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 ${\check{X}}_{9111}$.

arXiv comment55 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

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Cited by in corpus (6)

• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Tropically constructed Lagrangians in mirror quintic threefolds
• Lagrangian submanifolds from tropical hypersurfaces
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Toric vector bundles, non-abelianization, and spectral networks
• Symplectomorphisms of some Weinstein 4-manifolds