Advertisement

Communications in Mathematical Physics

, Volume 297, Issue 1, pp 95–127 | Cite as

Yukawa Couplings in Heterotic Compactification

  • Lara B. Anderson
  • James Gray
  • Dan Grayson
  • Yang-Hui He
  • André Lukas
Article

Abstract

We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automated investigation of Yukawa couplings for large classes of smooth heterotic compactifications a viable possibility.

Keywords

Vector Bundle Line Bundle Yukawa Coupling Chain Complex Cohomology Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Greene B.R., Kirklin K.H., Miron P.J., Ross G.G.: 27**3 Yukawa Couplings For A Three Generation Superstring Model. Phys. Lett. B 192, 111 (1987)CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Candelas P.: Yukawa Couplings Between (2,1) Forms. Nucl. Phys. B 298, 458 (1988)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Candelas P., Kalara S.: Yukawa couplings for a three generation superstring compactification. Nucl. Phys. B 298, 357 (1988)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    McOrist J., Melnikov I.V.: Summing the Instantons in Half-Twisted Linear Sigma Models. JHEP 0902, 026 (2009)CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Donagi, R., Reinbacher, R., Yau, S.T.: Yukawa couplings on quintic threefolds. http://arxiv.org/abs/hep-th/0605203v1, 2006
  6. 6.
    Donagi R., He Y.H., Ovrut B.A., Reinbacher R.: The particle spectrum of heterotic compactifications. JHEP 0412, 054 (2004)CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Berglund P., Parkes L., Hubsch T.: The Complete Matter Sector In A Three Generation Compactification. Commun. Math. Phys. 148, 57 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology. Cambridge: Cambridge Univ. Pr., 1987Google Scholar
  9. 9.
    Gabella M., He Y.H., Lukas A.: An Abundance of Heterotic Vacua. JHEP 0812, 027 (2008)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Anderson L.B., He Y.H., Lukas A.: Heterotic compactification, an algorithmic approach. JHEP 0707, 049 (2007)CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Candelas P., Dale A.M., Lutken C.A., Schimmrigk R.: Complete Intersection Calabi-Yau Manifolds. Nucl. Phys. B 298, 493 (1988)CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Okonek C., Schneider M., Spindler H.: Vector Bundles on Complex Projective Spaces. Birkhäuser Verlag, Basel (1988)Google Scholar
  13. 13.
    Anderson L.B., He Y.H., Lukas A.: Monad Bundles in Heterotic String Compactifications. JHEP 0807, 104 (2008)CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Anderson, L.B.: Heterotic and M-theory Compactifications for String Phenomenology. Oxford University DPhil Thesis, 2008, http://arxiv.org/abs/0808.3621v1[hep-th], 2008
  15. 15.
    Anderson, L.B., He, Y.H., Lukas, A.: Vector bundle stability in heterotic monad models. In preparationGoogle Scholar
  16. 16.
    Donaldson, S.K.: Some numerical results in complex differential geometry. http://arxiv.org/abs/math/0512625v1[math.DG], 2005. Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic. JHEP 0712, 083 (2007); Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical Calabi-Yau metrics. J. Math. Phys. 49, 032302 (2008). Braun, V., Brelidze, T., Douglas, M.R., Ovrut, B.A.: Calabi-Yau Metrics for Quotients and Complete Intersections. JHEP 0805, 080 (2008)Google Scholar
  17. 17.
    Blumenhagen R., Moster S., Weigand T.: Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds. Nucl. Phys. B 751, 186 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Blumenhagen R., Honecker G., Weigand T.: Loop-corrected compactifications of the heterotic string with line bundles. JHEP 0506, 020 (2005)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Distler J., Greene B.R.: Aspects of (2,0) String Compactifications. Nucl. Phys. B 304, 1 (1988)CrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Lukas A., Ovrut B.A., Waldram D.: On the four-dimensional effective action of strongly coupled heterotic string theory. Nucl. Phys. B 532, 43 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Lukas A., Ovrut B.A., Stelle K.S., Waldram D.: The universe as a domain wall. Phys. Rev. D 59, 086001 (1999)CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Hubsch T.: Calabi-Yau manifolds: A Bestiary for physicists. World Scientific, Singapore (1992)Google Scholar
  23. 23.
    Hartshorne, R.: Algebraic Geometry, Springer. GTM 52, Springer-Verlag, 1977; Griffith, P., Harris, J., Principles of algebraic geometry. New York: Wiley-Interscience, 1978Google Scholar
  24. 24.
    Grayson, D., Stillman, M.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
  25. 25.
    Greuel, G.-M., Pfister, G., Schönemann, H.: Singular: a computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern (2001). Available at http://www.singular.uni-kl.de/
  26. 26.
    Gray, J., He, Y.H., Ilderton, A., Lukas, A.: “STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology.” Comput. Phys. Commun. 180, 107–119 (2009); arXiv:0801.1508 [hep-th]. Gray, J., He, Y.H., Ilderton, A., Lukas, A.: “A new method for finding vacua in string phenomenology,” JHEP 0707 (2007) 023; Gray, J., He, Y.H., Lukas, A.: “Algorithmic algebraic geometry and flux vacua.” JHEP 0609 (2006) 031; The Stringvacua Mathematica package is available at: http://www-thphys.physics.ox.ac.uk/projects/Stringvacua/
  27. 27.
    Braun, V., He, Y.H., Ovrut, B.A., Pantev, T.: “A heterotic standard model.” Phys. Lett. B 618, 252 (2005); “The exact MSSM spectrum from string theory.” JHEP 0605, 043 (2006)Google Scholar
  28. 28.
    Donagi R., He Y.H., Ovrut B.A., Reinbacher R.: Moduli dependent spectra of heterotic compactifications. Phys. Lett. B 598, 279 (2004)CrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Bouchard V., Donagi R.: An SU(5) heterotic standard model. Phys. Lett. B 633, 783 (2006)CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Buchberger, B.: “An Algorithm for Finding the Bases Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal” (German), Phd thesis, Univ. of Innsbruck (Austria), 1965; B. Buchberger, “An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations” (German), Aequationes Mathematicae 4(3), 374–383,1970; English translation can be found in: Buchberger, B., Winkler, F., eds.: “Gröbner Bases and Applications.” Volume 251 of the L.M.S. series, Cambridge: Cambridge University Press, 1998; Proc. of the International Conference “33 Years of Gröbner bases”; See B. Buchberger, “Gröbner Bases: A Short Introduction for Systems Theorists.” p1-19 Lecture Notes in Computer Science, Computer Aided Systems Theory - EUROCAST 2001, Berlin-Heidelberg: Springer, 2001, pp. 1–19Google Scholar
  31. 31.
    Gray, J.: A Simple Introduction to Grobner Basis Methods in String Phenomenology. http://arxiv.org/abs/0901.1662v1[hep-th], 2009
  32. 32.
    Anderson L.B., Gray J., Lukas A., Ovrut B.: The Edge Of Supersymmetry: Stability Walls in Heterotic Theory. Phys. Lett B 677, 190–194 (2009)CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    Anderson L.B., Gray J., Lukas A., Ovrut B.: Stability Walls in Heterotic Theories. JHEP 0909, 026 (2009)CrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Avramov, L.L., Grayson, D.R.: Resolutions and cohomology over complete intersections, In: Computations in algebraic geometry with Macaulay 2, Algorithms Comput. Math., Vol. 8, Berlin: Springer, 2002, pp. 131–178Google Scholar
  35. 35.
    Boardman J.M.: The principle of signs. Enseignement Math. (2) 12, 191–194 (1966)MATHMathSciNetGoogle Scholar
  36. 36.
    Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitre 10. Algèbre homologique, Berlin: Springer-Verlag, 2007, (Reprint of the 1980 original [Paris: Masson])Google Scholar
  37. 37.
    Cartan H., Eilenberg S.: Homological algebra. Princeton University Press, Princeton, N. J. (1956)MATHGoogle Scholar
  38. 38.
    Godement, R.: Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Paris: Hermann, 1964Google Scholar
  39. 39.
    Grayson D.R.: Adams operations on higher K-theory. K-Theory 6(2), 97–111 (1992)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Swan R.G.: Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions. J. Pure Appl. Algebra 144(2), 169–211 (1999)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge: Cambridge University Press, 1994Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Lara B. Anderson
    • 1
    • 5
  • James Gray
    • 2
  • Dan Grayson
    • 3
  • Yang-Hui He
    • 2
    • 4
  • André Lukas
    • 2
  1. 1.Department of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordUK
  3. 3.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  4. 4.Merton CollegeOxfordUK
  5. 5.Institute for Advanced StudySchool of Natural SciencesPrincetonUSA

Personalised recommendations