Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations

TL;DR

This paper explores the connection between classical conformal blocks, isomonodromic deformations, and Painlevé VI equations, providing new methods to compute 4-point conformal blocks in 2D CFT.

Contribution

It identifies classical conformal blocks with Painlevé VI action and introduces a novel approach to calculate 4-point blocks via the isomonodromic $ au$-function expansion.

Findings

Recovered accessory parameter expansion from isomonodromic $ au$-function.

Linked classical conformal blocks to Painlevé VI and isomonodromic deformations.

Proposed a new method for calculating 4-point classical conformal blocks.

Abstract

Classical conformal blocks naturally appear in the large central charge limit of 2D Virasoro conformal blocks. In the $AdS_{3}/CFT_{2}$ 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 Painlev\'e 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.