Mathematical Problems from Geometric/Topological Quantum Field Theories

Louisiana State University, Baton Rouge LA

Investigators

Abstract

ABSTRACT for DMS 0201683 This project concerns problems at the juncture of geometry, topology, and physics. The mathematical challenge we address is :(i) to provide a rigorous framework for functional integrals in geometrical quantum field theories, and (ii) to use the rigorous framework to prove results which may be conjectured in the heuristic/informal setting suggested by physics. Specifically, we will be concerned with : (a) quantum Yang-Mills gauge theory over surfaces, (b) a three dimensional Chern-Simons gauge theory, and (c) other functional integrals arising from geometrical/topological quantum theories. The goal will be to give rigorous foundations to the mathematical expressions and objects which arise in these theories and then to use them to further develop mathematical ideas and solve specific problems. The mathematics involved includes geometry and topology connected with fiber bundles and Lie groups, as well as stochastic analysis and infinite-dimensional integration. Stochastic analysis has often been applied in the study of the geometry of manifolds. The present proposal addresses situations which promise deeper applications of stochastic analysis in settings enriched by geometry and topology. This research program will bring together geometry and topology with stochastic analysis to provide solid mathematical foundations for certain computations done in physics, solve certain specific problems and develop new mathematical ideas arising from quantum physics. The physics is quantum field theory, which describes the quantum behavior of forces that govern the interaction between elementary particles in nature such as those which make up protons and neutrons. This project is concerned with development of fundamental mathematics connected with an area of physics. There may be potential applications to surface physics and other phenomena involving geometry and stochastics, but this project itself is devoted primarily to foundations rather than application. History shows that mathematics arising through an investigation of fundamental notions associated to one particular context later finds application in areas far removed from the original context, this being an essential aspect of the centrality of mathematics to applied sciences.