# The Malgrange Form and Fredholm Determinants

## TL;DR

This paper links the Malgrange form to Fredholm determinants in the context of Riemann-Hilbert problems, defining a tau function that varies analytically with parameters and is related to integrable operators.

## Contribution

It introduces a novel construction of the tau function as a Fredholm determinant for parameter-dependent Riemann-Hilbert problems, emphasizing its geometric interpretation.

## Key findings

- Tau function is locally analytic on deformation space.
- Tau function expressed as a Fredholm determinant of integrable operators.
- Construction reveals the tau function as a section of a line bundle.

## 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.