Tropical Lagrangian multisections and toric vector bundles

It is well-known that toric line bundles on a toric variety correspond to piecewise linear functions on the fan. For toric vector bundles, Payne constructed a branched covering over the fan and a piecewise linear function on the domain. We think of these objects as the tropicalization of Lagrangian multi-sections and therefore, deserve the name tropical Lagrangian multi-sections. In this paper, we study the reconstruction problem of toric vector bundles from a given tropical Lagrangian multi-section. Those tropical Lagrangian multi-sections that arise from toric vector bundles are called unobstructed. We reformulate Kaneyama's classification of toric vector bundles in terms of the language of tropical Lagrangian multi-sections. We also provide a ``SYZ-type approach to construct toric vector bundles from tropical Lagrangian multi-sections. In dimension 2, such ``mirror-symmetric approach provides us a combinatorial condition for checking which rank 2 tropical Lagrangian multi-section is unobstructed.