Completed Cycles Leaky Hurwitz Numbers

TL;DR

This paper introduces a new class of leaky Hurwitz numbers called completed cycles, proves their piecewise polynomiality, and establishes their chamber structure and wall crossing formulas, extending previous results and connecting to Fock space perspectives.

Contribution

It defines $(r+1)$-completed cycles leaky Hurwitz numbers, proves their piecewise polynomiality, and derives wall crossing formulas, generalizing prior work and linking to Fock space models.

Findings

Proved piecewise polynomiality of completed cycles leaky Hurwitz numbers.

Established chamber polynomiality structure and wall crossing formulas.

Connected results to Fock space perspective, affecting genus zero and specific leaky parameters.

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.