Mathematical Problems from Geometric/Topological Quantum Field Theories

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ABSTRACT for DMS 0201683This project concerns problems at the juncture of geometry, topology, andphysics. The mathematical challenge we address is :(i) to provide arigorous framework for functional integrals in geometrical quantumfield theories, and (ii) to use the rigorous framework to prove resultswhich may be conjectured in the heuristic/informal setting suggested byphysics. Specifically, we will be concerned with : (a) quantum Yang-Millsgauge theory over surfaces, (b) a three dimensional Chern-Simons gaugetheory, and (c) other functional integrals arising fromgeometrical/topological quantum theories. The goal will be to giverigorous foundations to the mathematical expressions and objects whicharise in these theories and then to use them to further developmathematical ideas and solve specific problems. The mathematics involvedincludes geometry and topology connected with fiber bundles and Liegroups, as well as stochastic analysis and infinite-dimensionalintegration. Stochastic analysis has often been applied in the study ofthe geometry of manifolds. The present proposal addresses situationswhich promise deeper applications of stochastic analysis in settingsenriched by geometry and topology.This research program will bring together geometry and topology withstochastic analysis to provide solid mathematical foundations for certaincomputations done in physics, solve certain specific problems and developnew mathematical ideas arising from quantum physics. The physics is quantumfield theory, which describes the quantum behavior of forces that governthe interaction between elementary particles in nature such as those whichmake up protons and neutrons. This project is concerned with developmentof fundamental mathematics connected with an area of physics. There maybe potential applications to surface physics and other phenomena involvinggeometry and stochastics, but this project itself is devoted primarily tofoundations rather than application. History shows that mathematicsarising through an investigation of fundamental notions associated to oneparticular context later finds application in areas far removed from theoriginal context, this being an essential aspect of the centrality ofmathematics to applied sciences.