[Paper Review] Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants

Reinier Kramer, Danilo Lewański+1 more|arXiv (Cornell University)|Oct 26, 2016

Advanced Combinatorial Mathematics22 references12 citations

TL;DR

This paper proves quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers using the semi-infinite wedge formalism and A-operator techniques, confirming conjectures by Do-Karev and Do-Manescu. It establishes that these numbers are polynomials in the partition parameters multiplied by combinatorial factors, and shows this property is equivalent to the existence of a spectral curve representation underlying Chekhov-Eynard-Orantin topological recursion, with explicit verification for (g,n) = (0,1) and (0,2) cases.

ABSTRACT

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are necessary conditions for the Chekhov-Eynard-Orantin topological recursion.

Motivation & Objective

• To prove the conjectured quasi-polynomiality of monotone and strictly monotone orbifold Hurwitz numbers, which were previously unproven.
• To establish a connection between quasi-polynomiality and the Chekhov-Eynard-Orantin topological recursion via spectral curve representations.
• To provide a new combinatorial proof of quasi-polynomiality for usual orbifold Hurwitz numbers using A-operators in the semi-infinite wedge formalism.
• To verify the unstable cases (g,n) = (0,1) and (0,2) for the monotone case, completing the spectral curve data required for topological recursion.

Proposed method

• Uses the semi-infinite wedge formalism to define Hurwitz numbers as vacuum expectations, serving as the foundation for all computations.
• Introduces and manipulates A-operators analogous to Okounkov-Pandharipande’s, adapted to monotone and strictly monotone cases.
• Applies conjugation techniques to transform operators into a form suitable for extracting polynomial dependence on partition parameters.
• Employs the Euler operator in z-coordinates to derive differential equations for correlation functions, which are then expanded in x-variables.
• Performs explicit combinatorial calculations for (g,n) = (0,1) and (0,2) using generating functions and inclusion-exclusion principles.
• Verifies that the (0,1) and (0,2) generating functions match the expected forms ydx and B(z1,z2)−B(x1,x2), respectively, confirming spectral curve compatibility.

Experimental results

Research questions

• RQ1 Does the monotone orbifold Hurwitz number sequence exhibit quasi-polynomiality, as conjectured by Do and Karev?
• RQ2 Is there a spectral curve and topological recursion structure underlying monotone orbifold Hurwitz numbers, and how does it relate to quasi-polynomiality?
• RQ3 Can the quasi-polynomiality of strictly monotone orbifold Hurwitz numbers be rigorously proven, given their equivalence to Grothendieck’s dessins d’enfants?
• RQ4 What is the precise relationship between the generating functions of (0,1) and (0,2) correlation functions and the spectral curve data required for topological recursion?
• RQ5 Does the quasi-polynomiality of these Hurwitz numbers imply the existence of a symmetric n-differential on the spectral curve, and vice versa?

Key findings

• The paper proves that monotone orbifold Hurwitz numbers are quasi-polynomial: h◦,r,≤g;⃗µ = P⟨⃗µ⟩≤(μ1,…,μn) · ∏i (μi + [μi] choose μi), confirming a conjecture by Do and Karev.
• For strictly monotone orbifold Hurwitz numbers, the same quasi-polynomial structure holds: h◦,r,<g;⃗µ = P⟨⃗µ⟩<(μ1,…,μn) · ∏i (μi−1 choose [μi]), resolving a conjecture by Do and Manescu.
• The authors provide a new combinatorial proof of quasi-polynomiality for usual orbifold Hurwitz numbers, using A-operators and conjugation techniques.
• The (0,1) generating function for monotone Hurwitz numbers is shown to be equal to the expansion of ydx on the spectral curve x = z(1−zr), completing the spectral data.
• The (0,2) generating function for monotone Hurwitz numbers is proven to match the expansion of B(z1,z2)−B(x1,x2), confirming the spectral curve structure for the unstable case.
• The paper establishes that quasi-polynomiality is equivalent to the existence of a symmetric n-differential on the spectral curve x = z(1−zr), linking it directly to the Chekhov-Eynard-Orantin topological recursion.