A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list

Abstract

Kreuzer and Skarke famously produced the largest known database of CalabiYau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions [1]. 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 KreuzerSkarke 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.

Rigid limit for hypermultiplets and five-dimensional gauge theories

“…Let us consider CY which corresponds to geometry 1 of polytope 337 in the database [75]. It is defined by the data given in (E.1) and (E.2) and has h 1,1 = 4.…”

Patterns in Calabi–Yau Distributions

“…Of particular interest has been the investigation of further structures in the KreuzerSkarke database, including identification of "the tip" where Hodge numbers are small [21,35,46], the top bounding curves where Hodge numbers are large [43], identifying elliptically fibered threefolds [28,29,42,44], finding further fibrations such as K3-fibers [33,45], or a step-by-step construction of all possible smooth Calabi-Yau hypersurfaces from the reflexive polytope data [19], etc. Now, it should be emphasized that each of the some 473 million reflexive polytopes admits, as an ambient toric variety, many 7 so-called maximal projective crepant partial (MPCP) desingularization, each of which gives rise to a different Calabi-Yau threefold.…”

“…To facilitate this approach to a low-energy phenomenology derived from string theory, mathematicians and physicists have constructed large datasets of Calabi-Yau threefolds [7,[9][10][11][12][13][14][15][16][17][18][19][20][21][22] as well as various refined analyses of properties thereof [28][29][30][31][32][33][34][35]. By far the largest database was constructed in a tour de force of algebraic geometry, combinatorics, physics, and computer algorithms by Kreuzer and Skarke based on the theorems of Batyrev and Borisov [9][10][11][12][13][14]36,37].…”

“…Whereas K3 is essentially unique, we do not know how many Calabi-Yau threefolds there are. A systematic study to classify the desingularizations, to compute the necessary topological data, and to build an interactive online database [19] is under way. The moral is that there are almost certainly far more than half a billion Calabi-Yau threefolds!…”

Mirror Symmetry and Polar Duality of Polytopes

“…The 2015 paper [15] presents a state-of-the-art discussion of Figure 5, which includes data for other mirror pairs beyond those arising from 4-dimensional reflexive polytopes. That paper also describes the website [16] where the reader can find the most current version of Figure 5.…”