Integrable differential systems of topological type and reconstruction by the topological recursion

Abstract

Starting from a $d\times d$ rational Lax pair system of the form $\hslash {\partial }_x\Psi =L\Psi$ and $\hslash {\partial }_t\Psi =R\Psi$ 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 $\hslash$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painlevé systems.

arXiv commentPublished version in Annales Henri Poincaré

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Cited by in corpus (8)

• Weighted Hurwitz numbers and topological recursion: an overview
• Weighted Hurwitz numbers and topological recursion
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• From CFT to Ramond super-quantum curves
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• WKB solutions of difference equations and reconstruction by the topological recursion
• On tau-functions for the KdV hierarchy
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers