Tropically constructed Lagrangians in mirror quintic threefolds

Abstract

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians. We check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. We show that the weight of each of these Lagrangians equals to the multiplicity of the corresponding tropical curve.

arXiv comment75 pages, 13 figures, final version, to appear in Forum Math. Sigma, ancillary files with source code and documentation added

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Cited by in corpus (6)

• Lagrangian pairs of pants
• Tropical Lagrangian Hypersurfaces are Unobstructed
• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• Lagrangian submanifolds from tropical hypersurfaces
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Toric vector bundles, non-abelianization, and spectral networks