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Hersent, Kilian; Mathieu, Philippe; Wallet, Jean-Christophe

Algebraic structures in \(\kappa\)-Poincaré invariant gauge theories. (English) Zbl 1534.83067

Int. J. Geom. Methods Mod. Phys. 19, No. 5, Article ID 2250078, 20 p. (2022).
Summary: \(\kappa\)-Poincaré invariant gauge theories on \(\kappa\)-Minkowski space-time, which are noncommutative analogs of the usual \(U(1)\) gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the \(\kappa\)-Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermiticity condition obeyed by the gauge potential becomes twisted. Similarities between the present framework and algebraic features of twisted spectral triples are exhibited.

Cited in 2 Documents

MSC:

83C65 Methods of noncommutative geometry in general relativity
83C45 Quantization of the gravitational field
83E15 Kaluza-Klein and other higher-dimensional theories

Keywords:

noncommutative geometry; gauge theory; \(\kappa\)-Minkowski; \(\kappa\)-Poincaré
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[1] S. Doplicher, K. Fredenhagen and J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys. 172 (1995) 187; S. Doplicher, K. Fredenhagen and J. E. Roberts, Space-time quantization induced by classical gravity, Phys. Lett. B331 (1994) 39-44. · Zbl 0847.53051
[2] G. Amelino-Camelia, Quantum spacetime phenomenology, Living Rev. Relativ. 16 (2013) 5; S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Rev. Relativ. 16 (2013) 2. · Zbl 1320.83004
[3] Lukierski, J., Kappa-deformations: Historical developments and recent results, J. Phys. Conf. Ser.804 (2017) 012028.
[4] J. Lukierski, H. Ruegg, A. Nowicki and V. N. Tolstoï, q-deformation of Poincaré algebra, Phys. Lett. B264 (1991) 331; J. Lukierski, A. Nowicki and H. Ruegg, New quantum Poincaré algebra and \(\kappa \)-deformed field theory, Phys. Lett. B293 (1992) 344. · Zbl 0834.17022
[5] V. G. Drinfeld, Quantum groups, in Proc. Int. Congress of Mathematicians, Vols. 1, 2 (American Mathematical Society, Providence, RI, 1987), p. 798; L. A. Takhtadzhyan, Lectures on Quantum Groups, Nankai Lectures on Mathematical Physics, eds. M.-L. Ge and B.-H. Zhao (World Scientific, 1989).
[6] G. Amelino-Camelia, Doubly special relativity, Nature418 (2002) 34; G. Amelino-Camelia, G. Gubitosi, A. Marciano, P. Martinetti and F. Mercati, A no-pure boost uncertainity principle from spacetime noncommutativity, Phys. Lett. B671 (2009) 298; J. Kowalski-Glikman, Introduction to DSR in Planck scale Effects in Astrophysics and Cosmology, Lecture Notes in Physics, Vol. 669 (Springer, Berlin, 2005), p. 131.
[7] G. Amelino-Camelia, Testable scenario for relativity with minimum-length, Phys. Lett. B510 (2001) 255; J. Kowalski-Glikman, Introduction to doubly special relativity, Lect. Notes Phys.669 (2005) 131. · Zbl 1062.83540
[8] Majid, S. and Ruegg, H., Bicrossproduct structure of \(\kappa \)-Poincaré group and non-commutative geometry, Phys. Lett. B334 (1994) 348. · Zbl 1112.81328
[9] Dimitrijević, M., Jonke, L., Möller, L., Tsouchnika, E., Wess, J. and Wohlgennant, M., Deformed field theory on \(\kappa \)-spacetime, Eur. Phys. J. C31 (2003) 129. · Zbl 1465.81070
[10] A. Agostini, G. Amelino-Camelia, M. Arzano and F. D’Andrea, Action functional for kappa-Minkowski noncommutative spacetime, preprint (2004), arXiv:hep-th/0407227; A. Agostini, G. Amelino-Camelia and F. D’Andrea, Hopf-algebra description of noncommutative-spacetime symmetries, Int. J. Mod. Phys. A19 (2004) 5187. · Zbl 1078.81036
[11] A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marciano and R. Altair Tacchi, Generalizing the Noether theorem for Hopf-algebra spacetime symmetries, Mod. Phys. Lett. A22 (2007) 1779; G. Amelino-Camelia and M. Arzano, Coproduct and star-product in field theories on Lie-algebra noncommutative space-times, Phys. Rev. D65 (2002) 084044.
[12] S. Meljanac and A. Samsarov, Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance, Int. J. Mod. Phys. A26 (2011) 1439; E. Harikunmar, T. Jurić and S. Meljanac, Electrodynamics on \(\kappa \)-Minkowski space-time, Phys. Rev. D84 (2011) 085020; S. Meljanac, A. Samsarov, J. Trampetic and M. Wohlgenannt, Scalar field propagation in the \(\phi^4\) kappa-Minkowski model, J. High Energy Phys.12 (2011) 010; H. Grosse and M. Wohlgenannt, On \(\kappa \)-Deformation and UV/IR Mixing, Nucl. Phys. B748 (2006) 473. · Zbl 1306.81305
[13] Mercati, F. and Sergola, M., Pauli-Jordan function and scalar field quantization in \(\kappa \)-Minkowski noncommutative spacetime, Phys. Rev. D98 (2018) 045017.
[14] Poulain, T. and Wallet, J.-C., \( \kappa \)-Poincaré invariant quantum field theories with KMS weight, Phys. Rev. D98 (2018) 025002.
[15] Poulain, T. and Wallet, J.-C., \( \kappa \)-Poincaré invariant orientable field theories at 1-loop, J. High Energy Phys.01 (2019) 064. · Zbl 1409.83070
[16] H. Grosse and R. Wulkenhaar, Renormalisation of \(\varphi^4\)-theory on noncommutative \(\mathbb{R}^2\) in the matrix base, J. High Energy Phys.0312 (2003) 019; H. Grosse and R. Wulkenhaar, Renormalisation of \(\varphi^4\)-theory on noncommutative \(\mathbb{R}^4\) in the matrix base, Commun. Math. Phys.256 (2005) 305; H. Grosse and R. Wulkenhaar, Self-dual noncommutative \(\varphi^4\)-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory, Commun. Math. Phys.329 (2014) 1069. · Zbl 1305.81129
[17] H. Grosse and M. Wohlgenannt, Induced gauge theory on a noncommutative space, Eur. Phys. J. C52 (2007) 435; A. de Goursac, J.-C. Wallet and R. Wulkenhaar, Noncommutative induced gauge theory, Eur. Phys. J. C51 (2007) 977; J.-C. Wallet, Noncommutative induced gauge theories on Moyal spaces, J. Phys. Conf. Ser.103 (2008) 012007; P. Martinetti, P. Vitale and J.-C. Wallet, Noncommutative gauge theories on \(\mathbb{R}_\theta^2\) as matrix models, J. High Energy Phys.09 (2013) 051.
[18] E. Cagnache, T. Masson and J-C. Wallet, Noncommutative Yang-Mills-Higgs actions from derivation based differential calculus, J. Noncommut. Geom.5 (2011) 39-67; D. N. Blaschke, H. Grosse and J.-C. Wallet, Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences, J. High Energy Phys.06 (2013) 038. · Zbl 1226.81279
[19] A. de Goursac, J.-C. Wallet and R. Wulkenhaar, On the vacuum states for noncommutative gauge theory, Eur. Phys. J. C56 (2008) 293-304; A. de Goursac, A. Tanasa and J.-C. Wallet, Vacuum configurations for renormalizable noncommutative scalar models, Eur. Phys. J. C53 (2008) 459. · Zbl 1189.81214
[20] F. Vignes-Tourneret, Renormalisation of the orientable noncommutative Gross-Neveu model, Ann. H. Poincaré8 (2007) 427; A. de Goursac and J.-C. Wallet, Symmetries of noncommutative scalar field theory, J. Phys. A: Math. Theor.44 (2011) 055401; J.-C. Wallet, Connes distance by examples: Homothetic spectral metric spaces, Rev. Math. Phys.24 (2012) 1250027.
[21] J.-C. Wallet, Exact partition functions for gauge theories on \(\mathbb{R}_\lambda^3\), Nucl. Phys. B912 (2016) 354; A. Géré, T. Jurić and J.-C. Wallet, Noncommutative gauge theories on \(\mathbb{R}_\lambda^3\): Perturbatively finite models, J. High Energy Phys.12 (2015) 045; P. Vitale and J.-C. Wallet, Noncommutative field theories on \(\mathbb{R}_\lambda^3\): Toward UV/IR mixing freedom, J. High Energy Phys.04 (2013) 115. · Zbl 1387.81266
[22] Durhuus, B. and Sitarz, A., Star product realizations of kappa-Minkowski space, J. Noncommut. Geom.7 (2013) 605. · Zbl 1282.46063
[23] von Neumann, J., Die eindeutigkeit der Schrödingerschen operatoren, Math. Ann.104 (1931) 570. · Zbl 0001.24703
[24] Weyl, H., Quantenmechanik und gruppentheorie, Z. Phys.46 (1927) 1. · JFM 53.0848.02
[25] T. Jurić, T. Poulain and J.-C. Wallet, Involutive representations of coordinate algebras and quantum spaces, J. High Energy Phys.07 (2017) 116; T. Poulain and J.-C. Wallet, Quantum spaces, central extensions of Lie groups and related quantum field theories, J. Phys.: Conf. Ser.965 (2018) 012032.
[26] M. Dimitrijević, L. Jonke and L. Möller, U(1) gauge field theory on kappa-Minkowski space, J. High Energy Phys.09 (2005) 068; M. Dimitrijević, F. Meyer, L. Möller and J. Wess, Gauge theories on the kappa-Minkowski spacetime, Eur. Phys. J. C36 (2004) 117. · Zbl 1191.81204
[27] Dimitrijević, M., Jonke, L. and Pachol, A., Gauge theory on twisted \(\kappa \)-Minkowski: Old problems and possible solutions, Symmetry Integr. Geom.: Methods Appl.10 (2014) 063 · Zbl 1295.81125
[28] Mathieu, P. and Wallet, J.-C., Gauge theories on \(\kappa \)-Minkowski spaces: Twist and modular operators, J. High Energy Phys.05 (2020) 115. · Zbl 1437.83041
[29] Mathieu, Ph. and Wallet, J.-C., Single extra dimension from \(\kappa \)-Poincaré and gauge invariance, J. High Energy Phys.03 (2021) 209. · Zbl 1461.81071
[30] Mathieu, Ph. and Wallet, J.-C., Twisted BRST symmetry in gauge theories on \(\kappa \)-Minkowski, Phys. Rev. D103 (2021) 086018.
[31] K. Hersent, Ph. Mathieu and J.-C. Wallet, Quantum stability of gauge theories on \(\kappa \)-Minkowski space, preprint (2021), arXiv:2107.14462.
[32] Connes, A. and Moscovici, H., Type III and spectral triples, in Traces in Number Theory, Geometry and Quantum Fields, , Vol. 38 (Vieweg, Wiesbaden2008), p. 57. · Zbl 1159.46041
[33] G. Landi and P. Martinetti, Gauge transformations for twisted spectral triples, Lett. Math. Phys.108 (2018) 2589; P. Martinetti and J. Zanchettin, Twisted spectral triples without the first-order condition, preprint (2021), arXiv:2103.15643.
[34] Chamseddine, A. H., Connes, A. and van Suijlekom, W. D., Beyond the spectral standard model: Emergence of Pati-Salam unification, J. High Energy Phys.11 (2013) 132. · Zbl 1410.81032
[35] Takesaki, M., Theory of Operator Algebras I-III, , Vols. 124, 125, 127 (Springer2002). · Zbl 0990.46034
[36] Matassa, M., A modular spectral triple for \(\kappa \)-Minkowski space, J. Geom. Phys.76 (2014) 025011. · Zbl 1283.83028
[37] M. Matassa, On the spectral and homological dimension of \(\kappa \)-Minkowski space, preprint (2013), arXiv:1309.1054.
[38] Devastato, A. and Martinetti, P., Twisted spectral triple for the standard model and spontaneous breaking of the grand symmetry, Math. Phys. Anal. Geom.20 (2017) 2. · Zbl 1413.58003
[39] Filaci, M., Martinetti, P. and Pesco, S., Minimal twist for the Standard Model in noncommutative geometry: The field content, Phys. Rev. D104 (2021) 025011.
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