Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: The sl2 case

Abstract

In this paper, we show that it is always possible to deform a differential equation ∂xΨ(x) = L(x)Ψ(x) with L(x)∈sl2(C)(x) by introducing a small formal parameter ℏ in such a way that it satisfies the topological type properties of Bergère, Borot, and Eynard [Annales Henri Poincaré 16(12), 2713–2782 (2015)]. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce ℏ. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expressed in terms of intersections of tautological classes in the Deligne–Mumford compactification of the moduli space of surfaces. We present a few examples including any Fuchsian system of sl2(C)(x) as well as some elements of Painlevé hierarchies.