Abstract
The Poisson gauge algebra is a semi-classical limit of complete non- commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity Θab(x), especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the covariant derivatives defines the Poisson field strength which is covariant under the Poisson gauge transformations and reproduces the standard U(1) field strength in the commutative limit. We derive the corresponding Bianchi identities. The field equations for the gauge and the matter fields are obtained from the gauge invariant action. We consider different examples of linear in coordinates Poisson structures Θab(x), as well as non-linear ones, and obtain explicit expressions for all proposed constructions. Our model is unique up to invertible field redefinitions and coordinate transformations.
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N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
V. Schomerus, D-branes and deformation quantization, JHEP 06 (1999) 030 [hep-th/9903205] [INSPIRE].
C. Hull and R.J. Szabo, Noncommutative gauge theories on D-branes in non-geometric backgrounds, JHEP 09 (2019) 051 [arXiv:1903.04947] [INSPIRE].
J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000) 161 [hep-th/0001203] [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].
M. Dimitrijević, F. Meyer, L. Möller and J. Wess, Gauge theories on the kappa Minkowski space-time, Eur. Phys. J. C 36 (2004) 117 [hep-th/0310116] [INSPIRE].
D.V. Vassilevich, Twist to close, Mod. Phys. Lett. A 21 (2006) 1279 [hep-th/0602185] [INSPIRE].
R.J. Szabo, Symmetry, gravity and noncommutativity, Class. Quant. Grav. 23 (2006) R199 [hep-th/0606233] [INSPIRE].
M. Dimitrijević and L. Jonke, A Twisted look on kappa-Minkowski: U(1) gauge theory, JHEP 12 (2011) 080 [arXiv:1107.3475] [INSPIRE].
R. Blumenhagen, I. Brunner, V. Kupriyanov and D. Lüst, Bootstrapping non-commutative gauge theories from L∞ algebras, JHEP 05 (2018) 097 [arXiv:1803.00732] [INSPIRE].
M.D. Ćirić, G. Giotopoulos, V. Radovanović and R.J. Szabo, Braided L∞ -Algebras, Braided Field Theory and Noncommutative Gravity, arXiv:2103.08939 [INSPIRE].
V.G. Kupriyanov and R.J. Szabo, Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry, arXiv:2101.12618 [INSPIRE].
M. Kontsevich, Deformation quantization of Poisson manifolds. 1, Lett. Math. Phys. 66 (2003) 157 [q-alg/9709040] [INSPIRE].
V.G. Kupriyanov and D.V. Vassilevich, Star products made (somewhat) easier, Eur. Phys. J. C 58 (2008) 627 [arXiv:0806.4615] [INSPIRE].
V.G. Kupriyanov, L∞ -Bootstrap Approach to Non-Commutative Gauge Theories, Fortsch. Phys. 67 (2019) 1910010 [arXiv:1903.02867] [INSPIRE].
V.G. Kupriyanov, Non-commutative deformation of Chern-Simons theory, Eur. Phys. J. C 80 (2020) 42 [arXiv:1905.08753] [INSPIRE].
A. Weinstein, Symplectic groupoids and Poisson manifolds, Bul l. Am. Math. Soc. 16 (1987) 101.
M.V. Karasev, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 638.
V.G. Kupriyanov and R.J. Szabo, Symplectic realization of electric charge in fields of monopole distributions, Phys. Rev. D 98 (2018) 045005 [arXiv:1803.00405] [INSPIRE].
V.G. Kupriyanov, A hydrogen atom on curved noncommutative space, J. Phys. A 46 (2013) 245303 [arXiv:1209.6105] [INSPIRE].
V. Gáliková, S. Kováčik and P. Prešnajder, Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space, J. Math. Phys. 54 (2013) 122106 [arXiv:1309.4614] [INSPIRE].
P. Vitale and J.-C. Wallet, Noncommutative field theories on \( {\mathbb{Z}}_{\uplambda}^3 \): Toward UV/IR mixing freedom, JHEP 04 (2013) 115 [Addendum ibid. 03 (2015) 115] [arXiv:1212.5131] [INSPIRE].
P. Vitale, Noncommutative field theory on \( {\mathbb{Z}}_{\uplambda}^3 \) , Fortsch. Phys. 62 (2014) 825 [arXiv:1406.1372] [INSPIRE].
A. Géré, T. Jurić and J.-C. Wallet, Noncommutative gauge theories on \( {\mathbb{Z}}_{\uplambda}^3 \) : perturbatively finite models, JHEP 12 (2015) 045 [arXiv:1507.08086] [INSPIRE].
J. Lukierski, A. Nowicki and H. Ruegg, New quantum Poincaré algebra and k deformed field theory, Phys. Lett. B 293 (1992) 344 [INSPIRE].
E. Harikumar, T. Juric and S. Meljanac, Electrodynamics on κ-Minkowski space-time, Phys. Rev. D 84 (2011) 085020 [arXiv:1107.3936] [INSPIRE].
P. Aschieri, A. Borowiec and A. Pachoł, Observables and dispersion relations in κ-Minkowski spacetime, JHEP 10 (2017) 152 [arXiv:1703.08726] [INSPIRE].
F. Lizzi, M. Manfredonia and F. Mercati, The momentum spaces of κ-Minkowski noncommutative spacetime, Nucl. Phys. B 958 (2020) 115117 [arXiv:2001.08756] [INSPIRE].
P. Mathieu and J.-C. Wallet, Gauge theories on κ-Minkowski spaces: twist and modular operators, JHEP 05 (2020) 112 [arXiv:2002.02309] [INSPIRE].
V.G. Kupriyanov, Recurrence relations for symplectic realization of (quasi)-Poisson structures, J. Phys. A 52 (2019) 225204 [arXiv:1805.12040] [INSPIRE].
A.Y. Alekseev and A.Z. Malkin, Symplectic structures associated to Lie-Poisson groups, Commun. Math. Phys. 162 (1994) 147 [hep-th/9303038] [INSPIRE].
S. Gutt, An explicit ∗-product on the cotangent bundle of a Lie group, Lett. Math. Phys. 7 (1983) 249.
N. Durov, S. Meljanac, A. Samsarov and Z. Skoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, J. Algebra 309 (2007) 318.
V.G. Kupriyanov and P. Vitale, Noncommutative ℝd via closed star product, JHEP 08 (2015) 024 [arXiv:1502.06544] [INSPIRE].
V.G. Kupriyanov, M. Kurkov and P. Vitale, κ-Minkowski-deformation of U(1) gauge theory, JHEP 01 (2021) 102 [arXiv:2010.09863] [INSPIRE].
R. Blumenhagen, M. Brinkmann, V. Kupriyanov and M. Traube, On the Uniqueness of L∞ bootstrap: Quasi-isomorphisms are Seiberg-Witten Maps, J. Math. Phys. 59 (2018) 123505 [arXiv:1806.10314] [INSPIRE].
V.G. Kupriyanov and P. Vitale, A novel approach to non-commutative gauge theory, JHEP 08 (2020) 041 [arXiv:2004.14901] [INSPIRE].
M. Dimitrijević Ciric, N. Konjik, M.A. Kurkov, F. Lizzi and P. Vitale, Noncommutative field theory from angular twist, Phys. Rev. D 98 (2018) 085011 [arXiv:1806.06678] [INSPIRE].
L. Cornalba and R. Schiappa, Nonassociative star product deformations for D-brane world volumes in curved backgrounds, Commun. Math. Phys. 225 (2002) 33 [hep-th/0101219] [INSPIRE].
R. Blumenhagen and E. Plauschinn, Nonassociative Gravity in String Theory?, J. Phys. A 44 (2011) 015401 [arXiv:1010.1263] [INSPIRE].
D. Lüst, T-duality and closed string non-commutative (doubled) geometry, JHEP 12 (2010) 084 [arXiv:1010.1361] [INSPIRE].
D. Lüst, E. Malek, E. Plauschinn and M. Syväri, Open-String Non-Associativity in an R-flux Background, JHEP 05 (2020) 157 [arXiv:1903.05581] [INSPIRE].
D. Mylonas, P. Schupp and R.J. Szabo, Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds, JHEP 09 (2012) 012 [arXiv:1207.0926] [INSPIRE].
D. Mylonas, P. Schupp and R.J. Szabo, Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics, J. Math. Phys. 55 (2014) 122301 [arXiv:1312.1621] [INSPIRE].
V.G. Kupriyanov and R.J. Szabo, G2 -structures and quantization of non-geometric M-theory backgrounds, JHEP 02 (2017) 099 [arXiv:1701.02574] [INSPIRE].
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Kupriyanov, V.G. Poisson gauge theory. J. High Energ. Phys. 2021, 16 (2021). https://doi.org/10.1007/JHEP09(2021)016
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DOI: https://doi.org/10.1007/JHEP09(2021)016