Quantum curves and q-deformed Painleve equations

We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve, these are conjectured to be the q-difference Painleve equations as in Sakai's classification. More precisely, we propose that the tau functions of q-Painleve equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases, we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau functions in terms of the underlying quantum mirror curve. As a consequence, the zeroes of the tau functions compute the exact spectrum of the associated quantum integrable systems. We provide details of this construction for the local P1 x P1 case, which is related to q-difference Painleve with affine A1 symmetry, to SU( 2) Super Yang-Mills in five dimensions and to relativistic Toda system.