The Hodge Numbers of Divisors of Calabi-Yau Threefold Hypersurfaces

TL;DR

This paper provides a combinatorial formula and an efficient algorithm for computing Hodge numbers of divisors on Calabi-Yau threefold hypersurfaces, aiding the study of nonperturbative effects in string theory compactifications.

Contribution

It introduces a new formula and a CW complex construction for calculating Hodge numbers of divisors, enabling computations at large Hodge numbers for the first time.

Findings

Derived a formula for Hodge numbers in terms of combinatorial data.

Constructed a CW complex to compute sheaf cohomology.

Demonstrated the method on a threefold with h^{1,1}=491.

Abstract

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 $h^i({\cal{O}}_D)$. 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 $h^i({\cal{O}}_D)$ in terms of combinatorial data. Moreover, we construct a CW complex $\mathscr{P}_D$ such that $h^i({\cal{O}}_D)=h_i(\mathscr{P}_D)$. 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 $h^{1,1}=491$. Our results are a step toward a systematic computation of Euclidean brane superpotentials in Calabi-Yau hypersurfaces.