Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero-Gaudin system

Abstract

This paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ODE's on a torus. An isomonodromic analogue of the $\rmSU(2)$ Calogero-Gaudin system is used for a case study of this approach. A clue of this approach is a mapping to a finite number of points on the spectral curve of the isomonodromic Lax equation. The coordinates of these moving points give a new set of Darboux coordinates called the spectral Darboux coordinates. The system of isomonodromic deformations is thereby converted to a non-autonomous Hamiltonian system in the spectral Darboux coordinates. The Hamiltonians turn out to resemble those of a previously known isomonodromic system of a second order scalar ODE. The two isomonodromic systems are shown to be linked by a simple relation.

LaTex2e, 30 pages, no figure (v2) an error in the beginning of Introduction corrected; (v3) typos and an error in eq. (73) corrected; (v4) section 6 revised; (v5) final version for publication; (v6) serious errors in published version (JMP vol. 44, 2003, pp. 3979-3999) corrected

References in corpus (5)

• Separation of Variables in the Elliptic Gaudin Model
• Elliptic Calogero-Moser Systems and Isomonodromic Deformations
• Schlesinger transformations for elliptic isomonodromic deformations
• Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus
• Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations

Cited by in corpus (3)

• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus