Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the $mathfrak{sl}_2$ case

In this paper, we show that it is always possible to deform a differential equation $partial_x Psi(x) = L(x) Psi(x)$ with $L(x) in mathfrak{sl}_2(mathbb{C})(x)$ by introducing a small formal parameter $hbar$ in such a way that it satisfies the Topological Type properties of Berg`ere, Borot and Eynard. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce $hbar$. 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 $mathfrak{sl}_2(mathbb{C})(x)$ as well as some elements of Painleve hierarchies.