Quantum Geometry, Integrability, and Opers

TL;DR

This review explores recent advances in quantum geometry and integrable models, highlighting connections between algebraic structures, quantum systems, and geometric objects, and proposing conjectures and open problems in the field.

Contribution

It synthesizes recent progress in understanding integrable models through algebraic geometry, representation theory, and physics, and formulates new conjectures and open problems.

Findings

Connections between Calogero-Ruijsenaars systems and opers

Relations between quantum spin chains and double affine Hecke algebras

Enumerative geometry links to quiver varieties

Abstract

This review article discusses recent progress in understanding of various families of integrable models in terms of algebraic geometry, representation theory, and physics. In particular, we address the connections between soluble many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of opers, representations of double affine Hecke algebras, enumerative counts to quiver varieties, to name just a few. We formulate several conjectures and open problems. This is a contribution to the proceedings of the conference on Elliptic Integrable Systems and Representation Theory, which was held in August 2023 at University of Tokyo.