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

Abstract

Hurwitz numbers count genus g, degree d covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introduced as mixed double Hurwitz numbers and it was proved that these objects are piecewise polynomial in a certain sense. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d'enfant count. In this paper, we introduce a combinatorial interpolation between simple, monotone and strictly monotone double Hurwitz numbers as triply interpolated Hurwitz numbers. Our aim is twofold: Using a connection between triply interpolated Hurwitz numbers and tropical covers in terms of so-called monodromy graphs, we give algorithms to compute the polynomials for triply interpolated Hurwitz numbers in all genera using Erhart theory. We further use this approach to study the wall-crossing behaviour of triply interpolated Hurwitz numbers in genus 0 in terms of related Hurwitz-type counts. All those results specialise to the extremal cases of simple, monotone and Grothendieck dessins d'enfants Hurwitz numbers.

Tropical Jucys covers

“…In this section, we compare the tropical curves obtained in Theorem 4.1 to the ones derived in [12,13] by means of a particular example. Both these papers use the notion of monotone monodromy graphs associated to the data (g, μ, ν) (for a formal definition of such graphs we refer to the original papers).…”

“…Fig. 1 Comparing the tropical curves appearing in [12,13] (top) to the tropical curves appearing in Theorem 4.1 (bottom) for (g, μ, ν) = (0, (2, 1), (3))…”

“…These numbers have been intensively studied in terms of operators acting on the Fock space. Furthermore, special cases of these enumerations may be expressed by means of tropical geometry as well [12,13]. These expressions however differ fundamentally from the tropical covers involved in [10].…”

“…In fact we derive a new tropical expression for monotone and strictly monotone Hurwitz numbers that shares several features with the results of [10], as the tropical covers are once again weighted by local multiplicities given by Gromov-Witten invariants. This new interpretation has the advantage that the curves involved carry less non-geometric information than the ones in [12,13] and are closer to the tropical curves involved in [4,9]. This new interpretation allows applications in sequel work related to wall-crossing behaviour [15] and mirror symmetry [16].…”

Triply mixed coverings of arbitrary base curves: quasimodularity, quantum curves and a mysterious topological recursion

“…So, far, these have been studied for target surfaces of genus 0. Two of the most important cases are monotone and strictly monotone Hurwitz numbers, the latter of which are also called Grothendieck dessins d'enfants Hurwitz numbers [11,13,[21][22][23][24]32]. Monotone Hurwitz numbers appear as coefficients in the expansion of the HCIZ integral in random matrix theory [21], while strictly monotone Hurwitz numbers are equivalent to counting certain Grothendieck dessins d'enfants [1,32].…”