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

## TL;DR

This paper demonstrates how to deform $rak{sl}_2$ rational differential systems with a formal parameter to satisfy topological recursion properties, linking tau-functions to intersection theory on moduli spaces.

## Contribution

It introduces a method to incorporate a formal parameter into $rak{sl}_2$ systems, enabling topological recursion and tau-function expansion in terms of moduli space intersections.

## Key findings

- Deformation of $rak{sl}_2$ systems with $ o 0$ parameter satisfying topological properties.
- Tau-functions expanded via topological recursion linked to intersection theory.
- Application to Fuchsian systems and Painlevé hierarchies.

## Abstract

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 Painlev\'e hierarchies.