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 $({\mathbb{C}}^{\ast }{)}^n$ 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 ${\check{X}}_{9111}$.

55 pages, 24 figures. This is the final version of the paper accepted in Selecta Mathematica. arXiv admin note: substantial text overlap with arXiv:1904.06005