Classical conformal blocks and Painlevé VI

2013/09/30 by Alexey Litvinov, Sergei Lukyanov, Sergei L. Lukyanov +3 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Black Holes and Theoretical Physics #Conformal map #Connection (principal bundle) #Decoupling (probability) #Geometry #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Monodromy #Nonlinear Waves and Solitons #Pure mathematics #hep-th #math-ph #math.MP

paper · pdf · doi:10.1007/jhep07(2014)144

19 pages, 5 figures; references added

arxiv created 2013/10/13 · openalex publication_date 2014/07/01 · arxiv updated 2015/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations
• Small Instantons, Little Strings and Free Fermions
• Neue Beitr�ge zur Riemann'schen Functionentheorie
• Infinite conformal symmetry in two-dimensional quantum field theory
• Conformal symmetry in two-dimensional space: Recursion representation of conformal block
• Liouville Correlation Functions from Four-dimensional Gauge Theories

Cited by

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• BPS relations from spectral problems and blowup equations
• Scalar quasinormal modes of Kerr−AdS5
• Many-point classical conformal blocks and geodesic networks on the hyperbolic plane