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Published online by Cambridge University Press: 25 June 2025
The Hamiltonian approach to dual isomonodromic deformations in the setting of rational .R-matrix structures on loop algebras is reviewed. The construction of a particular class of solutions to the deformation equations, for which the isomonodromic r-functions are given by the Fredholm determinants of a special class of integrable integral operators, is shown to follow from the matrix Riemann-Hilbert approach of Its, Izergin, Korepin and Slavnov. This leads to an interpretation of the notion of duality in terms of the data defining the Riemann-Hilbert problem, and Laplace-Fourier transforms of the corresponding Fredholm integral operators.
1. Introduction la. Isomonodromic Deformation Equations. We consider rational covariant derivative operators on the punctured Riemann sphere, having the form They have regular singular points at and an irregular singularity at with Poincare index 1. If the residue matrices are deformed differentiably with respect to the parameters and the monodromy (including Stokes parameters and connection matrices) of the operator will be invariant under such deformations, as was shown in [Jimbo et al. 1980; Jimbo et al. 1981], provided the differential equations implied by the commutativity conditions.
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