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Classical conformal blocks and Painlevé VI

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

Abstract

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.

Keywords

Integrable Hierarchies Integrable Field Theories Differential and Algebraic Geometry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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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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