Mathematical Problems from Geometric/Topological Quantum Field Theories

本项目涉及几何学、拓扑学和物理学的结合点问题。数学挑战 我们的目标是：（一）提供一个严格的框架， 几何量子场论中的函数积分，以及（ii）使用严格的框架来证明结果，这些结果可以在物理学建议的启发式/非正式设置中实现。具体地说，我们将涉及：（a）量子杨米尔斯规范理论的表面，（B）三维 Chern-Simons规范理论，以及（c）其他由几何/拓扑量子理论产生的泛函积分。我们的目标将是为 这些理论中的数学表达式和对象，然后用它们来进一步发展数学思想和解决具体问题。所涉及的数学包括 几何学和拓扑学与纤维丛和李群，以及随机分析和无限维积分。随机分析经常被应用于流形几何的研究。本提案涉及 这些情况使随机分析在环境中有更深的应用， 几何学和拓扑学。这个研究项目将把几何学和拓扑学与随机分析结合起来，为物理学中的某些计算提供坚实的数学基础，解决某些特定的问题，并发展量子物理学中的新数学思想。物理学是量子场论，它描述了控制自然界中基本粒子（如构成质子和中子的粒子）之间相互作用的力的量子行为。这个项目 is concerned关心with development发展of fundamental基础mathematics数学connected连接with 物理学的一个领域。也许有潜在的应用表面物理学和其他现象涉及几何和随机，但这个项目本身主要致力于基础，而不是应用。 历史表明，通过对与一个特定背景相关的基本概念的调查而产生的数学，后来发现应用于远离原始背景的领域，这是数学对应用科学的中心地位的一个重要方面。

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.