Tropical reductive groups and principal bundles on metric graphs
Andreas Gross, Arne Kuhrs, Martin Ulirsch, Dmitry Zakharov
TL;DR
The paper develops an elementary tropical analogue of reductive groups by associating to each root datum $\Phi$ a tropical reductive group $\mathbf{G}^{\mathrm{trop}}=\check{M}_{\mathbb{R}}\rtimes W_{\Phi}$, and it provides tropical matrix descriptions for classical types (A, B, C, D) and the exceptional type $G_2$. It then constructs tropical principal $G$-bundles on metric graphs as torsors over the Weyl-extended harmonic sheaf and describes their moduli in terms of covers and line bundles with Weyl-symmetry data, yielding a decomposition into components indexed by Weyl torsors and further structure as torsors under tropical abelian varieties. Degree and stability notions from the algebraic theory are transported to the tropical setting, with explicit tropical analogues of slope, reductions to parabolics, and the semistable/stable stratifications, especially on tropical elliptic curves. A major result is that for Tate curves, the essential skeleton of the moduli space of semistable principal G-bundles is homeomorphic to the moduli of indecomposable tropical G^{trop}-bundles on the skeleton, establishing a non-Archimedean SYZ-type picture and connecting to previous tropicalization work for vector bundles. The framework thus unifies tropical reductive group theory, tropical bundles on graphs, and non-Archimedean geometry of moduli spaces, opening pathways to parallel tropicalizations across Dynkin types and to extensions for higher-genus Mumford curves.
Abstract
We propose an elementary tropical analogue of a reductive group that combines the datum of a Weyl group and the tropicalization of a fixed maximal torus. For the classical groups, as well as $G_2$, these tropical reductive groups admit descriptions as tropical matrix groups that resemble their classical counterparts. Employing this perspective, we introduce tropical principal bundles on metric graphs and study their explicit presentations as pushforwards of line bundles along covers with symmetries and extra data. Our main result identifies the essential skeleton of the moduli space of semistable principal bundles on a Tate curve with its tropical analogue.