Tropicalising hypergeometric $τ$-functions

TL;DR

This work develops a tropical geometry framework for weighted Hurwitz numbers arising from β-deformed hypergeometric τ-functions and proves a tropical correspondence theorem expressing these numbers as counts of tropical covers. It extends the framework to weighted elliptic Hurwitz numbers and establishes tropical mirror symmetry, including quasimodularity and a Feynman-integral formulation. The authors prove piecewise polynomiality and explicit wall-crossing formulas, generalizing known results for classical, monotone, strictly monotone, and completed cycles Hurwitz numbers. The methodology rests on the bosonification–tropicalisation viewpoint via the semi-infinite wedge formalism, enabling a uniform, combinatorial treatment across weight functions G and \tilde{G} and yielding a cohesive picture of the enumerative, modular, and diagrammatic structures involved.

Abstract

Weighted Hurwitz numbers arise as coefficients in the power sum expansion of deformed hypergeometric $τ$--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.

Figures (1)

• Figure 1: An illustration of the tropical projective line $\mathbb{P}^1_{\textit{trop}}$ on the left and the tropical elliptic curve $E_{\textit{trop}}$ on the right.

Theorems & Definitions (65)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: hahn2019tropicaljucyscovers
• Remark 2.6
• Lemma 2.7
• Theorem 2.8
• Definition 2.9
• Theorem 2.10