Resumen de Tropical Lagrangians in toric del-Pezzo surfaces

Jeffrey Hicks

• 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}.