ABSTRACT, MASON 97-00909 Abstract : PI will investigate the mathematical foundations of the theory of vertex operator algebras, their automorphism groups and representation theory, and related questions in the cohomology of groups and algebras and in the theory of modular forms. Techniques are based on past work of the PI and collaborators on the foundations of algebraic conformal field theory, including the theory of Zhu's associative algebras, quantum Galois theory, existence and modular-invariance of twisted sectors, and a cohomological theory of discrete torsion and Dijkgraaf-Witten cocycles. In recent years, mathematicians and physicists alike have been developing conformal field theory, which incorporates infinite-dimensional symmetry, in order to understand certain critical phenomena and also (via string theory) the first rigorous approach to quantum gravity. This effort is a highly intensive mathematical endeavour, and the PI will investigate a number of new phenomena in abstract algebra which are part of this program. So-called vertex operator algebras constitute a rigorous mathematical axiomatization of some of the main ideas of conformal field theory, and the PI will continue his research into the symmetries inherent in these very complicated objects in order to solidify the foundations of the subject and to uncover new phenomena.
