Dual Isomonodromic Deformations and Moment Maps to Loop Algebras

TL;DR

This paper explains the Hamiltonian structure of isomonodromic deformations using moment maps to loop algebras, revealing dual pairs of matrix differential operators and providing new examples related to Painleve equations and Bose gas correlations.

Contribution

It introduces a novel framework connecting isomonodromic deformations with moment maps to loop algebras, leading to dual pairs of differential operators and new integrable systems insights.

Findings

Hamiltonian structure of monodromy preserving equations explained via moment maps.

Construction of dual pairs of matrix differential operators with preserved monodromy.

New examples including generalizations of Painleve equations and Bose gas correlation functions.

Abstract

The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo {\it et al } is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to ``dual'' pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendents $P_{V}$ and $P_{VI}$.