Extend Fock-space tropical methods to recover full polynomiality for refined monotone Hurwitz numbers

Establish whether the Fock space–based tropical approach that recovers piecewise polynomiality on the resonance arrangement for classical monotone double Hurwitz numbers can be adapted to the CJT–refined setting to obtain the full (coarsest) piecewise polynomial structure for refined monotone Hurwitz numbers, rather than only a weaker piecewise quasipolynomiality with respect to a refined hyperplane arrangement.

Background

For complex double (and monotone) Hurwitz numbers, piecewise polynomiality with respect to the resonance arrangement is known, and tropical/Fock space techniques can recover this optimal structure.

In the refined (CJT) monotone setting developed here, the current tropical interpretation yields only piecewise quasipolynomiality with respect to a finer arrangement, not the coarsest resonance arrangement. The authors point to prior Fock space–based tropical methods that succeeded in the non-refined case and pose the question of extending them to the refined case.

Resolving this would align the refined theory’s structural behavior with its classical counterpart and sharpen the polynomiality results obtained in this work.