Chamber Structure of Double Hurwitz numbers

Understand

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.