Quantum curves and q-deformed Painlevé 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 Painlevé equations as in Sakai’s classification. More precisely, we propose that the tau functions of q-Painlevé 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×P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^1\times \mathbb {P}^1$$\end{document} case, which is related to q-difference Painlevé with affine A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_1$$\end{document} symmetry, to SU(2) Super Yang–Mills in five dimensions and to relativistic Toda system.