We show how to use Gromov-Witten invariants to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau manifolds X over Hirzebruch surfaces. To determine the representations of these matter multiplets under the gauge algebra \({\mathfrak {g}}\), we use toric methods to embed the weight lattice of \({\mathfrak {g}}\) into the integer homology lattice of X. We then apply mirror symmetry to determine whether classes in this lattice which correspond to weights of given representations are represented by irreducible curves. Applying mirror symmetry efficiently to such geometries requires obtaining good approximations to their Mori cones. We propose an algorithm for obtaining such approximations. When the algorithm yields a smooth cone, we find that the latter in fact coincides with the Mori cone of X and already contains information on the matter content of compactifications on X. Our algorithm relies on studying toric ambient spaces for the Calabi-Yau hypersurface X which are merely birationally equivalent to fibrations over Hirzebruch surfaces. We study the flops relating such varieties in detail.

我们展示了如何使用 Gromov-Witten 不变量来确定Hirzebruch 表面上椭圆纤维 Calabi-Yau 流形X上 F 理论紧化的物质含量。为了确定规范代数\({\mathfrak {g}}\)下这些物质多重态的表示 ，我们使用复曲面方法将\({\mathfrak {g}}\)的权格嵌入整数同调格的X. 然后，我们应用镜像对称性来确定该格子中对应于给定表示的权重的类是否由不可约曲线表示。将镜像对称有效地应用于此类几何图形需要获得与其森锥体的良好近似。我们提出了一种获得这种近似值的算法。当算法产生一个光滑锥时，我们发现后者实际上与X的 Mori 锥重合，并且已经包含关于X上紧化的物质含量的信息。我们的算法依赖于研究 Calabi-Yau 超曲面X 的复曲面环境空间，这些空间仅在双有理上等效于 Hirzebruch 曲面上的纤维化。我们详细研究了与此类品种相关的失败情况。