Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

Abstract : We 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 $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and 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 via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlev\'e VI equation.
Type de document :
Article dans une revue
Commun.Math.Phys., 2018, 363, pp.1-58. 〈10.1007/s00220-018-3224-7〉
Liste complète des métadonnées
Contributeur : Inspire Hep <>
Soumis le : mercredi 31 octobre 2018 - 10:44:39
Dernière modification le : mercredi 20 mars 2019 - 11:07:18

Lien texte intégral




P. Gavrylenko, O. Lisovyy. Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions. Commun.Math.Phys., 2018, 363, pp.1-58. 〈10.1007/s00220-018-3224-7〉. 〈hal-01909454〉



Consultations de la notice