Tropical Hurwitz numbers

Abstract

Hurwitz numbers count genus g, degree d covers of P 1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of Goulden et al. (in Adv. Math. 198:43-92, 2005) and Shadrin et al. (in Adv. Math. 217(1):79-96, 2008) on the piecewise polynomial structure of double Hurwitz numbers in genus 0.

A Monodromy Graph Approach to the Piecewise Polynomiality of Simple, Monotone and Grothendieck Dessins d’enfants Double Hurwitz Numbers

“…· |{σ ∈ S d : C(σ) = µ}| to obtain H ≤,<,(2) p,q,r;µ,ν . We can thus simplify this counting problem with a smart choice of σ 1 (see (crefequ:per), We translate the counting problem to a problem of counting monodromy graphs as in [CJM10], [CJM11] and [DK16]. In the latter, the choice of σ 1 as in Eq.…”

“…In [CJM11], these formulas were proved using so-called monodromy graphs which essentially express double Hurwitz numbers in terms of covers as they appear in tropical geometry. This description was developed in [CJM10] by giving a graph theoretic interpretation of factorisations in the symmetric group. There are several variants on the definition of Hurwitz numbers yielding so-called Hurwitz-type counts.…”

“…They are defined in the symmetric group setting and show up in the computation of the HCIZ integral. A tropical interpretation of certain monotone Hurwitz numbers in the flavour of [CJM10] was developed in [DK16], where a conjecture on the topological recursion for single monotone Hurwitz numbers was stated (for further literatue on monotone Hurwitz numbers and topological recursion, see e.g. [ALS16], [DDM14]).…”

The Double Gromov–Witten Invariants of Hirzebruch Surfaces are Piecewise Polynomial

“…In particular, Goulden, Jackson, and Vakil proved, among other things, that double Hurwitz numbers are piecewise polynomial [GJV05]. Later on, Cavalieri, Johnson, and Markwig used tropical geometry to give a new proof of this piecewise polynomiality [CJM10,CJM11]. In addition, they found a wall crossing formula giving the difference of double Hurwitz numbers between two adjacent chambers of polynomiality, generalizing the formula in genus 0 proved in [SSV08].…”

A graphical interface for the Gromov-Witten theory of curves

“…In [35], Okounkov and Pandharipande develop the GW/H correspondence, a substitution rule that relates Hurwitz numbers and stationary descendant Gromov-Witten invariants of P 1 . Tropical Hurwitz numbers are realized as weighted sums over appropriately decorated graphs: they were first introduced in particular cases in [10,11]; the general definition and correspondence theorem are due to Bertrand, Brugallé and Mikhalkin in [5].…”