Advertisement

The European Physical Journal C

, Volume 51, Issue 4, pp 977–987 | Cite as

Noncommutative induced gauge theory

  • A. de Goursac
  • J.-C. Wallet
  • R. Wulkenhaar
Regular Article - Theoretical Physics

Abstract

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ4-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

Keywords

Gauge Theory Gauge Transformation Noncommutative Geometry Gauge Potential Hermitian Structure 
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.
    M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001) [arXiv:hep-th/0106048]ADSCrossRefGoogle Scholar
  2. 2.
    R. Wulkenhaar, J. Geom. Phys. 56, 108 (2006)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Grossmann, G. Loupias, E.M. Stein, Ann. Inst. Fourier 18, 343 (1968)CrossRefGoogle Scholar
  4. 4.
    A. Connes, Noncommutative Geometry (Academic Press Inc., San Diego, 1994), available at http://www.alainconnes.org/downloads.htmlGoogle Scholar
  5. 5.
    A. Connes, M. Marcolli, A Walk in the Noncommutative Garden (2006), available at http://www.alainconnes.org/downloads.htmlGoogle Scholar
  6. 6.
    N. Seiberg, E. Witten, JHEP 9909, 032 (1999) [arXiv:hep-th/9908142]ADSCrossRefGoogle Scholar
  7. 7.
    V. Schomerus, JHEP 9906, 030 (1999) [arXiv:hep-th/9903205]ADSCrossRefGoogle Scholar
  8. 8.
    E. Witten, Nucl. Phys. B 268, 253 (1986)ADSCrossRefGoogle Scholar
  9. 9.
    A. Connes, M.R. Douglas, A.S. Schwarz, JHEP 9802, 003 (1998) [arXiv:hep-th/9711162]ADSCrossRefGoogle Scholar
  10. 10.
    V. Gayral, J.H. Jureit, T. Krajewski, R. Wulkenhaar, arXiv:hep-th/0612048Google Scholar
  11. 11.
    J.M. Gracia-Bondía, J.C. Várilly, J. Math. Phys. 29, 869 (1988)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    J.C. Várilly, J.M. Gracia-Bondía, J. Math. Phys. 29, 880 (1988)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Minwalla, M. Van Raamsdonk, N. Seiberg, JHEP 0002, 020 (2000) [arXiv:hep-th/9912072]ADSCrossRefGoogle Scholar
  14. 14.
    I. Chepelev, R. Roiban, JHEP 0005, 037 (2000) [arXiv:hep-th/9911098]ADSCrossRefGoogle Scholar
  15. 15.
    K.G. Wilson, J.B. Kogut, Phys. Rep. 12, 75 (1974)ADSCrossRefGoogle Scholar
  16. 16.
    J. Polchinski, Nucl. Phys. B 231, 269 (1984)ADSCrossRefGoogle Scholar
  17. 17.
    H. Grosse, R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005) [arXiv:hep-th/0401128]ADSCrossRefGoogle Scholar
  18. 18.
    H. Grosse, R. Wulkenhaar, Commun. Math. Phys. 254, 91 (2005) [arXiv:hep-th/0305066]ADSCrossRefGoogle Scholar
  19. 19.
    R. Gurau, J. Magnen, V. Rivasseau, F. Vignes-Tourneret, Commun. Math. Phys. 267, 515 (2006) [arXiv:hep-th/0512271]ADSCrossRefGoogle Scholar
  20. 20.
    B. Simon, Functional Integration and Quantum Physics (Academic Press, New York, San Francisco, London, 1994)Google Scholar
  21. 21.
    R. Gurau, V. Rivasseau, F. Vignes-Tourneret, Ann. Inst. Henri Poincare 7, 1601 (2006) [arXiv:hep-th/0512071]ADSCrossRefGoogle Scholar
  22. 22.
    E. Langmann, R.J. Szabo, K. Zarembo, JHEP 0401, 017 (2004) [arXiv:hep-th/0308043]ADSCrossRefGoogle Scholar
  23. 23.
    E. Langmann, R.J. Szabo, K. Zarembo, Phys. Lett. B 569, 95 (2003) [arXiv:hep-th/0303082]ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    H. Grosse, R. Wulkenhaar, JHEP 0312, 019 (2003) [arXiv:hep-th/0307017]ADSCrossRefGoogle Scholar
  25. 25.
    H. Grosse, H. Steinacker, Nucl. Phys. B 746, 202 (2006) [arXiv:hep-th/0512203]ADSCrossRefGoogle Scholar
  26. 26.
    H. Grosse, H. Steinacker, JHEP 0608, 008 (2006) [arXiv:hep-th/0603052]ADSCrossRefGoogle Scholar
  27. 27.
    H. Grosse, H. Steinacker, arXiv:hep-th/0607235Google Scholar
  28. 28.
    D.J. Gross, A. Neveu, Phys. Rev. D 10, 3235 (1974)ADSCrossRefGoogle Scholar
  29. 29.
    P.K. Mitter, P.H. Weisz, Phys. Rev. D 8, 4410 (1973)ADSCrossRefGoogle Scholar
  30. 30.
    C. Kopper, J. Magnen, V. Rivasseau, Commun. Math. Phys. 169, 121 (1995)ADSCrossRefGoogle Scholar
  31. 31.
    F. Vignes-Tourneret, arXiv:math-ph/0606069, to appear in Ann. H. PoincaréGoogle Scholar
  32. 32.
    F. Vignes-Tourneret, Renormalisation des théories de champs non commutatives, arXiv:math-ph/0612014, Ph.D. thesis, Université Paris 11Google Scholar
  33. 33.
    A. Lakhoua, F. Vignes-Tourneret, J.C. Wallet, arXiv:hep-th/0701170Google Scholar
  34. 34.
    E.T. Akhmedov, P. DeBoer, G.W. Semenoff, JHEP 0106, 009 (2001) [arXiv:hep-th/0103199]ADSCrossRefGoogle Scholar
  35. 35.
    E.T. Akhmedov, P. DeBoer, G.W. Semenoff, Phys. Rev. D 64, 065005 (2001) [arXiv:hep-th/0010003]ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    E. Langmann, R.J. Szabo, Phys. Lett. B 533, 168 (2002) [arXiv:hep-th/0202039]ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Hayakawa, Phys. Lett. B 478, 394 (2000) [arXiv:hep-th/9912094]ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    A. Matusis, L. Susskind, N. Toumbas, JHEP 0012, 002 (2000) [arXiv:hep-th/0002075]ADSCrossRefGoogle Scholar
  39. 39.
    H. Grosse, M. Wohlgenannt, arXiv:hep-th/0703169Google Scholar
  40. 40.
    M. Dubois-Violette, R. Kerner, J. Madore, J. Math. Phys. 31, 323 (1990)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    M. Dubois-Violette, T. Masson, J. Geom. Phys. 25, 104 (1998)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    T. Masson, J. Geom. Phys. 31, 142 (1999)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    T. Masson, E. Sérié, J. Math. Phys. 46, 123503 (2005)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    V. Gayral, Ann. Inst. Henri Poincare 6, 991 (2005) [arXiv:hep-th/0412233]ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    A. Connes, A.H. Chamseddine, J. Geom. Phys. 57, 1 (2006) [arXiv:hep-th/0605011]ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    M. Wodzicki, Invent. Math. 75, 143 (1984)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    O. Piguet, S.P. Sorella, Lect. Notes Phys. M 28, 1 (1995)CrossRefGoogle Scholar
  48. 48.
    H. Grosse, M. Wohlgenannt, J. Phys.: Conf. Ser. 53, 764 (2006) [arXiv:hep-th/0607208]Google Scholar
  49. 49.
    R. Gurau, V. Rivasseau, arXiv:math-ph/0606030Google Scholar
  50. 50.
    V. Rivasseau, A. Tanasa, arXiv:math-ph/0701034Google Scholar

Copyright information

© Springer-Verlag / Società Italiana di Fisica 2007

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, Bât. 210Université Paris XIOrsay CedexFrance
  2. 2.Mathematisches Institut der Westfälischen Wilhelms-UniversitätMünsterGermany

Personalised recommendations