From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

TL;DR

This paper demonstrates that wave functions derived from topological recursion on hyperelliptic spectral curves satisfy PDEs that serve as quantizations of the curves, linking topological recursion with quantum curves and isomonodromic systems.

Contribution

It extends the connection between topological recursion and quantum curves from genus zero to hyperelliptic curves, providing PDEs that quantize spectral curves.

Findings

Wave functions satisfy PDEs as quantizations of spectral curves.

Topological recursion wave functions coincide with WKB solutions of isomonodromic systems.

Generalization of quantum curve construction to hyperelliptic curves.

Abstract

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes this construction to hyperelliptic curves.