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

Tropical Lagrangian multi-sections and toric vector bundles