The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures

TL;DR

This paper derives spectral curves for various Hurwitz numbers, quantizes them to obtain Schroedinger equations, and confirms the conjecture of quantum curves in these generalized cases.

Contribution

It introduces a unified approach to derive and quantize spectral curves for multiple Hurwitz number types, confirming the quantum curve conjecture.

Findings

Spectral curves for double and simple Hurwitz numbers are derived.

Quantization leads to Schroedinger equations for the partition functions.

The existence of quantum curves in these cases is confirmed.

Abstract

We derive the spectral curves for $q$-part double Hurwitz numbers, $r$-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0,1)-geometry. We quantize this family of spectral curves and obtain the Schroedinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.