Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 063, 22 pages      arXiv:1403.1857      https://doi.org/10.3842/SIGMA.2014.063
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions

Marija Dimitrijević a, Larisa Jonke b and Anna Pachoł c
a) University of Belgrade, Faculty of Physics, Studentski trg 12, 11000 Beograd, Serbia
b) Division of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, Croatia
c) Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland

Received March 10, 2014, in final form June 05, 2014; Published online June 14, 2014

Abstract
We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $\kappa$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider ${U}(1)$ gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.

Key words: $\kappa$-Minkowski; twist; ${U}(1)$ gauge theory; Hodge dual.

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References

  1. Agostini A., Amelino-Camelia G., Arzano M., D'Andrea F., Action functional for kappa-Minkowski noncommutative spacetime, hep-th/0407227.
  2. Aschieri P., Castellani L., Noncommutative $D=4$ gravity coupled to fermions, J. High Energy Phys. 2009 (2009), no. 6, 086, 18 pages, arXiv:0902.3817.
  3. Aschieri P., Castellani L., Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map, J. High Energy Phys. 2012 (2012), no. 7, 184, 27 pages, arXiv:1111.4822.
  4. Aschieri P., Castellani L., Noncommutative gauge fields coupled to noncommutative gravity, Gen. Relativity Gravitation 45 (2013), 581-598, arXiv:1205.1911.
  5. Aschieri P., Dimitrijević M., Meyer F., Wess J., Noncommutative geometry and gravity, Classical Quantum Gravity 23 (2006), 1883-1911, hep-th/0510059.
  6. Ballesteros Á., Bruno N.R., Herranz F.J., A non-commutative Minkowskian spacetime from a quantum AdS algebra, Phys. Lett. B 574 (2003), 276-282, hep-th/0306089.
  7. Ballesteros Á., Herranz F.J., Bruno N.R., Quantum (anti)de Sitter algebras and generalizations of the kappa-Minkowski space, in Proceedings of 11th International Conference on Symmetry Methods in Physics (June 21-24, 2004, Prague), Editors C. Burdik, O. Navratil, S. Posta, Joint Institute for Nuclear Research, Dubna, 2004, 1-20, hep-th/0409295.
  8. Beggs E.J., Majid S., Quantization by cochain twists and nonassociative differentials, J. Math. Phys. 51 (2010), 053522, 32 pages, math.QA/0506450.
  9. Borowiec A., Lukierski J., Pachoł A., Twisting and $\kappa$-Poincaré, arXiv:1312.7807.
  10. Borowiec A., Pachoł A., $\kappa$-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009), 045012, 11 pages, arXiv:0812.0576.
  11. Bu J.-G., Kim H.-C., Lee Y., Vac C.H., Yee J.H., $\kappa$-deformed spacetime from twist, Phys. Lett. B 665 (2008), 95-99, hep-th/0611175.
  12. Burić M., Latas D., Radovanović V., Trampetić J., Chiral fermions in noncommutative electrodynamics: renormalizability and dispersion, Phys. Rev. D 83 (2011), 045023, 10 pages, arXiv:1009.4603.
  13. Chatzistavrakidis A., Jonke L., Matrix theory origins of non-geometric fluxes, J. High Energy Phys. 2013 (2013), no. 2, 040, 34 pages, arXiv:1207.6412.
  14. Connes A., Douglas M.R., Schwarz A., Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998 (1998), no. 2, 003, 35 pages, hep-th/9711162.
  15. Daszkiewicz M., Lukierski J., Woronowicz M., $\kappa$-deformed oscillators, the choice of star product and free $\kappa$-deformed quantum fields, J. Phys. A: Math. Theor. 42 (2009), 355201, 18 pages, arXiv:0807.1992.
  16. Dimitrijević M., Jonke L., A twisted look on kappa-Minkowski: ${\rm U}(1)$ gauge theory, J. High Energy Phys. 2011 (2011), no. 12, 080, 23 pages, arXiv:1107.3475.
  17. Dimitrijević M., Jonke L., Gauge theory on kappa-Minkowski revisited: the twist approach, J. Phys. Conf. Ser. 343 (2012), 012049, 14 pages, arXiv:1110.6767.
  18. Dimitrijević M., Jonke L., Möller L., ${\rm U}(1)$ gauge field theory on $\kappa$-Minkowski space, J. High Energy Phys. 2005 (2005), no. 9, 068, 15 pages, hep-th/0504129.
  19. Dimitrijević M., Jonke L., Möller L., Tsouchnika E., Wess J., Wohlgenannt M., Deformed field theory on $\kappa$-spacetime, Eur. Phys. J. C Part. Fields 31 (2003), 129-138, hep-th/0307149.
  20. Dimitrijević M., Meyer F., Möller L., Wess J., Gauge theories on the $\kappa$-Minkowski spacetime, Eur. Phys. J. C Part. Fields 36 (2004), 117-126, hep-th/0310116.
  21. Drinfel'd V.G., On constant quasiclassical solutions of the Yang-Baxter equations, Soviet Math. Dokl. 28 (1983), 667-671.
  22. Drinfel'd V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457.
  23. Felder G., Shoikhet B., Deformation quantization with traces, Lett. Math. Phys. 53 (2000), 75-86, math.QA/0002057.
  24. Gerstenhaber M., Giaquinto A., Schack S.D., Quantum symmetry, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 9-46.
  25. Giaquinto A., Zhang J.J., Bialgebra actions, twists, and universal deformation formulas, J. Pure Appl. Algebra 128 (1998), 133-151, hep-th/9411140.
  26. Govindarajan T.R., Gupta K.S., Harikumar E., Meljanac S., Meljanac D., Twisted statistics in $\kappa$-Minkowski spacetime, Phys. Rev. D 77 (2008), 105010, 6 pages, arXiv:0802.1576.
  27. Hoppe J., Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. Thesis, Massachusetts Institute of Technology, 1982.
  28. Hoppe J., Membranes and matrix models, hep-th/0206192.
  29. Jambor C., Sykora A., Realization of algebras with the help of star-products, hep-th/0405268.
  30. Jurčo B., Möller L., Schraml S., Schupp P., Wess J., Construction of non-abelian gauge theories on noncommutative spaces, Eur. Phys. J. C Part. Fields 21 (2001), 383-388, hep-th/0104153.
  31. Jurčo B., Schraml S., Schupp P., Wess J., Enveloping algebra-valued gauge transformations for non-abelian gauge groups on non-commutative spaces, Eur. Phys. J. C Part. Fields 17 (2000), 521-526, hep-th/0006246.
  32. Jurić T., Kovačević D., Meljanac S., $\kappa$-deformed phase space, Hopf algebroid and twisting, arXiv:1402.0397.
  33. Jurić T., Meljanac S., Strajn R., Differential forms and $\kappa$-Minkowski spacetime from extended twist, Eur. Phys. J. C 73 (2013), 2472, 8 pages, arXiv:1211.6612.
  34. Kim H.-C., Lee Y., Rim C., Yee J.H., Differential structure on the $\kappa$-Minkowski spacetime from twist, Phys. Lett. B 671 (2009), 398-401, arXiv:0808.2866.
  35. Kowalski-Glikman J., Introduction to doubly special relativity, in Planck Scale Effects in Astrophysics and Cosmology, Lecture Notes in Phys., Vol. 669, Springer, Berlin, 2005, 131-159, hep-th/0405273.
  36. Lu J.-H., Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996), 47-70, q-alg/9505024.
  37. Lukierski J., Nowicki A., Ruegg H., New quantum Poincaré algebra and $\kappa$-deformed field theory, Phys. Lett. B 293 (1992), 344-352.
  38. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., $q$-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  39. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-87.
  40. Madore J., An introduction to noncommutative differential geometry and its physical applications, London Mathematical Society Lecture Note Series, Vol. 257, 2nd ed., Cambridge University Press, Cambridge, 1999.
  41. Meljanac S., Krešić-Jurić S., Differential structure on $\kappa$-Minkowski space, and $\kappa$-Poincaré algebra, Internat. J. Modern Phys. A 26 (2011), 3385-3402, arXiv:1004.4647.
  42. Meljanac S., Samsarov A., Trampetić J., Wohlgenannt M., Scalar field propagation in the $\phi^4$ $\kappa$-Minkowski model, J. High Energy Phys. 2011 (2011), no. 12, 010, 23 pages, arXiv:1111.5553.
  43. Möller L., Second order expansion of action functionals of noncommutative gauge theories, J. High Energy Phys. 2004 (2004), no. 10, 063, 20 pages, hep-th/0409085.
  44. Mylonas D., Schupp P., Szabo R.J., Nonassociative geometry and twist deformations in non-geometric string theory, PoS Proc. Sci. (2014), PoS(ICMP2013), 007, 28 pages, arXiv:1402.7306.
  45. Ogievetsky O., Hopf structures on the Borel subalgebra of ${\rm sl}(2)$, Rend. Circ. Mat. Palermo (2) Suppl. (1994), 185-199.
  46. Ohn C., A $*$-product on ${\rm SL}(2)$ and the corresponding nonstandard quantum-${\rm U}({\mathfrak{sl}}(2))$, Lett. Math. Phys. 25 (1992), 85-88.
  47. Schenkel A., Noncommutative gravity and quantum field theory on noncommutative curved spacetimes, Ph.D. Thesis, University of Würzburg, 2011, arXiv:1210.1115.
  48. Schenkel A., Uhlemann C.F., Field theory on curved noncommutative spacetimes, SIGMA 6 (2010), 061, 19 pages, arXiv:1003.3190.
  49. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  50. Sitarz A., Noncommutative differential calculus on the $\kappa$-Minkowski space, Phys. Lett. B 349 (1995), 42-48, hep-th/9409014.
  51. Steinacker H., Noncommutative geometry and matrix models, PoS Proc. Sci. (2011), PoS(QGQGS2011), 004, 27 pages, arXiv:1109.5521.
  52. Tolstoy V.N., Twisted quantum deformations of Lorentz and Poincaré algebras, arXiv:0712.3962.
  53. Ülker K., Yapışkan B., Seiberg-Witten maps to all orders, Phys. Rev. D 77 (2008), 065006, 9 pages, arXiv:0712.0506.
  54. Xu P., Quantum groupoids, Comm. Math. Phys. 216 (2001), 539-581, math.QA/9905192.

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