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# Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

Published:2014-08-06
Printed: Dec 2014
• Renzo Cavalieri,
Colorado State University, Department of Mathematics, Weber Building, Fort Collins, CO 80523, U.S.A
• Steffen Marcus,
Department of Mathematics, University of Utah, E Room 233, Salt Lake City, UT 84112, U.S.A
 Format: LaTeX MathJax PDF

## Abstract

We describe double Hurwitz numbers as intersection numbers on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$. Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the $\psi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera $0$ and $1$).
 Keywords: double Hurwitz numbers, wall crossings, moduli spaces, ELSV formula
 MSC Classifications: 14N35 - Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

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