Elliptic Calogero-Moser Systems and Isomonodromic Deformations

Abstract

We show that various models of the elliptic Calogero-Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero-Moser system turns out to give a Lax representation of the non-autonomous system as well. This Lax representation ensures that the non-autonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the Lax pair is defined. A particularly interesting example is the ``extended twisted $B{C}_{\ell }$ model'' recently introduced along with some other models by Bordner and Sasaki, who remarked that this system is equivalent to Inozemtsev's generalized elliptic Calogero-Moser system. We use the ``root type'' Lax pair developed by Bordner et al. to formulate the associated isomonodromic system on the torus.

latex2e using amsfonts package, 50pages; (v2) typos corrected; (v3) typos in (3.35), (3.46), (3.48) and (B.26) corrected; (v4) errors in (1.7),(1.12),(3.46),(3.47) and (3.48) corrected; (v5) final version for publication, errors in (2.31),(2.35),(3.12),(3.30),(3.45),(4.16) and (4.37) corrected

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• Painleve-Calogero Correspondence Revisited
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Quantum spectral problems and isomonodromic deformations
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Inozemtsev System as Seiberg-Witten Integrable system
• $t{t}^{\ast }$ Toda equations for surface defects in $\mathcal{N}=2$ SYM and instanton counting for classical Lie groups
• Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero-Gaudin system
• Classical elliptic ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries
• An Explicit Characterization of Calogero--Moser Systems
• R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{BC}}_n$ type