Geodesic
The Malgrange Form and Fredholm Determinants
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Malgrange form; Fredholm determinants; tau function. 
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     author = {Marco Bertola},
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Marco Bertola. The Malgrange Form and Fredholm Determinants. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a45/

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