Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations

Abstract

Classical conformal blocks naturally appear in the large central charge limit of 2D Virasoro conformal blocks. In the $Ad{S}_3/CF{T}_2$ correspondence, they are related to classical bulk actions and are used to calculate entanglement entropy and geodesic lengths. In this work, we discuss the identification of classical conformal blocks and the Painlevé VI action showing how isomonodromic deformations naturally appear in this context. We recover the accessory parameter expansion of Heun's equation from the isomonodromic $\tau$-function. We also discuss how the $c=1$ expansion of the $\tau$-function leads to a novel approach to calculate the 4-point classical conformal block.

32+10 pages, 2 figures; v3: upgraded notation, discussion on moduli space and monodromies, numerical and analytic checks; v2: added refs, fixed email

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Cited by in corpus (14)

• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Perturbative connection formulas for Heun equations
• Kerr-de Sitter Quasinormal Modes via Accessory Parameter Expansion
• A slow review of the AGT correspondence
• Scalar quasinormal modes of Kerr-AdS${}_5$
• Riemann-Hilbert correspondence and blown up surface defects
• Accessory parameters in confluent Heun equations and classical irregular conformal blocks
• Quantum spectral problems and isomonodromic deformations
• Large-c superconformal torus blocks
• Resurgence, Painleve Equations and Conformal Blocks
• Higher rank isomonodromic deformations and W-algebras
• Expansions for semiclassical conformal blocks
• More on the SW-QNM correspondence
• Painlevé Kernels and Surface Defects at Strong Coupling