Cohomology of Line Bundles: Applications

TL;DR

This paper presents an algorithm for computing line bundle cohomology on toric varieties, with applications to string theory compactifications, including heterotic and Type II models, and provides cohomological interpretations of Batyrev's formula.

Contribution

We generalize our cohomology computation algorithm to equivariant cases and connect Batyrev's formula terms with cohomological interpretations, advancing string compactification methods.

Findings

Algorithm successfully computes line bundle cohomology on toric varieties.

Cohomological interpretation of Batyrev's formula terms established.

Applications demonstrated in heterotic and Type II string compactifications.

Abstract

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute equivariant cohomology for line bundles, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. This paper is considered the third in the row of arXiv:1003.5217 and arXiv:1006.2392.