Abstract
We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension D and at first order in the noncommutativity parameter θ. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives.
A simple explanation of the equality between noncommutative and commutative CS actions in D = 1 and D = 3 is obtained. The θ dependent terms are present for D ≥ 5 and give a higher derivative theory on commutative space reducing to classical CS theory for θ → 0. These terms depend on the field strength and not on the bare gauge potential.
In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension.
The Seiberg-Witten map on the D = 5 noncommutative CS theory is explored in more detail, and we give its second order θ-expansion for any gauge group. The example of extended D = 5 CS gravity, where the gauge group is SU(2, 2), is treated explicitly.
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References
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
B. Jurčo, S. Schraml, P. Schupp and J. Wess, Enveloping algebra valued gauge transformations for nonAbelian gauge groups on noncommutative spaces, Eur. Phys. J. C 17 (2000) 521 [hep-th/0006246] [INSPIRE].
B. Jurčo, L. Möller, S. Schraml, P. Schupp and J. Wess, Construction of nonAbelian gauge theories on noncommutative spaces, Eur. Phys. J. C 21 (2001) 383 [hep-th/0104153] [INSPIRE].
L. Möller, Second order of the expansions of action functionals of the noncommutative standard model, JHEP 10 (2004) 063 [hep-th/0409085] [INSPIRE].
P. Aschieri and L. Castellani, Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map, JHEP 07 (2012) 184 [arXiv:1111.4822] [INSPIRE].
P. Aschieri, L. Castellani and M. Dimitrijević, Noncommutative gravity at second order via Seiberg-Witten map, Phys. Rev. D 87 (2013) 024017 [arXiv:1207.4346] [INSPIRE].
M. Dimitrijević, V. Radovanović and H. Stefancic, AdS-inspired noncommutative gravity on the Moyal plane, Phys. Rev. D 86 (2012) 105041 [arXiv:1207.4675] [INSPIRE].
E. Di Grezia, G. Esposito, M. Figliolia and P. Vitale, The Seiberg-Witten map for non-commutative pure gravity and vacuum Maxwell theory, Int. J. Geom. Meth. Mod. Phys. 10 (2013) 1350023 [arXiv:1209.1331] [INSPIRE].
M. Dimitrijević and V. Radovanović, Noncommutative SO(2, 3) gauge theory and noncommutative gravity, Phys. Rev. D 89 (2014) 125021 [arXiv:1404.4213] [INSPIRE].
P. Schupp and J. You, UV/IR mixing in noncommutative QED defined by Seiberg-Witten map, JHEP 08 (2008) 107 [arXiv:0807.4886] [INSPIRE].
R. Horvat, A. Ilakovac, J. Trampetic and J. You, Self-energies on deformed spacetimes, JHEP 11 (2013) 071 [arXiv:1306.1239] [INSPIRE].
C.P. Martin, Computing the θ-exact Seiberg-Witten map for arbitrary gauge groups, Phys. Rev. D 86 (2012) 065010 [arXiv:1206.2814] [INSPIRE].
N.E. Grandi and G.A. Silva, Chern-Simons action in noncommutative space, Phys. Lett. B 507 (2001) 345 [hep-th/0010113] [INSPIRE].
B. Jurčo, P. Schupp and J. Wess, NonAbelian noncommutative gauge theory via noncommutative extra dimensions, Nucl. Phys. B 604 (2001) 148 [hep-th/0102129] [INSPIRE].
A.H. Chamseddine and J. Fröhlich, The Chern-Simons action in noncommutative geometry, J. Math. Phys. 35 (1994) 5195 [hep-th/9406013] [INSPIRE].
A.P. Polychronakos, Noncommutative Chern-Simons terms and the noncommutative vacuum, JHEP 11 (2000) 008 [hep-th/0010264] [INSPIRE].
S. Cacciatori and L. Martucci, Noncommutative AdS supergravity in three-dimensions, Phys. Lett. B 542 (2002) 268 [hep-th/0204152] [INSPIRE].
L. Castellani, Chern-Simons supergravities, with a twist, JHEP 07 (2013) 133 [arXiv:1305.1566] [INSPIRE].
A.P. Polychronakos, Seiberg-Witten map and topology, Annals Phys. 301 (2002) 174 [hep-th/0206013] [INSPIRE].
M. Nakahara, Geometry, Topology and Physics, Taylor and Francis (2003).
T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rept. 66 (1980) 213 [INSPIRE].
J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45 (1949) 99 [INSPIRE].
H.J. Groenewold, On the Principles of elementary quantum mechanics, Physica 12 (1946) 405 [INSPIRE].
P. Aschieri and L. Castellani, Noncommutative D = 4 gravity coupled to fermions, JHEP 06 (2009) 086 [arXiv:0902.3817] [INSPIRE].
K. Ulker and B. Yapiskan, Seiberg-Witten maps to all orders, Phys. Rev. D 77 (2008) 065006 [arXiv:0712.0506] [INSPIRE].
L.C.Q. Vilar, O.S. Ventura, R.L. P.G. Amaral, V.E.R. Lemes and L.O. Buffon, On The Complete Seiberg-Witten Map For Theories With Topological Terms, JHEP 04 (2007) 018 [hep-th/0612287] [INSPIRE].
A.H. Chamseddine, Topological Gauge Theory of Gravity in Five-dimensions and All Odd Dimensions, Phys. Lett. B 233 (1989) 291 [INSPIRE].
A.H. Chamseddine, Topological gravity and supergravity in various dimensions, Nucl. Phys. B 346 (1990) 213 [INSPIRE].
R. Troncoso and J. Zanelli, Gauge supergravities for all odd dimensions, Int. J. Theor. Phys. 38 (1999) 1181 [hep-th/9807029] [INSPIRE].
J. Zanelli, Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008), hep-th/0502193 [INSPIRE].
J. Zanelli, Chern-Simons Forms in Gravitation Theories, Class. Quant. Grav. 29 (2012) 133001 [arXiv:1208.3353] [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
F. Izaurieta, E. Rodriguez, P. Minning, P. Salgado and A. Perez, Standard General Relativity from Chern-Simons Gravity, Phys. Lett. B 678 (2009) 213 [arXiv:0905.2187] [INSPIRE].
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Aschieri, P., Castellani, L. Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map. J. High Energ. Phys. 2014, 103 (2014). https://doi.org/10.1007/JHEP11(2014)103
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DOI: https://doi.org/10.1007/JHEP11(2014)103