Topological Invariants and Fibration Structure of Complete Intersection Calabi-Yau Four-Folds

TL;DR

This paper systematically analyzes a large class of Calabi-Yau four-folds, computing their topological invariants and revealing that most admit elliptic fibrations, highlighting the rich fibration structures within this class.

Contribution

It provides the first comprehensive topological and fibration analysis of over 900,000 Calabi-Yau four-folds constructed as complete intersections.

Findings

At least 36,779 topologically distinct manifolds identified.

99.95% of manifolds can be described as elliptic fibrations.

Over 50 million elliptic fibrations found, with many admitting sections.

Abstract

We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there are at least 36,779 topologically distinct manifolds in our list. We also study the fibration structure of these manifolds and find that 99.95 percent can be described as elliptic fibrations. In total, we find 50,114,908 elliptic fibrations, demonstrating the multitude of ways in which many manifolds are fibered. A sub-class of 26,088,498 fibrations satisfy necessary conditions for admitting sections. The complete data set can be downloaded at http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html .