Skip to main content

Constraining the Kähler Moduli in the Heterotic Standard Model

Communications in Mathematical Physics volume 276, pages 1–21 (2007)Cite this article

Abstract

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitly, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Banks T. and Dine M. (1996). Couplings and Scales in Strongly Coupled Heterotic String Theory. Nucl. Phys. B 479: 173–196

    MATH  Article  ADS  MathSciNet  Google Scholar 

  2. Braun V., He Y-H., Ovrut B.A. and Pantev T. (2006). Vector Bundle Extensions, Sheaf Cohomology, and the Heterotic Standard Model. Adv. Theor. Math. Phys. 10: 4

    MathSciNet  Google Scholar 

  3. Braun V., He Y-H., Ovrut B.A. and Pantev T. (2006). Heterotic Standard Model Moduli. JHEP 0601: 025

    Article  ADS  MathSciNet  Google Scholar 

  4. Braun V., He Y-H., Ovrut B.A. and Pantev T. (2006). The Exact MSSM Spectrum from String Theory. JHEP 0605: 043

    Article  ADS  MathSciNet  Google Scholar 

  5. Braun V., Ovrut B.A., Pantev T. and Reinbacher R. (2004). Elliptic Calabi-Yau Threefolds with ℤ3 ×  ℤ3 Wilson Lines. JHEP 0412: 062

    Article  ADS  MathSciNet  Google Scholar 

  6. Curio G. and Krause A. (2001). Nucl.Phys. B 602: 172–200

    MATH  Article  ADS  MathSciNet  Google Scholar 

  7. Curio G. and Krause A. (2004). Nucl.Phys. B 693: 195–222

    MATH  Article  ADS  MathSciNet  Google Scholar 

  8. Donagi R. and Bouchard V. (2006). An SU(5) Heterotic Standard Model. Phys. Lett. B 633: 483–791

    Article  MathSciNet  Google Scholar 

  9. Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford: Oxford University Press, 1990

  10. Douglas M.R. (2003). The Statistics of String/M-Theory Vacua. JHEP 0305: 046

    Article  ADS  Google Scholar 

  11. Douglas M.R., Fiol B. and Römelsberger C. (2005). Stability and BPS branes. JHEP 0509: 006

    Article  ADS  Google Scholar 

  12. Grassi A. and Morrison D.R. (1993). Automorphisms and the Kähler cone of certain Calabi-Yau manifolds. Duke Math. J. 71: 831–838

    MATH  Article  MathSciNet  Google Scholar 

  13. Gukov S., Kachru S., Liu X. and McAllister L. (2004). Heterotic Moduli Stabilization with Fractional Chern- Simons Invariants. Phys. Rev. D 69: 086008

    Article  ADS  MathSciNet  Google Scholar 

  14. Hartshorne R. (1977). Algebraic Geometry. Graduate Texts in Mathematics, No. 52.. Springer-Verlag, New York

    Google Scholar 

  15. Hořava P. and Witten E. (1996). Eleven-dimensional supergravity on a manifold with boundary. Nucl. Phys. B 475: 94–114

    Article  ADS  Google Scholar 

  16. Joyce D.D. (2000). Compact Manifolds with Special Holonomy. Oxford University Press, Oxford

    MATH  Google Scholar 

  17. Looijenga E. (1981). Rational surfaces with an anti-canonical cycle. Ann. of Math. (2) 114: 267–322

    Article  MathSciNet  Google Scholar 

  18. Namikawa Yo. (1991). On the birational structure of certain Calabi-Yau threefolds. J. Math. Kyoto Univ. 31: 151–164

    MATH  MathSciNet  Google Scholar 

  19. Schoen C. (1988). On the fiber products of rational elliptic surfaces with sections. Math. Ann. 197: 177–199

    MATH  MathSciNet  Google Scholar 

  20. Sharpe E. (1998). Kähler Cone Substructure. Adv. Theor. Math. Phys. 2: 1441

    MATH  MathSciNet  Google Scholar 

  21. Wilson P.M.H. (1992). The Kähler cone on Calabi-Yau threefolds. Invent. Math. 107: 561–583

    MATH  Article  ADS  MathSciNet  Google Scholar 

  22. Witten E. (1996). Strong coupling expansion of Calabi-Yau compactification. Nucl. Phys. B 471: 135–158

    MATH  Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CSIC Instituto de Matemáticas y Fisica Fundamental, Madrid, 28006, Spain

    Tomás L. Gómez

  2. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, 08855-0849, USA

    Sergio Lukic

  3. Departamento de Álgebra, Facultad de Matemáticas UCM, Madrid, 28040, Spain

    Tomás L. Gómez & Ignacio Sols

Authors
  1. Tomás L. Gómez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergio Lukic
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ignacio Sols
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomás L. Gómez.

Additional information

Communicated by M.R. Douglas

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gómez, T.L., Lukic, S. & Sols, I. Constraining the Kähler Moduli in the Heterotic Standard Model. Commun. Math. Phys. 276, 1–21 (2007). https://doi.org/10.1007/s00220-007-0338-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0338-8

Keywords

  • Vector Bundle
  • Line Bundle
  • Minimal Supersymmetric Standard Model
  • Heterotic String
  • Holomorphic Vector Bundle