Research ArticlesNo Access

The Gribov problem in noncommutative gauge theory

Maxim Kurkov

Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

INFN, Sezione di Napoli, Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

E-mail Address: max.kurkov@gmail.com

Search for more papers by this author

and

Patrizia Vitale

http://orcid.org/0000-0002-5146-410X

Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

INFN, Sezione di Napoli, Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

E-mail Address: patrizia.vitale@na.infn.it

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219887818501190Cited by:6 (Source: Crossref)

Abstract

After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topology of the bundle of gauge connections, we show that there is a similar feature for noncommutative QED over Moyal space, despite the structure group being Abelian, and we exhibit an infinite number of solutions for the equation of Gribov copies. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.

Keywords:

AMSC: 81R60, 81T13

References

1. V. N. Gribov , Quantization of nonabelian gauge theories, Nucl. Phys. B 139 (1978) 1. Crossref, Web of Science, Google Scholar
2. I. M. Singer , Some remarks on the Gribov ambiguity, Commun. Math. Phys. 60 (1978) 7. Crossref, Web of Science, Google Scholar
3. M. S. Narasimhan and T. R. Ramadas , Geometry of SU(2) gauge fields, Commun. Math. Phys. 67 (1979) 121, https://doi.org/10.1007/BF01221361. Crossref, Web of Science, Google Scholar
4. F. Canfora, M. Kurkov, L. Rosa and P. Vitale , The Gribov problem in noncommutative QED, J. High Energy Phys. 1601 (2016) 014, arXiv: 1505.06342 [hep-th]. Crossref, Web of Science, Google Scholar
5. V. Parameswaran Nair , Quantum Field Theory. A Modern Perspective (Springer, 2004). Google Scholar
6. S. Galluccio, F. Lizzi and P. Vitale , Twisted noncommutative field theory with the Wick-Voros and Moyal products, Phys. Rev. D 78 (2008) 085007, https://doi.org/10.1103/PhysRevD.78.085007, arXiv: 0810.2095 [hep-th]. Crossref, Web of Science, Google Scholar
7. V. G. Kupriyanov and P. Vitale , Noncommutative $ℝ^{d}$ via closed star product, J. High Energy Phys. 1508 (2015) 024, https://doi.org/10.1007/JHEP08(2015)024, arXiv: 1502.06544 [hep-th]. Crossref, Web of Science, Google Scholar
8. I. E. Segal, Quantized differential forms, Topology 8 (1967) 147; Quantization of the de Rham complex, Proc. Sympos. Pure Math. 16 (1970) 205. Google Scholar
9. M. Dubois-Violette, Dérivations et calcul différentiel non commutatif, C. R. Acad. Sci. Paris, Série I 307 (1988) 403; M. Dubois-Violette and P. W. Michor, Dérivations et calcul différentiel non commutatif II, C. R. Acad. Sci. Paris, Série I 319 (1994) 927. Google Scholar
10. G. Landi and G. Marmo , Algebraic differential calculus for gauge theories, Nucl. Phys. Proc. Suppl. A 18 (1990) 171. Crossref, Google Scholar
11. J.-C. Wallet, Derivations of the Moyal algebra and noncommutative gauge theories, SIGMA 5 (2009) 013; E. Cagnache, T. Masson and J.-C. Wallet, Noncommutative Yang–Mills–Higgs actions from derivation based differential calculus, J. Noncommut. Geom. 5 (2011) 39, arXiv:0804.3061. Google Scholar
12. P. Martinetti, P. Vitale and J. C. Wallet , Noncommutative gauge theories on $ℝ_{𝜃}^{2}$ as matrix models, J. High Energy Phys. 1309 (2013) 051, https://doi.org/10.1007/JHEP09(2013)051, arXiv: 1303.7185 [hep-th]. Crossref, Web of Science, Google Scholar
13. A. Géré, P. Vitale and J. C. Wallet , Quantum gauge theories on noncommutative three-dimensional space, Phys. Rev. D 90 (2014) 045019, https://doi.org/10.1103/PhysRevD.90.045019, arXiv: 1312.6145 [hep-th]. Crossref, Web of Science, Google Scholar
14. G. Marmo, P. Vitale and A. Zampini , Noncommutative differential calculus for Moyal subalgebras, J. Geom. Phys. 56 (2006) 611. Crossref, Web of Science, Google Scholar
15. I. Chepelev and R. Roiban , Renormalization of quantum field theories on noncommutative $ℝ^{d}$ . 1. Scalars, J. High Energy Phys. 0005 (2000) 037, arXiv: hep-th/9911098. Crossref, Google Scholar
16. S. Minwalla, M. Van Raamsdonk and N. Seiberg , Noncommutative perturbative dynamics, J. High Energy Phys. 0002 (2000) 020, arXiv: hep-th/9912072. Crossref, Google Scholar
17. M. Hayakawa , Perturbative analysis on infrared aspects of noncommutative QED on $ℝ^{4}$ , Phys. Lett. B 478 (2000) 394, arXiv: hep-th/9912094. Crossref, Web of Science, Google Scholar
18. A. Matusis, L. Susskind and N. Toumbas , The IR/UV connection in the noncommutative gauge theories, J. High Energy Phys. 0012 (2000) 002, arXiv: hep-th/0002075. Crossref, Google Scholar
19. D. N. Blaschke, F. Gieres, E. Kronberger, M. Schweda and M. Wohlgenannt , Translation-invariant models for noncommutative gauge fields, J. Phys. A 41 (2008) 252002, arXiv: 0804.1914 [hep-th]. Crossref, Google Scholar
20. D. N. Blaschke, A. Rofner, R. I. P. Sedmik and M. Wohlgenannt , On non-commutative U*(1) gauge models and renormalizability, J. Phys. A 43 (2010) 425401, arXiv: 0912.2634 [hep-th]. Crossref, Google Scholar
21. D. N. Blaschke, F. Gieres, F. Heindl, M. Schweda and M. Wohlgenannt , BPHZ renor- malization and its application to non-commutative field theory, Eur. Phys. J. C 73 (2013) 2566 arXiv: 1307.4650 [hep-th]. Crossref, Web of Science, Google Scholar
22. R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A translation-invariant renormalizable noncommutative scalar model, Commun. Math. Phys. 287 (2009) 275, arXiv:0802.0791 [math-ph]; A. Tanasa and P. Vitale, Curing the UV/IR mixing for field theories with translation-invariant ⋆ products, Phys. Rev. D 81 (2010) 065008, doi:10.1103/PhysRevD.81.065008, arXiv:0912.0200 [hep-th]. Google Scholar
23. G. F. Dell’Antonio and D. Zwanziger, Ellipsoidal bound on the Gribov horizon contradicts the perturbative renormalization group, Nucl. Phys. B 326 (1989) 333; Every gauge orbit passes inside the Gribov horizon, Comm. Math. Phys. 138 (1991) 291–299. Google Scholar
24. M. de Cesare, G. Esposito and H. Ghorbani , Size of the Gribov region in curved spacetime, Phys. Rev. D 88 (2013) 087701. Crossref, Web of Science, Google Scholar
25. F. Canfora, A. Giacomini and J. Oliva , Gribov pendulum in the Coulomb gauge on curved spaces, Phys. Rev. D 84 (2011) 105019. Crossref, Web of Science, Google Scholar
26. D. Zwanziger Action from the Gribov horizon, Nucl. Phys. B 321 (1989) 591; Local and renormalizable action from the Gribov horizon, Nucl. Phys. B 323 (1989) 513; G. F. Dell’Antonio and D. Zwanziger, Every gauge orbit passes inside the Gribov horizon Comm. Math. Phys. 138 (1991) 291–299. Google Scholar
27. D. N. Blaschke , Aspects of perturbative quantum field theory on non-commutative spaces, PoS CORFU2015 104 (2016), arXiv: 1601.03109 [hep-th]. Google Scholar
28. A. Pinzul and A. Stern , Gauge theory of the star product, Nucl. Phys. B 791 (2008) 284, arXiv: 0705.1785 [hep-th]. Crossref, Web of Science, Google Scholar