Three Dimensional Einstein’s Gravity
General Relativity is a very complex theory whose quantization remains elusive. A reduced version of the theory exists: 3-dimensional Einstein’s gravity to which this lecture is dedicated. As a toy-model, it is a very useful framework thanks to which we can experiment some techniques and derive features, some of which extend to the physical 4d case.
In this lecture, we will review the typical properties of 3d gravity, which are mostly due to the vanishing of the Weyl curvature. Next we will turn to the AdS3 phase space: we will describe global features of AdS3 itself, give several elements on the Brown-Henneaux boundary conditions and the resulting asymptotic symmetry group, and finally discuss BTZ black holes. We will show that the asymptotically flat phase space can be obtained from the flat limit of the asymptotically AdS3 phase space. Finally, we will shortly present the Chern-Simons formulation of 3d gravity, which reduces the theory to the one of two non-abelian gauge vector fields.
- 4.E. Witten, (2+1)-Dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988). http://dx.doi.org/10.1016/0550-3213(88)90143-5
- 9.E. Witten, Three-dimensional gravity revisited (2007, Unpublished). http://arxiv.org/abs/0706.3359
- 10.A. Maloney, E. Witten, Quantum gravity partition functions in three dimensions. J. High Energy Phys. 02, 029 (2010). arXiv:0712.0155 [hep-th]; http://dx.doi.org/10.1007/JHEP02(2010)029
- 11.G. Compère, W. Song, A. Strominger, New boundary conditions for AdS3. J. High Energy Phys. 05, 152 (2013). arXiv:1303.2662 [hep-th]; http://dx.doi.org/10.1007/JHEP05(2013)152
- 13.G. Barnich, C. Troessaert, Aspects of the BMS/CFT correspondence. J. High Energy Phys. 1005, 062 (2010). arXiv:1001.1541 [hep-th]; http://dx.doi.org/10.1007/JHEP05(2010)062
- 14.G. Barnich, A. Gomberoff, H.A. Gonzalez, The flat limit of three dimensional asymptotically anti-de Sitter spacetimes. Phys. Rev. D 86, 024020 (2012). arXiv:1204.3288 [gr-qc]; http://dx.doi.org/10.1103/PhysRevD.86.024020
- 15.B. Oblak, BMS particles in three dimensions. PhD thesis, Brussels U., 2016. arXiv:1610.08526 [hep-th]; http://dx.doi.org/10.1007/978-3-319-61878-4; http://inspirehep.net/record/1494790/files/arXiv:1610.08526.pdf
- 17.G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions. J. High Energy Phys. 10, 095 (2012). arXiv:1208.4371 [hep-th]; http://dx.doi.org/10.1007/JHEP10(2012)095
- 18.A. Bagchi, S. Detournay, R. Fareghbal, J. Simón, Holography of 3D flat cosmological horizons. Phys. Rev. Lett. 110(14), 141302 (2013). arXiv:1208.4372 [hep-th]; http://dx.doi.org/10.1103/PhysRevLett.110.141302
- 21.J. de Boer, J.I. Jottar, Thermodynamics of higher spin black holes in AdS 3. J. High Energy Phys. 01, 023 (2014). arXiv:1302.0816 [hep-th]; http://dx.doi.org/10.1007/JHEP01(2014)023
- 22.G. Compère, P. Mao, A. Seraj, M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons. J. High Energy Phys. 01, 080 (2016). arXiv:1511.06079 [hep-th]; http://dx.doi.org/10.1007/JHEP01(2016)080