Classical conformal blocks and Painlevé VI

  • Alexey Litvinov
  • Sergei LukyanovEmail author
  • Nikita Nekrasov
  • Alexander Zamolodchikov
Open Access


We study the classical c → ∞ limit of the Virasoro conformal blocks. We point out that the classical limit of the simplest nontrivial null-vector decoupling equation on a sphere leads to the Painlevé VI equation. This gives the explicit representation of generic four-point classical conformal block in terms of the regularized action evaluated on certain solution of the Painlevé VI equation. As a simple consequence, the monodromy problem of the Heun equation is related to the connection problem for the Painlevé VI.


Integrable Hierarchies Integrable Field Theories Differential and Algebraic Geometry 


Open Access

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Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Alexey Litvinov
    • 1
  • Sergei Lukyanov
    • 1
    Email author
  • Nikita Nekrasov
    • 2
    • 3
    • 4
    • 5
  • Alexander Zamolodchikov
    • 1
    • 4
  1. 1.NHETC, Department of Physics and AstronomyRutgers UniversityPiscatawayU.S.A.
  2. 2.Simons Center for Geometry and PhysicsStony BrookU.S.A.
  3. 3.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  4. 4.Kharkevich Institute for Information Transmission ProblemsMoscowRussia
  5. 5.Alikhanov Institute of Theoretical and Experimental PhysicsMoscowRussia

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