Representation Theory as Gauge Theory

作为规范理论的表示论

• 批准号: 1705110
• 负责人: David Ben-Zvi
• 金额: $ 17.39万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1705110&HistoricalAwards=false
• 关键词: Representation; Theory; Gauge

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.

表征理论试图对对称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。表征理论几乎从一开始就是量子物理学的一个重要工具，例如提供了原子轨道的结构。规范理论，直接建立在局部对称结构之上的量子理论，是很多高能物理学的语言，尤其是标准模型，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。这个项目是关于表征理论和规范理论之间关系的反转：应用规范理论的结构作为抽象表征理论的一个强大的组织框架。在这一范式中，不同的表征理论被不同的规范理论模型编码，并从规范理论的可观测行为和缺陷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——特别是我们所知道的代数中一些最深奥的结构（朗兰兹程序，负责解决费马大定理）源于电和磁之间的对称性。这些联系和协同作用将在该项目下得到发展，包括在PI的研究和他广泛的解释性工作中，包括编写研究生文本，首次向更广泛的受众介绍这一范式。规范理论是直接建立在局域李群对称基础上的量子场论。相反，我们可以通过规范论的视角来看待李群表示理论的许多方面，规范论通过低维拓扑为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征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。第一个是利用几何表示理论中交换对称代数的新来源，该来源受到规范理论的Seiberg-Witten几何的启发，并在PI最近的工作中发现。PI将在各种情况下应用关于这些对称性的光谱分解，包括Lusztig的特征束理论，表面特征变体的同调，以及几何朗兰兹对应。第二个是PI引入的几何朗兰兹对应的新“贝蒂”（或拓扑）形式的发展，他打算通过证明一个“自同构Verlinde公式”将其简化为基本构建块。属一的情况似乎是可以理解的，并且在表示理论中有很多研究课题的含义。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。