Tropicalising hypergeometric $\tau$-functions

TL;DR

This paper introduces a tropical geometry framework to unify and analyze weighted Hurwitz numbers, extending structural results and establishing new properties like polynomiality, wall-crossing, and tropical mirror symmetry for elliptic variants.

Contribution

It develops a comprehensive tropical approach to study all weighted Hurwitz numbers simultaneously, generalizing known results and introducing elliptic weighted Hurwitz numbers with new symmetry properties.

Findings

Established a correspondence between weighted Hurwitz numbers and tropical covers.

Proved polynomiality and wall-crossing formulas for all weighted Hurwitz numbers.

Derived tropical mirror symmetry and quasimodularity for elliptic weighted Hurwitz numbers.

Abstract

Weighted Hurwitz numbers arise as coefficients in the power sum expansion of deformed hypergeometric $\tau$--functions. They specialise to essentially all known cases of Hurwitz numbers, including classical, monotone, strictly monotone and completed cycles Hurwitz numbers. In this work, we develop a tropical geometry framework for their study, thus enabling a simultaneous investigation of all these cases. We obtain a correspondence theorem expressing weighted Hurwitz numbers in terms of tropical covers. Using this tropical approach, we generalise most known structural results previously obtained for the aforementioned special cases to all weighted Hurwitz numbers. In particular, we study their polynomiality and derive wall--crossing formulae. Moreover, we introduce elliptic weighted Hurwitz numbers and derive tropical mirror symmetry for these new invariants, i.e. we prove that their generating function is quasimodular and that they may be expressed as Feynman integrals.