Mathematical Sciences: Geometric Structure of Conformal Field Theory

This research is concerned with conformal field theory and specifically with the study of vertex operator algebras. The principal investigator will study the following topics: vertex category theory; reconstruction of vertex operator algebras from vertex tensor categories; modular vertex tensor categories; vertex groups and vertex operator algebras; vertex groups and integrable systems; geometric construction of the Monster; cohomology theory for vertex operator algebras; and vertex manifolds. Conformal field theory is an important physical theory describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory also has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera, and knot theory, is revealed in the study of conformal field theory. It is believed that the study of the mathematics involved in conformal field theory will lead to new mathematical structures which will be important both in mathematics and theoretical physics.