Mirror symmetry for orbifold Hurwitz numbers

TL;DR

This paper explores the mirror symmetry of orbifold Hurwitz numbers, demonstrating their recursive structure, connection to the Eynard-Orantin theory, and the existence of a quantum curve, linking enumerative geometry and mathematical physics.

Contribution

It establishes the differential recursion for orbifold Hurwitz numbers and proves its equivalence to the Eynard-Orantin integral recursion with the r-Lambert spectral curve.

Findings

Laplace transform of orbifold Hurwitz numbers satisfies a differential recursion

Differential recursion is equivalent to Eynard-Orantin's integral recursion with r-Lambert curve

Mirror model admits a quantum curve

Abstract

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of [C3/(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.