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 $\partial_x Ψ(x) = L(x) Ψ(x)$ 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 Bergère, 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 Painlevé hierarchies.

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