Painlevé equations from Nakajima–Yoshioka blowup relations

  • M. BershteinEmail author
  • A. Shchechkin


Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal to the series of \(c=1\) Virasoro conformal blocks. We study similar series of \(c=-2\) conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima–Yoshioka blowup relations on Nekrasov partition functions. We also study series of q-deformed \(c=-2\) conformal blocks and relate it to q-Painlevé equation. As an application, we prove formula for the tau function of q-Painlevé \(A_7^{(1)'}\) equation.


Painleve equations Nekrasov partition functions Bilinear relations Conformal blocks Blowup relations q-difference equations 

Mathematics Subject Classification

81T60 34M55 39A13 14D21 



We thank B. Feigin, P. Gavrylenko, A. Grassi, A. Marshakov, A. Mironov, H. Nakajima for interest to our work and discussions. We are grateful to the referee for valuable comments. This work is partially supported by HSE University Basic Research Program and funded (partially) by the Russian Academic Excellence Project ‘5-100’ and by the RFBR Grant mol_a_ved 18-31-20062. Authors were also supported in part by Young Russian Mathematics award. The work of M.B. in Landau Institute has been funded by FANO assignment 0033-2018-0006.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Center for Advanced StudiesSkolkovo Institute of Science and TechnologyMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Institute for Information Transmission ProblemsMoscowRussia
  5. 5.Independent University of MoscowMoscowRussia

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