This research is concerned with the geometric structure of conformal field theory. The principal investigator will study the following topics: (1) binary trees and the Lie bracket; (2) determinant line bundles and vertex operator algebras; (3) a geometric proof of duality for more than two twisted elements; (4) an abstract form of the geometric definitions of finite dimensional Lie algebras, vertex operator algebras and higher dimensional quantum field theory; (5) exponentiating vertex operator algebras; (6) exponentiations of vertex operator algebras and elliptic cohomology. The research supported concerns an important focus of current mathematical activity, where group representation theory, quantum field theory, and algebraic geometry all come together. This work has important implications for theoretical physics, as well as several areas of mathematics.
