Journal of High Energy Physics

, 2020:76 | Cite as

\( \frac{1}{2} \) Calabi-Yau 3-folds, Calabi-Yau 3-folds as double covers, and F-theory with U(1)s

  • Yusuke KimuraEmail author
Open Access
Regular Article - Theoretical Physics


In this study, we introduce a new class of rational elliptic 3-folds, which we refer to as “1/2 Calabi-Yau 3-folds”. We construct elliptically fibered Calabi-Yau 3-folds by utilizing these rational elliptic 3-folds. The construction yields a novel approach to build elliptically fibered Calabi-Yau 3-folds of various Mordell-Weil ranks. Our construction of Calabi-Yau 3-folds can be considered as a three-dimensional generalization of the operation of gluing pairs of 1/2 K3 surfaces to yield elliptic K3 surfaces. From one to seven U(1)s form in six-dimensional N = 1 F-theory on the constructed Calabi-Yau 3-folds. Seven tensor multiplets arise in these models.


Differential and Algebraic Geometry F-Theory Gauge Symmetry Super- string 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]
    C. Vafa, Evidence for F-theory, Nucl. Phys.B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    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].
  3. [3]
    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].
  4. [4]
    D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, JHEP10 (2012) 128 [arXiv:1208.2695] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP03 (2013) 098 [arXiv:1211.6742] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    V. Braun, T.W. Grimm and J. Keitel, New Global F-theory GUTs with U(1) symmetries, JHEP09 (2013) 154 [arXiv:1302.1854] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, Elliptic fibrations for SU(5) × U(1) × U(1) F-theory vacua, Phys. Rev.D 88 (2013) 046005 [arXiv:1303.5054] [INSPIRE].ADSGoogle Scholar
  8. [8]
    M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections, JHEP06 (2013) 067 [arXiv:1303.6970] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    V. Braun, T.W. Grimm and J. Keitel, Geometric Engineering in Toric F-theory and GUTs with U(1) Gauge Factors, JHEP12 (2013) 069 [arXiv:1306.0577] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral Four-Dimensional F-theory Compactifications With SU(5) and Multiple U(1)-Factors, JHEP04 (2014) 010 [arXiv:1306.3987] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    M. Cvetič, D. Klevers and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Addendum, JHEP12 (2013) 056 [arXiv:1307.6425] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  12. [12]
    M. Cvetič, D. Klevers, H. Piragua and P. Song, Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1) × U(1) × U(1) gauge symmetry, JHEP03 (2014) 021 [arXiv:1310.0463] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S. Mizoguchi, F-theory Family Unification, JHEP07 (2014) 018 [arXiv:1403.7066] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    I. Antoniadis and G.K. Leontaris, F-GUTs with Mordell-Weil U(1)’s, Phys. Lett.B 735 (2014) 226 [arXiv:1404.6720] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  15. [15]
    M. Esole, M.J. Kang and S.-T. Yau, A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations, arXiv:1410.0003 [INSPIRE].
  16. [16]
    C. Lawrie, S. Schäfer-Nameki and J.-M. Wong, F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections, JHEP09 (2015) 144 [arXiv:1504.05593] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    M. Cvetič, D. Klevers, H. Piragua and W. Taylor, General U(1) × U(1) F-theory compactifications and beyond: geometry of unHiggsings and novel matter structure, JHEP11 (2015) 204 [arXiv:1507.05954] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    M. Cvetič, A. Grassi, D. Klevers, M. Poretschkin and P. Song, Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality, JHEP04 (2016) 041 [arXiv:1511.08208] [INSPIRE].ADSzbMATHGoogle Scholar
  19. [19]
    D.R. Morrison and D.S. Park, Tall sections from non-minimal transformations, JHEP10 (2016) 033 [arXiv:1606.07444] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    D.R. Morrison, D.S. Park and W. Taylor, Non-Higgsable abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces, Adv. Theor. Math. Phys.22 (2018) 177 [arXiv:1610.06929] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    M. Bies, C. Mayrhofer and T. Weigand, Gauge Backgrounds and Zero-Mode Counting in F-theory, JHEP11 (2017) 081 [arXiv:1706.04616] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    M. Cvetič and L. Lin, The Global Gauge Group Structure of F-theory Compactification with U(1)s, JHEP01 (2018) 157 [arXiv:1706.08521] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    M. Bies, C. Mayrhofer and T. Weigand, Algebraic Cycles and Local Anomalies in F-theory, JHEP11 (2017) 100 [arXiv:1706.08528] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Y. Kimura and S. Mizoguchi, Enhancements in F-theory models on moduli spaces of K 3 surfaces with ADE rank 17, PTEP2018 (2018) 043B05 [arXiv:1712.08539] [INSPIRE].
  25. [25]
    Y. Kimura, F-theory models on K 3 surfaces with various Mordell-Weil ranks — constructions that use quadratic base change of rational elliptic surfaces, JHEP05 (2018) 048 [arXiv:1802.05195] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    S.-J. Lee, D. Regalado and T. Weigand, 6d SCFTs and U(1) Flavour Symmetries, JHEP11 (2018) 147 [arXiv:1803.07998] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    S. Mizoguchi and T. Tani, Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications, JHEP03 (2019) 121 [arXiv:1808.08001] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Y. Kimura, Nongeometric heterotic strings and dual F-theory with enhanced gauge groups, JHEP02 (2019) 036 [arXiv:1810.07657] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    F.M. Cianci, D.K. Mayorga Peña and R. Valandro, High U(1) charges in type IIB models and their F-theory lift, JHEP04 (2019) 012 [arXiv:1811.11777] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    W. Taylor and A.P. Turner, Generic matter representations in 6D supergravity theories, JHEP05 (2019) 081 [arXiv:1901.02012] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Y. Kimura, Unbroken E7× E7nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals, arXiv:1902.00944 [INSPIRE].
  32. [32]
    Y. Kimura, F-theory models with 3 to 8 U(1) factors on K 3 surfaces, arXiv:1903.03608 [INSPIRE].
  33. [33]
    M. Esole and P. Jefferson, The Geometry of SO(3), SO(5) and SO(6) models, arXiv:1905.12620 [INSPIRE].
  34. [34]
    S.-J. Lee and T. Weigand, Swampland Bounds on the Abelian Gauge Sector, Phys. Rev.D 100 (2019) 026015 [arXiv:1905.13213] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) Tops with Multiple U(1)s in F-theory, Nucl. Phys.B 882 (2014) 1 [arXiv:1307.2902] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
  37. [37]
    G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, JHEP06 (2015) 061 [arXiv:1404.6300] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    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, JHEP01 (2015) 142 [arXiv:1408.4808] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  39. [39]
    V. Braun, T.W. Grimm and J. Keitel, Complete Intersection Fibers in F-theory, JHEP03 (2015) 125 [arXiv:1411.2615] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    T.W. Grimm, A. Kapfer and D. Klevers, The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle, JHEP06 (2016) 112 [arXiv:1510.04281] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    W. Taylor and A.P. Turner, An infinite swampland of U(1) charge spectra in 6D supergravity theories, JHEP06 (2018) 010 [arXiv:1803.04447] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  42. [42]
    Y. Kimura, F-theory models with U(1) ×2,4and transitions in discrete gauge groups, arXiv:1908.06621 [INSPIRE].
  43. [43]
    N. Nakayama, On Weierstrass Models, Algebraic Geometry and Commutative Algebra2 (1988) 405.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    I. Dolgachev and M. Gross, Elliptic Three-folds I: Ogg-Shafarevich Theory, J. Alg. Geom.3 (1994) 39 [alg-geom/9210009].zbMATHGoogle Scholar
  45. [45]
    M. Gross, Elliptic Three-folds II: Multiple Fibres, Trans. Am. Math. Soc.349 (1997) 3409.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys.15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — I, JHEP01 (2009) 058 [arXiv:0802.3391] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — II: Experimental Predictions, JHEP01 (2009) 059 [arXiv:0806.0102] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    R. Donagi and M. Wijnholt, Breaking GUT Groups in F-theory, Adv. Theor. Math. Phys.15 (2011) 1523 [arXiv:0808.2223] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    T.D. Brennan, F. Carta and C. Vafa, The String Landscape, the Swampland and the Missing Corner, PoS(TASI2017)015 (2017) [arXiv:1711.00864] [INSPIRE].Google Scholar
  51. [51]
    E. Palti, The Swampland: Introduction and Review, Fortsch. Phys.67 (2019) 1900037 [arXiv:1903.06239] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  52. [52]
    C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
  53. [53]
    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP06 (2007) 060 [hep-th/0601001] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys.B 766 (2007) 21 [hep-th/0605264] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    H.-C. Kim, G. Shiu and C. Vafa, Branes and the Swampland, Phys. Rev.D 100 (2019) 066006 [arXiv:1905.08261] [INSPIRE].ADSMathSciNetGoogle Scholar
  56. [56]
    V. Kumar and W. Taylor, A Bound on 6D N = 1 supergravities, JHEP12 (2009) 050 [arXiv:0910.1586] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP11 (2010) 118 [arXiv:1008.1062] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. [58]
    D.S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry, JHEP01 (2012) 141 [arXiv:1110.5916] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. [59]
    W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, arXiv:1104.2051 [INSPIRE].
  60. [60]
    Y. Kimura, Structure of stable degeneration of K 3 surfaces into pairs of rational elliptic surfaces, JHEP03 (2018) 045 [arXiv:1710.04984] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    T. Shioda, Mordell-Weil lattices and Galois representation, I, Proc. Japan Acad.A 65 (1989) 268.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [62]
    T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli39 (1990) 211.MathSciNetzbMATHGoogle Scholar
  63. [63]
    R. Wazir, Arithmetic on elliptic threefolds, Compos. Math.140 (2004) 567 [math/0112259].
  64. [64]
    R. Hartshorne, Algebraic Geometry, Springer (1977).Google Scholar
  65. [65]
    T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan24 (1972) 20.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  66. [66]
    J. Tate, Algebraic cycles and poles of zeta functions, in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row (1965), pp. 93–110.Google Scholar
  67. [67]
    J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki9 (1964–1966), Exposé no. 306, pp. 415–440.Google Scholar
  68. [68]
    M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys.B 481 (1996) 215 [hep-th/9605200] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  69. [69]
    K. Kodaira, On compact analytic surfaces II, Annals Math.77 (1963) 563.zbMATHCrossRefGoogle Scholar
  70. [70]
    K. Kodaira, On compact analytic surfaces III, Annals Math.78 (1963) 1.MathSciNetzbMATHCrossRefGoogle Scholar
  71. [71]
    A. Néron, Modèles minimaux des variétes abéliennes sur les corps locaux et globaux, Publ. Math. IHÉS21 (1964) 5.Google Scholar
  72. [72]
    J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable IV, Springer, Berlin (1975), pp. 33–52.zbMATHCrossRefGoogle Scholar
  73. [73]
    M. Schütt and T. Shioda, Elliptic Surfaces, in Algebraic Geometry in East Asia (Seoul 2008), Adv. Stud. Pure Math.60 (2010) 51 [arXiv:0907.0298].zbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2020

Authors and Affiliations

  1. 1.KEK Theory Center, Institute of Particle and Nuclear Studies, KEKIbarakiJapan

Personalised recommendations