### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 046, 12 pages      arXiv:1703.00046      https://doi.org/10.3842/SIGMA.2017.046

### The Malgrange Form and Fredholm Determinants

Marco Bertola ab
a) Department of Mathematics and Statistics, Concordia University, Montréal, Canada
b) Area of Mathematics SISSA/ISAS, Trieste, Italy

Received March 12, 2017, in final form June 17, 2017; Published online June 22, 2017

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.

Key words: Malgrange form; Fredholm determinants; tau function.

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