Failing to keep the balance: explicit formulae and topological recursion for leaky Hurwitz numbers

TL;DR

This paper extends the understanding of leaky Hurwitz numbers by proving their polynomiality, deriving explicit formulas in genus zero, and establishing their connection to topological recursion through spectral curves.

Contribution

It generalizes polynomiality results for leaky Hurwitz numbers, finds explicit formulas in genus zero, and links these invariants to topological recursion via spectral curves.

Findings

Polynomiality of leaky Hurwitz numbers is generalized and proven.

Explicit formulas for genus zero one-part and two-part leaky Hurwitz numbers are derived.

Leaky Hurwitz numbers satisfy topological recursion under certain conditions.

Abstract

Recently a new family of enumerative invariants called leaky Hurwitz numbers was introduced by Cavalieri-Markwig-Ranganathan in the context of logarithmic intersection theory. They admit an interpretation via tropical covers where the balancing condition fails. We employ tropical geometry to prove a generalisation of the piecewise polynomiality of Accadia-Karev-Lewanski for leaky completed cycles Hurwitz numbers, and a different wall crossing that is cubic instead of quadratic. Using tropical combinatorics and generatingfunctionology, we also find closed formulae for one-part and two-part completed cycles leaky Hurwitz numbers in genus $0$. Working more generally with a view towards topological recursion, we use Hamiltonian flows to associate spectral curves to very general cut-and-join operators. Under mild analytic constraints, we find the appropriate spectral curves, and in case the leakiness is fixed, we show that the resulting enumerative invariants satisfy topological recursion. This provides a partial inverse to recent work of Alexandrov-Bychkov-Dunin-Barkowski-Kazarian-Shadrin producing differentials satisfying topological recursion for KP $\tau$-functions. In particular these results specialise to completed cycles leaky Hurwitz numbers.