Interrelations between dualities in classical integrable systems and classical-classical version of quantum-classical duality

TL;DR

The paper investigates how three classical dualities in integrable systems—Ruijsenaars p-q duality, spectral duality between Gaudin models and spin chains, and the quantum-classical duality—are interrelated. By introducing a fictitious spectral parameter through a gauge transform L -> L(z), the authors recast many-body Lax matrices into Gaudin-like forms and apply spectral duality to obtain models gauge-equivalent to the Ruijsenaars p-q duals, across rational and trigonometric families. They establish explicit correspondences for both CM/RS and XXZ XXX formulations and derive a classical-classical analogue of quantum-classical duality via Schlesinger-type constructions, linking moving poles to CM coordinates. The work unifies these dualities under a common spectral-curve and gauge-transformation framework, highlighting the deep algebraic structures shared by classical and quantum integrable systems and offering a path to analyze quantum-classical correspondences from purely classical data.

Abstract

We describe the Ruijsenaars' action-angle duality in classical many-body integrable systems through the spectral duality transformation relating the classical spin chains and Gaudin models. For this purpose, the Lax matrices of many-body systems are represented in the multi-pole (Gaudin-like) form by introducing a fictitious spectral parameter. This form of Lax matrices is also interpreted as classical-classical version of quantum-classical duality.