Journal of High Energy Physics

, 2015:142 | Cite as

F-theory on all toric hypersurface fibrations and its Higgs branches

  • Denis Klevers
  • Damián Kaloni Mayorga Peña
  • Paul-Konstantin Oehlmann
  • Hernan Piragua
  • Jonas Reuter
Open Access
Regular Article - Theoretical Physics


We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in \( {\mathrm{\mathbb{P}}}^2,{\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^1 \) and the recently studied \( {\mathrm{\mathbb{P}}}^2\left(1,\;1,\;2\right) \), yield F-theory realizations of SUGRA theories with discrete gauge groups \( {\mathbb{Z}}_3 \), \( {\mathbb{Z}}_2 \) and \( {\mathbb{Z}}_4 \). This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I 2-fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. We also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP 1 has a U(1)-gauge field induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q = 3.


F-Theory Differential and Algebraic Geometry Discrete and Finite Symmetries Supergravity Models 


Open Access

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  1. [1]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSCrossRefMathSciNetGoogle 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].ADSCrossRefMathSciNetGoogle Scholar
  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].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    R. Donagi and M. Wijnholt, Model building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theoryI, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    H. Hayashi, R. Tatar, Y. Toda, T. Watari and M. Yamazaki, New aspects of heterotic-F theory duality, Nucl. Phys. B 806 (2009) 224 [arXiv:0805.1057] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory - II: experimental predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    K. Hori and D. Tong, Aspects of non-abelian gauge dynamics in two-dimensional N = (2, 2) theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Nonabelian 2D gauge theories for determinantal Calabi-Yau varieties, JHEP 11 (2012) 166 [arXiv:1205.3192] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, fluxes and compact three-generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [arXiv:1312.5746] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, arXiv:1407.6359 [INSPIRE].
  14. [14]
    J.J. Heckman, More on the matter of 6D SCFTs, arXiv:1408.0006 [INSPIRE].
  15. [15]
    M.R. Douglas and W. Taylor, The landscape of intersecting brane models, JHEP 01 (2007) 031 [hep-th/0606109] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    M. Cvetič, J. Halverson, D. Klevers and P. Song, On finiteness of Type IIB compactifications: magnetized branes on elliptic Calabi-Yau threefolds, JHEP 06 (2014) 138 [arXiv:1403.4943] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    A.P. Braun, Y. Kimura and T. Watari, On the classification of elliptic fibrations modulo isomorphism on K3 surfaces with large Picard number, arXiv:1312.4421 [INSPIRE].
  19. [19]
    M.R. Douglas, D.S. Park and C. Schnell, The Cremmer-Scherk mechanism in f-theory compactifications on K3 manifolds, JHEP 05 (2014) 135 [arXiv:1403.1595] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    V. Kumar, D.R. Morrison and W. Taylor, Mapping 6D N = 1 supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    T.W. Grimm and W. Taylor, Structure in 6D and 4D N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    D.S. Park and W. Taylor, Constraints on 6D supergravity theories with abelian gauge symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    F. Bonetti, T.W. Grimm and T.G. Pugh, Non-supersymmetric F-theory compactifications on Spin(7) manifolds, JHEP 01 (2014) 112 [arXiv:1307.5858] [INSPIRE].CrossRefGoogle Scholar
  27. [27]
    F. Bonetti, T.W. Grimm, E. Palti and T.G. Pugh, F-theory on Spin(7) manifolds: weak-coupling limit, JHEP 02 (2014) 076 [arXiv:1309.2287] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, arXiv:1404.6300 [INSPIRE].
  31. [31]
    T.W. Grimm and T. Weigand, On abelian gauge symmetries and proton decay in global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [INSPIRE].ADSGoogle Scholar
  32. [32]
    S. Krause, C. Mayrhofer and T. Weigand, G 4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Cvetič, T.W. Grimm and D. Klevers, Anomaly cancellation and abelian gauge symmetries in F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP 03 (2013) 098 [arXiv:1211.6742] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    V. Braun, T.W. Grimm and J. Keitel, New Global F-theory GUTs with U(1) symmetries, JHEP 09 (2013) 154 [arXiv:1302.1854] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    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
  39. [39]
    M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections, JHEP 06 (2013) 067 [arXiv:1303.6970] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    T.W. Grimm, A. Kapfer and J. Keitel, Effective action of 6D F-theory with U(1) factors: rational sections make Chern-Simons terms jump, JHEP 07 (2013) 115 [arXiv:1305.1929] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    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].ADSCrossRefGoogle Scholar
  42. [42]
    M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors, JHEP 04 (2014) 010 [arXiv:1306.3987] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    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].ADSCrossRefMathSciNetGoogle Scholar
  44. [44]
    M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: addendum, JHEP 12 (2013) 056 [arXiv:1307.6425] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    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, JHEP 03 (2014) 021 [arXiv:1310.0463] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, Mordell-Weil torsion and the global structure of gauge groups in F-theory, JHEP 1410 (2014) 16 [arXiv:1405.3656] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  47. [47]
    G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, New branches of string compactifications and their F-theory duals, Nucl. Phys. B 492 (1997) 119 [hep-th/9607121] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, Nucl. Phys. Proc. Suppl. 58 (1997) 177 [hep-th/9607139] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  49. [49]
    A. Klemm, M. Kreuzer, E. Riegler and E. Scheidegger, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 (2005) 023 [hep-th/0410018] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  50. [50]
    M. Esole, J. Fullwood and S.-T. Yau, D 5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory, arXiv:1110.6177 [INSPIRE].
  51. [51]
    T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive abelian gauge symmetries and fluxes in F-theory, JHEP 12 (2011) 004 [arXiv:1107.3842] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    A.P. Braun, A. Collinucci and R. Valandro, The fate of U(1)’s at strong coupling in F-theory, JHEP 07 (2014) 028 [arXiv:1402.4054] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    P.S. Aspinwall and D.R. Morrison, Nonsimply connected gauge groups and rational points on elliptic curves, JHEP 07 (1998) 012 [hep-th/9805206] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  54. [54]
    V. Braun and D.R. Morrison, F-theory on genus-one fibrations, JHEP 08 (2014) 132 [arXiv:1401.7844] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
  56. [56]
    L.B. Anderson, I. García-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M - and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M.-x. Huang, A. Klemm, J. Reuter and M. Schiereck, Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit, arXiv:1401.4723 [INSPIRE].
  59. [59]
    K. Kodaira, On compact analytic surfaces: II, Ann. Math. 77 (1963) 563.CrossRefzbMATHGoogle Scholar
  60. [60]
    J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular functions of one variable IV, B.J. Birch and W. Kuyk, Springer, Germany (1975).Google Scholar
  61. [61]
    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  62. [62]
    S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tates algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    C. Lawrie and S. Schäfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    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].ADSCrossRefMathSciNetGoogle Scholar
  65. [65]
    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].CrossRefMathSciNetGoogle Scholar
  66. [66]
    M. Kuntzler and S. Schäfer-Nameki, Tate trees for elliptic fibrations with rank one Mordell-Weil group, arXiv:1406.5174 [INSPIRE].
  67. [67]
    L. Lin and T. Weigand, Towards the standard model in F-theory, arXiv:1406.6071 [INSPIRE].
  68. [68]
    L.M. Krauss and F. Wilczek, Discrete gauge symmetry in continuum theories, Phys. Rev. Lett. 62 (1989) 1221 [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    T. Banks, Effective lagrangian description of discrete gauge symmetries, Nucl. Phys. B 323 (1989) 90 [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    L.E. Ibanez and G.G. Ross, Should discrete symmetries be anomaly free?, CERN-TH-6000-91 (1991) [INSPIRE].
  71. [71]
    J.H. Silverman, The arithmetic of elliptic curves, Graduate texts in mathematics volume 106, Springer, Germant (2009).Google Scholar
  72. [72]
    S. Lang and A. Neron, Rational points of abelian varieties over function fields, Am. J. Math. 81 (1959) 95.CrossRefzbMATHMathSciNetGoogle Scholar
  73. [73]
    B. Mazur, Modular curves and the eisenstein ideal, Inst. Hautes Études Sci. Publ. 47 (1977) 33.CrossRefzbMATHMathSciNetGoogle Scholar
  74. [74]
    B. Mazur and D. Goldfeld, Rational isogenies of prime degree, Inv. Math. 44 (1978) 129.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  75. [75]
    N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra II, H. Hijikata and M. Nagata eds., Kinokuniya (1988).Google Scholar
  76. [76]
    S.Y. An et al., Jacobians of genus one curves, J. Number Theory 90 (2001) 304.CrossRefzbMATHMathSciNetGoogle Scholar
  77. [77]
    D.R. Morrison and D.S. Park, F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  78. [78]
    T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  79. [79]
    F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [INSPIRE].
  80. [80]
    W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [INSPIRE].
  81. [81]
    E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  82. [82]
    S.H. Katz, D.R. Morrison and M.R. Plesser, Enhanced gauge symmetry in type-II string theory, Nucl. Phys. B 477 (1996) 105 [hep-th/9601108] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  83. [83]
    R. Hartshorne, Algebraic geometry, Springer, Germany (1977).CrossRefzbMATHGoogle Scholar
  84. [84]
    W. Fulton, Introduction to toric varieties, Princeton University Press, Princeton U.S.A. (1993).zbMATHGoogle Scholar
  85. [85]
    D.A. Cox, J.B. Little and H.K. Schenck, Toric varieties, American Mathematical Society, U.S.A. (2011).zbMATHGoogle Scholar
  86. [86]
    D.A. Cox, The homogeneous coordinate ring of a toric variety, revised version, alg-geom/9210008 [INSPIRE].
  87. [87]
    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
  88. [88]
    W. Decker, G.M. Greuel, G. Pfister and H. Schönemann, Singular 3-1-6A computer algebra system for polynomial computations, (2012).
  89. [89]
    T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].ADSGoogle Scholar
  90. [90]
    H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box graphs and singular fibers, JHEP 05 (2014) 048 [arXiv:1402.2653] [INSPIRE].ADSCrossRefGoogle Scholar
  91. [91]
    R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  92. [92]
    G. Honecker and M. Trapletti, Merging heterotic orbifolds and K3 compactifications with line bundles, JHEP 01 (2007) 051 [hep-th/0612030] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  93. [93]
    A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  94. [94]
    K. Intriligator, H. Jockers, P. Mayr, D.R. Morrison and M.R. Plesser, Conifold transitions in M-theory on Calabi-Yau Fourfolds with background fluxes, Adv. Theor. Math. Phys. 17 (2013) 601 [arXiv:1203.6662] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  95. [95]
    N.C. Bizet, A. Klemm and D.V. Lopes, Landscaping with fluxes and the E8 Yukawa point in F-theory, arXiv:1404.7645 [INSPIRE].
  96. [96]
    P. Berglund and P. Mayr, Heterotic string/F theory duality from mirror symmetry, Adv. Theor. Math. Phys. 2 (1999) 1307 [hep-th/9811217] [INSPIRE].MathSciNetGoogle Scholar
  97. [97]
    G.R. Farrar and P. Fayet, Phenomenology of the production, decay and detection of new hadronic states associated with supersymmetry, Phys. Lett. B 76 (1978) 575 [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    S. Dimopoulos, S. Raby and F. Wilczek, Proton decay in supersymmetric models, Phys. Lett. B 112 (1982) 133 [INSPIRE].ADSCrossRefGoogle Scholar
  99. [99]
    L.E. Ibáñez and G.G. Ross, Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model, Nucl. Phys. B 368 (1992) 3 [INSPIRE].ADSCrossRefGoogle Scholar
  100. [100]
    H.K. Dreiner, C. Luhn and M. Thormeier, What is the discrete gauge symmetry of the MSSM?, Phys. Rev. D 73 (2006) 075007 [hep-ph/0512163] [INSPIRE].ADSGoogle Scholar
  101. [101]
    H.M. Lee et al., A unique Z 4R symmetry for the MSSM, Phys. Lett. B 694 (2011) 491 [arXiv:1009.0905] [INSPIRE].ADSCrossRefGoogle Scholar
  102. [102]
    E. Dudas and E. Palti, Froggatt-Nielsen models from E 8 in F-theory GUTs, JHEP 01 (2010) 127 [arXiv:0912.0853] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  103. [103]
    S. Krippendorf, D.K. Mayorga Pena, P.-K. Oehlmann and F. Ruehle, Rational F-theory GUTs without exotics, JHEP 07 (2014) 013 [arXiv:1401.5084] [INSPIRE].ADSCrossRefGoogle Scholar
  104. [104]
    A. Grassi, J. Halverson and J.L. Shaneson, Matter from geometry without resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  105. [105]
    A. Grassi, J. Halverson and J.L. Shaneson, Non-abelian gauge symmetry and the Higgs mechanism in F-theory, arXiv:1402.5962 [INSPIRE].
  106. [106]
    H. Hayashi, C. Lawrie and S. Schäfer-Nameki, Phases, flops and F-theory: SU(5) gauge theories, JHEP 10 (2013) 046 [arXiv:1304.1678] [INSPIRE].ADSCrossRefGoogle Scholar
  107. [107]
    M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases, arXiv:1402.6331 [INSPIRE].
  108. [108]
    M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases II, arXiv:1407.1867 [INSPIRE].
  109. [109]
    A.P. Braun and S. Schäfer-Nameki, Box graphs and resolutions I, arXiv:1407.3520 [INSPIRE].
  110. [110]
    M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math. Phys. 17 (2013) 1195 [arXiv:1107.0733] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  111. [111]
    J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yaus, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].ADSCrossRefGoogle Scholar
  112. [112]
    P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four folds, Nucl. Phys. B 494 (1997) 489 [hep-th/9610162] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

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© The Author(s) 2015

Authors and Affiliations

  • Denis Klevers
    • 1
    • 2
  • Damián Kaloni Mayorga Peña
    • 3
  • Paul-Konstantin Oehlmann
    • 3
  • Hernan Piragua
    • 1
  • Jonas Reuter
    • 3
  1. 1.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Theory Group, Physics DepartmentCERNGeneva 23Switzerland
  3. 3.Bethe Center for Theoretical Physics, Physikalisches Institut der Universität BonnBonnGermany

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