nLab D=3 quantum gravity

Idea

Solving the problem of quantization of systems including gravity – quantum gravity – is notoriously hard. It does however simplify drastically in very low dimensions. While the theory in dimension 4 (or higher) is evidently relevant for the phenomenology of the world that we perceive, the Einstein-Hilbert action and its variants that defines the theory of gravity (at least as an effective QFT) makes sense in any dimension.

The case of dimension 3 is noteworthy, because in this case the quantum theory can be and has been fairly completely understood and is nevertheless non-trivial. Informally, this is due to the fact that behaviour of gravity in 3-dimensions is much simpler than in higher dimensions: there cannot be gravitational waves in 3-dimensions, hence no “local excitations”. Accordingly, the theory turns out to have a finite dimensional covariant phase space.

More formally, one finds that in 3-dimensions, the Einstein-Hilbert action, in the first order formulation of gravity (Cartan connection), becomes equivalent to the action functional of a very well studied 3-dimensional field theory, namely Chern-Simons theory for gauge group the Poincaré group $Iso(2,1)$ (for vanishing cosmological constant) or the de Sitter group/anti de Sitter group $Iso(2,2)$ or $Iso(3,1)$ (for non-vanishing cosmological constant). See at de Sitter gravity and at Chern-Simons Gravity.

This means that one can take the quantization of $Iso(2,1)$-Chern-Simons theory as the definition of 3d quantum gravity. This has first been noticed and successfully carried out in (Witten88). One should note here that this means that one allows degenerate vielbein/pseudo-Riemannian metric tensors as field configurations of gravity. In fact, as (Witten88) discusses in detail, this compactification of configuration space can be seen as the source of the reasons why 3d quantum gravity makes exists.

Based on this striking situation in 3-dimensions it is natural to wonder if it makes sense to consider $Iso(n,1)$-higher dimensional Chern-Simons theory, or variants thereof, as theories of quantum gravity in higher dimensions. The for $n \gt 2$ the corresponding action functionals differ from the Einstein-Hilbert action by higher curvature terms, but in suitable limits of the theory these can be argued to play a negligible role. For more on this see the entry Chern-Simons gravity .

Variants

One can add additional terms arriving at what is called massive 3d gravity models . Very relevant for its study is the AdS3/CFT2 correspondence.

Related concepts

• BTZ black hole

• D=3 TQFT

• D=3 supergravity

• quantum gravity

• Chern-Simons gravity

• Boulatov model

• D=2 quantum gravity

• volume conjecture

• piecewise flat spacetime

References

• Nathan Benjamin, Scott Collier, Alexander Maloney, Pure Gravity and Conical Defects, Journal of High Energy Physics 2020 34 (2020) [arXiv:2004.14428, doi:10.1007/JHEP09(2020)034]

• Mauricio Leston, et al., 3d Quantum Gravity Partition Function at 3 Loops: a brute force computation [arXiv:2307.03830]

Survey:

• Steve Carlip, Quantum Gravity in 2+1 Dimensions, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.12596]

3d Gravity and Chern-Simons theory

On 3-dimensional (quantum) gravity (general relativity) with cosmological constant, and its (non-)relation to Chern-Simons theory with non-compact gauge groups:

The original articles on 3d gravity, discussing its formulation as a Chern-Simons theory and discovering its holographic relation to a 2d CFT boundary field theory (well before AdS/CFT was conceived from string theory):

• Stanley Deser, Roman Jackiw, Gerard 't Hooft, Three-dimensional Einstein gravity: Dynamics of flat space, Ann. Phys. 152 (1984) 220 (doi:10.1016/0003-4916(84)90085-X)

• Stanley Deser, Roman Jackiw, Three-dimensional cosmological gravity: Dynamics of constant curvature, Annals of Physics, Volume 153, Issue 2, 1 April 1984, Pages 405-416 (doi:10.1016/0003-4916(84)90025-3, spire:192694)

• Roman Jackiw, Lower dimensional gravity, Nuclear Physics B Volume 252, 1985, Pages 343-356 (doi:10.1016/0550-3213(85)90448-1, spire:204694)

• J. D. Brown, Marc Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity, Commun. Math. Phys. (1986) 104: 207 (doi:10.1007/BF01211590)

• A. Achucarro, Paul Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories, Phys. Lett. B180 (1986) 89 (doi:10.1016/0370-2693(86)90140-1, spire:21208)

• Steven Carlip, Inducing Liouville theory from topologically massive gravity, Nuclear Physics B Volume 362, Issues 1–2, 16 September 1991, Pages 111-124 (doi:10.1016/0550-3213(91)90558-F)

• O. Coussaert, Marc Henneaux, P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961-2966 (arXiv:gr-qc/9506019)

The corresponding non-perturbative quantization of 3-dimensional gravity, via quantization of 3d Chern-Simons theory:

• Edward Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46. [doi:10.1016/0550-3213(88)90143-5

• Herman Verlinde, Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space, Nuclear Physics B 337 3 (1990) 652-680 [doi:10.1016/0550-3213(90)90510-K]

Review:

• Steven Carlip, Lectures in (2+1)-Dimensional Gravity, J. Korean Phys. Soc. 28: S447-S467, 1995 (arXiv:gr-qc/9503024)

• Steven Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge Monographs on Mathematical Physics (2003) (publisher)

• Steven Carlip, Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe, Living Rev. Rel. 8:1, 2005 (arXiv:gr-qc/0409039)

• Steven Carlip, My Research – (2+1)-Dimensional quantum gravity

• Laura Donnay, Asymptotic dynamics of three-dimensional gravity, Proceedings of Eleventh Modave Summer School in Mathematical Physics, POS 271 (2016) $[$arXiv:1602.09021, doi:10.22323/1.271.0001$]$

• Wout Merbis, Chern-Simons-like Theories of Gravity (arXiv:1411.6888)

• Chen-Te Ma, $AdS_3$ Einstein Gravity and Boundary Description: Pedagogical Review $[$arXiv:2310.04665$]$

Further developments:

• Edward Witten, Three-dimensional gravity revisited (2007) [arXiv:0706.3359]

• Paul Townsend, Massive 3d (super)gravity, slides, (pdf)

• Gaston Giribet, Black hole physics and AdS3/CFT2 correspondence, lectures at Croatian Black Hole School 2010

• Alan Garbarz, Gaston Giribet, Yerko Vásquez, Asymptotically AdS$_3$ solutions to topologically massive gravity at special values of the coupling constants, arxiv/0811.4464

• Rudranil Basu, Samir K Paul, Consistent 3D Quantum Gravity on Lens Spaces (arXiv:1109.0793)

• Marc Henneaux, Wout Merbis, Arash Ranjbar, Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions (arXiv:1912.09465)

• Viraj Meruliya, Sunil Mukhi, Palash Singh, Poincaré Series, 3d Gravity and Averages of Rational CFT (arXiv:2102.03136)

• Scott Collier, Lorenz Eberhardt, Mengyang Zhang, 3d gravity and Chern-Simons theory – references $[$arXiv:2304.13650$]$

See also:

• Nathan Benjamin, Scott Collier, Alexander Maloney, Pure Gravity and Conical Defects, Journal of High Energy Physics 2020 34 (2020) $[$arXiv:2004.14428, doi:10.1007/JHEP09(2020)034$]$

Wilson lines computing holographic entropy in $AdS_3/CFT_2$

Discussion of BTZ black hole entropy and more generally of holographic entanglement entropy in 3d quantum gravity/AdS3/CFT2 via Wilson line observables in Chern-Simons theory:

• Martin Ammon, Alejandra Castro, Nabil Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110 (arXiv:1306.4338)

• Jan de Boer, Juan I. Jottar, Entanglement Entropy and Higher Spin Holography in $AdS_3$, JHEP 1404:089, 2014 (arXiv:1306.4347)

• Alejandra Castro, Stephane Detournay, Nabil Iqbal, Eric Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 (arXiv:1405.2792)

• Mert Besken, Ashwin Hegde, Eliot Hijano, Per Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099 (arXiv:1603.07317)

• Andreas Blommaert, Thomas G. Mertens, Henri Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (arXiv:1806.07765)

• Ashwin Dushyantha Hegde, Role of Wilson Lines in 3D Quantum Gravity, 2019 (spire:1763572)

• Xing Huang, Chen-Te Ma, Hongfei Shu, Quantum Correction of the Wilson Line and Entanglement Entropy in the $AdS_3$ Chern-Simons Gravity Theory (arXiv:1911.03841)

• Eric D'Hoker, Per Kraus, Gravitational Wilson lines in $AdS_3$ (arXiv:1912.02750)

• Marc Henneaux, Wout Merbis, Arash Ranjbar, Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions (arXiv:1912.09465)

and similarly for 3d flat-space holography:

• Arjun Bagchi, Rudranil Basu, Daniel Grumiller, Max Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114, 111602 (2015) (arXiv 1410.4089)

• Rudranil Basu, Max Riegler, Wilson Lines and Holographic Entanglement Entropy in Galilean Conformal Field Theories, Phys. Rev. D 93, 045003 (2016) (arXiv:1511.08662)

• Wout Merbis, Max Riegler, Geometric actions and flat space holography (arXiv:1912.08207)

Discussion for 3d de Sitter spacetime:

• Alejandra Castro, Philippe Sabella-Garnier, Claire Zukowski, Gravitational Wilson Lines in 3D de Sitter (arXiv:2001.09998)

Via AdS/CFT duality

Approaching 3d quantum gravity on asymptotically $AdS_3$-spacetimes via AdS/CFT duality as described by statistical ensembles of 2d CFTs:

• Alexandre Belin, Jan de Boer, Daniel Louis Jafferis, Pranjal Nayak, Julian Sonner: Approximate CFTs and Random Tensor Models [arXiv:2308.03829]

• Daniel L. Jafferis, Liza Rozenberg, Gabriel Wong: 3d Gravity as a random ensemble [arXiv:2407.02649]