Skip to main content
SpringerLink
Log in

Deformed Gauge Theory: Twist Versus Seiberg–Witten Approach

  • Chapter
  • First Online:
Book cover Noncommutative Spacetimes

Part of the book series: Lecture Notes in Physics ((LNP,volume 774))

  • 1108 Accesses

In this chapter we discuss two possible ways of introducing gauge theories on noncommutative spaces. In the first approach the algebra of gauge transformations is unchanged, but the Leibniz rule is changed (compared with gauge theories on commutative space). Consistency of the equations of motion requires enveloping algebravalued gauge fields, which leads to new degrees of freedom. In the second approach we have to go to the enveloping algebra again if we want noncommutative gauge transformations to close in the algebra. However, no new degrees of freedom appear here because of the Seiberg–Witten map. This map enables one to express noncommutative gauge parameters and fields in terms of the corresponding commutative variables.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 69.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Aschieri, M. Dimitrijević, F. Meyer, S. Schraml and J. Wess, Twisted gauge theories, Lett. Math. Phys. 78, 61–71 (2006), [hep-th/0603024].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. D. V. Vassilevich, Twist to close, Mod. Phys. Lett. A 21, 1279 (2006), [hep-th/0602185].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 9909, 032 (1999), [hep-th/9908142].

    Article  ADS  MathSciNet  Google Scholar 

  4. B. Jurčo, S. Schraml, P. Schupp and J. Wess, Enveloping algebra valued gauge transformations for non-Abelian gauge groups on non-commutative spaces, Eur. Phys. J. C17, 521 (2000), [hep-th/0006246].

    ADS  Google Scholar 

  5. B. Jurčo, L. Möller, S. Schraml, P. Schupp and J. Wess, Construction of non-Abelian gauge theories on noncommutative spaces, Eur. Phys. J. C21, 383 (2001), [hep-th/0104153 ].

    ADS  Google Scholar 

  6. B. L. Cerchiai, A. F. Pasqua and B. Zumino, The Seiberg-Witten map for noncommutative gauge theories, hep-th/0206231.

    Google Scholar 

  7. L. Bonora, M. Schnabl, M. M. Sheikh-Jabbari and A. Tomasiello, Noncommutative \(SO(n)\) and \(Sp(n)\) gauge theories, Nucl. Phys. B 589, 461 (2000), [hep-th/0006091].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. I. Bars, M. M. Sheikh-Jabbari and M. A. Vasiliev, Noncommutative \(o^*(N)\) and \(usp^*(2N)\) algebras and the corresponding gauge field theories, Phys. Rev. D 64, 086004 (2001), [hep-th/0103209].

    Article  ADS  MathSciNet  Google Scholar 

  9. X. Calmet, B. Jurčo, P. Schupp, J. Wess and M. Wohlgenannt, The standard model on noncommutative spacetime, Eur. Phys. J. C23, 363 (2002), [hep-ph/0111115].

    ADS  Google Scholar 

  10. P. Aschieri, B. Jurčo, P. Schupp and J. Wess, Noncommutative GUTs, standard model and C, P, T, Nucl. Phys. B 651, 45 (2003), [hep-th/0205214].

    Article  MATH  ADS  Google Scholar 

  11. W. Behr, N. G. Deshpande, G. Duplancić, P. Schupp, J. Trampetić, and J. Wess, The \(Z^0 \to \gamma \gamma,\, gg\) decays in the noncommutative standard model, Eur. Phys. J. C29, 441 (2003), [hep-ph/0202121].

    ADS  Google Scholar 

  12. B. Melić, K. Pasek-Kimericki, P. Schupp, J. Trampetić and M. Wohlgennant, The standard model on non-commutative space-time: electroweak currents and higgs sector, Eur. Phys. J. C 42, 483–497 (2005), [hep-ph/0502249].

    ADS  Google Scholar 

  13. Y. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Commun. Math. Phys. 123, 163 (1989).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. J. Wess, Deformed Coordinate Spaces; Derivatives, in Proceedings of the BW2003 Workshop, Vrnjacka Banja, Serbia (2003), [hep-th/0408080].

    Google Scholar 

  15. J. Wess, Gauge theory beyond gauge theory Fortsch.Phys. 49, 377–385 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. J. Lukierski, A. Nowicki, H. Ruegg and V. N. Tolstoy, Q -deformation of Poincaré algebra Phys. Lett. B264, 331 (1991).

    ADS  MathSciNet  Google Scholar 

  17. J. Lukierski, A. Nowicki and H. Ruegg, New quantum Poincaré algebra and \(\kappa\) -deformed field theory, Phys. Lett. B293, 344 (1992).

    ADS  MathSciNet  Google Scholar 

  18. J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane, Nucl. Phys. Proc. Suppl. 18B, 302–312 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  19. A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Springer, Berlin (1997).

    MATH  Google Scholar 

  20. H. Weyl, Quantenmechenik und Gruppentheorie, Z. Phys. 46, 1 (1927).

    Article  ADS  Google Scholar 

  21. J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45, 99 (1949).

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Giller, C. Gonera, P. Kosinski and P. Maslanka, On the consistency of twisted gauge theory, Phys. Lett. B 655, 80–83 (2007), [hep-th/0701014].

    ADS  MathSciNet  Google Scholar 

  23. C. Gonera, P. Kosinski, P. Maslanka and S. Giller, Space-time symmetry of noncommutative field theory, Phys. Lett. B 622, 192 (2005), [hep-th/0504132].

    Article  ADS  MathSciNet  Google Scholar 

  24. P. Aschieri, L. Castellani and M. Dimitrijević, Dynamical noncommutativity and Noether theorem in twisted \(\phi^{\star\,4}\) theory, Lett. Math. Phys. 85, 39–53 (2008), [0803.4325 [hep-th]].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. S. Minwalla, M. Van Raamsdonk and N. Seiberg, Noncommutative perturbative dynamics, JHEP 0002, 020 (2000), [hep-th/9912072].

    Article  ADS  Google Scholar 

  26. M. Van Raamsdonk and N. Seiberg, Comments on noncommutative perturbative dynamics, JHEP 0003, 035 (2000), [hep-th/0002186].

    Article  Google Scholar 

  27. D. Bahns, S. Doplicher, K. Fredenhagen and G. Piacitelli, Ultraviolet finite quantum field theory on quantum spacetime, Commun. Math. Phys. 237 221 (2003), [hep-th/0301100].

    MATH  ADS  MathSciNet  Google Scholar 

  28. G. Piacitelli, DFR perturbative quantum field theory on quantum space time, and wick reduction, Prog. Math. 251, 225 (2007), [hep-th/0511282].

    Article  MathSciNet  Google Scholar 

  29. H. Grosse and R. Wulkenhaar, Renormalization of \(\phi^4\)-theory on noncommutative \(\mathbb{R}^4\) in the matrix base, Commun. Math. Phys. 256, 305 (2005), [hep-th/0401128].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. H. Grosse and R. Wulkenhaar, Renormalization of \(\phi^4\) -theory on noncommutative \(\mathbb{R}^4\) to all orders, Lett. Math. Phys. 71, 13 (2005), [hep-th/0403232].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. L. Álvarez-Gaumé and M. A. Vázquez-Mozo, General properties of noncommutative field theories, Nucl. Phys. B668, 293 (2003), [hep-th/0305093].

    Article  ADS  Google Scholar 

  32. N. Nekrasov and A. S. Schwarz, Instantons on noncommutative \(R^4\), and \((2,0)\) superconformal six dimensional theory, Comm. Math. Phys. 198, 689 (1998), [hep-th/9802068].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. D. Gross and N. Nekrasov, Monopoles and strings in noncommutative gauge theory, JHEP 0007, 034 (2000), [hep-th/0005204].

    Article  ADS  MathSciNet  Google Scholar 

  34. N. Nekrasov, Trieste lectures on solitons in noncommutative gauge theories, hep-th/0011095.

    Google Scholar 

  35. L. Möller, Second order of the expansions of action functionals of the noncommutative standard model, JHEP 0410, 063 (2004) [hep-th/0409085].

    Article  Google Scholar 

  36. G. Barnich, F. Brandt and M. Grigoriev, Local BRST cohomology and Seiberg-Witten maps in noncommutative Yang-Mills theory, Nucl. Phys. B 677, 503–534 (2004), [hep-th/0308092].

    Article  ADS  MathSciNet  Google Scholar 

  37. R. Wulkenhaar, Non-renormalizability of \(\theta\) -expanded noncommutative QED, JHEP 0203, 024 (2002), [hep-th/0112248].

    Article  ADS  MathSciNet  Google Scholar 

  38. H. Grosse and M. Wohlgenannt, Noncommutative QFT and renormalization, J. Phys. Conf. Ser. 53, 764 (2006), [hep-th/0607208].

    Article  Google Scholar 

  39. M. Burić, D. Latas and V. Radovanović, Renormalizability of noncommutative \(SU(N)\) gauge theory, JHEP 0602, 046 (2006), [hep-th/0510133].

    ADS  Google Scholar 

  40. J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C16, 161 (2000), [hep-th/0001203].

    ADS  MathSciNet  Google Scholar 

  41. M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66, 157–216 (2003), [q-alg/9709040].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. B. Jurčo, P. Schupp and J. Wess, Noncommutative gauge theory for Poisson manifolds, Nucl. Phys. B 584, 784 (2000), [hep-th/0005005].

    Article  MATH  ADS  Google Scholar 

  43. B. Jurčo, P. Schupp and J. Wess, Nonabelian noncommutative gauge theory via noncommutative extra dimensions, Nucl. Phys. B 604, 148 (2001), [hep-th/0102129].

    Article  MATH  ADS  Google Scholar 

  44. A. Duenas-Vidal and M. A. Vazquez-Mozo, Twisted invariances of noncommutative gauge theories, 0802.4201[hep-th].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INFN Torino, Alessandria, Italy

    Marija Dimitrijević

Authors
  1. Marija Dimitrijević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marija Dimitrijević .

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer

About this chapter

Cite this chapter

Dimitrijević, M. (2009). Deformed Gauge Theory: Twist Versus Seiberg–Witten Approach. In: Noncommutative Spacetimes. Lecture Notes in Physics, vol 774. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89793-4_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-89793-4_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89792-7

  • Online ISBN: 978-3-540-89793-4

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation