Mathematical Problems in Low Dimensional Gauge Theories

建议：DMS-9800955首席研究员：Ambar Sengupta这个项目是研究几何量子场理论中的某些无限维泛函积分和测量，以及它们与二维和三维自然几何对象的关系。在二维中，所讨论的对象是曲面上平面连接的模空间。这个空间出现在与代数几何和量子规范理论一样多样的上下文中：它是曲面基本群表示的模空间，它出现在研究三维流形的某些不变量中，在某些上下文中，它是曲面上某些类型丛的模空间。森古普塔博士通过研究无限维度量的一种极限经典形式，研究了模空间的结构和性质，这种形式出现在规范场的量子理论研究中。这项研究将研究无限维量子场理论度量，了解它是如何凝聚成模空间上的自然体积度量的，并提取关于该空间各层的结构和体积的信息。本文认为，在杨-米尔斯概率度量中包含了许多关于平面连通模空间的拓扑信息。希望这个项目将提供一个富有成效的例子，说明如何利用概率论来理解这种背景下的拓扑学。在三维空间中，这个项目的目的是对在Chern-Simons理论中产生的量子场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。这些研究还将提供新的具体的非线性问题和技术，涉及无限维高斯分析中的李群上的随机过程。从广义上讲，这个项目研究的是几何、拓扑学和物理学之间的深层次关系所产生的数学问题。可以用几何学来理解空间、时间和物质的想法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、米尔斯等人的工作发现，几何和拓扑在调解自然基本力方面发挥着更深更微妙的作用。近年来，许多数学家将注意力集中在拓扑学和几何学中的深层次问题上，这些问题源于基础粒子物理中的问题。这里的数学通常是以粗略和直观的水平理解的。这个项目致力于将这些数学问题中的一些放在坚实的基础上，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.