Three-family particle physics models from global F-theory compactifications

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


We construct four-dimensional, globally consistent F-theory models with three chiral generations, whose gauge group and matter representations coincide with those of the Minimal Supersymmetric Standard Model, the Pati-Salam Model and the Trinification Model. These models result from compactification on toric hypersurface fibrations X with the choice of base \( {\mathrm{\mathbb{P}}}^3 \). We observe that the F-theory conditions on the G4-flux restrict the number of families to be at least three. We comment on the phenomenology of the models, and for Pati-Salam and Trinification models discuss the Higgsing to the Standard Model. A central point of this work is the construction of globally consistent G4-flux. For this purpose we compute the vertical cohomology HV(2,2)(X) in each case and solve the conditions imposed by matching the M- and F-theoretical 3D Chern-Simons terms. We explicitly check that the expressions found for the G4-flux allow for a cancellation of D3-brane tadpoles. We also use the integrality of 3D Chern-Simons terms to ensure that our G4-flux solutions are adequately quantized.


Beyond Standard Model F-Theory Supersymmetric Standard Model GUT 


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].MathSciNetCrossRefADSGoogle Scholar
  2. [2]
    R. Donagi and M. Wijnholt, Model building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theoryI, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  4. [4]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theoryII: experimental predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  5. [5]
    R. Donagi and M. Wijnholt, Breaking GUT Groups in F-theory, Adv. Theor. Math. Phys. 15 (2011) 1523 [arXiv:0808.2223] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory compactifications for supersymmetric GUTs, JHEP 08 (2009) 030 [arXiv:0904.3932] [INSPIRE].CrossRefADSGoogle Scholar
  7. [7]
    R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  8. [8]
    T.W. Grimm, S. Krause and T. Weigand, F-theory GUT vacua on compact Calabi-Yau fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  9. [9]
    E. Dudas and E. Palti, Froggatt-Nielsen models from E 8 in F-theory GUTs, JHEP 01 (2010) 127 [arXiv:0912.0853] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  10. [10]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1)(P Q), JHEP 04 (2010) 095 [arXiv:0912.0272] [INSPIRE].CrossRefADSGoogle Scholar
  11. [11]
    E. Dudas and E. Palti, On hypercharge flux and exotics in F-theory GUTs, JHEP 09 (2010) 013 [arXiv:1007.1297] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  12. [12]
    M.J. Dolan, J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory GUTs with U(1) symmetries: generalities and survey, Phys. Rev. D 84 (2011) 066008 [arXiv:1102.0290] [INSPIRE].ADSGoogle Scholar
  13. [13]
    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].MathSciNetCrossRefADSGoogle Scholar
  14. [14]
    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].MathSciNetADSGoogle Scholar
  15. [15]
    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].CrossRefADSGoogle Scholar
  16. [16]
    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].CrossRefADSGoogle Scholar
  17. [17]
    R. Blumenhagen, Gauge coupling unification in F-theory grand unified theories, Phys. Rev. Lett. 102 (2009) 071601 [arXiv:0812.0248] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  18. [18]
    M.J. Dolan, J. Marsano and S. Schäfer-Nameki, Unification and Phenomenology of F-theory GUTs with U(1)(P Q), JHEP 12 (2011) 032 [arXiv:1109.4958] [INSPIRE].CrossRefADSGoogle Scholar
  19. [19]
    K.-S. Choi and T. Kobayashi, Towards the MSSM from F-theory, Phys. Lett. B 693 (2010) 330 [arXiv:1003.2126] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  20. [20]
    K.-S. Choi, On the standard model group in F-theory, Eur. Phys. J. C 74 (2014) 2939 [arXiv:1309.7297] [INSPIRE].CrossRefADSGoogle Scholar
  21. [21]
    L. Lin and T. Weigand, Towards the standard model in F-theory, Fortsch. Phys. 63 (2015) 55 [arXiv:1406.6071] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  22. [22]
    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].CrossRefADSGoogle Scholar
  23. [23]
    A. Grassi, J. Halverson, J. Shaneson and W. Taylor, Non-higgsable QCD and the standard model spectrum in F-theory, JHEP 01 (2015) 086 [arXiv:1409.8295] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  24. [24]
    M. Cvetič, T. Li and T. Liu, Supersymmetric Pati-Salam models from intersecting D6-branes: a road to the standard model, Nucl. Phys. B 698 (2004) 163 [hep-th/0403061] [INSPIRE].CrossRefADSGoogle Scholar
  25. [25]
    S. Förste, H.P. Nilles, P.K.S. Vaudrevange and A. Wingerter, Heterotic brane world, Phys. Rev. D 70 (2004) 106008 [hep-th/0406208] [INSPIRE].ADSGoogle Scholar
  26. [26]
    T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4D string models with a Pati-Salam symmetry: orbifold grand unified theories from heterotic string compactification on a Z(6) orbifold, Nucl. Phys. B 704 (2005) 3 [hep-ph/0409098] [INSPIRE].
  27. [27]
    T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  28. [28]
    A. Maharana and E. Palti, Models of particle physics from type IIB string theory and F-theory: a review, Int. J. Mod. Phys. A 28 (2013) 1330005 [arXiv:1212.0555] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  29. [29]
    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].MathSciNetCrossRefADSGoogle Scholar
  30. [30]
    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].MathSciNetCrossRefADSGoogle Scholar
  31. [31]
    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].MathSciNetCrossRefADSGoogle Scholar
  32. [32]
    S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tates algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].CrossRefADSGoogle Scholar
  33. [33]
    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].MathSciNetCrossRefADSGoogle Scholar
  34. [34]
    P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].MathSciNetMATHGoogle Scholar
  35. [35]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  36. [36]
    D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  37. [37]
    J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yaus, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].CrossRefADSGoogle Scholar
  38. [38]
    S. Krause, C. Mayrhofer and T. Weigand, Gauge fluxes in F-theory and type IIB orientifolds, JHEP 08 (2012) 119 [arXiv:1202.3138] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  39. [39]
    D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].ADSGoogle Scholar
  40. [40]
    D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].CrossRefADSMATHGoogle Scholar
  41. [41]
    G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, JHEP 06 (2015) 061 [arXiv:1404.6300] [INSPIRE].CrossRefADSGoogle Scholar
  42. [42]
    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].
  43. [43]
    M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    M. Del Zotto, J.J. Heckman, D.R. Morrison and D.S. Park, 6D SCFTs and gravity, JHEP 06 (2015) 158 [arXiv:1412.6526] [INSPIRE].CrossRefADSGoogle Scholar
  45. [45]
    J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].CrossRefGoogle Scholar
  46. [46]
    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].CrossRefADSGoogle Scholar
  47. [47]
    A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [INSPIRE].
  48. [48]
    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].MathSciNetCrossRefADSGoogle Scholar
  49. [49]
    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
  50. [50]
    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].MathSciNetCrossRefADSGoogle Scholar
  51. [51]
    T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  52. [52]
    D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].CrossRefADSGoogle Scholar
  53. [53]
    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].
  54. [54]
    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].CrossRefADSGoogle Scholar
  55. [55]
    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
  56. [56]
    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].CrossRefADSGoogle Scholar
  57. [57]
    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].MathSciNetCrossRefADSGoogle Scholar
  58. [58]
    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].CrossRefADSGoogle Scholar
  59. [59]
    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].MathSciNetCrossRefADSGoogle Scholar
  60. [60]
    M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: addendum, JHEP 12 (2013) 056 [arXiv:1307.6425] [INSPIRE].CrossRefADSGoogle Scholar
  61. [61]
    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].CrossRefADSGoogle Scholar
  62. [62]
    C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, Mordell-Weil torsion and the global structure of gauge groups in F-theory, JHEP 10 (2014) 16 [arXiv:1405.3656] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  63. [63]
    V. Braun and D.R. Morrison, F-theory on genus-one fibrations, JHEP 08 (2014) 132 [arXiv:1401.7844] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  64. [64]
    D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
  65. [65]
    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].CrossRefADSGoogle Scholar
  66. [66]
    C. Mayrhofer, E. Palti, O. Till and T. Weigand, Discrete gauge symmetries by higgsing in four-dimensional F-theory compactifications, JHEP 12 (2014) 068 [arXiv:1408.6831] [INSPIRE].CrossRefADSGoogle Scholar
  67. [67]
    I. García-Etxebarria, T.W. Grimm and J. Keitel, Yukawas and discrete symmetries in F-theory compactifications without section, JHEP 11 (2014) 125 [arXiv:1408.6448] [INSPIRE].CrossRefADSGoogle Scholar
  68. [68]
    C. Mayrhofer, E. Palti, O. Till and T. Weigand, On discrete symmetries and torsion homology in F-theory, JHEP 06 (2015) 029 [arXiv:1410.7814] [INSPIRE].CrossRefADSGoogle Scholar
  69. [69]
    M. Cvetič, R. Donagi, D. Klevers, H. Piragua and M. Poretschkin, F-theory vacua with Z 3 gauge symmetry, Nucl. Phys. B 898 (2015) 736 [arXiv:1502.06953] [INSPIRE].CrossRefGoogle Scholar
  70. [70]
    M. Kreuzer and H. Skarke, On the classification of reflexive polyhedra, Commun. Math. Phys. 185 (1997) 495 [hep-th/9512204] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  71. [71]
    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].MathSciNetCrossRefMATHGoogle Scholar
  72. [72]
    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].MathSciNetCrossRefADSGoogle Scholar
  73. [73]
    P. Candelas, E. Perevalov and G. Rajesh, Toric geometry and enhanced gauge symmetry of F-theory/heterotic vacua, Nucl. Phys. B 507 (1997) 445 [hep-th/9704097] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  74. [74]
    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
  75. [75]
    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].MathSciNetCrossRefADSGoogle Scholar
  76. [76]
    A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys. B 518 (1998) 515 [hep-th/9701023] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  77. [77]
    T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing brane and flux superpotentials in F-theory compactifications, JHEP 04 (2010) 015 [arXiv:0909.2025] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  78. [78]
    B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys. 173 (1995) 559 [hep-th/9402119] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  79. [79]
    E. Witten, Nonperturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [INSPIRE].CrossRefADSGoogle Scholar
  80. [80]
    T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  81. [81]
    E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].MathSciNetCrossRefADSMATHGoogle Scholar
  82. [82]
    S. Sethi, C. Vafa and E. Witten, Constraints on low dimensional string compactifications, Nucl. Phys. B 480 (1996) 213 [hep-th/9606122] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  83. [83]
    S. Gukov, C. Vafa and E. Witten, CFTs from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE].
  84. [84]
    H. Jockers, P. Mayr and J. Walcher, On N = 1 4D effective couplings for F-theory and heterotic vacua, Adv. Theor. Math. Phys. 14 (2010) 1433 [arXiv:0912.3265] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  85. [85]
    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].MathSciNetCrossRefMATHGoogle Scholar
  86. [86]
    N.C. Bizet, A. Klemm and D.V. Lopes, Landscaping with fluxes and the E8 Yukawa point in F-theory, arXiv:1404.7645 [INSPIRE].
  87. [87]
    M. Haack and J. Louis, M theory compactified on Calabi-Yau fourfolds with background flux, Phys. Lett. B 507 (2001) 296 [hep-th/0103068] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  88. [88]
    D. Belov and G.W. Moore, Classification of abelian spin chern-Simons theories, hep-th/0505235 [INSPIRE].
  89. [89]
    A. Kapustin and N. Saulina, Topological boundary conditions in abelian Chern-Simons theory, Nucl. Phys. B 845 (2011) 393 [arXiv:1008.0654] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  90. [90]
    T.W. Grimm and R. Savelli, Gravitational instantons and fluxes from M/F-theory on Calabi-Yau fourfolds, Phys. Rev. D 85 (2012) 026003 [arXiv:1109.3191] [INSPIRE].ADSGoogle Scholar
  91. [91]
    A.J. Niemi and G.W. Semenoff, Axial anomaly induced fermion fractionization and effective gauge theory actions in odd dimensional space-times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  92. [92]
    A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].MathSciNetADSGoogle Scholar
  93. [93]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  94. [94]
    T.W. Grimm, D. Klevers and M. Poretschkin, Fluxes and warping for gauge couplings in F-theory, JHEP 01 (2013) 023 [arXiv:1202.0285] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  95. [95]
    T.W. Grimm and A. Kapfer, Anomaly cancelation in field theory and F-theory on a circle, arXiv:1502.05398 [INSPIRE].
  96. [96]
    F. Bonetti and T.W. Grimm, Six-dimensional (1, 0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  97. [97]
    M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  98. [98]
    A. Sagnotti, A note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].CrossRefADSGoogle Scholar
  99. [99]
    E. Witten, An SU(2) anomaly, Phys. Lett. B 117 (1982) 324 [INSPIRE].CrossRefADSGoogle Scholar
  100. [100]
    M. Bies, C. Mayrhofer, C. Pehle and T. Weigand, Chow groups, Deligne cohomology and massless matter in F-theory, arXiv:1402.5144 [INSPIRE].
  101. [101]
    A. Font and L.E. Ibáñez, Yukawa structure from U(1) fluxes in F-theory grand unification, JHEP 02 (2009) 016 [arXiv:0811.2157] [INSPIRE].CrossRefADSGoogle Scholar
  102. [102]
    J.J. Heckman and C. Vafa, Flavor hierarchy from F-theory, Nucl. Phys. B 837 (2010) 137 [arXiv:0811.2417] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  103. [103]
    A. Font, L.E. Ibáñez, F. Marchesano and D. Regalado, Non-perturbative effects and Yukawa hierarchies in F-theory SU(5) unification, JHEP 03 (2013) 140 [Erratum ibid. 1307 (2013) 036] [arXiv:1211.6529] [INSPIRE].
  104. [104]
    R. Barbier et al., R-parity violating supersymmetry, Phys. Rept. 420 (2005) 1 [hep-ph/0406039] [INSPIRE].
  105. [105]
    I. Antoniadis and G.K. Leontaris, A supersymmetric SU(4) × O(4) model, Phys. Lett. B 216 (1989) 333 [INSPIRE].CrossRefADSGoogle Scholar
  106. [106]
    S.F. King and Q. Shafi, Minimal supersymmetric SU(4) × SU(2)L × SU(2)R, Phys. Lett. B 422 (1998) 135 [hep-ph/9711288] [INSPIRE].
  107. [107]
    G. Lazarides, C. Panagiotakopoulos and Q. Shafi, Supersymmetric unification without proton decay, Phys. Lett. B 315 (1993) 325 [Erratum ibid. B 317 (1993) 661] [hep-ph/9306332] [INSPIRE].
  108. [108]
    G.R. Dvali and Q. Shafi, Gauge hierarchy in SU(3)(C) × SU(3)(L) × SU(3)R and low-energy implications, Phys. Lett. B 326 (1994) 258 [hep-ph/9401337] [INSPIRE].
  109. [109]
    G. Dvali and Q. Shafi, Supersymmetric trinification and low energy consequences, in the proceedings of 1996 Summer school in High Energy Physics and Cosmology, E. Gava et al. eds., World Scientific, Singapore (1996).Google Scholar
  110. [110]
    T.W. Grimm, Axion inflation in F-theory, Phys. Lett. B 739 (2014) 201 [arXiv:1404.4268] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  111. [111]
    I. García-Etxebarria, T.W. Grimm and I. Valenzuela, Special points of inflation in flux compactifications, arXiv:1412.5537 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Mirjam Cvetič
    • 1
  • Denis Klevers
    • 2
  • Damián Kaloni Mayorga Peña
    • 3
  • Paul-Konstantin Oehlmann
    • 4
  • Jonas Reuter
    • 4
  1. 1.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Theory Group, Physics DepartmentCERNGeneva 23Switzerland
  3. 3.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  4. 4.Bethe Center for Theoretical PhysicsPhysikalisches Institut der Universität BonnBonnGermany

Personalised recommendations