Bilinear equations on Painleve tau functions from CFT

TL;DR

This paper proves a conjecture linking Painleve VI tau functions with Liouville conformal blocks, using bilinear relations and algebra embeddings, and interprets these relations through instanton counting in gauge theory.

Contribution

It establishes the validity of a conjectured explicit expression for Painleve VI tau functions in terms of conformal blocks, confirming a deep connection between integrable systems and conformal field theory.

Findings

Proved the bilinear equations for Painleve VI tau functions

Validated the explicit conformal block expression for tau functions

Connected bilinear relations to instanton counting via AGT correspondence

Abstract

In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlev\'e VI $\tau$~function in terms of the Liouville conformal blocks with central charge $c=1$. We prove that proposed expression satisfies Painlev\'e VI $\tau$~function bilinear equations (and therefore prove the conjecture). The proof reduces to the proof of bilinear relations on conformal blocks. These relations were studied using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra. In the framework of the AGT correspondence the bilinear equations on the conformal blocks can be interpreted in terms of instanton counting on the minimal resolution of $\mathbb{C}^2/\mathbb{Z}_2$ (similarly to Nakajima-Yoshioka blow-up equations).