Tropical Lagrangians in toric del-Pezzo surfaces

Abstract: We look at how one can construct from the data of a dimer model a Lagrangian submanifold in $(\mathbb{C}^*)^n$ 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 $L_{T^2}$ 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 $(\mathbb{CP}^2\setminus E, W), \check X_{9111}$. We find a symplectomorphism of $\mathbb{CP}^2\setminus E$ interchanging $L_{T^2}$ and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier-Mukai transform on $\check X_{9111}$.