Journal of High Energy Physics

, 2015:158 | Cite as

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

  • Ross Altman
  • James Gray
  • Yang-Hui He
  • Vishnu Jejjala
  • Brent D. NelsonEmail author
Open Access
Regular Article - Theoretical Physics


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 [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, 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.


Differential and Algebraic Geometry Superstring Vacua 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].MathSciNetGoogle Scholar
  2. [2]
    T. Kaluza, On the problem of unity in physics, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1921 (1921) 966 [INSPIRE].Google Scholar
  3. [3]
    O. Klein, Quantum theory and five-dimensional theory of relativity (in German and English), Z. Phys. 37 (1926) 895 [Surveys High Energ. Phys. 5 (1986) 241] [INSPIRE].
  4. [4]
    P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A standard model from the E 8 × E 8 heterotic superstring, JHEP 06 (2005) 039 [hep-th/0502155] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  7. [7]
    V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    V. Braun, P. Candelas, R. Davies and R. Donagi, The MSSM spectrum from (0, 2)-deformations of the heterotic standard embedding, JHEP 05 (2012) 127 [arXiv:1112.1097] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    L.B. Anderson, J. Gray, Y.-H. He and A. Lukas, Exploring positive monad bundles and a new heterotic standard model, JHEP 02 (2010) 054 [arXiv:0911.1569] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic line bundle standard models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    T. Hubsch, Calabi-Yau manifolds: motivations and constructions, Commun. Math. Phys. 108 (1987) 291 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].CrossRefADSGoogle Scholar
  14. [14]
    P. Green and T. Hubsch, Calabi-Yau manifolds as complete intersections in products of complex projective spaces, Commun. Math. Phys. 109 (1987) 99 [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  15. [15]
    P. Candelas, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds. 2. Three generation manifolds, Nucl. Phys. B 306 (1988) 113 [INSPIRE].CrossRefADSGoogle Scholar
  16. [16]
    I. Brunner, M. Lynker and R. Schimmrigk, Unification of M-theory and F-theory Calabi-Yau fourfold vacua, Nucl. Phys. B 498 (1997) 156 [hep-th/9610195] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    J. Gray, A.S. Haupt and A. Lukas, All complete intersection Calabi-Yau four-folds, JHEP 07 (2013) 070 [arXiv:1303.1832] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    J. Gray, A.S. Haupt and A. Lukas, Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds, JHEP 09 (2014) 093 [arXiv:1405.2073] [INSPIRE].CrossRefADSGoogle Scholar
  19. [19]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].CrossRefADSzbMATHGoogle Scholar
  20. [20]
    L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    M. Kreuzer and H. Skarke, Reflexive polyhedra, weights and toric Calabi-Yau fibrations, Rev. Math. Phys. 14 (2002) 343 [math/0001106] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    P. Berglund and T. Hubsch, A generalized construction of mirror manifolds, Nucl. Phys. B 393 (1993) 377 [hep-th/9201014] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  23. [23]
    R. Blumenhagen, X. Gao, T. Rahn and P. Shukla, A note on poly-instanton effects in type IIB orientifolds on Calabi-Yau threefolds, JHEP 06 (2012) 162 [arXiv:1205.2485] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    X. Gao and P. Shukla, On classifying the divisor involutions in Calabi-Yau threefolds, JHEP 11 (2013) 170 [arXiv:1307.1139] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  25. [25]
    X. Gao and P. Shukla, F-term stabilization of odd axions in LARGE volume scenario, Nucl. Phys. B 878 (2014) 269 [arXiv:1307.1141] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    M. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, D-branes at del Pezzo singularities: global embedding and moduli stabilisation, JHEP 09 (2012) 019 [arXiv:1206.5237] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    M. Cicoli, M. Kreuzer and C. Mayrhofer, Toric K3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications, JHEP 02 (2012) 002 [arXiv:1107.0383] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  28. [28]
    M. Cicoli, J.P. Conlon and F. Quevedo, General analysis of LARGE volume scenarios with string loop moduli stabilisation, JHEP 10 (2008) 105 [arXiv:0805.1029] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    M. Cicoli et al., Explicit de Sitter flux vacua for global string models with chiral matter, JHEP 05 (2014) 001 [arXiv:1312.0014] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    V. Batyrev and M. Kreuzer, Integral cohomology and mirror symmetry for Calabi-Yau 3-folds, math/0505432 [INSPIRE].
  31. [31]
    V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].zbMATHMathSciNetGoogle Scholar
  32. [32]
    M. Kreuzer and H. Skarke, PALP: a package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun. 157 (2004) 87 [math/0204356] [INSPIRE].CrossRefADSzbMATHMathSciNetGoogle Scholar
  33. [33]
    M. Kreuzer and H. Skarke, Calabi-Yau data webpage,
  34. [34]
    R. Altman, Toric Calabi-Yau threefold database webpage,
  35. [35]
    A.P. Braun and N.-O. Walliser, A new offspring of PALP, arXiv:1106.4529 [INSPIRE].
  36. [36]
    Y.-H. He, S.-J. Lee and A. Lukas, Heterotic models from vector bundles on toric Calabi-Yau manifolds, JHEP 05 (2010) 071 [arXiv:0911.0865] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    Y.-H. He, M. Kreuzer, S.-J. Lee and A. Lukas, Heterotic bundles on Calabi-Yau manifolds with small Picard number, JHEP 12 (2011) 039 [arXiv:1108.1031] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    Y.-H. He, S.-J. Lee, A. Lukas and C. Sun, Heterotic model building: 16 special manifolds, JHEP 06 (2014) 077 [arXiv:1309.0223] [INSPIRE].CrossRefADSGoogle Scholar
  39. [39]
    A.P. Braun, J. Knapp, E. Scheidegger, H. Skarke and N.-O. Walliser, PALPa user manual, arXiv:1205.4147 [INSPIRE].
  40. [40]
    Sage Development Team collaboration, W.A. Stein et al., Sage mathematics software (version 5.12),, (2013).
  41. [41]
    V. Braun, J. Whitney and M. Hampton, Triangulations of a point configuration, configuration.html.
  42. [42]
    K. Matsuki, Introduction to the Mori program, Springer, Germany (2002).CrossRefzbMATHGoogle Scholar
  43. [43]
    S. Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Annal. 280 (1988) 521.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [44]
    M. Gross, D. Huybrechts and D. Joyce eds., Calabi-Yau manifolds and related geometries: lectures at a summer school in Nordfjordeid Norway June 2001, Springer, Germany (2003).Google Scholar
  45. [45]
    I.M. Gelfand, M.M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston U.S.A. (1994).CrossRefzbMATHGoogle Scholar
  46. [46]
    C.W. Lee, Regular triangulations of convex polytopes, American Mathematical Soc., U.S.A. (1990).Google Scholar
  47. [47]
    R.R. Thomas, Lectures in geometric combinatorics, volume 33, American Mathematical Soc., U.S.A. (2006).Google Scholar
  48. [48]
    C. Haase and B. Nill, Lattices generated by skeletons of reflexive polytopes, J. Combinat. Theor. A 115 (2008) 340.CrossRefMathSciNetGoogle Scholar
  49. [49]
    B. Nill, Complete toric varieties with reductive automorphism group, Math. Z. 252 (2006) 767 [math/0407491].CrossRefzbMATHMathSciNetGoogle Scholar
  50. [50]
    T. Oda, Convex bodies and algebraic geometryan introduction to the theory of toric varieties, in A series of modern surveys in mathematics 15, Springer Germany (1985).Google Scholar
  51. [51]
    J. Rambau. TOPCOM: triangulations of point configurations and oriented matroids, in Mathematical softwareICMS 2002, A.M. Cohen, X.-S. Gao and N. Takayama eds., World Scientific, Singapore (2002), pg. 330.Google Scholar
  52. [52]
    C. Long, L. McAllister and P. McGuirk, Heavy tails in Calabi-Yau moduli spaces, JHEP 10 (2014) 187 [arXiv:1407.0709] [INSPIRE].CrossRefADSGoogle Scholar
  53. [53]
    L.J. Billera, P. Filliman and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990) 155.CrossRefzbMATHMathSciNetGoogle Scholar
  54. [54]
    P. Berglund, S.H. Katz and A. Klemm, Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties, Nucl. Phys. B 456 (1995) 153 [hep-th/9506091] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  55. [55]
    T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  56. [56]
    S. Reffert, The geometers toolkit to string compactifications, arXiv:0706.1310 [INSPIRE].
  57. [57]
    B. Szendröi, On a conjecture of Cox and Katz, Math. Z. 240 (2002) 233.CrossRefzbMATHMathSciNetGoogle Scholar
  58. [58]
    B. Szendröi, On the ample cone of an ample hypersurface, Asian J. Math. 7 (2003) 001.Google Scholar
  59. [59]
    D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, in Mathematical Surveys and Monographs 68, American Mathematical Soc., U.S.A. (1999).Google Scholar
  60. [60]
    C.T.C. Wall, Classification problems in differential topology. V, Invent. Math. 1 (1966) 355.CrossRefADSzbMATHGoogle Scholar
  61. [61]
    Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83 (1996) 461.CrossRefzbMATHMathSciNetGoogle Scholar
  62. [62]
    P. Berglund, S.H. Katz, A. Klemm and P. Mayr, New Higgs transitions between dual N = 2 string models, Nucl. Phys. B 483 (1997) 209 [hep-th/9605154] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  63. [63]
    R. Altman, J. Gray, Y. He, V. Jejjala, B. Nelson and J. Simon, Exploring the landscape of large volume Calabi-Yau minima, manuscript in preparation, (2014).Google Scholar
  64. [64]
    J. Knapp and M. Kreuzer, Toric methods in F-theory model building, Adv. High Energy Phys. 2011 (2011) 513436 [arXiv:1103.3358] [INSPIRE].CrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Ross Altman
    • 1
  • James Gray
    • 2
  • Yang-Hui He
    • 3
    • 4
    • 5
  • Vishnu Jejjala
    • 6
  • Brent D. Nelson
    • 1
    • 7
    Email author
  1. 1.Department of PhysicsNortheastern UniversityBostonU.S.A.
  2. 2.Physics Department, Robeson Hall, Virginia TechBlacksburgU.S.A.
  3. 3.Department of MathematicsCity UniversityLondonU.K.
  4. 4.School of PhysicsNanKai UniversityTianjinP.R. China
  5. 5.Merton CollegeUniversity of OxfordOxfordU.K.
  6. 6.Centre for Theoretical Physics, NITheP, and School of PhysicsUniversity of the WitwatersrandJohannesburgSouth Africa
  7. 7.International Center for Theoretical PhysicsTriesteItaly

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