Cohomology of Line Bundles: Applications

TL;DR

The paper develops and extends an algorithm for computing line-bundle cohomology on toric varieties, enabling precise determination of massless spectra in heterotic and Type II string compactifications. It generalizes to equivariant cohomology, essential for orientifolds and orbifolds, by leveraging explicit rationom representatives and lifted group actions, and connects cohomology to Batyrev-type Hodge-number formulas through Koszul sequences. The framework unifies ambient-space line-bundle cohomology with Calabi–Yau hypersurfaces and complete intersections, providing algorithmic tools for Hodge diamonds of 4- and 5-fold toric subspaces and for complete-intersection Calabi–Yau manifolds, including CICYs via Cayley polytopes. These advances offer a practical, scalable path for computing spectrum data in realistic string-model building and reveal a close combinatorial interpretation of geometric and non-geometric contributions in mirror-symmetric settings.

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.