Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations

Abstract

The Hamiltonian approach to the theory of dual isomonodromic deformations is developed within the framework of rational classical R-matrix structures on loop algebras. Particular solutions to the isomonodromic deformation equations appearing in the computation of correlation functions in integrable quantum field theory models are constructed through the Riemann-Hilbert problem method. The corresponding $τ$-functions are shown to be given by the Fredholm determinant of a special class of integral operators.