Dualities for rational multi-particle Painlev\'e systems: Spectral versus Ruijsenaars

TL;DR

This paper explores dualities in rational multi-particle Painlevé systems, linking spectral curve descriptions with Ruijsenaars duality, and extends the Painlevé-Calogero correspondence to multi-particle contexts.

Contribution

It introduces dual systems for rational multi-particle Painlevé equations via Hamiltonian reduction and compares spectral and Ruijsenaars dualities.

Findings

Constructed dual systems for Painlevé I, II, and IV.

Described spectral curve and Ruijsenaars duality relationship.

Extended Painlevé-Calogero correspondence to multi-particle systems.

Abstract

The extension of the Painlev\'e-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlev\'e equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlev\'e systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlev\'e systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. We describe this duality in terms of the spectral curve of non-reduced system in comparison to the Ruijsenaars duality.