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.

中文翻译：

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热带构造的拉格朗日在镜像五重三次

我们使用热带曲线和复曲面退化技术在很多Calabi-Yau 三倍体中构造封闭嵌入的拉格朗日有理同调球。同调球与导致 Gromov-Witten (GW) 不变量的全纯曲线镜像对偶。根据乔伊斯猜想，这些拉格朗日量应该有特殊的拉格朗日代表，从而解决了卡拉比丘三重上的特殊拉格朗日枚举问题。我们将这种构造应用于从五次卡拉比丘三重上的 2,875 条线获得的热带曲线。每个可接受的热带曲线在相应的镜像五次三次中给出一个拉格朗日有理同调球，并且这些拉格朗日中的每一个的乔伊斯权重等于相应热带曲线的多重性。作为应用程序，$>300$互不相交的曲线（因此拉格朗日）出现。Dehn 沿着这些拉格朗日数扭曲生成具有该等级的辛映射类群的阿贝尔子群。