Topological recursion for leaky Hurwitz numbers

Establish topological recursion in the sense of Eynard and Orantin for a variation of leaky Hurwitz numbers, and, if possible, identify an associated spectral curve or quantum curve and determine the modifications required in the recursive procedure.

Background

Topological recursion is one of the major structural properties studied across modern Hurwitz theories, and the paper lists it among the important directions for understanding such enumerative invariants. The results here establish piecewise polynomiality, wall-crossing formulae, and cut-and-join equations, but do not establish topological recursion for the leaky theories.

The authors specifically identify the presence of non-zero-energy operators in the middle of the vacuum expectation value as a feature whose effect on topological recursion is not understood. A solution would need to determine both the appropriate leaky Hurwitz variation and the spectral data governing its recursion.