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Fredholm determinant representations for isomonodromic tau functions I

Le : 27/05/2016 11h00
Par : Oleg Lisovyy (Laboratoire de Mathématiques et Physique Théorique, Tours)
Lieu : I001
Lien web :
Résumé : We will derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N\lb n-3\rb $ copies of $L^2\lb S^1\rb$. Its kernel is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier in the framework of two-dimensional conformal field theory. Further specialization to $n=4$ gives a series representation of the general Painlevé VI transcendent.