The Malgrange Form and Fredholm Determinants

Abstract

We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function $\tau$ which is locally analytic on the space of deformations and that is expressed as a Fredholm determinant of an operator of "integrable" type in the sense of Its-Izergin-Korepin-Slavnov. The construction is not unique and the non-uniqueness highlights the fact that the tau function is really the section of a line bundle.

References in corpus (2)

• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Monodromy dependence and connection formulae for isomonodromic tau functions

Cited by in corpus (2)

• Fredholm determinant representation of the Painlevé II $\tau$-function
• Painlevé/CFT correspondence on a torus