Status  Confirmed 
Seminar Series  WORKCONF 
Subjects  mathph 
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 3point Fuchsian systems whose monodromy is determined by the monodromy of the associated linear problem via a decomposition of the 4punctured 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 selfdual $\Omega$background. 
arXiv Preprint Number  
Comments  Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage https://indico.in2p3.fr/event/14720/ 
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