In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlevé VI τ function in terms of the Liouville conformal blocks with central charge c = 1. We prove that proposed expression satisfies Painlevé 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 C 2 /Z 2 (similarly to Nakajima-Yoshioka blow-up equations).
We propose q-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé τ function. Namely we propose the formula for τ function for q-difference Painlevé equation corresponding to A(1)′ 7 surface (and A (1) 1 symmetry) in Sakai classification. In this formula τ function equals the series of q-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure SU (2) 5d theory). [arXiv:q-alg/9605002v2].
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