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We prove that the topological recursion formalism can be used to compute theWKB expansion of solutions of second order differential operators obtained byquantization of any hyper-elliptic curve. We express this quantum curve interms of spectral Darboux coordinates on the moduli space of meromorphic$\mathfrak{sl}_2$-connections on $\mathbb{P}^1$ and argue that the topologicalrecursion produces a $2g$-parameter family of associated tau functions, where$2g$ is the dimension of the moduli space considered. We apply this procedureto the 6 Painlev\'e equations which correspond to $g=1$ and consider a $g=2$example.

Quantization of hyper-elliptic curves from isomonodromic systems and topological recursion