Statut Confirmé Série WORK-CONF Domaines math-ph Date Jeudi 17 Août 2017 Heure 09:30 Institut LPTENS Salle Room Conf. IV Nom de l'orateur Lisovyi Prenom de l'orateur Oleg Addresse email de l'orateur Institution de l'orateur Université de Tours Titre Painlevé functions, Fredholm determinants and combinatorics Résumé I am going to explain the explicit construction of general solutions to isomonodromy equations, with the main focus on the Painlevé VI equation. I will start by deriving a Fredholm déterminant representation of the Painlevé VI tau function. The corresponding integral operator acts in the direct sum of two copies of $L^2(S^1)$. Its kernel is expressed in terms of hypergeometric fundamental solutions of two auxiliary 3-point Fuchsian systems whose monodromy is determined by the monodromy of the associated linear problem via a decomposition of the 4-punctured sphere into two pairs of pants. In the Fourier basis, this kernel is given by an infinite Cauchy matrix. I will explain how the principal minor expansion of the Fredholm determinant yields a combinatorial series representation for the general solution to Painlevé VI in the form of a sum over pairs of Young diagrams. The latter series coincides with the dual Nekrasov partition function of the $\mathcal N=2$ $N_f=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background. Numéro de preprint arXiv Commentaires Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage https://indico.in2p3.fr/event/14720/ Fichiers attachés

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