Mathematics

We propose q-deformation of the Gamayun-Iorgov-Lisovyy formula for Painleve tau function. Namely we propose formula for tau function for q-difference Painleve equation corresponding to A^(1)_7 surface (and A^(1)_1 symmetry) in Sakai classification. In this formula tau function equals the series of q-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure SU(2) 5d theory). Tau functions of q-difference Painleve equation should be consistent with the action of symmetry group of the equation. For our formula this reduces to (conjectural) bilinear relations on Nekrasov partition functions and fiber-base duality of topological vertex