Monotone orbifold Hurwitz numbers

TL;DR

This paper introduces monotone orbifold Hurwitz numbers, extending existing theories by deriving recursive formulas, identifying a governing quantum curve, and proposing a conjecture linking them to topological recursion.

Contribution

It initiates the study of monotone orbifold Hurwitz numbers, deriving a cut-and-join recursion, identifying a quantum curve, and conjecturing their relation to topological recursion.

Findings

Derived a cut-and-join recursion for monotone orbifold Hurwitz numbers.

Determined a quantum curve that governs their wave function.

Proposed a conjecture linking these numbers to topological recursion.

Abstract

In general, Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. In this paper, we initiate the study of monotone orbifold Hurwitz numbers. These are simultaneously variations of the orbifold case and generalisations of the monotone case, both of which have been previously studied in the literature. We derive a cut-and-join recursion for monotone orbifold Hurwitz numbers, determine a quantum curve governing their wave function, and state an explicit conjecture relating them to topological recursion.