Mathematical Problems in Low Dimensional Gauge Theories

本项目主要研究由几何量子场论产生的无限维泛函积分和测度及其与二维和三维自然几何物体的关系。在二维空间中，所讨论的对象是平面上的平连接的模空间。这个空间出现在各种各样的背景下，如代数几何和量子规范理论：它是曲面基本群表示的模空间，它出现在三维流形的某些不变量的研究中，在某些背景下，它是曲面上某些类型束的模空间。森古普塔博士通过对规范场量子理论研究中出现的无限维测度的极限经典形式的研究，被引导研究模空间的结构和性质。本研究将研究无限维量子场论测度，了解其在模空间上凝聚成自然体积测度的方式，并提取有关该空间各层的结构和体积的信息。作者认为，关于平面连接模空间的许多拓扑信息包含在Yang-Mills概率测度中。希望这个项目将提供一个富有成效的例子，说明概率论如何在这种情况下用于理解拓扑。在三维空间中，本项目旨在对chen - simons理论中出现的量子场理论泛函积分进行严格的分析，并在严格的水平上理解它们与拓扑不变量的关系。这些研究也将提供新的具体的非线性问题和技术，涉及李群上的随机过程，在无限维高斯分析中。从广义上讲，这个项目研究了几何学、拓扑学和物理学之间的深刻关系所产生的数学问题。空间、时间和物质可以用几何学来理解，这一思想构成了爱因斯坦广义相对论的基础。后来，随着量子理论和基本粒子研究的发展，通过杨、米尔斯等人的工作发现，几何和拓扑在调节自然界的基本力量方面发挥着更深刻、更微妙的作用。近年来，许多数学家把注意力集中在由基本粒子物理问题引起的拓扑和几何的深层问题上。这里的数学通常是在一个粗略和直观的水平上理解的。这个项目致力于将其中一些数学问题建立在坚实的基础上，并在数学环境中发现新的结果。

Abstract Proposal: DMS-9800955 Principal Investigator: Ambar Sengupta This project is an investigation of certain infinite dimensional functional integrals and measures arising from geometric quantum field theories and their relationship to natural geometric objects in two and three dimensions. In two dimensions the object in question is the moduli space of flat connections over a surface. This space appears in contexts as diverse as algebraic geometry and quantum gauge theory: it is the moduli space of representations of the fundamental group of a surface, it has arisen in the study of certain invariants of three-dimensional manifolds, and in certain contexts it is the moduli space of certain types of bundles over a surface. Dr. Sengupta has been led to study the structure and nature of the moduli space through his investigation of a limiting classical form of an infinite dimensional measure which arises in the quantum theoretic study of gauge fields. This investigation will study the infinite-dimensional quantum field theoretic measure, understand the way in which it condenses onto a natural volume measure on the moduli space, and extract information about the structure and volumes of the various strata of this space. This investigator believes that much topological information about the moduli space of flat connections is contained in the Yang-Mills probability measure. It is hoped that this project will provide a fruitful example of how probability theory can be used to understand topology in this context. In three dimensions, this project aims to develop a rigorous analysis of the quantum-field theoretic functional integrals arising in Chern-Simons theory and understand, at a rigorous level, their relationship with topological invariants. These investigations will also provide new concrete non-linear problems and techniques, involving stochastic processes on Lie groups, in infinite-dimensional Gaussian analysis. In broad terms, this project inv estigates mathematical questions that arise from the deep relationship between geometry, topology, and physics. The idea that space, time, and matter can be understood in terms of geometry forms the foundation of Einstein's general theory of relativity. Later, as the study of quantum theory and elementary particles developed, it was discovered through the works of Yang, Mills, and others, that geometry and topology play an even deeper and more subtle role in mediating the fundamental forces of nature. In recent years, many mathematicians have focused their attention on deep problems in topology and geometry which arise from questions in fundamental particle physics. The mathematics here is often understood at a rough and intuitive level. This project is devoted to putting some of these mathematical issues on firm foundations as well as discovering new results in the mathematical setting.