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

TL;DR

The paper extends the complete classification of complete intersection Calabi-Yau four-folds (CICY4s) by computing a comprehensive set of topological invariants and exhaustively characterizing their elliptic fibrations. It derives explicit formulas for Chern classes and uses index theorems to relate Hodge numbers, establishing a lower bound of 36,779 distinct topologies and revealing systematic structure in the Hodge data and mirror behavior. A major result is the near-ubiquity of elliptic fibrations: over 50 million fibrations across more than 900k manifolds, with many admitting sections under a necessary condition, enabling vast potential for F-theory compactifications. The work provides a detailed, machine-readable data set of invariants, fibrations, and base types, facilitating further mathematical and physical exploration of Calabi-Yau geometry and string phenomenology.

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 .

Figures (7)

• Figure 1: Distribution of the Euler characteristic $\chi$ in the CICY four-fold list (excluding product manifolds), as a logarithmic plot. The values lie in the range $288 \leq \chi \leq 2610$.
• Figure 2: Logarithmic plots of the abundance of Hodge numbers in the CICY four-fold list (excluding product manifolds). Here, $N$ is the number of times a given value of the Hodge number appears in the CICY four-fold list.
• Figure 3: The canonical two-dimensional sections of the space spanned by the Hodge numbers. The colouring encodes the abundance with which the particular combination of Hodge numbers occurs in the CICY four-fold list (excluding product manifolds).
• Figure 4: Density histogram of the pair $(h^{3,1}, h^{2,2})$ in the CICY four-fold list (excluding product manifolds) overlaid with the linear equation $h^{2,2} \approx 4 h^{3,1} + 82.8$ (orange curve). The purely empirical origin of this apparent linear relation between $h^{3,1}$ and $h^{2,2}$ is explained on page \ref{['rel_h31_h22_origin']}.
• Figure 5: A plot of $(h^{1,1} + h^{3,1})$ against $(h^{1,1} - h^{3,1})$. The dashed lines bound the region $h^{1,1} \geq 0$, $h^{3,1} \geq 0$.