All Weight Systems for Calabi–Yau Fourfolds from Reflexive Polyhedra

  • Friedrich SchöllerEmail author
  • Harald Skarke


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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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.


  1. 1.
    Candelas P., Horowitz G.T., Strominger A., Witten E.: “Vacuum Configurations for Superstrings”.Nucl. Phys. B 258, 46–74 (1985)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Candelas P., Dale A.M., Lutken C.A., Schimmrigk R.: Complete intersection Calabi–Yau manifolds. Nucl. Phys. B 298, 493 (1988)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Klemm A., Schimmrigk R.: Landau-Ginzburg string vacua. Nucl. Phys. B411, 559–583 (1994) arXiv:hep-th/9204060 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kreuzer M., Skarke H.: No mirror symmetry in Landau–Ginzburg spectra!. Nucl. Phys. B388, 113–130 (1992) arXiv:hep-th/9205004 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kreuzer M., Skarke H.: All Abelian symmetries of Landau–Ginzburg potentials. Nucl. Phys. B405, 305–325 (1993) arXiv:hep-th/9211047 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493–545 (1994) arXiv:alg-geom/9310003 [alg-geom]MathSciNetzbMATHGoogle Scholar
  7. 7.
    Batyrev V.V., Borisov L.A.: Dual cones and mirror symmetry for generalized Calabi–Yau manifolds. arXiv:alg-geom/9402002
  8. 8.
    Witten E.: Phases of N = 2 theories in two-dimensions. Nucl. Phys. B403, 159–222 (1993) arXiv:hep-th/9301042 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Skarke, H.: Weight systems for toric Calabi–Yau varieties and reflexivity of Newton polyhedra. Mod. Phys. Lett. A11, 1637–1652 (1996). arXiv:alg-geom/9603007 [alg-geom]
  10. 10.
    Avram A.C., Kreuzer M., Mandelberg M., Skarke H.: The Web of Calabi–Yau hypersurfaces in toric varieties. Nucl. Phys. B505, 625–640 (1997) arXiv:hep-th/9703003 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kreuzer M., Skarke H.: Complete classification of reflexive polyhedra in four-dimensions. Adv. Theor. Math. Phys. 4, 1209–1230 (2002) arXiv:hep-th/0002240 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Skarke, H.: How to Classify Reflexive Gorenstein Cones. In: Rebhan, A., Katzarkov, L., Knapp, J., Rashkov, R., Scheidegger E. (Eds.) Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, pp. 443–458 (2012). arXiv:1204.1181 [hep-th]
  13. 13.
    Vafa C.: Evidence for F theory. Nucl. Phys. B469, 403–418 (1996) arXiv:hep-th/9602022 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lynker, M., Schimmrigk, R., Wisskirchen, A.: Landau-Ginzburg vacua of string, M theory and F theory at c =  12. Nucl. Phys. B550, 123–150 (1999). arXiv:hep-th/9812195 [hep-th]
  15. 15.
    Gray J., Haupt A.S., Lukas A.: All complete intersection Calabi–Yau four-folds. JHEP 07, 070 (2013) arXiv:1303.1832 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kreuzer M., Skarke H.: Classification of reflexive polyhedra in three-dimensions. Adv. Theor. Math. Phys. 2, 853–871 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Batyrev, V.V., Dais, D.I.: Strong McKay correspondence, string theoretic Hodge numbers and mirror symmetry. arXiv:alg-geom/9410001 [alg-geom]
  18. 18.
    Kreuzer, M., Skarke, H.: On the classification of reflexive polyhedra. Commun. Math. Phys. 185, 495–508 (1997). arXiv:hep-th/9512204 [hep-th]
  19. 19.
    Kreuzer M., Skarke H.: Reflexive polyhedra, weights and toric Calabi–Yau fibrations. Rev. Math. Phys. 14, 343–374 (2002) arXiv:math/0001106 [math-ag]MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kreuzer M., Skarke H.: PALP: a package for analyzing lattice polytopes with applications to toric geometry. Comput. Phys. Commun. 157, 87–106 (2004) arXiv:math/0204356 [math.NA]ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Braun, A.P., Knapp, J., Scheidegger, E., Skarke, H., Walliser, N.-O.: PALP - a User Manual. In: Rebhan, A., Katzarkov, L., Knapp, J., Rashkov, R., Scheidegger E. (Eds.) Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, pp. 461–550 (2012). arXiv:1205.4147 [math.AG]
  22. 22.
    Kreuzer, M., Skarke, H.: Calabi–Yau data.”
  23. 23.
    Kreuzer M., Skarke H.: Calabi–Yau four folds and toric fibrations. J. Geom. Phys. 26, 272–290 (1998) arXiv:hep-th/9701175 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Candelas P., Constantin A., Skarke H.: An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts. Commun. Math. Phys. 324, 937–959 (2013) arXiv:1207.4792 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Candelas, P., Font, A.: Duality between the webs of heterotic and type II vacua. Nucl. Phys. B511, 295–325 (1998). arXiv:hep-th/9603170 [hep-th]
  26. 26.
    Candelas P., Skarke H.: F theory, SO(32) and toric geometry. Phys. Lett. B413, 63–69 (1997) arXiv:hep-th/9706226 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Mavlyutov, A.R.: Mirror symmetry for Calabi–Yau complete intersections in Fano toric varieties. arXiv:1103.2093 [math.AG]
  28. 28.
    Batyrev, V.: The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality. arXiv:1707.02602 [math.AG]

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Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität WienViennaAustria

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