Quantum Integrable Systems and Geometry

可积系统是数学和物理学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论的背景下，由量子场论的研究激发的可积系统的复兴。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跨学科课程。该项目包含本科生和研究生的研究课题，旨在研究和教育的垂直整合。PI还将共同组织为早期职业研究人员设计的几次会议。该项目由几何分析计划，代数和数论计划以及刺激竞争研究的既定计划（EPSCoR）共同资助。该项目涉及量子可积系统的Bethe anomaly方法的几种观点的几何实现。一种方法使用量子Knizhnik-Zamolodchikov方程自然出现在Nakajima的枚举几何中。另一个依赖于QQ系统的研究，推广了巴克斯特算子所满足的关系。从射影直线上oper联络的差分方程形式的性质出发，导出了QQ系统的几何实现，推广了Gaudin模型下oper联络与Bethe方程之间的对应关系. PI计划在量子q-朗兰兹对应的框架内，展示这些几何观点如何在贝特安托上结合在一起。该奖项反映了NSF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.