Skip to main content
SpringerLink
Log in

Tate’s algorithm and F-theory

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  4. M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  6. D.R. Morrison, TASI lectures on compactification and duality, hep-th/0411120 [SPIRES].

  7. F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [SPIRES].

  8. R. Donagi and M. Wijnholt, Model building with F-theory, arXiv:0802.2969 [SPIRES].

  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].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  11. T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  12. M. Graña, Flux compactifications in string theory: a comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [SPIRES].

    Article  ADS  Google Scholar 

  13. M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  15. V. Kumar and W. Taylor, A bound on 6DN =1 supergravities, JHEP 12 (2009) 050 [arXiv:0910.1586] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  16. V. Kumar, D.R. Morrison and W. Taylor, Mapping 6DN =1 supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  18. W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [SPIRES].

  19. K. Kodaira, On compact analytic surfaces. II, Ann. of Math. 77 (1963) 563.

    Article  MATH  Google Scholar 

  20. K. Kodaira, On compact analytic surfaces. III, Ann. of Math. 78 (1963) 1.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964) 5.

    Article  Google Scholar 

  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.

  23. R. Donagi and M. Wijnholt, Higgs bundles and UV completion in F-theory, arXiv:0904.1218 [SPIRES].

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  26. V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  27. D.R. Morrison and W. Taylor, Matter and singularities, to appear.

  28. M. Auslander and D.A. Buchsbaum, Unique factorization in regular local rings, Proc. Nat. Acad. Sci. U.S.A. 45 (1959) 733.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  30. A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, to appear.

  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].

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL, 61801, U.S.A.

    Sheldon Katz

  2. Department of Mathematics, University of California, Santa Barbara, CA, 93106, U.S.A.

    David R. Morrison

  3. Department of Physics, University of California, Santa Barbara, CA, 93106, U.S.A.

    David R. Morrison & James Sully

  4. Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

    David R. Morrison

  5. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106, U.S.A.

    Sakura Schäfer-Nameki

  6. Department of Mathematics, King’s College, University of London, The Strand, WC2R 2LS, London, U.K.

    Sakura Schäfer-Nameki

Authors
  1. Sheldon Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David R. Morrison
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sakura Schäfer-Nameki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. James Sully
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sakura Schäfer-Nameki.

Additional information

ArXiv ePrint:1106.3854

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katz, S., Morrison, D.R., Schäfer-Nameki, S. et al. Tate’s algorithm and F-theory. J. High Energ. Phys. 2011, 94 (2011). https://doi.org/10.1007/JHEP08(2011)094

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP08(2011)094

Keywords

Search

Navigation