All Weight Systems for Calabi–Yau Fourfolds from Reflexive Polyhedra

Abstract

For any given dimension d, all reflexive d-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of (d + 1)-tuples of integers (weights), or combinations of k-tuples of weights with k < d + 1. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322,383,760,930 such weight systems. 185,269,499,015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi–Yau fourfolds. These leads to 532,600,483 distinct sets of Hodge numbers.

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Acknowledgements

The authors thank Victor Batyrev for email correspondence and Roman Schönbichler for helpful discussions. We are grateful to the Vienna Scientific Cluster for unbureaucratically providing computing time and in particular to Ernst Haunschmid for explanations on how to use these resources. F.S. has been supported by the Austrian Science Fund (FWF), projects P 27182-N27 and P 28751-N27.

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Correspondence to Friedrich Schöller.

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Schöller, F., Skarke, H. All Weight Systems for Calabi–Yau Fourfolds from Reflexive Polyhedra. Commun. Math. Phys. 372, 657–678 (2019). https://doi.org/10.1007/s00220-019-03331-9

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