Mathematical Problems from Geometric/Topological Quantum Field Theories

DMS 0201683本课题涉及几何、拓扑和物理的交叉问题。我们解决的数学挑战是：(i)为几何量子场理论中的函数积分提供严格的框架，以及（ii）使用严格的框架来证明可能在物理学建议的启发式/非正式设置中推测的结果。具体来说，我们将关注：(a)曲面上的量子Yang-Millsgauge理论，(b)三维chen - simons gauge理论，以及(c)由几何/拓扑量子理论产生的其他泛函积分。我们的目标是为这些理论中出现的数学表达式和对象提供坚实的基础，然后利用它们进一步发展数学思想和解决具体问题。所涉及的数学包括与纤维束和李群相关的几何和拓扑，以及随机分析和无限维积分。随机分析在流形的几何研究中经常被应用。目前的建议解决的情况下，承诺更深入的应用随机分析的设置丰富的几何和拓扑。这个研究项目将把几何和拓扑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，并从量子物理学中产生新的数学思想。物理学就是量子场理论，它描述了控制自然界基本粒子（如组成质子和中子的粒子）之间相互作用的力的量子行为。这个项目涉及到与物理学领域相关的基础数学的发展。表面物理和其他涉及几何和随机的现象可能有潜在的应用，但这个项目本身主要致力于基础而不是应用。历史表明，通过对与某一特定背景相关的基本概念的研究，数学化后来在与原始背景相去甚远的领域得到了应用，这是数学在应用科学中处于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.