Combinatorial Algebraic Geometry:Vector bundles as multidivisors on metric graphs

We introduce and discuss the notion of a (tropical) vector bundle on a metric graph (equivalently, an abstract tropical curve). We show that unlike the classical setting, we can express any vector bundle as the pushforward of a line bundle on a free cover of our graph. This lends itself to a natural notion of "multidivisors": a tropical incarnation of Weil's notion of a matrix divisor from Généralisation des fonctions abéliennes. We then use these notions to comment on the moduli space of vector bundles on a tropical curve, and prove analogues of the Narasimhan-Seshadri and Weil-Riemann-Roch theorem.