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

TL;DR

This paper demonstrates how topological recursion can be used to compute the WKB expansion of solutions to quantum curves derived from hyper-elliptic curves, linking spectral geometry, isomonodromic systems, and tau functions.

Contribution

It establishes a method to quantize hyper-elliptic curves using topological recursion and relates it to moduli spaces of meromorphic connections and tau functions.

Findings

Topological recursion computes WKB expansions for quantum hyper-elliptic curves.

Quantum curves are expressed via spectral Darboux coordinates on moduli spaces.

Application to Painlevé equations and higher genus examples.

Abstract

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