Quantum Geometry of Moduli Spaces and Motives

该项目涉及表征理论、代数几何和数学物理交叉领域的研究。自从伽罗瓦研究多项式的根以来，对称原理在数学中发挥了至关重要的作用，自从爱因斯坦发展狭义相对论以来，对称原理在物理学中发挥了至关重要的作用。在过去的50年里，对称性概念的新实例已经被发现：超对称和后来的量子对称，以量子群的形式表达。表征理论作为理解量子对称概念的一种方式进入了人们的视野，但量子对称的概念及其几何起源仍然难以捉摸。沿着这些思路，本项目关注的是具有代数系数和任意奇点的单复变量线性微分方程系统的研究。研究的主要对象之一是这类微分方程（及其推广）的解的空间，Stokes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征理论和理论物理的联系。Stokes数据模空间的量子几何为研究量子对称性提供了一种新的几何途径。它将量子群及其关键属性嵌入到Stokes数据的量子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，使用理论簇泊松变分作为主要工具，并将其应用于代数几何、表示理论和数学物理。此外，PI将研究经典和量子多对数以及量子霍奇场论。本课题的主要工作重点如下：a)综合处理黎曼曲面上可能具有不规则奇点的亚纯连接的模空间的簇结构，并将结果应用于这些模空间的量化。b)发展单位根的聚类量子化，并将其应用于deccini - kac量子群和三倍不变量的表示。c)发展了量子多重多对数理论，提供了刺穿射影线的动力基本群的前单能补全周期的量子变形。d)进一步发展量子霍奇场论。该奖项反映了美国国家科学基金会的法定使命，并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。

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.