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Communications in Mathematical Physics

, Volume 363, Issue 1, pp 1–58 | Cite as

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

  • P. Gavrylenko
  • O. LisovyyEmail author
Article

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 GL \({(N,\mathbb{C})}\). The corresponding operator acts in the direct sum of N (n − 3) copies of L2 (S1). 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 Painlevé VI equation.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussian Federation
  2. 2.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  3. 3.Center for Advanced StudiesSkolkovo Institute of Science and TechnologyMoscowRussian Federation
  4. 4.Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350Université de ToursToursFrance

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