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}.
Highlights
1.1 Homological mirror symmetry and tropical geometryTropical geometry plays an important role in mirror symmetry, a duality proposed in [10] between symplectic geometry on a space X, and complex geometry on a mirror space X
A proposed mechanism for constructing pairs of mirror geometries comes from SYZ mirror symmetry [37] where X and Xhave dual Lagrangian torus fibrations over a common affine manifold Q
Mirror symmetry is recovered by degenerating the symplectic geometry of X and complex geometry of Xto tropical geometry on the base Q
Summary
Tropical geometry plays an important role in mirror symmetry, a duality proposed in [10] between symplectic geometry on a space X , and complex geometry on a mirror space X. An expectation is that homological and SYZ mirror symmetry interact by relating Lagrangian torus fibers of val : X → Q to skyscraper sheaves of points on X , and sections of the Lagrangian torus fibration to line bundles of X This intuition was used in [1] which proved that the Fukaya–Seidel category Fuk((C∗)n, W ) is equivalent to Db Coh(X ), the derived category of coherent sheaves on a mirror toric manifold. In [19], it was shown that the tropical-Lagrangian and tropical-complex correspondences are compatible with this mirror functor, in the sense that when a tropical hypersurface V is approximated by val(D) for a divisor D, the Lagrangian L(V ) is mirror to the sheaf OD This extends the relation between homological and SYZ mirror symmetry to sheaves beyond line bundles and skyscrapers of points
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