Complete classification of reflexive polyhedra in four dimensions

By Maximilian Kreuzer, H. Skarke. 2000

555 citations recorded by the source corpus.

Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory. We describe how we obtained all 473,800,776 reflexive polyhedra that exist in four dimensions and the 30,108 distinct pairs of Hodge numbers of the resulting Calabi-Yau manifolds. As a by-product we show that all these spaces (and hence the corresponding string vacua) are connected via a chain of singular transitions.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/44c6ab611d9a47a87cc803342d4ae01694e8fa54

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Citations in the corpus

Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds (2011, background): “[54, 55], although this requires developing a number of technical tools [56, 57].”
The F-theory geometry with most flux vacua (2015, methodology): “Comparing to the largest values of h3,1 appearing in the Kreuzer-Skarke database, it turns out that the values 243731, 261857, 278140, 281581, 299707 and 303148 in [55] all appear in Table 1.”
An algorithm for the classification of smooth Fano polytopes (2007, background): “To our knowledge smooth Fano d-polytopes have been classified in the following cases: • When the number of vertices is d + 1, d + 2 or d + 3 ([9],[2]).”
The statistics of supersymmetric D-brane models (2004, background): “1 Numbers of this order appeared for instance in the classification of CY fourfolds [2].”
Moduli Stabilisation versus Chirality for MSSM like Type IIB Orientifolds (2007, background): “To this end, it would be very important to know which of the toric Calabi-Yau manifolds in the list of [68] have a swiss-cheese like structure, respectively can lead to large volume moduli stabilisation.”
Four-dimensional string compactifications with D-branes, orientifolds and fluxes (2006, background): “This is an amazingly large number, which is may orders of magnitudes larger than the ensembles of string vacua people have dealt with before, like for instance toric varieties [551], where numbers like 4 · 10(9) occurred (86).”
D-branes at del Pezzo singularities: global embedding and moduli stabilisation (2012, methodology): “In addition, this class is already quite large [27] and one assumes that the properties of these CY manifolds represent those of all CY manifolds.”
Dirichlet Branes and Mirror Symmetry (2009, background): “Since there is a huge number of reflexive polytopes (in four dimensions, Kreuzer and Skarke obtained 473, 800, 776 reflexive polytopes [315]), this supplies a rich collection of examples of non-compact integral affine manifolds.”
The Kreuzer-Skarke axiverse (2018, unknown): “The Kreuzer-Skarke database of four-dimensional reflexive polytopes [1] is a fount of data on Calabi-Yau compactifications.”
Building a better racetrack (2004, background): “and study the physics of each, and “classification” and “mapping” results, such as the classification of of CalabiYau’s which are toric hypersurfaces, and the detailed study of their moduli spaces, for example [15].”
Building an explicit de Sitter (2012, background): “In this section, we give the results of our scan for the maximal gauge group indicator Nlg in the 97,036 models of the F-theory type contained in the classification of [50].”
Constraints on axion inflation from the weak gravity conjecture (2015, background): “This is a very common occurence in the toric-projective Calabi-Yau manifolds famously classified in [33].”
Unimodality problems in Ehrhart theory (2015, methodology): “It is known from work of Lagarias and Ziegler [49] that there a r only finitely many reflexive polytopes (up to unimodular equivalence) in e ach dimension, with one reflexive in dimension one, 16 in dimension two, 4 ,319 in dimension three, and 473,800,776 in dimension four according to computations”