Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The research is focused on the representation theory of quantum groups at roots of 1, on invariants of 3-manifolds with flat connections, and on some aspects of statistical mechanics. Quantum groups emerged from the study of integrable models in quantum field theory. These models though quite simplistic relative to their realistic counterparts, exhibit properties which are hard to study using conventional methods. Another class of simplistic but extremely interesting quantum field theories are topological quantum field theories. In these theories the "dynamics" is absent and all they "know" in the topology of the "underlying space-time". The theories provided new topological invariants. "Physical" formulation involves integration over the infinite dimensional space of connections, mathematical formulation involves representations of quantum groups. One of the major directions of the project is to reconcile two approaches. Another direction of research is related to random discrete surfaces and related structures such as random partition, random tilings of a plane, random spanning trees on a planar graph etc. Some of the questions in this direction are closely related to the limit when a topological quantum field theory known as Chern-Simons theory turns into a topological string theory. Part of the project is focused on the study of quantum groups at roots of unity. The goal is to investigate the category of generic modules as a monoidal category fibered over the corresponding group, and when it is the case, as as a braided monoidal category fibered over a braided group. The results will be applied to construct and study invariants of 3-manifolds with flat connections. This direction is closely related to the possible relation between the A-polynomial and Jones invariant of knots, which was discussed in the literature in the last few years. The research will also be focused on dimer models and on related models in statistical mechanics. For example, the limit shapes for the 6-vertex model may have singular boundaries, and one of the goals is to study the fluctuations such singularities.