Quantum Integrable Systems and Geometry

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.