Integrable Fredholm Operators and Dual Isomonodromic Deformations

Abstract:

The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators \(\), having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators \(\), having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the rôles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators \(\) are analogously associated to an integral operator \(\) whose Fredholm determinant is equal to that of K.