Conformal field theory of Painlevé VI

TL;DR

This work establishes a bridge between Painlevé VI isomonodromic tau functions and $c=1$ conformal field theory by interpreting $\tau(t)$ as a four-point correlator of monodromy fields with dimensions $\Delta_\nu=\theta_\nu^2$. It leverages the AGT representation to produce a complete, explicit near-singularity expansion: a sum over all monodromy channels labeled by $n\in\mathbb{Z}$, each weighted by a conformal block $\mathcal{B}(\boldsymbol{\theta},\sigma_{0t}+n;t)$ and a structure constant $C_n(\boldsymbol{\theta},\boldsymbol{\sigma})$ expressed via Barnes $G$-functions, with a braid-phase factor ensuring consistency under analytic continuation. The authors provide a concrete recurrence for the coefficients, verify the expansion up to high levels, and illustrate how several classical PVI solutions (Riccati, Chazy, algebraic, Picard) arise as simplified blocks within the general framework, including explicit formulas in special cases. The results illuminate a deep link between isomonodromic deformation theory and Virasoro representation theory, opening avenues for applying AGT techniques to monodromy problems and suggesting extensions to more points, higher rank, and higher genus.

Abstract

Generic Painlevé VI tau function τ(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 τ(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 Painlevé VI.

Figures (1)

• Figure :

Theorems & Definitions (8)

• Claim
• Proposition 1
• Remark 2
• Remark 3
• Remark 4
• Example 5
• Example 6
• Example 7