Quantum Integrable Systems and Geometry

可积系统是数学和物理多个子领域的基本数学工具，更重要的是，它有助于在看似不同的领域之间建立桥梁。这个项目涉及最近在代数几何和表示理论背景下的可积系统的复兴，受到量子场论研究的推动。这一研究结果有望阐明最初在理论物理中发现的三维镜像对称现象。更广泛的影响包括为数学和物理学生建立一个跨学科的项目。该项目包含本科生和研究生的研究课题，旨在实现研究与教育的垂直整合。PI还将共同组织几次为早期职业研究人员设计的会议。该项目由几何分析项目、代数和数论项目以及促进竞争研究的既定项目（EPSCoR）共同资助。该项目涉及在量子可积系统的Bethe ansatz方法上的几个观点的几何实现。一种方法是利用中岛颤振变种的枚举几何中自然出现的量子Knizhnik-Zamolodchikov方程。另一种依赖于对qq系统的研究，推广了Baxter算子所满足的关系。从射影线上开放连接的差分方程的性质出发，推导出了qq -系统的几何实现，推广了Gaudin模型中开放连接与Bethe方程的对应关系。PI计划展示这些关于Bethe ansatz的几何观点是如何在量子q-Langlands对应的框架内结合在一起的。这有望在三维镜像对称的数学公式研究中产生新的结果。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。

Integrable systems are a fundamental mathematical tool in multiple subfields of mathematics and physics, and more importantly, are instrumental in building bridges between seemingly disparate areas. This project deals with the recent resurgence of integrable systems in the context of algebraic geometry and representation theory, motivated by the study of quantum field theories. The results of the research are expected to elucidate the phenomenon of three-dimensional mirror symmetry originally discovered in theoretical physics. Broader impacts include establishing an interdisciplinary program for mathematics and physics students. The project contains undergraduate and graduate student research topics and aims for vertical integration of research and education. The PI will also co-organize several conferences designed for early-career researchers. The project is jointly funded by the Geometric Analysis program, the Algebra and Number Theory program, and the Established Program to Stimulate Competitive Research (EPSCoR).The project deals with the geometric realization of several points of view on the Bethe ansatz approach to quantum integrable systems. One approach uses quantum Knizhnik-Zamolodchikov equations emerging naturally in the enumerative geometry of Nakajima quiver varieties. Another one relies on the study of QQ-systems, generalizing the relations satisfied by the Baxter operators. The geometric realization of QQ-systems is derived from the properties of the difference equation version of oper connections on the projective line, generalizing the correspondence between oper connections and Bethe equations for the Gaudin model. The PI plans to show how these geometric points of view on the Bethe ansatz are bonded together within the framework of the quantum q-Langlands correspondence. This is expected to lead to new results in the study of the mathematical formulation of three-dimensional mirror symmetry.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.