Combinatorics of pruned Hurwitz numbers

TL;DR

This paper develops the combinatorial theory of pruned Hurwitz numbers, a variant of classical Hurwitz numbers derived from topological recursion, providing new formulas and a tropical correspondence for their enumeration.

Contribution

It introduces two new combinatorial formulas for pruned Hurwitz numbers and establishes a tropical correspondence theorem for their enumeration.

Findings

Pruned Hurwitz numbers can be expressed via Hurwitz mobiles.

A tropical correspondence theorem for pruned Hurwitz numbers is proved.

Pruned Hurwitz numbers are smaller but retain maximal information.

Abstract

Hurwitz numbers enumerate branched morphisms between Riemannn surfaces with fixed numerical data. They represent important objects in enumerative geometry that are accessible by combinatorial techniques. In the past decade, many variants of Hurwitz numbers have appeared in the literature. In this paper, we focus on an exciting such variant that arises naturally from the theory of topological recursion: Pruned Hurwitz numbers. These are defined as an enumeration of a relevant subset of branched morphisms between Riemann surfaces, that yield smaller numbers than their classical counterparts while retaining maximal information. Thus, pruned Hurwitz numbers may be viewed as the core of the Hurwitz problem. In this paper, we develop the combinatorial theory of pruned Hurwitz numbers. In particular, motivated by the successful application of combinatorial techniques to classical Hurwitz numbers, we derive two new combinatorial expressions of pruned Hurwitz numbers. Firstly, we show that they may be expressed in terms of Hurwitz mobiles which are tree-like structure that arise from the theory of random planar maps. Secondly, we prove a tropical correspondence theorem which allows the enumeration of pruned Hurwitz numbers in terms of tropical covers.