Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion

Starting from a d×d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\times d$$\end{document} rational Lax pair system of the form ħ∂xΨ=LΨ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \partial _x \Psi = L\Psi $$\end{document} and ħ∂tΨ=RΨ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \partial _t \Psi =R\Psi $$\end{document}, we prove that, under certain assumptions (genus 0 spectral curve and additional conditions on R and L), the system satisfies the “topological type property.” A consequence is that the formal ħ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar $$\end{document}-WKB expansion of its determinantal correlators satisfies the topological recursion. This applies in particular to all (p, q) minimal models reductions of the KP hierarchy, or to the six Painlevé systems.