Tropical Lagrangians in toric del-Pezzo surfaces
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Springer International Publishing
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Funder: University of Cambridge
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 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), Xˇ9111. 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 Xˇ9111.
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 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), Xˇ9111. 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 Xˇ9111.