The Malgrange form and Fredholm determinants

Marco Bertola

Published 2017-02-25Version 1

We consider the factorization problem of matrix symbols depending analytically on parameters on a closed contour (i.e. a Riemann--Hilbert problem). 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.