Chamber Structure of Double Hurwitz numbers

TL;DR

This paper extends the understanding of double Hurwitz numbers by analyzing their chamber structure and wall crossing behavior across all genera, using graph-based and hyperplane arrangement techniques.

Contribution

It generalizes previous genus-zero results to arbitrary genus, providing a unified approach and new insights into the chamber structure of double Hurwitz numbers.

Findings

Extended wall crossing formulas to all genera.

Identified graph labels with lattice points in hyperplane chambers.

Connected wall crossing behavior to Varchenko's work.

Abstract

Double Hurwitz numbers count covers of the projective line by genus g curves with assigned ramification profiles over 0 and infinity, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0. This paper gives a unified approach to these results and strengthens them in several ways -- the most important being the extension of the results of Shapiro, Shadrin and Vainshtein to arbitrary genus. The main tool is the authors' previous work expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which are well known to give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko, and could have broader applications.