Conformal field theory of Painlev\'e VI

TL;DR

This paper establishes a connection between Painlevé VI tau functions and 2D conformal field theory, providing explicit expansions and exploring special solutions through CFT techniques.

Contribution

It introduces a conformal field theory framework for Painlevé VI, utilizing AGT representation to derive explicit tau function expansions and analyze special solutions.

Findings

Explicit expansion of tau(t) near singular points

Identification of conformal blocks from special Painlevé VI solutions

Validation of the CFT approach through known solutions

Abstract

Generic Painlev\'e VI tau function \tau(t) can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c=1. Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely explicit expansion of \tau(t) near the singular points. After a check of this expansion, we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic solutions of Painlev\'e VI.