Bilinear Equations on Painlevé τ Functions from CFT

TLDR: In 2012, Gamayun, Iorgov, and Lisovyy conjectured an explicit expression for the Painleve VI τ function in terms of the Liouville conformal blocks with central charge c = 1.

Abstract: In 2012, Gamayun, Iorgov, and Lisovyy conjectured an explicit expression for the Painleve VI τ function in terms of the Liouville conformal blocks with central charge c = 1. We prove that the proposed expression satisfies Painleve VI τ 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).