Vertex algebras and geometry of manifolds

University Of Southern California, Los Angeles CA

Investigators

Abstract

The purpose of this project is to further the understanding of the relation between vertex algebras and the geometry of manifolds. This relation, discovered relatively recently, is an infinite dimensional version of the classical relation between theoretical physics and mathematics. Indeed, quantum mechanics of a particle on a manifold is a part of analysis on manifolds. Similarly, super-symmetric quantum mechanics is essentially equivalent to the study of the de Rham complex on manifolds. Attempts to carry this over to the case of the space of loops on a manifold have created an entirely new Universe of mathematical and physical concepts. One such concept, vertex algebra, a child of string theory and infinite dimensional representation theory, has already proved indispensable in areas as remote form mathematical physics as abstract finite group theory. Relatively recently, thanks to work of Gorbounov and Malikov among others, vertex algebras have been fruitfully applied to several chapters of geometry including elliptic genus and mirror symmetry. String theory, a part of quantum field theory and the only existing candidate for the "theory of everything", while having not yet achieved its ultimate goal, has enriched modern mathematics and physics with a dazzling array of new concepts and new relations among old concepts. Vertex algebra, a concept that simultaneously generalizes the notions of Lie and commutative algebra, is one conspicuous example. It has long since found applications in areas as old and diverse as representation theory, group theory, combinatorics, and number theory. The present project is devoted to the exploration of a recently discovered relation to geometry of higher dimensional spaces.