A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List

TL;DR

This paper reviews how to extract and catalog topological and geometric data of Calabi-Yau threefolds from the Kreuzer-Skarke list, providing a comprehensive online database for physicists and mathematicians.

Contribution

It introduces a detailed online database of Calabi-Yau threefolds derived from the Kreuzer-Skarke list, including topological invariants and geometric properties, facilitating physical and mathematical research.

Findings

Compiled a detailed inventory of Calabi-Yau threefolds from reflexive polyhedra.

Provided explicit calculations of Chern classes, intersection numbers, and cones.

Enabled identification of multiple Calabi-Yau realizations from a single polytope.

Abstract

Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, in a companion online database (see http://nuweb1.neu.edu/cydatabase), a detailed inventory of these quantities which are of interest to physicists. Many of the singular ambient spaces described by the Kreuzer-Skarke list can be smoothed out into multiple distinct toric ambient spaces describing different Calabi-Yau threefolds. We provide a list of the different Calabi-Yau threefolds which can be obtained from each polytope, up to current computational limits. We then give the details of a variety of quantities associated to each of these Calabi-Yau such as Chern classes, intersection numbers, and the K\"ahler and Mori cones, in addition to the Hodge data. This data forms a useful starting point for a number of physical applications of the Kreuzer-Skarke list.