The Zoo of Opers and Dualities

TL;DR

This paper explores the structure of $SL(r+1)$-opers, their deformations, and the quantum/classical dualities linking quantum spin chains with classical many-body systems, revealing deep geometric correspondences.

Contribution

It introduces new connections between oper spaces, symplectic duality, and bispectral duality, advancing understanding of quantum/classical dualities in integrable systems.

Findings

Established dualities between different oper spaces.

Linked quantum spin chains with classical systems via dualities.

Connected oper space correspondences with symplectic and bispectral dualities.

Abstract

We investigate various spaces of $SL(r+1)$-opers and their deformations. For each type of such opers, we study the quantum/classical duality, which relates quantum integrable spin chains with classical solvable many body systems. In this context, quantum/classical dualities serve as an interplay between two different coordinate systems on the space of opers. We also establish correspondences between the underlying oper spaces, which recently had multiple incarnations in symplectic duality and bispectral duality.