Representation Theory as Gauge Theory

表示论试图对对称性的可能实现进行分类和描述，并通过提供一种将对称结构分解为基本成分的工具来利用对称性。表示论几乎从一开始就是量子物理学中的一个重要工具，例如提供原子轨道的结构。规范理论，直接建立在局域对称结构之上的量子理论，是许多高能物理学的语言，特别是标准模型，它描述了除引力之外的所有基本力。规范理论反过来又对低维拓扑和几何产生了巨大的影响。这个项目关注的是表征理论和规范理论之间关系的逆转：应用规范理论的结构作为表征理论的一个强大的组织框架。在这种范式中，不同的表象理论由不同的规范理论模型编码，并从规范理论中的可观测量和缺陷的行为中继承了一个全新的统一结构。因此，自然界的基本对称性成为理解代数和分析中最抽象问题的有力工具-特别是我们在代数中所知道的一些最深刻的结构（朗兰兹程序，负责解决费马大定理）来自电和磁之间的对称性。这些联系和协同作用将在这个项目下发展，无论是在PI的研究中，还是在他广泛的临时工作中，包括写一本研究生教材，第一次向更广泛的读者介绍这种范式。规范理论是直接从局域李群对称性建立起来的量子场论。相反，人们可以通过规范理论的透镜来观察李群表示论的许多方面，规范理论通过低维拓扑为表示论提供了一个强有力的组织原则。这个项目的目的是发展和传播表征理论作为规范理论的观点。该项目详细介绍了规范理论发展的两个主要研究项目。第一个是利用交换对称代数的一个新的来源在几何表示理论的启发塞伯格-威滕几何规范理论和发现在PI的最近的工作。PI将在各种背景下对这些对称性应用谱分解，包括Lusztig的特征层理论，曲面特征簇的同源性和几何朗兰兹对应。第二个是由PI引入的几何朗兰兹对应的新“贝蒂”（或拓扑）形式的发展，他打算通过证明“自守Verlinde公式”将其简化为基本的构建块。“类属1的情况似乎是可以理解的，并且对表示论中许多研究的主题有影响。此外，PI打算从事广泛的论文写作，包括研究生文本和理论高能物理学家的跨学科论文工作。

Representation theory seeks to classify and describe the possible realizations of symmetries, and to exploit symmetry by providing a tool to decompose symmetric structures into elementary constituents. Representation theory has been an essential tool in quantum physics almost from its inception, providing for example the structure of atomic orbitals. Gauge theories, quantum theories built directly out of the structure of local symmetry, are the language of much of high energy physics, in particular the Standard Model, which describes all the fundamental forces besides gravity. Gauge theory in turn has had a tremendous impact on low dimensional topology and geometry. This project is concerned with the reversal of the relationship between representation theory and gauge theory: applying the structure of gauge theory as a powerful organizing framework for representation theory in the abstract. In this paradigm, different representation theories are encoded by different models of gauge theory, and inherit a radically new and uniform structure from the behavior of observables and defects in gauge theory. Thus the fundamental symmetries of nature become powerful tools to understand the most abstract questions in algebra and analysis -- in particular some of the deepest structures we know in algebra (the Langlands program, responsible for the resolution of Fermat's Last Theorem) derive from the symmetry between electricity and magnetism. These connections and synergies will be developed under this project, both in the PI's research and in his extensive expository work, including writing a graduate text introducing this paradigm to a broader audience for the first time.Gauge theories are quantum field theories built directly out of local Lie group symmetry. Conversely, one can view many aspects of representation theory of Lie groups through the lens of gauge theory, which provides a powerful organizing principle for representation theory through the medium of low-dimensional topology. The object of this project is to develop and disseminate the perspective of representation theory as gauge theory. The project details two primary research projects inspired by developments in gauge theory. The first is the exploitation of a new source of commutative symmetry algebras in geometric representation theory inspired by Seiberg-Witten geometry of gauge theory and uncovered in the PI's recent work. The PI will apply spectral decomposition with respect to these symmetries in a variety of contexts, including Lusztig's theory of character sheaves, the homology of character varieties of surfaces, and the geometric Langlands correspondence. The second is the development of the new "Betti" (or topological) form of the Geometric Langlands Correspondence introduced by the PI, which he intends to reduce to elementary building blocks by proving an "automorphic Verlinde formula." The case of genus one appears accessible, and has implications for much-studied topics in representation theory. In addition the PI intends to engage in extensive expository writing, including a graduate text and interdisciplinary expository work with theoretical high energy physicists.