Tate’s algorithm and F-theory

  • Sheldon Katz
  • David R. Morrison
  • Sakura Schäfer-Nameki
  • James Sully
Article

Abstract

The “Tate forms” for elliptically fibered Calabi-Yau manifolds are reconsidered in order to determine their general validity. We point out that there were some implicit assumptions made in the original derivation of these “Tate forms” from the Tate algorithm. By a careful analysis of the Tate algorithm itself, we deduce that the “Tateforms” (without any futher divisiblity assumptions) do not hold in some instances and have to be replaced by a new type of ansatz. Furthermore, we give examples in which the existence of a “Tate form” can be globally obstructed, i.e., the change of coordinates does not extend globally to sections of the entire base of the elliptic fibration. These results have implications both for model-building and for the exploration of the landscape of F-theory vacua.

Keywords

F-Theory D-branes 

References

  1. [1]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds — I, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau Threefolds — II, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    M. Bershadsky, A. Johansen, T. Pantev, V. Sadov and C. Vafa, F-theory, geometric engineering and N =1 dualities, Nucl. Phys. B 505 (1997) 153 [hep-th/9612052] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    D.R. Morrison, TASI lectures on compactification and duality, hep-th/0411120 [SPIRES].
  7. [7]
    F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [SPIRES].
  8. [8]
    R. Donagi and M. Wijnholt, Model building with F-theory, arXiv:0802.2969 [SPIRES].
  9. [9]
    C. Beasley, J.J. Heckman and C. Vafa, GUT s and exceptional branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    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] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  12. [12]
    M. Graña, Flux compactifications in string theory: a comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [SPIRES].CrossRefADSGoogle Scholar
  13. [13]
    M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  14. [14]
    F. Denef, M.R. Douglas and S. Kachru, Physics of string flux compactifications, Ann. Rev. Nucl. Part. Sci. 57 (2007) 119 [hep-th/0701050] [SPIRES].CrossRefADSGoogle Scholar
  15. [15]
    V. Kumar and W. Taylor, A bound on 6DN =1 supergravities, JHEP 12 (2009) 050 [arXiv:0910.1586] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    V. Kumar, D.R. Morrison and W. Taylor, Mapping 6DN =1 supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6DN =1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [SPIRES].
  19. [19]
    K. Kodaira, On compact analytic surfaces. II, Ann. of Math. 77 (1963) 563.CrossRefMATHGoogle Scholar
  20. [20]
    K. Kodaira, On compact analytic surfaces. III, Ann. of Math. 78 (1963) 1.CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964) 5.CrossRefGoogle Scholar
  22. [22]
    J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular functions of one variable, IV — Proc. Internat. Summer School, Univ. Antwerp, Antwerp, (1972), Lecture Notes in Math. 476, Springer, Berlin Germany (1975) 33.Google Scholar
  23. [23]
    R. Donagi and M. Wijnholt, Higgs bundles and UV completion in F-theory, arXiv:0904.1218 [SPIRES].
  24. [24]
    J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, fluxes and compact three-generation F-theory GUT s, JHEP 08 (2009) 046 [arXiv:0906.4672] [SPIRES].CrossRefADSGoogle Scholar
  25. [25]
    O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N =1 six-dimensional E 8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [SPIRES].ADSMathSciNetGoogle Scholar
  27. [27]
    D.R. Morrison and W. Taylor, Matter and singularities, to appear.Google Scholar
  28. [28]
    M. Auslander and D.A. Buchsbaum, Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A. 45 (1959) 733.CrossRefMATHADSMathSciNetGoogle Scholar
  29. [29]
    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, to appear.Google Scholar
  31. [31]
    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] [SPIRES].MATHMathSciNetGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Sheldon Katz
    • 1
  • David R. Morrison
    • 2
    • 3
    • 4
  • Sakura Schäfer-Nameki
    • 5
    • 6
  • James Sully
    • 3
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaU.S.A.
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraU.S.A.
  3. 3.Department of PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  4. 4.Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan
  5. 5.Kavli Institute for Theoretical PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  6. 6.Department of Mathematics, King’s CollegeUniversity of LondonLondonU.K.

Personalised recommendations