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Generalised Chern-Simons actions for 3d gravity and \(\kappa \)-Poincaré symmetry. (English) Zbl 1192.83011

Summary: We consider Chern-Simons theories for the Poincaré, de Sitter and anti-de Sitter groups in three dimensions which generalise the Chern-Simons formulation of 3d gravity. We determine conditions under which \(\kappa \)-Poincaré symmetry and its de Sitter and anti-de Sitter analogues can be associated to these theories as quantised symmetries. Assuming the usual form of those symmetries, with a timelike vector as deformation parameter, we find that such an association is possible only in the de Sitter case, and that the associated Chern-Simons action is not the gravitational one. Although the resulting theory and 3d gravity have the same equations of motion for the gauge field, they are not equivalent, even classically, since they differ in their symplectic structure and the coupling to matter. We deduce that \(\kappa \)-Poincaré symmetry is not associated to either classical or quantum gravity in three dimensions. Starting from the (non-gravitational) Chern-Simons action we explain how to construct a multi-particle model which is invariant under the classical analogue of \(\kappa \)-de Sitter symmetry, and carry out the first steps in that construction.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
16T05 Hopf algebras and their applications
22E70 Applications of Lie groups to the sciences; explicit representations
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[1] Lukierski, J.; Nowicki, A.; Ruegg, H.; Tolstoi, V., \(q\)-deformation of Poincaré algebra, Phys. Lett. B, 264, 331-338 (1991)
[2] Lukierski, J.; Nowicki, A.; Ruegg, H., New quantum Poincaré algebra and \(κ\)-deformed field theory, Phys. Lett. B, 293, 344-352 (1992) · Zbl 0834.17022
[3] Majid, S.; Ruegg, H., Bicrossproduct structure of the \(κ\)-Poincaré group and non-commutative geometry, Phys. Lett. B, 334, 348 (1994) · Zbl 1112.81328
[4] Amelino-Camelia, G.; Smolin, L.; Starodubtsev, A., Quantum symmetry, the cosmological constant and Planck scale phenomenology, Class. Quantum Grav., 21, 3095-3110 (2004) · Zbl 1061.83025
[5] Achucarro, A.; Townsend, P., A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180, 85-100 (1986)
[6] Witten, E., \(2 + 1\) dimensional gravity as an exactly soluble system, Nucl. Phys. B, 311, 46-78 (1988) · Zbl 1258.83032
[7] Ballesteros, A.; Herranz, F. J.; Del Olmo, M. A.; Santander, M., Quantum \((2 + 1)\) kinematical algebras: A global approach, J. Phys. A: Math. Gen., 27, 1283-1298 (1994) · Zbl 0819.17015
[8] Ballesteros, A.; Bruno, N. R.; Herranz, F. J., Non-commutative relativistic spacetimes and wordlines from \(2 + 1\) quantum (anti) de Sitter groups
[9] Freidel, L.; Kowalski-Glikman, J.; Smolin, L., \(2 + 1\) gravity and doubly special relativity, Phys. Rev. D, 69, 044001 (2004)
[10] Chari, V.; Pressley, A., A Guide to Quantum Groups (1994), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0839.17009
[11] Majid, S., Foundations of Quantum Group Theory (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0857.17009
[12] Kosmann-Schwarzbach, Y., Lie bi-algebras, Poisson-Lie groups, and dressing transformations, Lect. Notes Phys., 638, 107-173 (2004)
[13] Schlieker, M.; Weich, W.; Weixler, R., Inhomogeneous quantum groups, Z. Phys. C, 79-82 (1992) · Zbl 0793.17006
[14] Majid, S., Braided momentum in \(q\)-Poincaré group, J. Math. Phys., 34, 2045-2058 (1993) · Zbl 0786.17013
[15] Zakrzewski, S., Quantum Poincaré group related to \(κ\)-Poincaré algebra, J. Phys. A, 2075-2082 (1994) · Zbl 0834.17024
[16] Kosinski, P.; Maslanka, P., The duality between \(κ\)-Poincaré group algebra and \(κ\)-Poincaré group · Zbl 1058.81558
[17] Fock, V. V.; Rosly, A. A., Poisson structures on moduli of flat connections on Riemann surfaces and \(r\)-matrices, ITEP preprint (1992) 72-92
[18] Alekseev, A. Yu.; Malkin, A. Z., Symplectic structure of the moduli space of flat connections on a Riemann surface, Commun. Math. Phys., 169, 99-119 (1995) · Zbl 0829.53028
[19] Semenov-Tian-Shansky, M. A., Dressing transformations and Poisson-Lie group actions, Publ. Res. Instrum. Math. Sci. Kyoto University, 21, 1237-1260 (1986) · Zbl 0673.58019
[20] Buffenoir, E.; Noui, K.; Roche, P., Hamiltonian quantization of Chern-Simons theory with \(SL(2, C)\) group, Class. Quantum Grav., 19, 4953-5016 (2002) · Zbl 1021.83016
[21] Meusburger, C.; Schroers, B. J., Poisson structure and symmetry in the Chern-Simons formulation of \((2 + 1)\)-dimensional gravity, Class. Quantum Grav., 20, 2193-2233 (2003) · Zbl 1029.83036
[22] Meusburger, C.; Schroers, B. J., Boundary conditions and symplectic structure in the Chern-Simons formulation of \(2 + 1\) dimensional gravity, Class. Quantum Grav., 22, 3689-3724 (2005) · Zbl 1153.83378
[23] Meusburger, C., Geometrical \((2 + 1)\)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant, Commun. Math. Phys., 273, 705 (2007) · Zbl 1148.83016
[24] Bonzom, V.; Livine, E. R., A Immirzi-like parameter for 3d quantum gravity · Zbl 1151.83323
[25] Meusburger, C.; Schroers, B. J., Quaternionic and Poisson-Lie structures in 3d gravity: The cosmological constant as deformation parameter · Zbl 1152.81561
[26] Meusburger, C.; Schroers, B. J., Phase space structure of Chern-Simons theory with a non-standard puncture, Nucl. Phys. B, 738, 425-456 (2006) · Zbl 1109.81054
[27] Meusburger, C., Grafting and Poisson structure in \((2 + 1)\)-gravity with vanishing cosmological constant, Commun. Math. Phys., 266, 735-775 (2006) · Zbl 1114.53066
[28] Meusburger, C., Dual generators of the fundamental group and the moduli space of flat connections, J. Phys. A, 39, 14781-14832 (2006) · Zbl 1107.53057
[29] Kowalski-Glikman, J.; Nowak, S., Doubly special relativity theories as different basis for \(κ\)-Poincaré algebra, Phys. Lett. B, 539, 126-132 (2002) · Zbl 0996.83004
[30] Bais, F. A.; Muller, N. M.; Schroers, B. J., Quantum group symmetry and particle scattering in \((2 + 1)\)-dimensional quantum gravity, Nucl. Phys. B, 640, 3-45 (2002) · Zbl 0997.83021
[31] Meusburger, C.; Schroers, B. J., The quantisation of Poisson structures arising in Chern-Simons theory with gauge group \(G <imes g^\ast \), Adv. Theor. Math. Phys., 7, 1003-1043 (2004) · Zbl 1063.81078
[32] Schroers, B. J., Lessons from \((2 + 1)\)-dimensional quantum gravity, in: Proceedings PoS (QG-Ph) 035 for workshop “From Quantum to Emergent Gravity: Theory and Phenomenology”, Trieste 2007
[33] Majid, S.; Schroers, B. J., \(q\)-deformation and semidualisation in 3d quantum gravity · Zbl 1187.83033
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