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Tropically constructed Lagrangians in mirror quintic threefolds. (English) Zbl 07289293
Summary: We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds.
We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve.
As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $$>300$$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14T20 Geometric aspects of tropical varieties 14T90 Applications of tropical geometry 53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category 57R17 Symplectic and contact topology in high or arbitrary dimension 53D12 Lagrangian submanifolds; Maslov index
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