Mirror symmetry and elliptic Calabi-Yau manifolds

  • Yu-Chien HuangEmail author
  • Washington Taylor
Open Access
Regular Article - Theoretical Physics


We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface B that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface \( \tilde{B} \) that is related through toric geometry to the line bundle −6KB. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also factorizes. Since recent evidence suggests that most Calabi-Yau threefolds are elliptic or genus one fibered, this points to a new way of understanding mirror symmetry that may apply to a large fraction of smooth Calabi-Yau threefolds. The factorization structure identified here can also apply for CalabiYau manifolds of higher dimension.


Differential and Algebraic Geometry F-Theory String Duality Superstring Vacua 


Open Access

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  1. [1]
    P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
  2. [2]
    K. Hori et al. eds., Clay Mathematics Monographs. Vol. 1: Mirror Symmetry, AMS Press, Providence U.S.A. (2003).Google Scholar
  3. [3]
    P. Candelas, M. Lynker and R. Schimmrigk, Calabi-Yau Manifolds in Weighted P(4), Nucl. Phys. B 341 (1990) 383 [INSPIRE].
  4. [4]
    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].MathSciNetzbMATHGoogle Scholar
  5. [5]
    M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
  7. [7]
    W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].
  8. [8]
    P. Candelas, A. Constantin and H. Skarke, An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts, Commun. Math. Phys. 324 (2013) 937 [arXiv:1207.4792] [INSPIRE].
  9. [9]
    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].ADSCrossRefGoogle Scholar
  10. [10]
    S.B. Johnson and W. Taylor, Calabi-Yau threefolds with large h 2,1, JHEP 10 (2014) 23 [arXiv:1406.0514] [INSPIRE].
  11. [11]
    L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Tools for CICYs in F-theory, JHEP 11 (2016) 004 [arXiv:1608.07554] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Multiple Fibrations in Calabi-Yau Geometry and String Dualities, JHEP 10 (2016) 105 [arXiv:1608.07555] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY Threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    L.B. Anderson, J. Gray and B. Hammack, Fibrations in Non-simply Connected Calabi-Yau Quotients, JHEP 08 (2018) 128 [arXiv:1805.05497] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Y.-C. Huang and W. Taylor, Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers, JHEP 02 (2019) 087 [arXiv:1805.05907] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    Y.-C. Huang and W. Taylor, On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds, JHEP 03 (2019) 014 [arXiv:1809.05160] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, Searching for K3 fibrations, Nucl. Phys. B 494 (1997) 567 [hep-th/9610154] [INSPIRE].
  18. [18]
    P. Berglund and P. Mayr, Heterotic string/F theory duality from mirror symmetry, Adv. Theor. Math. Phys. 2 (1999) 1307 [hep-th/9811217] [INSPIRE].
  19. [19]
    A. Grassi and V. Perduca, Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts, Adv. Theor. Math. Phys. 17 (2013) 741 [arXiv:1201.0930] [INSPIRE].
  20. [20]
    M. Cvetič, A. Grassi and M. Poretschkin, Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry, JHEP 06 (2017) 156 [arXiv:1607.03176] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D. Klevers, D.K. Mayorga Pena, P.-K. Oehlmann, H. Piragua and J. Reuter, F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tor, Duke Math. J. 69 (1993) 349.Google Scholar
  23. [23]
    M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    F. Rohsiepe, Fibration structures in toric Calabi-Yau fourfolds, hep-th/0502138 [INSPIRE].
  25. [25]
    V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification of tops, Adv. Theor. Math. Phys. 7 (2003) 205 [hep-th/0303218] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    V. Braun, Toric Elliptic Fibrations and F-theory Compactifications, JHEP 01 (2013) 016 [arXiv:1110.4883] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    V. Braun, T.W. Grimm and J. Keitel, Geometric Engineering in Toric F-theory and GUTs with U(1) Gauge Factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE].
  28. [28]
    H. Skarke, String dualities and toric geometry: An Introduction, Chaos Solitons Fractals 10 (1999) 543 [hep-th/9806059] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A.P. Braun and M. Del Zotto, Mirror Symmetry for G 2 -Manifolds: Twisted Connected Sums and Dual Tops, JHEP 05 (2017) 080 [arXiv:1701.05202] [INSPIRE].
  30. [30]
    P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].
  31. [31]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
  32. [32]
    A. Braun, W. Taylor and Y. Wang, unpublished notes (2016).Google Scholar
  33. [33]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
  34. [34]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
  35. [35]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
  36. [36]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
  37. [37]
    D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
  38. [38]
    W. Taylor and Y.-N. Wang, Scanning the skeleton of the 4D F-theory landscape, JHEP 01 (2018) 111 [arXiv:1710.11235] [INSPIRE].
  39. [39]
    V. Braun, T.W. Grimm and J. Keitel, Complete Intersection Fibers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    P.-K. Oehlmann, J. Reuter and T. Schimannek, Mordell-Weil Torsion in the Mirror of Multi-Sections, JHEP 12 (2016) 031 [arXiv:1604.00011] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].
  42. [42]
    M.-x. Huang, S. Katz and A. Klemm, Topological String on elliptic CY 3-folds and the ring of Jacobi forms, JHEP 10 (2015) 125 [arXiv:1501.04891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
  44. [44]
    N. Mekareeya, K. Ohmori, H. Shimizu and A. Tomasiello, Small instanton transitions for M5 fractions, JHEP 10 (2017) 055 [arXiv:1707.05785] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Theoretical Physics, Department of PhysicsMassachusetts Institute of TechnologyCambridgeU.S.A.

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