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 (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 (CP 2 \ E, W ), X 9111 . We find a symplectomorphism of CP 2 \E interchanging L T 2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier-Mukai transform on X 9111 .