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

Ross Altman, James Gray, Yang-Hui He, Vishnu Jejjala, Brent D. Nelson

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 Kähler 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.

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

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 Kähler 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.

Paper Structure

This paper contains 52 sections, 86 equations, 29 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods: Calabi--Yau Threefolds from Toric Construction
  3. Kreuzer--Skarke Database
  4. Parsing the Database using Sage
  5. Stringy Hodge Numbers and Euler Number
  6. MPCP Desingularization and Triangulation of $\Delta^{*}$
  7. Weight Matrix
  8. The Chow Group and Intersection Numbers
  9. Linear Ideal and Stanley--Reisner Ideal
  10. Linear Ideal and Stanley--Reisner Ideal
  11. Chow Group
  12. Intersection Numbers
  13. Favorability
  14. Mori and Kähler Cones
  15. Gluing of Kähler Cones
  16. ...and 37 more sections

Figures (29)

  • Figure 1: Field-Based Search
  • Figure 2: SQL-Based Search
  • Figure 3: Properties and Column Names in SQL Tables
  • Figure :
  • Figure :
  • ...and 24 more figures