Principal bundles on metric graphs: the $\mathrm{GL}_n$ case

TL;DR

This paper develops a tropical analogue of principal G-bundles on metric graphs, focusing on vector bundles for G=GL(n), and establishes key theorems and correspondences in this new setting.

Contribution

It introduces a tropical framework for principal bundles on metric graphs, characterizes vector bundles via multidivisors, and proves analogues of classical theorems.

Findings

Tropical principal G-bundles are defined via root data.

Characterization of vector bundles using multidivisors.

Isomorphism between non-Archimedean skeletons and tropical moduli spaces.

Abstract

Using the notion of a root datum of a reductive group $G$ we propose a tropical analogue of a principal $G$-bundle on a metric graph. We focus on the case $G=\mathrm{GL}_n$, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove analogues of the Weil--Riemann--Roch theorem and the Narasimhan--Seshadri correspondence. We proceed by studying the process of tropicalization. In particular, we show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to a certain component of the moduli space of semistable tropical vector bundles on its dual metric graph.