Mathematical Problems in Low Dimensional Gauge Theories

摘要 提案：DMS-9800955主要研究者：安巴尔Sengupta 该项目是对几何量子场论中的某些无穷维函数积分和测度及其与二维和三维自然几何对象的关系的研究。 在二维中，所讨论的对象是曲面上平坦连接的模空间。 这个空间出现在代数几何和量子规范理论等不同的背景中：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的丛的模空间。 森古普塔博士通过研究规范场的量子理论研究中出现的无限维测度的极限经典形式，研究了模空间的结构和性质。 本研究将研究无限维量子场论测度，了解它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结构和体积的信息。 研究者认为，关于平坦联络模空间的许多拓扑信息包含在杨-米尔斯概率测度中。 希望这个项目将提供一个富有成效的例子，如何概率论可以用来理解拓扑结构在这种情况下。 在三维空间中，该项目旨在对Chern-Simons理论中产生的量子场论泛函积分进行严格分析，并在严格的水平上理解它们与拓扑不变量的关系。 这些调查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。 从广义上讲，这个项目研究了从几何，拓扑和物理之间的深层关系中产生的数学问题。 空间、时间和物质可以用几何学来理解的观点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，人们通过杨振宁、米尔斯和其他人的工作发现，几何和拓扑学在调节自然界的基本力方面发挥着更深刻、更微妙的作用。 近年来，许多数学家把注意力集中在拓扑学和几何学中的深层次问题上，这些问题源于基本粒子物理学中的问题。 这里的数学通常是在粗略和直观的水平上理解的。 该项目致力于将这些数学问题中的一些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.