Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers

TL;DR

This paper extends topological recursion to transalgebraic spectral curves with exponential singularities, enabling the computation of Atlantes Hurwitz numbers and establishing a quantum curve correspondence for this broader class.

Contribution

It introduces a new framework for topological recursion on transalgebraic spectral curves, including exponential singularities, and applies it to Atlantes Hurwitz numbers.

Findings

Atlantes Hurwitz numbers satisfy the extended topological recursion.

A quantum curve is constructed directly from the extended recursion.

The framework is compatible with limits of spectral curves.

Abstract

Given a spectral curve with exponential singularities (which we call a "transalgebraic spectral curve"), we extend the definition of topological recursion to include contributions from the exponential singularities in a way that is compatible with limits of sequences of spectral curves. This allows us to prove the topological recursion/quantum curve correspondence for a large class of transalgebraic spectral curves. As an application, we find that Atlantes Hurwitz numbers, which were previously thought to fall outside the scope of topological recursion, satisfy (our extended version of) topological recursion, and we construct the corresponding quantum curve directly from topological recursion.