very active in their teaching, writing, organizing of conferences and seminars, and their other activities designed to increase and enhance the understanding in the mathematical community of the very rich field in which they work. They will also continue to try to convey to members of the theoretical physics community, through writing and their speaking, their enthusiasm for the richness and utility of their approach to vertex operator algebra theory and mathematical conformal field theory. The theory of vertex operator algebras arose naturally in the study of certain infinite-dimensional symmetries and in the solution of fundamental problems relating the ``Monster,'' a certain finite symmetry group, with number theory, and this theory is fundamental to a wide range of problems in both mathematics and theoretical physics. Vertex operator algebras are crucial ingredients in the mathematical study of conformal field theory, a physical theory that arose in both condensed matter physics and string theory, which in turn is a physical theory attempting to unify all the interactions in the universe. Conformal field theory is in the process of being rapidly developed into a rich and beautiful mathematical theory. This development has already led to new ideas, surprising results and the solutions of many mathematical problems, and it is expected to lead to many more. The mathematical development of conformal field theory has also been applied to the study of physical phenomena such as disorder sytems, condensed matter physics on surfaces with boundaries and certain higher-dimensional physical objects called D-branes in string theory.