Representation Theory as Gauge Theory

表征理论试图对对称性的可能实现进行分类和描述，并通过提供将对称结构分解为基本成分的工具来利用对称性。表象理论几乎从一开始就是量子物理中的一个重要工具，它提供了例如原子轨道的结构。规范理论是直接建立在局部对称性结构之外的量子理论，是许多高能物理的语言，特别是标准模型，它描述了引力以外的所有基本力。规范理论反过来又对低维拓扑和几何产生了巨大的影响。这个项目涉及表象理论和规范理论之间关系的颠倒：从抽象上应用规范理论的结构作为表象理论的强大组织框架。在这个范式中，不同的表示理论由不同的规范理论模型编码，并从规范理论中可见的行为和缺陷中继承了一种全新而统一的结构。因此，自然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--特别是我们所知的代数中一些最深层的结构(朗兰兹计划，负责费马最后定理的解决)源于电和磁之间的对称性。这些联系和协同效应将在这个项目下得到发展，无论是在PI的研究中，还是在他广泛的说明性工作中，包括写一篇研究生课文首次向更广泛的受众介绍这一范式。Gauge理论是直接建立在局部李群对称性基础上的量子场论。相反，人们可以通过规范理论的镜头来看待李群表示理论的许多方面，这为通过低维拓扑来组织表示理论提供了强有力的组织原则。本项目的目的是发展和传播表象理论作为规范理论的观点。该项目详细介绍了受规范理论发展启发的两个主要研究项目。第一个是从规范理论的Seiberg-Witten几何中得到启发，并在PI最近的工作中发现的几何表示理论中一个新的交换对称代数来源的开发。PI将在不同的上下文中针对这些对称性应用谱分解，包括Lusztig的特征轴理论、特征曲面的特征变体的同调以及几何朗兰兹对应。第二是发展了PI引入的几何朗兰兹对应的新的“Betti”(或拓扑学)形式，他打算通过证明一个“自同构弗林德公式”将其简化为基本的积木。属一的情况似乎是可以理解的，并对表象理论中的许多研究主题具有启示意义。此外，PI打算从事广泛的说明性写作，包括研究生文本和与理论高能物理学家的跨学科说明性工作。

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.