Tropical reductive groups and principal bundles on metric graphs

TL;DR

This paper develops a tropical analogue of reductive groups and principal bundles on metric graphs, providing new combinatorial tools to study moduli spaces of bundles in tropical geometry.

Contribution

It introduces tropical reductive groups and principal bundles, linking classical algebraic groups with tropical geometry and describing their moduli spaces explicitly.

Findings

Tropical reductive groups resemble classical groups as tropical matrix groups.

Principal bundles on metric graphs can be described as pushforwards of line bundles.

The moduli space's essential skeleton matches its tropical analogue.

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.