Vertex Operator Algebras and Mathematical Conformal Field Theory

Abstract for 0070800 - Lepowsky / Huang

The investigators study 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. Vertex operator algebra theory arose naturally in the representation theory of infinite-dimensional Lie algebras and in the construction of the infinite-dimensional 'moonshine module' for the Monster finite simple group, and vertex operator algebras are basic ingredients in the mathematical construction of conformal field theories, which arose in both condensed matter physics and in string theory. The investigators use algebraic, geometric and analytic methods and ideas. Huang develops an analytic and geometric theory underlying conformal field theory and applies the results obtained to the study of geometry and topology. Lepowsky continues his investigations into the relations between vertex operator algebra theory and number theory and other topics. The investigators also study algebraic problems underlying geometric uniformization. The various topics studied are in fact deeply connected with one another, and the solution of certain of these problems is expected to be useful in the analysis and solution of other problems. The long-term goal of the investigators 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.

The theory of 'vertex operator algebras' arose naturally in the study of both continuous symmetries and a very special large finite group of symmetries called the 'Monster.' This theory is fundamental to a wide range of problems in many branches of mathematics and in theoretical physics. Some years ago, it was very surprising to mathematicians that the Monster appeared to be deeply connected to number theory. Equally remarkably, this purely mathematical connection, at first only speculative, was considerably clarified by the introduction of new mathematical ideas related to a completely different theory---a physical theory called 'string theory,' which aims at unifying all the fundamental forces in nature, including gravity, electricity and magnetism, and nuclear forces. One result of this mathematical progress has been a richly developed theory of vertex operator algebras, a theory that continues to find new and surprising applications at a rapid rate. A very important theme is that vertex operator algebras are basic ingredients of the mathematical construction of physical theories called 'conformal quantum field theories,' which arose both in the study of the properties of solids and fluids and in string theory. Conformal field theory is in the process of being rapidly developed into a rich and beautiful mathematical theory. This development is expected to continue to yield solutions to many mathematical problems, involving symmetry, geometry, topology, algebra and number theory, and to yield further applications to the deeper understanding of nature and, it is hoped, the development of technology involving solids and fluids. The proposed project uses a variety of mathematical ideas to deepen the understanding of vertex operator algebra theory and of conformal field theory, and to develop a range of new applications.