We prove a formula for the Hodge numbers of square‐free divisors of Calabi‐Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi‐Yau compactifications of type IIB string theory, M‐theory, and F‐theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor D requires counting fermion zero modes, which depend on the Hodge numbers . Suppose that X is a smooth Calabi‐Yau threefold hypersurface in a toric variety V, and let D be the restriction to X of a square‐free divisor of V. We give a formula for in terms of combinatorial data. Moreover, we construct a CW complex such that . We describe an efficient algorithm that makes possible for the first time the computation of sheaf cohomology for such divisors at large h^{1, 1}. As an illustration we compute the Hodge numbers of a class of divisors in a threefold with . Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi‐Yau hypersurfaces.

中文翻译：

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Calabi-Yau三重超曲面的除数的霍奇数

我们证明了复曲面变种中Calabi-Yau三重超曲面的无平方除数的Hodge数的公式。欧氏黄铜包裹除数会影响IIB型弦理论，M理论和F理论的Calabi‐Yau压实体的真空结构。要确定除数D上的欧几里得布朗引起的非摄动耦合，需要计算费米子零模，这取决于霍奇数。假设X是复曲面V中的光滑Calabi-Yau三重超曲面，并且令D为V的无平方除数对X的限制。我们根据组合数据给出一个公式。此外，我们构建了CW复合体这样。我们描述了一种有效的算法，该算法首次使在大h ^1，1处的此类除数的捆同调计算成为可能。作为说明，我们用的三倍计算一类除数的霍奇数。我们的结果迈出了对Calabi-Yau超曲面中欧几里德麸质超势进行系统计算的一步。