Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

摘要

We describe double Hurwitz numbers as intersection numbers on themoduli space of curves $overline{mathcal{M}}_{g,n}$. Using a result on thepolynomiality of intersection numbers of psi classes with the DoubleRamification Cycle, our formula explains the polynomiality in chambersof double Hurwitz numbers, and the wall crossing phenomenon in termsof a variation of correction terms to the $psi$ classes. Weinterpret this as suggestive evidence for polynomiality of the DoubleRamification Cycle (which is only known in genera $0$ and $1$).