Abstract
It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagrangian of the scalar field propagating in a curved noncommutative space. In this interpretation, renormalizability of the model is related to the interaction with the background curvature which introduces explicit coordinate dependence in the action. In this paper we construct the U 1 gauge field on the same noncommutative space: since covariant derivatives contain coordinates, the Yang-Mills action is again coordinate dependent. To obtain a two-dimensional model we reduce to a subspace, which results in splitting of the degrees of freedom into a gauge and a scalar. We define the gauge fixing and show the BRST invariance of the quantum action.
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References
M. Dubois-Violette, J. Madore and R. Kerner, Gauge bosons in a noncommutative geometry, Phys. Lett. B 217 (1989) 485 [SPIRES].
M. Dubois-Violette, J. Madore and R. Kerner, Classical bosons in a noncommutative geometry, Class. Quant. Grav. 6 (1989) 1709 [SPIRES].
M. Dubois-Violette, R. Kerner and J. Madore, Noncommutative Differential Geometry and New Models of Gauge Theory, J. Math. Phys. 31 (1990) 323 [SPIRES].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [SPIRES].
I. Chepelev and R. Roiban, Convergence theorem for non-commutative Feynman graphs and renormalization, JHEP 03 (2001) 001 [hep-th/0008090] [SPIRES].
I. Chepelev and R. Roiban, Renormalization of quantum field theories on noncommutative R d . I: Scalars, JHEP 05 (2000) 037 [hep-th/9911098] [SPIRES].
H. Grosse and R. Wulkenhaar, Renormalisation of ϕ 4 theory on noncommutative R 2 in the matrix base, JHEP 12 (2003) 019 [hep-th/0307017] [SPIRES].
H. Grosse and R. Wulkenhaar, Renormalisation of ϕ 4 theory on noncommutative R 4 in the matrix base, Commun. Math. Phys. 256 (2005) 305 [hep-th/0401128] [SPIRES].
H. Grosse and R. Wulkenhaar, Renormalisation of ϕ 4 -theory on non-commutative R 4 to all orders, Lett. Math. Phys. 71 (2005) 13 [SPIRES].
E. Langmann and R.J. Szabo, Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B 533 (2002) 168 [hep-th/0202039] [SPIRES].
R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, A translation-invariant renormalizable non-commutative scalar model, Commun. Math. Phys. 287 (2009) 275 [arXiv:0802.0791] [SPIRES].
R. Gurau, J. Magnen, V. Rivasseau and F. Vignes-Tourneret, Renormalization of non-commutative ϕ 44 field theory in x space, Commun. Math. Phys. 267 (2006) 515 [hep-th/0512271] [SPIRES].
D.N. Blaschke et al., On the Problem of Renormalizability in Non-Commutative Gauge Field Models. A Critical Review, Fortschr. Phys. 58 (2010) 364 [arXiv:0908.0467] [SPIRES].
A. de Goursac, J.C. Wallet and R. Wulkenhaar, Noncommutative induced gauge theory, Eur. Phys. J. C 51 (2007) 977 [hep-th/0703075] [SPIRES].
H. Grosse and M. Wohlgenannt, Induced Gauge Theory on a Noncommutative Space, Eur. Phys. J. C 52 (2007) 435 [hep-th/0703169] [SPIRES].
A. de Goursac, J.-C. Wallet and R. Wulkenhaar, On the vacuum states for noncommutative gauge theory, Eur. Phys. J. C 56 (2008) 293 [arXiv:0803.3035] [SPIRES].
H. Grosse and R. Wulkenhaar, 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory, arXiv:0709.0095 [SPIRES].
D.N. Blaschke, H. Grosse and M. Schweda, Non-commutative U(1) gauge theory on R 4(⊝) with oscillator term and BRST symmetry, Europhys. Lett. 79 (2007) 61002 [arXiv:0705.4205] [SPIRES].
D.N. Blaschke, A. Rofner, M. Schweda and R.I.P. Sedmik, Improved Localization of a Renormalizable Non-Commutative Translation Invariant U(1) Gauge Model, Europhys. Lett. 86 (2009) 51002 [arXiv:0903.4811] [SPIRES].
M. Burić and M. Wohlgenannt, Geometry of the Grosse-Wulkenhaar Model, JHEP 03 (2010) 053 [arXiv:0902.3408] [SPIRES].
J. Madore, An introduction to noncommutative differential geometry and itsphysical applications, Lond. Math. Soc. Lect. Note Ser. 257 (2000) 1 [SPIRES].
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [SPIRES].
A. Sitarz, Noncommutative differential calculus on the kappa Minkowski space, Phys. Lett. B 349 (1995) 42 [hep-th/9409014] [SPIRES].
G. Amelino-Camelia, A. Marciano and D. Pranzetti, On the 5D differential calculus and translation transformations in 4D kappa-Minkowski noncommutative spacetime, Int. J. Mod. Phys. A 24 (2009) 5445 [arXiv:0709.2063] [SPIRES].
G. Fiore and J. Madore, Leibniz Rules and Reality Conditions, [math/9806071].
E. Cagnache, T. Masson and J.-C. Wallet, Noncommutative Yang-Mills-Higgs actions from derivation- based differential calculus, arXiv:0804.3061 [SPIRES].
A. Connes , Noncommutative Geometry, Academic Press (1994).
H. Aoki et al., Noncommutative Yang-Mills in IIB matrix model, Nucl. Phys. B 565 (2000) 176 [hep-th/9908141] [SPIRES].
J. Ambjørn, Y.M. Makeenko, J. Nishimura and R.J. Szabo, Finite N matrix models of noncommutative gauge theory, JHEP 11 (1999) 029 [hep-th/9911041] [SPIRES].
H. Grosse and J. Madore, A Noncommutative version of the Schwinger model, Phys. Lett. B 283 (1992) 218 [SPIRES].
H. Steinacker, Quantized gauge theory on the fuzzy sphere as random matrix model, Nucl. Phys. B 679 (2004) 66 [hep-th/0307075] [SPIRES].
S. Iso, Y. Kimura, K. Tanaka and K. Wakatsuki, Noncommutative gauge theory on fuzzy sphere from matrix model, Nucl. Phys. B 604 (2001) 121 [hep-th/0101102] [SPIRES].
Y. Kimura, Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model, Prog. Theor. Phys. 106 (2001) 445 [hep-th/0103192] [SPIRES].
A.Y. Alekseev, A. Recknagel and V. Schomerus, Brane dynamics in background fluxes and non-commutative geometry, JHEP 05 (2000) 010 [hep-th/0003187] [SPIRES].
A.H. Chamseddine and J. Fröhlich, The Chern-Simons action in noncommutative geometry, J. Math. Phys. 35 (1994) 5195 [hep-th/9406013] [SPIRES].
C.-S. Chu, Induced Chern-Simons and WZW action in noncommutative spacetime, Nucl. Phys. B 580 (2000) 352 [hep-th/0003007] [SPIRES].
N.E. Grandi and G.A. Silva, Chern-Simons action in noncommutative space, Phys. Lett. B 507 (2001) 345 [hep-th/0010113] [SPIRES].
A.P. Polychronakos, Noncommutative Chern-Simons terms and the noncommutative vacuum, JHEP 11 (2000) 008 [hep-th/0010264] [SPIRES].
H. Steinacker, Emergent Gravity from Noncommutative Gauge Theory, JHEP 12 (2007) 049 [arXiv:0708.2426] [SPIRES].
A. D. Sakharov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Sov. Phys. Dokl. 12 (1968) 1040 [Dokl. Akad. Nauk Ser. Fiz. 177 (1967) 70] [Sov. Phys. Usp. 34 (1991) 394].
D.N. Blaschke, A. Rofner, R.I.P. Sedmik and M. Wohlgenannt, On Non-Commutative U*(1) Gauge Models and Renormalizability, arXiv:0912.2634 [SPIRES].
D.N. Blaschke, H. Grosse, E. Kronberger, M. Schweda and M. Wohlgenannt, Loop Calculations for the Non-Commutative U(1) Gauge Field Model with Oscillator Term, Eur. Phys. J. C 67 (2010) 575 [arXiv:0912.3642] [SPIRES].
J. Madore, Kaluza-Klein aspects of noncommutative geometry, in Differential Geometric Methods in Theoretical Physics, A.I. Solomon ed., World Scientific Publishing, Chester U.K. (1988), pp. 243–252.
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Burić, M., Grosse, H. & Madore, J. Gauge fields on noncommutative geometries with curvature. J. High Energ. Phys. 2010, 10 (2010). https://doi.org/10.1007/JHEP07(2010)010
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DOI: https://doi.org/10.1007/JHEP07(2010)010