Gauge theories on quantum spaces. (English) Zbl 1523.83080
Summary: We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.
MSC:
| 83F05 | Relativistic cosmology |
| 83C45 | Quantization of the gravitational field |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 83C65 | Methods of noncommutative geometry in general relativity |
| 51B20 | Minkowski geometries in nonlinear incidence geometry |
| 81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 46L87 | Noncommutative differential geometry |
| 14D15 | Formal methods and deformations in algebraic geometry |
Keywords:
gauge theories; quantum spaces; noncommutative geometry; quantum space-times; quantum symmetries; quantum gravity; kappa-Minkowski space-time; Moyal spaces; quantum fluctuations; renormalization; noncommutative field theories; noncommutative differential geometry; deformations theoryReferences:
| [1] | A. Connes, Noncommutative Geometry, Academic Press, 1994, alainconnes.org.; Gracia-Bondía, J. M.; Várilly, J. C.; Figueroa, H., (Elements of Noncommutative Geometry. Elements of Noncommutative Geometry, Birkhaüser Advanced Texts (2001), Birkhaüser Boston: Birkhaüser Boston Basel, Berlin) |
| [2] | Gracia-Bondía, J. M.; Lizzi, F.; Marmo, G.; Vitale, P., Infinitely many star-products to play with, J. High Energy Phys., 04, 026 (2002), arXiv:hep-th/0112092 |
| [3] | Snyder, H. S., Quantized space-time, Phys. Rev., 71, 38 (1947) · Zbl 0035.13101 |
| [4] | Witten, E., Non-commutative geometry and string field theory, Nuclear Phys. B, 268, 253-294 (1986), Academia.edu |
| [5] | Connes, A.; Douglas, M. R.; Schwarz, A., Noncommutative geometry and Matrix theory, J. High Energy Phys., 1998, 003 (1998), arXiv:hep-th/9711162 |
| [6] | Seiberg, N.; Witten, E., String theory and noncommutative geometry, J. High Energy Phys., 09, 032 (1999), arXiv:hep-th/9908142, see also; Schomerus, V., D-branes and deformation quantization, JHEP, 06, 030 (1999), arXiv:hep-th/9903205 · Zbl 0957.81085 |
| [7] | Doplicher, S.; Fredenhagen, K.; Roberts, J. E., The quantum structure of space-time at the Planck scale and quantum fields, Comm. Math. Phys., 172, 187 (1995), arXiv:hep-th/0303037 · Zbl 0847.53051 |
| [8] | Gracia-Bondia, J. M.; Várilly, J. C., Algebras of distributions suitable for phase-space quantum mechanics I, J. Math. Phys., 29, 869 (1988), researchgate.net; Várilly, J. C.; Gracia-Bondia, J. M., Algebras of distributions suitable for phase-space quantum mechanics II: Topologies on the moyal algebra, J. Math. Phys., 29, 880 (1988) · Zbl 0652.46027 |
| [9] | Lukierski, J., \( \kappa \)-Deformations: historical developments and recent results, J. Phys. Conf. Ser., 804, Article 012028 pp. (2017), arXiv:1611.10213 |
| [10] | Lukierski, J.; Ruegg, H.; Nowicki, A.; Tolstoy, V. N., \(q\)-Deformation of Poincare algebrá, Phys. Lett. B, 264, 331-338 (1991) |
| [11] | Majid, S.; Ruegg, H., Bicrossproduct structure of \(\kappa \)-Poincare group and non-commutative geometry, Phys. Lett. B, 334, 348-354 (1994), arXiv:hep-th/9405107 · Zbl 1112.81328 |
| [12] | Oriti, D., The group field theory approach to quantum gravity: some recent results, (Kowalski-Glikman, J.; Kowalski-Glikman, J.; Durka, R.; Szczachor, M., THE PLANCK SCALE: Proceedings of the XXV Max Born Symposium. THE PLANCK SCALE: Proceedings of the XXV Max Born Symposium, AIP: conference proceedings (2009)), arXiv:0912.2441 |
| [13] | Addazi, A., Quantum gravity phenomenology at the dawn of the multi-messenger era - A review, Prog. Part. Nucl. Phys., 125, Article 103948 pp. (2022), arXiv:2111.05659 |
| [14] | Douglas, M. R.; Nekrasov, N. A., Noncommutative field theory, Rev. Modern Phys., 73, 977 (2001), arXiv:hep-th/0106048; Szabo, R. J., Quantum field theory on noncommutative spaces, Phys. Rep., 378, 207 (2003), arXiv:hep-th/0109162 |
| [15] | Minwalla, S.; Van Raamsdonk, M.; Seiberg, N., Noncommutative perturbative dynamics, J. High Energy Phys., 02, 020 (2000), arXiv:hep-th/9912072 · Zbl 0959.81108 |
| [16] | For deformation quantization of sympletic and Poisson manifolds see; Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, A., Deformation theory and quantization I: Deformations of symplectic structures, Ann. Physics. Ann. Physics, Ann. Phys., 111, 111 (1978), Academia.edu · Zbl 0377.53025 |
| [17] | De Wilde, M.; Lecomte, P. B., Existence of star-products and formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys., 7, 487 (1983) · Zbl 0526.58023 |
| [18] | Rieffel, M. A., Deformation quantization of Heisenberg manifolds, Comm. Math. Phys.. Comm. Math. Phys., Amer. J. Math., 112, 657 (1990), projecteuclid.org · Zbl 0713.58054 |
| [19] | Kontsevich, M., Deformation quantization of Poisson manifolds I, Lett. Math. Phys., 66, 157 (2003), arXiv:q-alg/9709040 · Zbl 1058.53065 |
| [20] | Drinfeld, V. G., Quasi-Hopf algebras, Lengingrad Math. J., 1, 1419-1457 (1990) · Zbl 0718.16033 |
| [21] | Dubois-Violette, M., (Lectures on Graded Differential Algebras and Noncommutative Geometry. Lectures on Graded Differential Algebras and Noncommutative Geometry, Noncommutative Differential Geometry and its Applications to Physics (2001), Springer Netherlands), 245-306, arXiv:math/9912017 · Zbl 1038.58004 |
| [22] | Dubois-Violette, M.; Michor, P. W., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys., 20, 218-232 (1996), arXiv:q-alg/9503020 · Zbl 0867.53023 |
| [23] | Madore, J., Gravity on fuzzy space-time, ArXiv (1997), arXiv:gr-qc/9709002 · Zbl 0957.83525 |
| [24] | von Neumann, J., Die eindeutigkeit der Schrödingerschen operatoren (in German), Math. Ann., 104, 570 (1931) · Zbl 0001.24703 |
| [25] | Weyl, H., Quantenmechanik und gruppentheorie (in German), Z. Phys., 46, 1 (1927) · JFM 53.0848.02 |
| [26] | Klimyk, A.; Schmüdgen, K., Quantum Groups and their Representations (1997), Springer Science & Business Media · Zbl 0891.17010 |
| [27] | Majid, S., Foundations of Quantum Group Theory (1995), Cambridge University Press · Zbl 0857.17009 |
| [28] | Oeckl, R., Untwisting noncommutative and the equivalence of quantum field theories, Nuclear Phys. B, 581, 559-574 (2000), arXiv:hep-th/0003018 · Zbl 0984.81164 |
| [29] | Aschieri, P.; Schenkel, A., Noncommutative connections on bimodules and Drinfeld twist deformation, ArXiv (2012), arXiv:1210.0241 · Zbl 1317.14008 |
| [30] | Chaichian, M.; Kulish, P. P.; Nishijima, K.; Tureanu, A., On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT, Phys. Lett. B, 604, 98-102 (2004), arXiv:hep-th/0408069 · Zbl 1247.81518 |
| [31] | Chaichian, M.; Prešnajder, P.; Tureanu, A., New concept of relativistic invariance in noncommutative space-time: Twisted poincaré symmetry and its implications, Phys. Rev. Lett., 94 (2005), arXiv:hep-th/0409096 |
| [32] | Dubois-Violette, M., Dérivations et calcul différentiel non commutatif (in French), C. R. Acad. Sci. Paris, Sér. I, 307, 403-408 (1988), Academie des Sciences · Zbl 0661.17012 |
| [33] | Koszul, J. L., Lectures on Fiber Bundles and Differential Geometry (1960), Tata Institute of Fundamental Research publications: Tata Institute of Fundamental Research publications Bombay, math.tifr.res.in |
| [34] | Koszul, J. L., Homologie et cohomologie des algèbres de Lie (in French), Bull. Soc. Math., 78, 65 (1950), numdam.org · Zbl 0039.02901 |
| [35] | Dubois-Violette, M.; Michor, P. W., Dérivations et calcul différentiel non commutatif II (in French), C. R. Acad. Sci. Paris, Sér. I, 319, 927-931 (1994), Académie des Sciences, CERN Document Server · Zbl 0829.16028 |
| [36] | Majid, S.; Oeckl, R., Twisting of quantum differentials and the Planck scale Hopf algebra, Comm. Math. Phys., 205, 617-655 (1999), arXiv:math/9811054 · Zbl 0939.58007 |
| [37] | Sitarz, A., Twists and spectral triples for isospectral deformations, Lett. Math. Phys., 58, 69-79 (2001), arXiv:math/0102074 · Zbl 1020.17012 |
| [38] | Aschieri, P.; Dimitrijević, M.; Meyer, F.; Wess, J., Noncommutative geometry and gravity, Classical Quantum Gravity, 23, 1883-1911 (2006), arXiv:hep-th/0510059 · Zbl 1091.83022 |
| [39] | Borowiec, A.; Pachoł, A., \( \kappa \)-Minkowski space-time as the result of Jordanian twist deformation, Phys. Rev. D, 79 (2009), arXiv:0812.0576 |
| [40] | Grosse, H.; Wulkenhaar, R., Power-counting theorem for non-local matrix models and renormalisation, Comm. Math. Phys., 254, 91 (2005), arXiv:hep-th/0305066; Grosse, H.; Wulkenhaar, R., Renormalisation of \(\phi^4\)-theory on noncommutative \(\mathbb{R}^2\) in the matrix base, JHEP, 0312, 019 (2003), arXiv:hep-th/0307017; Grosse, H.; Wulkenhaar, R., Renormalisation of \(\phi^4\)-theory on noncommutative \(\mathbb{R}^4\) in the matrix base, Commun. Math. Phys., 256, 305 (2005), arXiv:hep-th/0401128 · Zbl 1075.82005 |
| [41] | Marmo, G.; Vitale, P.; Zampini, A., Noncommutative differential calculus for Moyal subalgebras, J. Geom. Phys., 56, 611 (2006), arXiv:hep-th/0411223 · Zbl 1100.46044 |
| [42] | Wallet, J.-C., Derivations of the Moyal algebra and noncommutative gauge theories, SIGMA, 5, 013 (2009), arXiv:0811.3850 · Zbl 1160.81470 |
| [43] | Cagnache, E.; Masson, T.; Wallet, J-C., Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus, J. Noncommut. Geom., 5, 39-67 (2011), arXiv:0804.3061 · Zbl 1226.81279 |
| [44] | Martin, C. P.; Ruiz Ruiz, F.; dominance, Paramagnetic., The sign of the beta function and UV/IR mixing in non-commutative \(U ( 1 )\), Nuclear Phys. B, 597, 197 (2001), arXiv:hep-th/0007131 · Zbl 0972.81185 |
| [45] | Armoni, A., Comments on perturbative dynamics of non-commutative Yang-Mills theory, Nuclear Phys. B, 593, 229 (2001), arXiv:hep-th/0005208 · Zbl 0971.81522 |
| [46] | Armoni, A.; Lopez, E., UV/IR mixing via closed strings and tachyonic instabilities, Nuclear Phys. B, 632, 240 (2002), arXiv:hep-th/0110113 · Zbl 0995.81510 |
| [47] | Martin, C. P.; Sanchez-Ruiz, D., The BRS invariance of noncommutative \(U ( N )\) yang-mills theory at the one-loop level, Nuclear Phys. B, 598, 348 (2001), arXiv:hep-th/0012024 · Zbl 1046.81528 |
| [48] | Liao, Y., One-loop renormalization of spontaneously broken \(U ( 2 )\) gauge theory on noncommutative space-time, J. High Energy Phys., 11, 067 (2001), arXiv:hep-th/0110112 |
| [49] | Aoki, H.; Ishibashi, N.; Iso, S.; Kawai, H.; Kitazawa, Y.; Tada, T., Noncommutative Yang-Mills in IIB matrix model, Nuclear Phys. B, 565, 176 (2000), arXiv:hep-th/9908141 · Zbl 0956.81089 |
| [50] | Steinacker, H., Emergent gravity from noncommutative gauge theory, J. High Energy Phys., 2007, 049 (2007), arXiv:0708.2426 · Zbl 1246.81153 |
| [51] | Madore, J.; Schraml, S.; Schupp, P.; Wess, J., Gauge theory on noncommutative spaces, Eur. Phys. J. C, 16, 161 (2000), arXiv:hep-th/0001203 |
| [52] | Grosse, H.; Steinacker, H.; Wohlgenannt, M.; Gravity, Emergent., Matrix models and UV/IR mixing, J. High Energy Phys., 04, 023 (2008), arXiv:0802.0973 · Zbl 1246.81162 |
| [53] | Bichl, A. A.; Grimstrup, J. M.; Popp, L.; Schweda, M.; Wulkenhaar, R., Perturbative analysis of the Seiberg-Witten Map, Internat. J. Modern Phys. A, 17, 2219 (2002), arXiv:hep-th/0102044, and; Wulkenhaar, R., Non-renormalizability of \(\theta \)-expanded noncommutative QED, JHEP, 03, 024 (2002), arXiv:hep-th/0112248 · Zbl 1099.81520 |
| [54] | de Goursac, A.; Wallet, J.-C.; Wulkenhaar, R., Noncommutative induced gauge theory, Eur. Phys. J. C, 51, 977 (2007), arXiv:hep-th/0703075 · Zbl 1189.81215 |
| [55] | de Goursac, A.; Wallet, J.-C.; Wulkenhaar, R., On the vacuum states for noncommutative gauge theory, Eur. Phys. J. C, 56, 293 (2008), arXiv:0803.3035 · Zbl 1189.81214 |
| [56] | Hayakawa, M., Perturbative analysison infrared aspects of noncommutative QED on \(\mathbb{R}^4\), Phys. Lett. B, 478, 394 (2000), arXiv:hep-th/9912094 · Zbl 1050.81719 |
| [57] | Matusis, A.; Susskind, L.; Toumbas, N., The IR/UV connection in the non-commutative gauge theories, J. High Energy Phys., 12, 002 (2000), arXiv:hep-th/0002075 · Zbl 0990.81549 |
| [58] | Blaschke, D. N.; Kronberger, E.; Rofner, A.; Schweda, M.; Sedmik, R. I.P.; M. Wohlgenannt, T., On the problem of renormalizability in non-commutative gauge field models - A critical review, Fortschr. Phys., 58, 364 (2010), arXiv:0908.0467 · Zbl 1191.81200 |
| [59] | Ruiz Ruiz, F., Gauge-fixing independence of IR divergences in non-commutative u(1), perturbative tachyonic instabilities and supersymmetry, Phys. Lett. B, 502, 274 (2001), ArXiv:Hep-Th/0012171 · Zbl 0977.81156 |
| [60] | Attems, M.; Blaschke, D. N.; Ortner, M.; Schweda, M.; Stricker, S.; Weiretmayr, M., Gauge independence of IR singularities in non-commutative QFT - and interpolating gauges, J. High Energy Phys., 7, 071 (2005), arXiv:hep-th/0506117 |
| [61] | Van Raamsdonk, M., The meaning of infrared singularities in noncommutative gauge theories, J. High Energy Phys., 11, 006 (2001), arXiv:hep-th/0110093 |
| [62] | Armoni, A., Comments on perturbative dynamics of non-commutative Yang-Mills theory, Nuclear Phys. B, 593, 229 (2001), https://arxiv.org/abs/hep-th/0005208 · Zbl 0971.81522 |
| [63] | Martin, C. P.; Ruiz Ruiz, F.; dominance, Paramagnetic., The sign of the beta function and UV/IR mixing in non-commutative \(U ( 1 )\), Nuclear Phys. B, 597, 197 (2001), arXiv:hep-th/0007131 · Zbl 0972.81185 |
| [64] | Blaschke, D. N.; Grosse, H.; Wallet, J.-C.; identities, Slavnov-Taylor., Non-commutative gauge theories and infrared divergences, J. High Energy Phys., 2013, 38 (2013), arXiv:1302.2903 |
| [65] | Landi, G.; Lizzi, F.; Szabo, R. J., String geometry and the noncommutative torus, Comm. Math. Phys., 206, 603 (1999), arXiv:hep-th/9806099 · Zbl 0949.46037 |
| [66] | Landi, G.; Lizzi, F.; Szabo, R. J., From large \(N\) matrices to the noncommutative torus, Comm. Math. Phys., 217, 181 (2001), arXiv:hep-th/9912130 · Zbl 0982.58004 |
| [67] | Krajewski, T.; Wulkenhaar, R., Perturbative quantum gauge fields on the noncommutative torus, Internat. J. Modern Phys. A, 15, 1011 (2000), arXiv:hep-th/9903187 · Zbl 0963.81055 |
| [68] | Slavnov, A. A., Consistent non-commutative quantum gauge theories?, Phys. Lett. B, 565, 246 (2003), arXiv:hep-th/0304141 · Zbl 1072.81556 |
| [69] | Blaschke, D. N.; Gieres, F.; Piguet, O.; Schweda, M., A vector supersymmetry in noncommutative \(U ( 1 )\) gauge theory with the Slavnov term, J. High Energy Phys., 05, 059 (2006), arXiv:hep-th/0604154 |
| [70] | Blaschke, D. N.; Hohenegger, S., A generalization of Slavnov-extended non-commutative gauge theories, J. High Energy Phys., 08, 032 (2007), arXiv:0705.3007 · Zbl 1326.81218 |
| [71] | Blaschke, D. N.; Hohenegger, S.; Schweda, M., Divergences in non-commutative gauge theories with the Slavnov term, J. High Energy Phys., 11, 041 (2005), arXiv:hep-th/0510100 |
| [72] | Blaschke, D. N.; Hohenegger, S., A generalization of Slavnov-extended non-commutative gauge theories, J. High Energy Phys., 08, 032 (2007), arXiv:0705.3007 · Zbl 1326.81218 |
| [73] | Blaschke, D. N.; Grosse, H.; Schweda, M., Non-commutative \(U ( 1 )\) gauge theory on \(\mathbb{R}_\Theta^4\) with oscillator term and BRST symmetry, Europhys. Lett., 79, 61002 (2007), arXiv:0705.4205 |
| [74] | Langmann, E.; Szabo, R. J., Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B, 533, 168 (2002), arXiv:hep-th/0202039 · Zbl 0994.81116 |
| [75] | Disertori, M.; Gurau, R.; Magnen, J.; Rivasseau, V., Vanishing of beta function of non commutative \(\varphi_4^{\star 4}\) theory to all orders, Phys. Lett. B, 649, 95 (2007), arXiv:hep-th/0612251 · Zbl 1248.81253 |
| [76] | Grosse, H.; Wulkenhaar, R., Self-dual noncommutative \(\varphi^4\)-theory in four dimensions is a non-perturbatively solvable and nontrivial quantum field theory, Comm. Math. Phys., 329, 1069 (2014), arXiv:1205.0465 · Zbl 1305.81129 |
| [77] | de Goursac, A.; Wallet, J.-C., Symmetries of noncommutative scalar field theory, J. Phys. A, 44, Article 055401 pp. (2011), arXiv:0911.2645 · Zbl 1208.81192 |
| [78] | Blaschke, D. N.; Grosse, H.; Kronberger, E.; Schweda, M.; Wohlgenannt, M., Loop calculations for the non-commutative \(U_\star ( 1 )\) gauge field model with oscillator term, Eur. Phys. J. C, 67, 575-582 (2010), arXiv:0912.3642 |
| [79] | Gurau, R.; Magnen, J.; Rivasseau, V.; Tanasa, A., A translation-invariant renormalizable non-commutative scalar model, Comm. Math. Phys., 287, 275 (2009), arXiv:0802.0791 · Zbl 1170.81041 |
| [80] | Blaschke, D. N.; Gieres, F.; Kronberger, E.; Schweda, M.; Wohlgenannt, M., Translation-invariant models for non-commutative gauge fields, J. Phys. A, 41, Article 252002 pp. (2008), arXiv:0804.1914 · Zbl 1192.81326 |
| [81] | Blaschke, D. N.; Rofner, A.; Schweda, M.; Sedmik, R. I.P., One-loop calculations for a translation invariant non-commutative gauge model, Eur. Phys. J. C, 62, 433 (2009), arXiv:0901.1681 · Zbl 1188.81155 |
| [82] | Blaschke, D. N.; Rofner, A.; Sedmik, R. I.P.; Wohlgenannt, M., On non-commutative \(U_\star ( 1 )\) gauge models and renormalizability, J. Phys. A, 43, Article 425401 pp. (2010), arXiv:0912.2634 · Zbl 1201.81101 |
| [83] | Vilar, L. C.Q.; Ventura, O. S.; Tedesco, D. G.; Lemes, V. E.R., On the renormalizability of noncommutative \(U ( 1 )\) gauge theory – an algebraic approach, J. Phys. A, 43, Article 135401 pp. (2010), arXiv:0902.2956 · Zbl 1188.81157 |
| [84] | Zwanziger, D., Local and renormalizable action from the Gribov horizon, Nuclear Phys. B, 323, 513-544 (1989), Indiana.edu |
| [85] | Gribov, V. N., Quantization of non-Abelian gauge theories, Nuclear Phys. B, 139, 1 (1978), Indiana.edu |
| [86] | Canfora, F.; Kurkov, M.; Rosa, L.; Vitale, P., The gribov problem in noncommutative QED, J. High Energy Phys., 01, 014 (2016), arXiv:1505.06342 · Zbl 1388.83521 |
| [87] | Blaschke, D. N.; Rofner, A.; Schweda, M.; Sedmik, R. I.P., Improved localization of a renormalizable non-commutative translation invariant \(U ( 1 )\) Gauge Model, Europhys. Lett., 86, 51002 (2009), arXiv:0903.4811 |
| [88] | Blaschke, D. N.; Rofner, A.; Sedmik, R. I.P., One-loop calculations and detailed analysis of the localized non-commutative \(1 / p^2 U ( 1 )\) gauge model, SIGMA, 6, 037 (2010), arXiv:0908.1743 · Zbl 1188.81130 |
| [89] | Holanda, O.; Guimaraes, M. S.; Rosa, L.; Vitale, P., Gribov horizon in noncommutative QED, Nuclear Phys. B, 974, Article 115624 pp. (2022), arXiv:2106.03600 · Zbl 1483.81104 |
| [90] | Hohm, O.; Zwiebach, B., \( L_\infty\) Algebras and field theory, Fortschr. Phys., 65 (2017), arXiv:1701.08824 · Zbl 1371.81261 |
| [91] | Blumenhagen, R.; Brunner, I.; Kupriyanov, V. G.; Lust, D., Bootstrapping non-commutative gauge theories from \(L_\infty\) algebras, ArXiv (2018), arXiv:1803.00732 · Zbl 1391.81144 |
| [92] | Kupriyanov, V. G., Non-commutative deformation of Chern-Simons theory, Eur. Phys. J. C, 80 (2020), arXiv:1905.08753 |
| [93] | Kurkov, M.; Vitale, P., Four-dimensional noncommutative deformations of u(1) gauge theory and \(L_\infty\) bootstrap, J. High Energy Phys., 01, 032 (2022), arXiv:2108.04856 · Zbl 1521.81411 |
| [94] | Dimitrijević-Ćirić, M.; Konjik, N.; Radovanović, V.; Szabo, R. J.; Toman, M., \( L_\infty \)-Algebra of braided electrodynamics, ArXiv (2022), arXiv:2204.06448 |
| [95] | Burić, M.; Radovanović, V.; Trampetić, J., The one-loop renormalization of the gauge sector in the \(\theta \)-expanded noncommutative standard model, J. High Energy Phys., 03, 030 (2007), arXiv:hep-th/0609073 |
| [96] | Jurčo, B.; Schupp, P.; Wess, J., Noncommutative gauge theory for Poisson manifolds, Nuclear Phys. B, 584, 784-794 (2000), arXiv:hep-th/0005005 · Zbl 0984.81167 |
| [97] | Carlson, C. E.; Carone, C. D.; Zobin, N., Noncommutative gauge theory without Lorentz violation, Phys. Rev. D, 66 (2002), arXiv:hep-th/0206035 |
| [98] | Martín, C. P.; Sánchez-Ruiz, D.; Tamarit, C., The noncommutative \(U ( 1 )\) higgs-kibble model in the enveloping-algebra formalism and its renormalizability, J. High Energy Phys., 02, 065 (2007), arXiv:hep-th/0612188 |
| [99] | Bichl, A.; Grimstrup, J.; Popp, L.; Schweda, M.; Grosse, H.; Wulkenhaar, R., Renormalization of the noncommutative photon self-energy to all orders via Seiberg-Witten map, J. High Energy Phys., 06, 013 (2001), arXiv:hep-th/0104097 |
| [100] | Burić, M.; Latas, D.; Radovanović, V., Renormalizability of noncommutative \(S U ( N )\) gauge theory, J. High Energy Phys., 02, 046 (2006), arXiv:hep-th/0510133 |
| [101] | Latas, D.; Radovanović, V.; Trampetić, J., Non-commutative \(S U ( N )\) gauge theories and asymptotic freedom, Phys. Rev. D, 76 (2007), arXiv:hep-th/0703018 |
| [102] | Burić, M.; Latas, D.; Radovanović, V.; Trampetić, J., The absence of the \(4 \psi\) divergence in noncommutative chiral models, Phys. Rev. D, 77 (2008), arXiv:0711.0887 |
| [103] | Tamarit, C.; Trampetić, J., Noncommutative fermions and quarkonia decays, Phys. Rev. D, 79 (2009), arXiv:0812.1731 |
| [104] | Martín, C. P.; Tamarit, C., Renormalizability of noncommutative GUT inspired field theories with anomaly safe groups, J. High Energy Phys., 12, 042 (2009), arXiv:0910.2677 |
| [105] | Calmet, X.; Jurčo, B.; Schupp, P.; Wess, J.; Wohlgenannt, M., The standard model on non-commutative space-time, Eur. Phys. J. C, 23, 363-376 (2002), arXiv:hep-ph/0111115 · Zbl 1099.81575 |
| [106] | Aschieri, P.; Jurčo, B.; Schupp, P.; Wess, J., Noncommutative GUTs, standard model and C, P, T, Nuclear Phys. B, 651, 45-70 (2003), arXiv:hep-th/0205214 · Zbl 1008.81098 |
| [107] | Melić, B.; Passek-Kumerički, K.; Trampetić, J.; Schupp, P.; Wohlgenannt, M., The standard model on non-commutative space-time: electroweak currents and the Higgs sector, Eur. Phys. J. C, 42, 483-497 (2005), arXiv:hep-ph/0502249 · Zbl 1189.81255 |
| [108] | Melić, B.; Passek-Kumerički, K.; Trampetić, J.; Schupp, P.; Wohlgenannt, M., The standard model on non-commutative space-time: strong interactions included, Eur. Phys. J. C, 42, 499-504 (2005), arXiv:hep-ph/0503064 · Zbl 1189.81256 |
| [109] | Martín, C. P., Computing the \(\theta \)-exact Seiberg-Witten map for arbitrary gauge groups, Phys. Rev. D, 86 (2012), arXiv:1206.2814 |
| [110] | Barnich, G.; Brandt, F.; Grigoriev, M., Seiberg-Witten maps in the context of the antifield formalism, Fortschr. Phys., 50, 825-830 (2002), arXiv:hep-th/0201139 · Zbl 1008.81076 |
| [111] | Cerchiai, B. L.; Pasqua, A. F.; Zumino, B., The Seiberg-Witten map for noncommutative gauge theories (2002), arXiv arXiv:hep-th/0206231 · Zbl 1055.81587 |
| [112] | Barnich, G.; Brandt, F.; Grigoriev, M., Seiberg-Witten maps and noncommutative Yang-Mills theories for arbitrary gauge groups, J. High Energy Phys., 2002, 023 (2002), arXiv:hep-th/0206003 · Zbl 1226.81120 |
| [113] | Schupp, P.; You, J., UV/IR mixing in noncommutative QED defined by Seiberg-Witten map, J. High Energy Phys., 2008, 107 (2008), arXiv:0807.4886 |
| [114] | Martín, C. P.; ad J., Trampetić; You, J., UV/IR mixing in Noncommutative \(S U ( N )\) Yang-Mills theory (2020), arXiv arXiv:2012.09119 |
| [115] | Kupriyanov, V. G.; Vitale, P., A novel approach to non-commutative gauge theory, J. High Energy Phys., 2020 (2020), arXiv:2004.14901; Kupriyanov, V. G., Poisson gauge theory, J. High Energy Phys., 09, 016 (2021), arXiv:2105.14965; Kupriyanov, V. G.; Szabo, R. J., Symplectic embeddings, homotopy algebras and almost poisson gauge symmetry, J. Phys. A, 55, 3, 035201 (2022), arXiv:2101.12618 · Zbl 1499.81072 |
| [116] | Kupriyanov, V. G.; Kurkov, M. A.; Vitale, P., Poisson gauge models and Seiberg-Witten map, J. High Energy Phys., 11, 062 (2022), arXiv2209.13044 · Zbl 1536.81222 |
| [117] | Hewett, J. L.; Petriello, F. J.; Rizzo, T. G., Signals for noncommutative interactions at linear colliders, Phys. Rev. D, 64 (2001), arXiv:hep-ph/0010354 |
| [118] | Hinchliffe, I.; Kersting, N., \( g - 2\) From noncommutative geometry (2001), arXiv arXiv:hep-ph/0109224 |
| [119] | Schupp, P.; Trampetić, J.; Wess, J.; Raffelt, G., The photon-neutrino interaction induced by non-commutativity and astrophysical bounds, Eur. Phys. J. C., 36, 405-410 (2004), arXiv:hep-ph/0212292 |
| [120] | Minkowski, P.; Schupp, P.; Trampetic, J., Neutrino dipole moments and charge radii in noncommutative space-time (2003), arXiv arXiv:hep-th/0302175 |
| [121] | Horvat, R.; Trampetić, J., Constraining space-time noncommutativity with primordial nucleosynthesis, Phys. Rev. D, 79 (2009), arXiv:0901.4253 |
| [122] | Carlson, C. E.; Carone, C. D.; Lebed, R. F., Bounding noncommutative QCD, Phys. Lett. B, 518, 201-206 (2001), arXiv:hep-ph/0107291 · Zbl 0971.81180 |
| [123] | Wallet, J.-C., Noncommutative induced gauge theories on moyal spaces, J. Phys. Conf. Ser., 103, Article 012007 pp. (2008), arXiv:0708.2471 |
| [124] | Wohlgenannt, M., Induced gauge theory on a noncommutative space, Fortschr. Phys., 56, 547 (2008), arXiv:0804.1259 · Zbl 1145.81050 |
| [125] | Weyl, H., The Classical Groups (1946), Princeton University Press · JFM 65.0058.02 |
| [126] | de Goursac, A.; Tanasa, A.; Wallet, J.-C., Vacuum configurations for renormalizable non-commutative scalar models, Eur. Phys. J. C, 53, 459 (2008), arXiv:0709.3950 · Zbl 1189.81213 |
| [127] | Martinetti, P.; Vitale, P.; Wallet, J.-C., Noncommutative gauge theories on \(\mathbb{R}_\theta^2\) as matrix models, J. High Energy Phys., 09, 051 (2013), arXiv:1303.7185 · Zbl 1342.81309 |
| [128] | Géré, A.; Wallet, J.-C., Spectral theorem in noncommutative field theories: Jacobi dynamics, J. Phys.: Conf. Ser., 634, Article 012006 pp. (2015), arXiv:1402.6976 |
| [129] | Ginsparg, P. H., Matrix models of 2-d gravity (1991), arXiv arXiv:hep-th/9112013 |
| [130] | Kostov, I. K., Exact solution of the six vertex model on a random lattice, Nuclear Phys. B, 575, 513 (2000), arXiv:hep-th/9911023 · Zbl 1037.82509 |
| [131] | Hammou, A. B.; Lagraa, M.; Sheikh-Jabbari, M. M., Coherent state induced star-product on \(R_\lambda^3\) and the fuzzy sphere, Phys. Rev. D, 66, Article 025025 pp. (2002), arXiv:hep-th/0110291 |
| [132] | Wick-Voros, A., Wentzel-Kramers-Brillouin method in the Bargmann representation, Phys. Rev. A, 40, 6814 (1989) |
| [133] | Kustaanheimo, P.; Stiefel, E., Perturbation theory of Kepler motion based on spinor regularization, J. R. Angew. Math., 218, 204 (1965), degruyter.com · Zbl 0151.34901 |
| [134] | Vitale, P., Noncommutative field theories on \(\mathbb{R}_\lambda^3\), Fortschr. Phys., 62, 910, 825 (2014), arXiv:1406.1372 |
| [135] | Géré, A.; Vitale, P.; Wallet, J.-C., Quantum gauge theories on noncommutative 3-d space, Phys. Rev. D, 90, Article 045019 pp. (2014), arXiv:1312.6145 |
| [136] | Felder, G.; Shoikhet, B., Deformation quantization with traces, Lett. Math. Phys., 53, 75 (2000), arXiv:math/0002057 · Zbl 0983.53065 |
| [137] | Kupriyanov, V. G.; Vitale, P., Noncommutative \(\mathbb{R}^d\) via closed star-product, J. High Energy Phys., 08, 024 (2015), arXiv:1502.06544 · Zbl 1388.81419 |
| [138] | Jurić, T.; Poulain, T.; Wallet, J.-C., Involutive representaions of coordinate algebras and quantum spaces, J. High Energy Phys., 07, 116 (2017), arXiv:1702.06348 · Zbl 1380.83095 |
| [139] | Poulain, T.; Wallet, J.-C., Quantum spaces, central extensions of Lie groups and related quantum field theories, J. Phys.: Conf. Ser., 965, Article 012032 pp. (2018), arXiv:1707.03474 |
| [140] | Jurić, T.; Poulain, T.; Wallet, J.-C., Closed star product on noncommutative \(\mathbb{R}^3\) and scalar field dynamics, J. High Energy Phys., 05, 146 (2016), arXiv:1603.09122 · Zbl 1388.83538 |
| [141] | Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc., 70, 28 (1951) · Zbl 0042.12701 |
| [142] | Duflo, M., Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sc. Ec. Norm. Sup., 10, 107 (1977), numdam.org · Zbl 0353.22009 |
| [143] | Deitmar, A.; Echterhoff, S., Principles of Harmonic Analysis (2009), Springer: Springer New York · Zbl 1158.43001 |
| [144] | Stora, R.; Thuillier, F.; Wallet, J.-C., Algebraic structure of cohomological field theory models and equivariant cohomology, (Infinite Dimensional Geometry, Noncommutative Geometry, Operator Algebras, Fundamental Interactions (1995), Cambridge Press), 266-297, numdam.org |
| [145] | Stora, R., Algebraic structure and topological origin of anomalies, (Lehmann, G.; etal., Recent Progress in Gauge Theories (1984), Plenum: Plenum New York); Stora, R., Differential algebras in field theory, (Quantum Mechanics of Fundamental Systems, Vol. 2 (1989), Springer), iaea.org |
| [146] | Géré, A.; Jurić, T.; Wallet, J.-C., Noncommutative gauge theories on \(\mathbb{R}_\lambda^3\): Perturbatively finite models, J. High Energy Phys., 12, 045 (2015) · Zbl 1387.81266 |
| [147] | Wallet, J.-C., Exact Partition Functions for Gauge Theories on \(\mathbb{R}_\lambda^3\), Nuclear Phys. B, 912, 354 (2016), arXiv:1603.05045 · Zbl 1349.81173 |
| [148] | Vitale, P., Noncommutative field theories on \(\mathbb{R}_\lambda^3\), Fortschr. Phys., 62, 825 (2014), arXiv:1406.1372 |
| [149] | Madore, J., The fuzzy sphere, Classical Quantum Gravity, 9, 69 (1992) · Zbl 0742.53039 |
| [150] | Balachandran, A. P.; Kurkcuoglu, S.; Vaidya, S., Lectures on Fuzzy and Fuzzy SUSY Physics (2007), World Scientific Publishing, arXiv:hep-th/0511114 · Zbl 1132.81001 |
| [151] | Das, S. R.; Michelson, J.; Shapere, A. D., Fuzzy spheres in pp wave matrix string theory, Phys. Rev. D, 70, Article 026004 pp. (2004), arXiv:hep-th/0306270 |
| [152] | Langmann, E.; Szabo, R. J.; Zarembo, K., Exact solution of quantum field theory on noncommutative phase spaces, J. High Energy Phys., 01, 017 (2004), arXiv:hep-th/0308043 · Zbl 1243.81205 |
| [153] | Langmann, E.; Szabo, R. J., Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B, 533, 168 (2002), arXiv:hep-th/0202039 · Zbl 0994.81116 |
| [154] | Vitale, P.; Wallet, J.-C., Noncommutative field theories on \(\mathbb{R}_\lambda^3\): Toward UV/IR mixing freedom, J. High Energy Phys., 04, 115 (2013), arXiv:1212.5131 · Zbl 1342.81641 |
| [155] | Douglas, M. R.; Hull, C., D-branes and the noncommutative torus, J. High Energy Phys., 02, 008 (1998), arXiv:hep-th/9711165 · Zbl 0957.81017 |
| [156] | Cheung, Y.-K. E.; Krogh, M., Non-commutative geometry from 0-branes in a background \(B\)-field, Nuclear Phys. B, 528, 185-196 (1998), arXiv:hep-th/9803031 · Zbl 0951.81091 |
| [157] | Schomerus, V., D-branes and deformation quantization, J. High Energy Phys., 06, 030 (1999), arXiv:hep-th/9903205 · Zbl 0961.81066 |
| [158] | Steinacker, H., Quantized gauge theory on the fuzzy sphere as random matrix model, Nuclear Phys. B, 679, 66-98 (2004), arXiv:hep-th/0307075 · Zbl 1045.81526 |
| [159] | Castro-Villarreal, P.; Delgadillo-Blando, R.; Ydri, B., A gauge invariant UV/IR mixing and the corresponding phase transition for \(U ( 1 )\) fields on the fuzzy sphere, Nuclear Phys. B, 704, 111-153 (2005), arXiv:hep-th/0405201 · Zbl 1198.81180 |
| [160] | Oriti, D., The microscopic dynamics of quantum space as a group field theory (2011), arXiv arXiv:1110.5606 · Zbl 1223.83025 |
| [161] | Oriti, D., Group field theory as the 2nd quantization of Loop Quantum Gravity (2013), arXiv arXiv:1310.7786 · Zbl 1298.83058 |
| [162] | Freidel, L.; Livine, E. R., Ponzano-Regge model revisited: III. Feynman diagrams and effective field theory, Classical Quantum Gravity, 23, 2021-2061 (2006), arXiv:hep-th/0502106 · Zbl 1091.83503 |
| [163] | Freidel, L.; Livine, E. R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett., 96 (2006), arXiv:hep-th/0512113 · Zbl 1211.83015 |
| [164] | Fairbairn, W.; Livine, E., 3D spinfoam quantum gravity: Matter as a phase of the group field theory, Classical Quantum Gravity, 24, 5277-5297 (2007), arXiv:gr-qc/0702125 · Zbl 1126.83303 |
| [165] | Fairbairn, W.; Livine, E., Matrix models as non-commutative field theories on \(R^3\), Classical Quantum Gravity, 26, Article 195014 pp. (2009), arXiv:0811.1462 · Zbl 1178.83025 |
| [166] | Girelli, F.; Livine, E., A deformed poincare invariance for group field theories, Classical Quantum Gravity, 27, Article 245018 pp. (2010), arXiv:1001.2919 · Zbl 1206.83074 |
| [167] | Freidel, L.; Majid, S., Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity, Classical Quantum Gravity, 25, Article 045006 pp. (2008), arXiv:hep-th/0601004 · Zbl 1190.83070 |
| [168] | Joung, E.; Mourad, J.; Noui, K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys., 50, Article 052503 pp. (2009), arXiv:0806.4121 · Zbl 1187.58008 |
| [169] | Lukierski, J.; Lyakhovsky, V.; Mozrzymas, M., \( \kappa \)-Deformations of D=4 Weyl and conformal symmetries, Phys. Lett. B, 538, 375-384 (2002), arXiv:hep-th/0203182 · Zbl 0995.17006 |
| [170] | Lukierski, J.; Nowicki, A., Doubly Special Relativity versus \(\kappa \)-defomartion of relativistic kinematics, Int. J. Mod. Phys. A, 18, 7-18 (2003), arXiv:hep-th/0203065 · Zbl 1021.81024 |
| [171] | Amelino-Camelia, G., Relativity in spacetimes with short-distance structure governed by an observer-independant (Plankian) length scale, Int. J. Mod. Phys. D, 11, 35-59 (2002), arXiv:gr-qc/0012051 · Zbl 1062.83500 |
| [172] | Kowalski-Glikman, J., Introduction To Doubly Special Relativity, 131-159 (2005), Springer-Verlag, arXiv:hep-th/0405273 |
| [173] | Amelino-Camelia, G., Special treatment, Nature, 418, 34-35 (2002), arXiv:gr-qc/0207049 |
| [174] | Dimitrijevic, M.; Jonke, L.; Möller, L.; Tsouchnika, E.; Wess, J.; Wohlgenannt, M., Deformed Field Theory on \(\kappa \)-space-time, Eur. Phys. J. C, 31, 129-138 (2004), arXiv:hep-th/0307149 · Zbl 1032.81529 |
| [175] | Dimitrijevic, M.; Jonke, L.; Möller, L.; Tsouchnika, E.; Wess, J.; Wohlgenannt, M., Field Theory on \(\kappa \)-space-time, Czech J. Phys., 54, 1243-1248 (2004), arXiv:hep-th/0407187 · Zbl 1465.81070 |
| [176] | Dimitrijevic, M.; Möller, L.; Tsouchnika, E., Derivatives, forms and vector fields on the \(\kappa \)-deformed Euclidean space, J. Phys. A, 37, 9749-9770 (2004), arXiv:hep-th/0404224 · Zbl 1073.81053 |
| [177] | Dimitrijevic, M.; Meyer, F.; Möller, L.; Wess, J., Gauge theories on the \(\kappa \)-Minkowski space-time, Eur. Phys. J. C, 36, 117-126 (2004), arXiv:hep-th/0310116 · Zbl 1191.81204 |
| [178] | Dimitrijevic, M.; Jonke, L.; Möller, L., \( U ( 1 )\) Gauge field theory on \(\kappa \)-Minkowski space, J. High Energy Phys., 2005 (2005), arXiv:hep-th/0504129 |
| [179] | Sitarz, A., Noncommutative differential calculus on the \(\kappa \)-Minkowski space, Phys. Lett. B, 349, 42-48 (1995), arXiv:hep-th/9409014 |
| [180] | Dimitrijevic, M.; Jonke, L., A twisted look on \(\kappa \)-Minkowski: \( U ( 1 )\) gauge theory, J. High Energy Phys., 2011 (2011), arXiv:1107.3475 · Zbl 1306.81046 |
| [181] | Dimitrijevic, M.; Jonke, L.; Pachol, A., Gauge theory on twisted \(\kappa \)-Minkowski: Old problems and possible solutions, SIGMA, 10 (2014), arXiv:1403.1857 · Zbl 1295.81125 |
| [182] | Bu, J.-G.; Kim, H.-C.; Lee, Y.; Vac, C. H.; Yee, J. H., \( \kappa \)-Deformed space-time from twist, Phys. Lett. B, 665, 95-99 (2008), arXiv:hep-th/0611175 · Zbl 1328.81133 |
| [183] | Meljanac, S.; Samsarov, A.; Stojić, M.; Gupta, K. S., \( \kappa \)-Minkowski space-time and the star product realizations, Eur. Phys. J. C, 53, 295-309 (2007), arXiv:0705.2471 · Zbl 1189.81115 |
| [184] | Borowiec, A.; Lukierski, J.; Pachoł, A., Twisting and \(\kappa \)-Poincaré, J. Phys. A, 47, Article 405203 pp. (2014), arXiv:1312.7807 · Zbl 1301.81084 |
| [185] | Aschieri, P.; Castellani, L., Noncommutative gauge fields coupled to noncommutative gravity, Gen. Relativity Gravitation, 45, 581-598 (2012), arXiv:1205.1911 · Zbl 1262.83039 |
| [186] | Mathieu, P.; Wallet, J.-C., Gauge theories on \(\kappa \)-Minkowski spaces: Twist and modular operators, J. High Energy Phys., 05, 112 (2020), arXiv:2002.02309 · Zbl 1437.83041 |
| [187] | Mathieu, P.; Wallet, J.-C., Single extra dimension from \(\kappa \)-Poincaré and gauge invariance, J. High Energy Phys., 03, 209 (2021), arXiv:2007.14187 · Zbl 1461.81071 |
| [188] | Hersent, K.; Mathieu, P.; Wallet, J.-C., Algebraic structures in \(\kappa \)-Poincaré invariant gauge theories, Int. J. Geom. Methods Mod. Phys., 19, Article 2250078 pp. (2022), arXiv:2110.10763 · Zbl 1534.83067 |
| [189] | Durhuus, B.; Sitarz, A., Star-product realizations of \(\kappa \)-Minkowski space, J. Noncommut. Geom., 7, 605 (2013), arXiv:1104.0206 · Zbl 1282.46063 |
| [190] | Poulain, T.; Wallet, J.-C., \( \kappa \)-Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight, Phys. Rev. D, 98 (2018), arXiv:1801.02715 |
| [191] | Poulain, T.; Wallet, J.-C., \( \kappa \)-Poincaré invariant orientable field theories at one-loop, J. High Energy Phys., 2019 (2019), arXiv:1808.00350 · Zbl 1409.83070 |
| [192] | Khalil, I., Sur l’analyse harmonique du groupe affine de la droite, Studia Math., 51, 139 (1974), https://eudml.org/, matwbn.icm.edu.pl · Zbl 0294.43007 |
| [193] | Kustermans, J., KMS-weights on C*-algebras (1997), arXiv:funct-an/9704008; Combes, F., Poids sur une c*-algèbre, J. Math. Pures Appl., 47, 57 (1968), See also · Zbl 0165.15401 |
| [194] | Takesaki, M., Theory of operator algebras I-III, (EMS Vols. 124, 125, 127 (2002), Springer) · Zbl 0990.46034 |
| [195] | Connes, A.; Rovelli, C., Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories, Classical Quantum Gravity, 11, 2899 (1994), arXiv:gr-qc/9406019 · Zbl 0821.46086 |
| [196] | Connes, A.; Moscovici, H., Type III and spectral triples, (Traces in Number Theory, Geometry and Quantum Fields. Traces in Number Theory, Geometry and Quantum Fields, Aspects of Math. E38 (2008), Vieweg: Vieweg Wiesbaden), 57, arXiv:math/0609703 · Zbl 1159.46041 |
| [197] | Landi, G.; Martinetti, P., Gauge transformations for twisted spectral triples, Lett. Math. Phys., 108, 2589 (2018), arXiv:1704.06212; Martinetti, P.; Zanchettin, J., Twisted spectral triples without the first-order condition (2021), arXiv arXiv:2103.15643 · Zbl 1404.58015 |
| [198] | Devastato, A.; Martinetti, P., Twisted spectral triple for the standard model and spontaneous breaking of the grand symmetry, Math. Phys. Anal. Geom., 20, 2 (2017) · Zbl 1413.58003 |
| [199] | Filaci, M.; Martinetti, P.; Pesco, S., Minimal twist for the Standard Model in noncommutative geometry : the field content, Phys. Rev. D, 104, Article 025011 pp. (2021), arXiv:2008.01629 |
| [200] | Rowen, L. H., Ring theory, (Pure and Applied Mathematics 127, Vol. 1 (1988), Academic Press Inc.: Academic Press Inc. Boston MA) · Zbl 0651.16001 |
| [201] | Bäck, P.; Richter, J.; Silvestrov, S., Hom-associative Ore extensions and weak unitalizations, Int. Electron. J. Algebra, 24, 174 (2018), and references therein, arXiv:1710.04190 · Zbl 1446.17001 |
| [202] | de Goursac, A.; Masson, T.; Wallet, J.-C., Noncommutative epsilon-graded connections, J. Noncommut. Geom., 6, 343 (2012), arXiv:1710.04190 · Zbl 1275.58003 |
| [203] | Mathieu, P.; Wallet, J.-C., Twisted BRST symmetry in gauge theories on the \(\kappa \)-Minkowski space-time, Phys. Rev. D, 103 (2021), arXiv:2102.10860 |
| [204] | Hooper, D.; Profumo, S., Dark matter and collider phenomenology of universal extra dimensions, Phys. Rep., 453, 29 (2007), arXiv:hep-ph/0701197 |
| [205] | Datta, A.; Kong, K.; Matchev, K. T., Minimal universal extra dimensions in CalcHEP/CompHEP, New J. Phys., 12, Article 075017 pp. (2010), arXiv:1002.4624 |
| [206] | Hersent, K.; Mathieu, P.; Wallet, J.-C., Quantum instability of gauge theories on \(\kappa \)-Minkowski space, Phys. Rev. D, 105 (2022), arXiv:2107.14462 |
| [207] | Khorrami, M.; Fatollahi, A.; Shariati, A., Gauge theory on a space with linear Lie type fuzziness, Internat. J. Modern Phys. A, 28 (2013), arXiv:1305.5410 · Zbl 1262.81176 |
| [208] | Kupriyanov, V. G.; Kurkov, M.; Vitale, P., \( \kappa \)-Minkowski-deformation of \(U ( 1 )\) gauge theory, J. High Energy Phys., 2021 (2021), arXiv:2010.09863 · Zbl 1459.83028 |
| [209] | Pachol, A.; Vitale, P., \( \kappa \)-Minkowski star-product in any dimension from symplectic realization, J. Phys. A, 48 (2015), arXiv:1507.03523 · Zbl 1327.53111 |
| [210] | Dimitrijević Ćirić, M.; Konjik, N.; Samsarov, A., Noncommutative scalar quasinormal modes of the Reissner-Nordström black hole, Classical Quantum Gravity, 35, Article 175005 pp. (2018), arXiv:1708.04066 · Zbl 1409.83089 |
| [211] | Dimitrijević Ćirić, M.; Konjik, N.; Kurkov, M. A.; Lizzi, F.; Vitale, P., Noncommutative field theory from angular twist, Phys. Rev. D, 98 (2018), arXiv:1806.06678 |
| [212] | Lizzi, F.; Scala, L.; Vitale, P., Localization and observers in \(\varrho \)-Minkowski space-time, Phys. Rev. D, 106 (2022), arXiv:2205.10862 |
| [213] | Müller-Hoissen, F., Noncommutative geometries and gravity, AIP Conf. Proc. (2008), arXiv:0710.4418 · Zbl 1183.83002 |
| [214] | Chamseddine, A. H.; Connes, A., The spectral action principle, Comm. Math. Phys., 186, 731-750 (1997), arXiv:hep-th/9606001 · Zbl 0894.58007 |
| [215] | Chamseddine, A. H.; Connes, A.; Marcolli, M., Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys., 11, 991-1089 (2007), arXiv:hep-th/0610241 · Zbl 1140.81022 |
| [216] | Da̧browski, L.; Sitarz, A.; Zalecki, P., Spectral metric and Einstein functionals (2022), arXiv, arXiv:2206.02587 |
| [217] | Connes, A.; Moscovici, H., Modular curvature for noncommutative two-tori, J. Amer. Math. Soc., 27, 639-684 (2014), arXiv:1110.3500 · Zbl 1332.46070 |
| [218] | Fathizadeh, F.; Khalkhali, M., Scalar curvature for the noncommutative two torus, J. Noncommut. Geom., 7, 1145-1183 (2013), arXiv:1110.3511 · Zbl 1295.46053 |
| [219] | Fathizadeh, F.; Khalkhali, M., Scalar curvature for noncommutative four-tori, J. Noncommut. Geom., 9, 473-503 (2015), arXiv:1301.6135 · Zbl 1332.46071 |
| [220] | Lesch, M.; Moscovici, H., Modular curvature and Morita equivalence, Geom. Funct. Anal., 26, 818-873 (2016), arXiv:1505.00964 · Zbl 1375.46053 |
| [221] | Chamseddine, A. H.; Felder, G.; Fröhlich, J., Gravity in non-commutative geometry, Comm. Math. Phys., 155, 205-217 (1993), arXiv:hep-th/9209044 · Zbl 0818.58008 |
| [222] | Landi, G.; Ai, Viet Nguyen; Wali, K. C., Gravity and electromagnetism in noncommutative geometry, Phys. Lett. B, 326, 45-50 (1994), arXiv:hep-th/9402046 |
| [223] | Sitarz, A., Gravity from non-commutative geometry, Classical Quantum Gravity, 11, 2127-2134 (1994), arXiv:hep-th/9401145 · Zbl 0812.53082 |
| [224] | Chamseddine, A. H.; Connes, A.; Mukhanov, V., Quanta of geometry: Noncommutative aspects, Phys. Rev. Lett., 114 (2015), arXiv:1409.2471 |
| [225] | Chamseddine, A. H., Complexified gravity in noncommutative spaces, Comm. Math. Phys., 2, 283-292 (2001), arXiv:hep-th/0005222 · Zbl 1042.83023 |
| [226] | Moffat, J. W., Noncommutative quantum gravity, Phys. Lett. B, 491, 345-352 (2000), arXiv:hep-th/0007181 · Zbl 0961.83030 |
| [227] | Nishino, H.; Rajpoot, S., Teleparallel complex gravity as foundation for noncommutative gravity, Phys. Lett. B, 532, 334-344 (2002), arXiv:hep-th/0107216 · Zbl 0994.83041 |
| [228] | Chamseddine, A. H., An invariant action for noncommutative gravity in four dimensions, J. Math. Phys., 44, 2534 (2003), arXiv:hep-th/0202137 · Zbl 1062.83061 |
| [229] | Mourad, J., Linear connections in non-commutative geometry, Classical Quantum Gravity, 12, 965-974 (1995), arXiv:hep-th/9410201 · Zbl 0822.58006 |
| [230] | Madore, J.; Masson, T.; Mourad, J., Linear connections on matrix geometries, Classical Quantum Gravity, 6, 1429-1440 (1995), arXiv:hep-th/9411127 · Zbl 0824.58008 |
| [231] | Da̧browski, L.; Hajac, P. M.; Landi, G.; Siniscalco, P., Metrics and pairs of left and right connections on bimodules, J. Math. Phys., 37, 4635-4646 (1996), arXiv:q-alg/9602035 · Zbl 0863.58004 |
| [232] | Dubois-Violette, M.; Madore, J.; Masson, T.; Mourad, J., Linear connections on the quantum plane, Lett. Math. Phys., 35, 351-358 (1995), arXiv:hep-th/9410199 · Zbl 0835.58003 |
| [233] | Georgelin, Y.; Wallet, J.-C.; Masson, T., Linear connections on the two-parameter quantum plane, Rev. Math. Phys., 8, 1055 (1996), arXiv:q-alg/9507032 · Zbl 0872.58007 |
| [234] | Bhowmick, J.; Goswami, D.; Landi, G., Levi-Civita connections and vector fields for noncommutative differential calculi, Internat. J. Math., 31, Article 2050065 pp. (2020), arXiv:2001.01545 · Zbl 1512.58005 |
| [235] | Bhowmick, J.; Goswami, D.; Landi, G., On the koszul formula in noncommutative geometry, Rev. Math. Phys., 32, Article 2050032 pp. (2020), arXiv:1910.09306 · Zbl 1461.58003 |
| [236] | Brzeziński, T.; Majid, S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys., 157, 591-638 (1993), arXiv:hep-th/9208007 · Zbl 0817.58003 |
| [237] | Aschieri, P., Cartan structure equations and levi-civita connection in braided geometry (2020), arXiv, arXiv:2006.02761 |
| [238] | Weber, T., Braided Cartan calculi and submanifold algebras, J. Geom. Phys., 150, Article 103612 pp. (2020), arXiv:1907.13609 · Zbl 1437.18010 |
| [239] | Weber, T., Braided commutative geometry and drinfel’d twist deformations (2020), arXiv, arXiv:2002.11478 |
| [240] | Aschieri, P.; Blohmann, C.; Dimitrijević, M.; Meyer, F.; Schupp, P.; Wess, J., A gravity theory on noncommutative spaces, Classical Quantum Gravity, 22, 3511-3532 (2005), arXiv:hep-th/0504183 · Zbl 1129.83011 |
| [241] | Wess, J., Deformed gauge theories, J. Phys. Conf. Ser., 53, 752-763 (2006), arXiv:hep-th/0608135 |
| [242] | Kobakhidze, A., Diffeomorphism-invariant noncommutative gravity with twisted local Lorentz invariance (2010), arXiv, arXiv:0905.3427 · Zbl 1255.81220 |
| [243] | Aschieri, P.; Borowiec, A.; Pachoł, A., Dispersion relations in \(\kappa \)-noncommutative cosmology, J. Cosmol. Astropart. Phys., 2021, 025 (2021), arXiv:2009.01051 · Zbl 1485.83031 |
| [244] | S. Kürkçüoglu, C. Sämann, Drinfeld twist and general relativity with fuzzy spaces, 24 (2006) 291-311. http://dx.doi.org/10.1088/0264-9381/24/2/003, arXiv:hep-th/0606197. · Zbl 1133.53311 |
| [245] | Dimitrijević Ćirić, M.; Giotopoulos, G.; Radovanović, V.; Szabo, R. J., Braided \(L_\infty \)-algebras, braided field theory and noncommutative gravity, Lett. Math. Phys., 111 (2021), arXiv:2103.08939 · Zbl 1484.18025 |
| [246] | Dimitrijević Ćirić, M.; Giotopoulos, G.; Radovanović, V.; Szabo, R. J., \( L_\infty \)-Algebras of Einstein-Cartan-Palatini gravity, J. Math. Phys., 61, Article 112502 pp. (2020), arXiv:2003.06173 · Zbl 1455.83021 |
| [247] | Barnes, G. E.; Schenkel, A.; Szabo, R. J., Working with nonassociative geometry and field theory, Proc. Sci., 081 (2016), arXiv:1601.07353 |
| [248] | Madore, J.; Mourad, J., Quantum space-time and classical gravity, J. Math. Phys., 39, 423-442 (1998), arXiv:gr-qc/9607060 · Zbl 0938.58006 |
| [249] | Madore, J., An Introduction to Noncommutative Differential Geometry and its Physical Applications (1999), Cambridge University Press · Zbl 0942.58014 |
| [250] | Madore, J., Kaluza-Klein aspects of noncommutative geometry (2015), arXiv, arXiv:1510.03915 |
| [251] | Maceda, M.; Madore, J.; Manousselis, P.; Zoupanos, G., Can non-commutativity resolve the big-bang singularity?, Eur. Phys. J. C, 36, 529-534 (2004), arXiv:hep-th/0306136 · Zbl 1191.83030 |
| [252] | Burić, M.; Madore, J., Noncommutative 2-dimensional models of gravity (2005), arXiv, arXiv:hep-th/0406232 |
| [253] | Burić, M.; Madore, J.; Grammatikopoulos, T.; Zoupanos, G., Gravity and the structure of noncommutative algebras, J. High Energy Phys., 2006, 054 (2006), arXiv:hep-th/0603044 |
| [254] | Szabo, R. J., Symmetry, gravity and noncommutativity, Classical Quantum Gravity, 23, R199-R242 (2006), arXiv:hep-th/0606233 · Zbl 1117.83001 |
| [255] | Aschieri, P.; Di Grezia, E.; Esposito, G., Non-commutative Einstein equations and Seiberg-Witten map, Int. J. Mod. Phys.: Conf. Ser., 03, 143-149 (2011), arXiv:1103.3348 |
| [256] | Cacciatori, S.; Chamseddine, A. H.; Klemm, D.; Martucci, L.; Sabra, W. A.; Zanon, D., Noncommutative gravity in two dimensions, Classical Quantum Gravity, 19, 4029-4042 (2002), arXiv:hep-th/0203038 · Zbl 1003.83026 |
| [257] | Cacciatori, S.; Klemm, D.; Martucci, L.; Zanon, D., Noncommutative Einstein-AdS gravity in three dimensions, Phys. Lett. B, 536, 101-106 (2002), arXiv:hep-th/0201103 · Zbl 0995.83047 |
| [258] | Cemsinan, D., Noncommutative gravity in six dimensions (2006), arXiv, arXiv:hep-th/0607096 |
| [259] | Balachandran, A. P.; Govindarajan, T. R.; Gupta, K. S.; Kürkçüo AĂglu, S., Noncommutative two-dimensional gravities, Classical Quantum Gravity, 23, 5799-5810 (2006), arXiv:hep-th/0602265 · Zbl 1133.83372 |
| [260] | Álvarez-Gaumé, L.; Meyer, F.; Vázquez-Mozo, M. A., Comments on noncommutative gravity, Nuclear Phys. B, 753, 92-117 (2006), arXiv:hep-th/0605113 · Zbl 1215.83040 |
| [261] | Castellani, L., Chern-Simons supergravities, with a twist, J. High Energy Phys., 2013, 1-13 (2013), arXiv:abs/1305.1566 · Zbl 1342.83457 |
| [262] | Aschieri, P.; Castellani, L., Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map, J. High Energy Phys., 2014 (2014), arXiv:1406.4896 · Zbl 1333.81229 |
| [263] | Đorđević, D.; Gočanin, D., Noncommutative \(D = 5\) Chern-Simons gravity: Kaluza-Klein reduction and chiral gravitational anomaly (2022), arXiv, arXiv:2203.05020 |
| [264] | M.A. Cardella, D. Zanon, Noncommutative deformation of four-dimensional Einstein gravity, 20 (2003) L95-L103. http://dx.doi.org/10.1088/0264-9381/20/8/101, arXiv:hep-th/0212071. · Zbl 1035.83020 |
| [265] | Bonora, L.; Schnabl, M.; Sheikh-Jabbari, M. M.; Tomasiello, A., Noncommutative and gauge theories, Nuclear Phys. B, 589, 461-474 (2000), arXiv:hep-th/0006091 · Zbl 1060.81607 |
| [266] | Bars, I.; Sheikh-Jabbari, M. M.; Vasiliev, M. A., Noncommutative \(o_{\star ( N )}\) and \(u s p_\star ( 2 N )\) algebras and the corresponding gauge field theories, Phys. Rev. D, 64 (2001), arXiv:hep-th/0103209 |
| [267] | Jurco, 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, 17, 521-526 (2000), arXiv:hep-th/0006246 · Zbl 1099.81525 |
| [268] | Vacaru, S. I., Gauge and Einstein gravity from non-Abelian gauge models on noncommutative spaces, Phys. Lett. B, 498, 74-82 (2001), arXiv:hep-th/0009163 · Zbl 0972.83047 |
| [269] | Sardanashvily, G., Gauge gravitation theory: Gravity as a Higgs field, Int. J. Geom. Methods Mod. Phys., 13, Article 1650086 pp. (2016), arXiv:1602.06776 · Zbl 1345.83033 |
| [270] | Dimitrijević, M.; Radovanović, V.; Štefančić, H., Ads-inspired noncommutative gravity on the Moyal plane, Phys. Rev. D, 86 (2012), arXiv:1207.4675 |
| [271] | Dimitrijević, M.; Radovanović, V., Noncommutative \(S O ( 2 , 3 )\) gauge theory and noncommutative gravity, Phys. Rev. D, 89 (2014), arXiv:1404.4213 |
| [272] | Stelle, K. S.; West, P. C., De sitter gauge invariance and the geometry of the Einstein-Cartan theory, J. Phys. A, 12, L205-L210 (1979) |
| [273] | Stelle, K. S.; West, P. C., Spontaneously broken de Sitter symmetry and the gravitational holonomy group, Phys. Rev. D, 21, 1466-1488 (1980) |
| [274] | Dimitrijević Ćirić, M.; Đorđević, D.; Gočanin, D.; Nikolić, B.; Radovanović, V., Noncommutative \(S O ( 2 , 3 )_\star\) Gauge theory of gravity (2022), arXiv, arXiv:2208.02152 |
| [275] | Chamseddine, A. H., Deforming Einstein’s gravity, Phys. Lett. B, 504, 33-37 (2001), arXiv:hep-th/0009153 · Zbl 0977.83065 |
| [276] | Isham, C. J.; Salam, A.; Strathdee, J., \( S L ( 6 , \mathbb{C} )\) Gauge invariance of Einstein-like Lagrangians, Lett. Nuovo Cimento, 5, 969-972 (1972), Research Gate |
| [277] | Chamseddine, A. H., \( S L ( 2 , \mathbb{C} )\) Gravity with a complex vierbein and its noncommutative extension, Phys. Rev. D, 69 (2004), arXiv:hep-th/0309166 |
| [278] | Aschieri, P.; Castellani, L., Noncommutative \(D = 4\) gravity coupled to fermions, J. High Energy Phys., 2009, 086 (2009), arXiv:0902.3817 |
| [279] | Di Grezia, E.; Esposito, G.; Vital, P., Self-dual road to noncommutative gravity with twist: A new analysis, Phys. Rev. D, 89 (2014), arXiv:abs/1312.1279 |
| [280] | Miao, Y.-G.; Zhang, S.-J., \( S L_2 ( \mathbb{C} )\) Gravity on noncommutative space with Poisson structure, Phys. Rev. D, 82 (2010), arXiv:1004.2118 |
| [281] | Chaichian, M.; Oksanen, M.; Tureanu, A.; Zet, G., Noncommutative gauge theory using a covariant star-product defined between Lie-valued differential forms, Phys. Rev. D, 81 (2010), arXiv:1001.0508 · Zbl 1194.83035 |
| [282] | de Cesare, M.; Sakellariadou, M.; Vitale, P., Noncommutative gravity with self-dual variables, Classical Quantum Gravity, 35, Article 215009 pp. (2018), arXiv:1806.04666 · Zbl 1431.83130 |
| [283] | García-Compeán, H.; Obregón, O.; Ramírez, C.; Sabido, M., Noncommutative topological theories of gravit, Phys. Rev. D, 68 (2003), arXiv:hep-th/0210203 · Zbl 1244.83025 |
| [284] | Plebański, J. F., On the separation of Einsteinian substructures, J. Math. Phys., 18, 2511-2520 (1977) · Zbl 0368.53032 |
| [285] | García-Compeán, H.; Obregón, O.; Ramírez, C.; Sabido, M., Noncommutative self-dual gravity, Phys. Rev. D, 68 (2003), arXiv:hep-th/0302180 · Zbl 1244.83025 |
| [286] | Estrada-Jiménez, S.; García-Compeán, H.; Obregón, O.; Ramírez, C., Twisted covariant noncommutative self-dual gravity, Phys. Rev. D, 78 (2008), arXiv:0808.0211 |
| [287] | Bañados, M.; Chandía, O.; Grandi, N.; Schaposnik, F.; Silva, G., Chern-Simons formulation of noncommutative gravity in three dimensions, Phys. Rev. D, 64 (2001), arXiv:hep-th/0104264 |
| [288] | Chatzistavrakidis, A.; Jonke, L.; Jurman, D.; Manolakos, G.; Manousselis, P.; Zoupanos, G., Noncommutative Gauge theory and gravity in three dimensions, Fortschr. Phys., 66, Article 1800047 pp. (2018), arXiv:1802.07550 · Zbl 1535.83084 |
| [289] | Dobrski, M., Some models of geometric noncommutative general relativity, Phys. Rev. D, 84 (2011), arXiv:1011.0165 |
| [290] | Dobrski, M., Background independent noncommutative gravity from fedosov quantization of endomorphism bundle, Classical Quantum Gravity, 34, Article 075004 pp. (2017), arXiv:1512.04504 · Zbl 1368.83050 |
| [291] | Langmann, E.; Szabo, R. J., Teleparallel gravity and dimensional reductions of noncommutative gauge theory, Phys. Rev. D, 64 (2001), arXiv:hep-th/0105094 |
| [292] | Sakharov, A. D., Vacuum quantum fluctuations in curved space and the theory of gravitation, Dokl. Akad. Nauk. Dokl. Akad. Nauk, Sov. Phys. Uspekhi. Dokl. Akad. Nauk. Dokl. Akad. Nauk, Sov. Phys. Uspekhi, Gen. Relativity and Gravitation, 32, 365-367-71 (2000), Google Scholar · Zbl 0971.83005 |
| [293] | Visser, M., Sakharov’s indueced gravity: a modenr perspective, Modern Phys. Lett. A, 17, 977-991 (2002), arXiv:gr-qc/0204062 · Zbl 1083.83544 |
| [294] | Klammer, D.; Steinacker, H., Fermions and emergent noncommutative gravity, J. High Energy Phys., 2008, 074 (2008), arXiv:0805.1157 · Zbl 1270.83021 |
| [295] | Rivelles, V. O., Noncommutative field theories and gravity, Phys. Lett. B, 558, 191-196 (2003), arXiv:hep-th/0212262 · Zbl 1011.81074 |
| [296] | Yang, H. S., On the correspondance between noncommutative field theory and gravity, Modern Phys. Lett. A, 22, 1119-1132 (2007), arXiv:hep-th/0612231 · Zbl 1117.83082 |
| [297] | Cortese, I.; García, J. A., Emergent noncommutative gravity from a consistent deformation of gauge theory, Phys. Rev. D, 81 (2010), arXiv:1001.4180 |
| [298] | Gutt, S., An explicit \(\star \)-product on the cotangent bundle of a Lie group, Lett. Math. Phys., 7, 249 (1983) · Zbl 0522.58019 |
| [299] | Rivasseau, V., The Tensor Track, III, Fortschr. Phys., 62, 81 (2014), arXiv:1311.1461; Rivasseau, V., Quantum gravity and renormalization: the tensor track, AIP Conf. Proc., 1444, 18 (2012), arXiv:1112.5104 · Zbl 1264.82056 |
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