Quantum Geometry of Moduli Spaces and Motives

这个项目涉及表现论、代数几何和数学物理的交叉研究。自伽罗瓦研究多项式的根以来，对称原理在数学中起着至关重要的作用，而自爱因斯坦发展狭义相对论以来，对称原理在物理学中也起到了至关重要的作用。在过去的50年里，发现了对称性概念的新实例：超对称性，后来的量子对称性，以量子群的形式表示。表象理论作为理解量子对称性概念的一种方式出现在画面中，但量子对称性的概念及其几何起源仍然难以捉摸。沿着这些思路，这个项目涉及一个复变量线性微分方程组的研究，它具有代数系数和任意奇性。主要的研究对象之一是这类微分方程解的空间(及其推广)，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示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方法。它将量子群及其关键性质嵌入到Stokes数据的量子化模空间的更一般的几何框架中。这个项目将为研究生提供在这一研究领域的培训和研究机会。更详细地说，PI将研究各种模空间的量子几何，以理论簇Poisson簇为主要工具，并应用于代数几何、表示论和数学物理。此外，PI还将学习经典和量子多对数以及量子霍奇场理论。本项目的主要工作如下：a)对黎曼曲面上可能具有不规则奇点的亚纯联络的模空间的簇结构进行了全面的研究，并将结果应用于这些模空间的量子化。B)在单位根上发展团簇量子化，并将其应用于DeConcini-Kac量子群和三重不变量的表示。C)发展量子多重多对数理论，给出被删截射影直线的动机基本群的原统一完备化周期的量子形变。D)进一步发展量子霍奇场理论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.