On singular fibres in F-theory

  • Andreas P. BraunEmail author
  • Taizan Watari


In this paper, we propose a connection between the field theory local model (Katz-Vafa field theory) and the type of singular fibre in flat crepant resolutions of elliptic Calabi-Yau fourfolds, a class of fourfolds considered by Esole and Yau. We review the analysis of degenerate fibres for models with gauge groups SU(5) and SO(10) in detail, and observe that the naively expected fibre type is realized if and only if the Higgs vev in the field theory local model is unramified. To test this idea, we implement a linear (unramified) Higgs vev for the “E 6” Yukawa point in a model with gauge group SU(5) and verify that this indeed leads to a fibre of Kodaira type IV*. Based on this observation, we argue i) that the singular fibre types appearing in the fourfolds studied by Esole-Yau are not puzzling at all, (so that this class of fourfolds does not have to be excluded from the candidate of input data of some yet-unknown formulation of F-theory) and ii) that such fourfold geometries also contain more information than just the eigenvalues of the Higgs field vev configuration in the field theory local models.


F-Theory String Duality 


  1. [1]
    M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, arXiv:1107.0733 [INSPIRE].
  2. [2]
    T. Hübsch, Calabi-Yau manifolds. A bestiary for physicists, World Scientific, Singapore (1991).Google 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].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    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
  6. [6]
    K. Kodaira, On compact analytic surfaces: ii, Ann. Math. 77 (1963) 563.zbMATHCrossRefGoogle Scholar
  7. [7]
    K. Kodaira, On compact analytic surfaces, iii, Ann. Math. 78 (1963) 1.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    W. Barth, C. Peters and A. van de Ven, Compact complex surfaces, Springer-Verlag, Heidelberg Germany (1984).zbMATHCrossRefGoogle Scholar
  9. [9]
    J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Lecture Notes in Mathematics. Vol. 476: Modular Functions of One Variable IV, B. Birch and W. Kuyk eds., Springer, Berlin Germany (1975).Google Scholar
  10. [10]
    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
  11. [11]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  13. [13]
    C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    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].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    H. Hayashi, T. Kawano, R. Tatar and T. Watari, Codimension-3 Singularities and Yukawa Couplings in F-theory, Nucl. Phys. B 823 (2009) 47 [arXiv:0901.4941] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, Flavor structure in F-theory compactifications, JHEP 08 (2010) 036 [arXiv:0910.2762] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-branes and monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    T. Kawano, Y. Tsuchiya and T. Watari, A note on Kähler potential of charged matter in F-theory, Phys. Lett. B 709 (2012) 254 [arXiv:1112.2987] [INSPIRE].MathSciNetADSGoogle Scholar
  19. [19]
    B. Andreas and G. Curio, On discrete twist and four flux in N = 1 heterotic/F theory compactifications, Adv. Theor. Math. Phys. 3 (1999) 1325 [hep-th/9908193] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  20. [20]
    V.V. Batyrev, Birational Calabi-Yau n-folds have equal Betti numbers, in New trends in algebraic geometry (Warwick, 1996). Vol. 264, Cambridge University Press, Cambridge U.K. (1999), pg. 1 [alg-geom/9710020].
  21. [21]
    C. Lawrie and S. Schfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    R. Tatar and W. Walters, GUT theories from Calabi-Yau 4-folds with SO(10) singularities, arXiv:1206.5090 [INSPIRE].
  23. [23]
    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].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    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].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    A. Collinucci and R. Savelli, On flux quantization in F-theory, JHEP 02 (2012) 015 [arXiv:1011.6388] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    J. Knapp, M. Kreuzer, C. Mayrhofer and N.-O. Walliser, Toric construction of global F-theory GUTs, JHEP 03 (2011) 138 [arXiv:1101.4908] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    A. Collinucci and R. Savelli, On flux quantization in F-theory II: unitary and symplectic gauge groups, JHEP 08 (2012) 094 [arXiv:1203.4542] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory compactifications for supersymmetric GUTs, JHEP 08 (2009) 030 [arXiv:0904.3932] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    P.S. Aspinwall and M. Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B 387 (1996) 735 [hep-th/9605131] [INSPIRE].MathSciNetADSGoogle Scholar
  31. [31]
    H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, More on dimension-4 proton decay problem in F-theory - spectral surface, discriminant locus and monodromy, Nucl. Phys. B 840 (2010) 304 [arXiv:1004.3870] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    R. Donagi and M. Wijnholt, Breaking GUT groups in F-theory, Adv. Theor. Math. Phys. 15 (2011) 1523 [arXiv:0808.2223] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  33. [33]
    R. Tatar and T. Watari, Proton decay, Yukawa couplings and underlying gauge symmetry in string theory, Nucl. Phys. B 747 (2006) 212 [hep-th/0602238] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    R. Donagi and D. Gaitsgory, The Gerbe of Higgs bundles, Transform. Groups 7 (2002) 109 [math/0005132] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    R. Tatar, Y. Tsuchiya and T. Watari, Right-handed neutrinos in F-theory compactifications, Nucl. Phys. B 823 (2009) 1 [arXiv:0905.2289] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    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
  37. [37]
    K.-S. Choi and H. Hayashi, U(n) spectral covers from decomposition, JHEP 06 (2012) 009 [arXiv:1203.3812] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    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].ADSCrossRefGoogle Scholar
  39. [39]
    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
  40. [40]
    A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl. Phys. B 856 (2012) 129 [arXiv:1107.5337] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    V. Bouchard, J.J. Heckman, J. Seo and C. Vafa, F-theory and neutrinos: Kaluza-Klein dilution of flavor hierarchy, JHEP 01 (2010) 061 [arXiv:0904.1419] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    J.J. Heckman, A. Tavanfar and C. Vafa, The point of E 8 in F-theory GUTs, JHEP 08 (2010) 040 [arXiv:0906.0581] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    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
  44. [44]
    S. Cecotti, M.C. Cheng, J.J. Heckman and C. Vafa, Yukawa couplings in F-theory and non-commutative geometry, arXiv:0910.0477 [INSPIRE].
  45. [45]
    J.P. Conlon and E. Palti, Aspects of flavour and supersymmetry in F-theory GUTs, JHEP 01 (2010) 029 [arXiv:0910.2413] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Department of MathematicsKing‘s CollegeLondonU.K.
  2. 2.Institute for the Physics and Mathematics of the UniverseUniversity of TokyoTokyoJapan

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