Tropical Lagrangians in toric del-Pezzo surfaces
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Jeffrey Hicks [1]
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[1] University of Cambridge
University of Cambridge
Cambridge District, Reino Unido
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 1, 2021
- Idioma: inglés
- DOI: 10.1007/s00029-020-00614-1
- Enlaces
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Resumen
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}.
