Tropical Lagrangian multi-sections and toric vector bundles

Abstract: We introduce the notion of tropical Lagrangian multi-sections over a fan and study its relation with toric vector bundles. We also introduce a "SYZ-type" construction for toric vector bundles which gives a reinterpretation of Kaneyama's linear algebra data. In dimension 2, such "mirror-symmetric" approach provides us a pure combinatorial condition for checking which rank 2 tropical Lagrangian multi-section arises from toric vector bundles.