Communications in Mathematical Physics

, Volume 365, Issue 2, pp 741–772 | Cite as

Tau Functions as Widom Constants

  • M. Cafasso
  • P. Gavrylenko
  • O. LisovyyEmail author


We define a tau function for a generic Riemann–Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann–Hilbert problem. In the case of one circle, the tau function coincides with Widom’s determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo–Miwa–Ueno tau function for Riemann–Hilbert problems of isomonodromic origin (Painlevé VI, V, III, Garnier system, etc) and the Sato–Segal–Wilson tau function for integrable hierarchies such as Gelfand–Dickey and Drinfeld–Sokolov.


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The authors would like to thank M. Bertola, T. Grava, Y. Haraoka, N. Iorgov, A. Its, K. Iwaki, H. Nagoya, A. Prokhorov, and V. Roubtsov for useful discussions. The present work was supported by the PHC Sakura Grant No. 36175WA and CNRS/PICS project, “Isomonodromic deformations and conformal field theory”. The work of P.G. was partially supported the Russian Academic Excellence Project ‘5-100’ and by the RSF Grant No. 16-11-10160. In particular, results of Subsection 3.1 have been obtained using support of Russian Science Foundation. P.G. is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. M.C. acknowledges the support of the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant No. 778010.


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Copyright information

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

Authors and Affiliations

  1. 1.LAREMAUniversité d’AngersAngersFrance
  2. 2.Center for Advanced StudiesSkolkovo Institute of Science and TechnologyMoscowRussia
  3. 3.Department of Mathematics and International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussia
  4. 4.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  5. 5.Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350Université de ToursToursFrance

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