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Statut |
Confirmé |
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Série |
WORK-CONF |
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Domaines |
math-ph |
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Date |
Jeudi 17 Août 2017 |
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Heure |
09:30 |
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Institut |
LPTENS |
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Salle |
Room Conf. IV |
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Nom de l'orateur |
Lisovyi |
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Prenom de l'orateur |
Oleg |
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Addresse email de l'orateur |
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Institution de l'orateur |
Université de Tours |
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Titre |
Painlevé functions, Fredholm determinants and combinatorics |
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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. |
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Numéro de preprint arXiv |
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Commentaires |
Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage
https://indico.in2p3.fr/event/14720/ |
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Fichiers attachés |
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