Completed Cycles Leaky Hurwitz Numbers

Abstract

We introduce $(r+1)$-completed cycles $k$-leaky Hurwitz numbers and prove piecewise polynomiality as well as establishing their chamber polynomiality structure and their wall crossing formulae. For $k=0$ the results recover previous results of Shadrin-Spitz-Zvonkine. The specialization for $r=1$ recovers Hurwitz numbers that are close to the ones studied by Cavalieri-Markwig-Ranganathan and Cavalieri-Markwig-Schmitt. The ramifications differ by a lower order torus correction, natural from the Fock space perspective, not affecting the genus zero enumeration, nor the enumeration for leaky parameter values $k=\pm 1$ in all genera.

v1 22 pages, 2 figures

References in corpus (10)

• Hurwitz numbers and intersections on moduli spaces of curves
• The structure of 2D semi-simple field theories
• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• On double Hurwitz numbers in genus 0
• Ramifications of Hurwitz theory, KP integrability and quantum curves
• On double Hurwitz numbers with completed cycles
• Chiodo formulas for the r-th roots and topological recursion
• Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson-Pandharipande-Tseng formula
• Towards an orbifold generalization of Zvonkine's $r$-ELSV formula
• An intersection-theoretic proof of the Harer-Zagier formula