Classical conformal blocks and Painlevé VI

Alexey Litvinov, Sergei Lukyanov, Nikita Nekrasov, Alexander Zamolodchikov

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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.

Original languageEnglish (US)
Article number144
JournalJournal of High Energy Physics
Issue number7
StatePublished - Jul 2014

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


  • Differential and Algebraic Geometry
  • Integrable Field Theories
  • Integrable Hierarchies

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