Given a smooth projective toric variety X with a polarization, we consider the mirror family of maps Wt: ( C⋆ )n → Rn , and associated tropical degeneration of the hypersurface M = W-10 (0) ⊂ ( C⋆ )n. We define a relative Fukaya category Fuk(( C⋆ )n, M) consisting of compact Lagrangian submanifolds of ( C⋆ )n with boundary on M. Using tropical geometry, we describe an explicit subcategory of Fuk(( C⋆ )n, M) consisting of sections of the Lagrangian torus fibration ( C⋆ )n → Rn . Using work of Fukaya and Oh relating Floer and Morse theory, classical toric techniques relating Cech cohomology and singular cohomology, and a new explicit Ainfinity map from cellular to Morse chains, we prove that this subcategory is quasi-equivalent to the DG category of line bundles on X.
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