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.
Categories: math.SG math.AG math.DG