Dualities for rational multi-particle Painlev\'e systems: Spectral versus Ruijsenaars
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.
Keywords
Cite
@article{arxiv.1912.12588,
title = {Dualities for rational multi-particle Painlev\'e systems: Spectral versus Ruijsenaars},
author = {Ilia Gaiur and Vladimir Rubtsov},
journal= {arXiv preprint arXiv:1912.12588},
year = {2020}
}
Comments
25 pages, 1 figure. Few misprints and a computational mistake in the interactive term of PII are corrected. The Appendix with technical computations is added