The Malgrange Form and Fredholm Determinants

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.