Tropical Lagrangians in toric del-Pezzo surfaces

We look at how one can construct from the data of a dimer model a Lagrangian submanifold in (C∗)n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori LT2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair (CP²\ E, W) , Xˇ ₉₁₁₁. We find a symplectomorphism of CP²\ E interchanging LT2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on Xˇ ₉₁₁₁.