Abstract
AbstractWe 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.
Highlights
Special Lagrangian submanifolds of Calabi-Yau threefolds have received much attention due to their role in mirror symmetry
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
Based on Thomas [57] and Thomas and Yau [58], Dominic Joyce [31] conjectured that a Lagrangian submanifold admits a special Lagrangian representative if is a stable object in the derived Fukaya category with respect to an appropriate Bridgeland stability condition
Summary
Special Lagrangian submanifolds of Calabi-Yau threefolds have received much attention due to their role in mirror symmetry. If there is a Lagrangian torus fibration for a Calabi-Yau manifold and a tropical curve in the base integral affine manifold such that all edges of have weight one, it is easy to construct for each edge of a Lagrangian torus times interval lying above , and for each trivalent vertex of a Lagrangian pairs of pants times torus lying above a small neighbourhood of These local pieces can be constructed in a way that can be patched together smoothly, resulting in a Lagrangian submanifold ◦. Unless we are talking about complex tori, in practice there are singular torus fibres in these bundles for Euler characteristic reasons, and we will get back to this Note that this toy model gives insight on how a complex submanifold ought to become a Lagrangian submanifold of the mirror dual (see Subsection 6.3 of [3]). Most Calabi-Yau threefolds permit degenerations to a reducible union of toric varieties, introducing the toric techniques we lay out for the quintic and its mirror dual
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