Quantum Geometry of Moduli Spaces and Motives

模空间和动机的量子几何

• 批准号: 2153059
• 负责人: Alexander Goncharov
• 金额: $ 32万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153059&HistoricalAwards=false
• 关键词: Quantum; Geometry; Moduli; Spaces; Motives

This project concerns research at the crossroads of representation theory, algebraic geometry and mathematical physics. The principle of symmetry has played a crucial role in mathematics since Galois's study of roots of polynomials, and in physics since Einstein’s development of special relativity. Over the past 50 years, new instances of the very idea of symmetry have been discovered: supersymmetry, and later quantum symmetry, expressed in the form of the quantum groups. Representation theory comes into the picture as the way to understand the notion of quantum symmetry, but the very idea of quantum symmetry and its geometric origins remain elusive. ALong these lines, this project concerns the study on systems of linear differential equations of one complex variable, with algebraic coefficients and arbitrary singularities. One of the main objects of study is the space of solutions of such differential equations (and their generalizations), Stokes data, reflecting asymptotic properties of solutions. The PI will investigate quantization of these moduli spaces and its connections with representation theory and theoretical physics. Quantum geometry of moduli spaces of Stokes data provides a new geometric approach to quantum symmetries. It embeds quantum groups and their key properties into a much more general and geometric framework of the quantized moduli spaces of Stokes data. This project will provide training and research opportunities for graduate students in this area of research.In more detail, the PI will study the quantum geometry of various moduli spaces, using the theory cluster Poisson varieties as the main tool, and applications to algebraic geometry, representation Theory, and mathematical physics. In addition, the PI will study classical and quantum polylogarithms and quantum Hodge field theory. The main priorities of the project are the following: a) To give a comprehensive treatment of the cluster structure of moduli spaces of meromorphic connections with possibly irregular singularities on Riemann surfaces and apply the results to quantization of these moduli spaces. b) To develop cluster quantization at roots of unity with applications to representations of DeConcini-Kac quantum groups and invariants of threefolds. c) To develop the theory of quantum multiple polylogarithms, providing a quantum deformation of the periods of the pro-unipotent completion of the motivic fundamental group of the punctured projective line. d) To further develop quantum Hodge field theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目涉及研究在十字路口的表示理论，代数几何和数学物理。自从伽罗瓦研究多项式的根以来，对称性原理在数学中起着至关重要的作用，自从爱因斯坦发展狭义相对论以来，对称性原理在物理学中起着至关重要的作用。在过去的50年里，对称性概念的新实例被发现：超对称性，以及后来的量子对称性，以量子群的形式表达。表象论作为理解量子对称性概念的一种方法出现了，但量子对称性的概念及其几何起源仍然难以捉摸。沿着这些路线，这个项目涉及一个复杂的变量的线性微分方程系统的研究，代数系数和任意奇异性。研究的主要对象之一是这种微分方程（及其推广）的解的空间，斯托克斯数据，反映解的渐近性质。PI将研究这些模空间的量子化及其与表示论和理论物理的联系。Stokes数据模空间的量子几何为量子对称性提供了一种新的几何方法。它将量子群及其关键性质嵌入到斯托克斯数据的量化模空间的更一般和几何框架中。该项目将为该研究领域的研究生提供培训和研究机会。更详细地说，PI将研究各种模空间的量子几何，使用理论簇Poisson簇作为主要工具，并应用于代数几何，表示论和数学物理。此外，PI将研究经典和量子多面体和量子霍奇场论。该项目的主要优先事项如下：a）全面处理黎曼曲面上可能具有不规则奇点的亚纯联络模空间的簇结构，并将结果应用于这些模空间的量子化。B）发展在单位根上的团簇量子化，并应用于DeConcini-Kac量子群的表示和三重不变量。c）发展量子多重多项式理论，提供了被穿孔的射影直线的motivic基本群的pro-unipotent完备化的周期的量子变形。d）进一步发展量子霍奇场论。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

The Inverse Spectral Map for Dimers

二聚体的逆谱图

• DOI: 10.1007/s11040-023-09466-5
• 发表时间: 2023
• 期刊: Analysis and Geometry
• 作者: George, T.;Goncharov, A. B.;Kenyon, R.
• 通讯作者: Kenyon, R.

Cluster construction of the second motivic Chern class

第二届陈省身班集群建设

• DOI: 10.1007/s00029-023-00854-x
• 发表时间: 2023
• 期刊: Selecta Mathematica
• 作者: Goncharov, Alexander B.;Kislinskyi, Oleksii
• 通讯作者: Kislinskyi, Oleksii