Quantum curves and q-deformed Painlevé equations

Abstract

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 $$\mathbb {P}^1\times \mathbb {P}^1$$ case, which is related to q-difference Painleve with affine $$A_1$$ symmetry, to SU(2) Super Yang–Mills in five dimensions and to relativistic Toda system.