Status Confirmed Seminar Series WORK-CONF Subjects math-ph Date Thursday 17 August 2017 Time 09:30 Institute LPTENS Seminar Room Room Conf. IV Speaker's Last Name Lisovyi Speaker's First Name Oleg Speaker's Email Address Speaker's Institution Université de Tours Title Painlevé functions, Fredholm determinants and combinatorics Abstract 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. arXiv Preprint Number Comments Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage https://indico.in2p3.fr/event/14720/ Attachments

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