# Line bundle cohomologies on CICYs with Picard number two

## TL;DR

This paper studies line bundle cohomologies on specific Calabi-Yau manifolds, revealing that their dimensions are often polynomial or recursive functions of charges, with regional variations and links to manifold configurations.

## Contribution

It provides new analytic and recursive descriptions of line bundle cohomologies on CICYs with Picard number two, highlighting regional and configuration-based patterns.

## Key findings

- Cohomology dimensions are polynomial functions of charges in many cases.
- Recursive relationships can determine cohomologies more succinctly for some manifolds.
- Regions where polynomial fits fail are linked to irregular index behaviour.

## Abstract

We analyse line bundle cohomologies on all favourable co-dimension two Complete Intersection Calabi Yau (CICY) manifolds of Picard number two. Our results provide further evidence that the cohomology dimensions of such line bundles are given by analytic expressions, which change between regions in the line bundle charge space. This agrees with recent observations of CY line bundles presented in Refs [1,2]. In many cases, the expressions for bundle cohomology dimensions are polynomial functions of the line bundle charges (of degree at most 3), and the regions are cones. A more novel observation is that for some CICY manifolds, the cohomologies are more succinctly determined by recursive relationships. There can also be boundaries between regions where a polynomial fit fails, and we link these exceptional cases to irregular behaviour of the index of the line bundle. Finally, our observations provide evidence for similarities in the line bundle cohomologies for CICY manifolds that share rows in the configuration matrix. Among such related CICY manifolds, we find both that the line bundle charge space is partitioned in the same manner, and that the same, or closely related, analytical descriptions apply for the cohomology dimensions in these regions.