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Types of quantum field thories
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 .
One can add additional terms arriving at what is called massive 3d gravity models . Very relevant for its study is the AdS3/CFT2 correspondence.
A classical article on 3d gravity is
The first correct (complete, i.e. non-perturbative) quantization of 3-dimensional gravity, on manifolds of the product form $\Sigma \times \mathbb{R}$ appears in
A textbook account discussing this and a variety of approaches to quantization of 3d gravity is
see also the further pointers here on Carlip’s webpage.
More recent developments include
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)
Reviews include
Steven Carlip, Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe (living reviews)
Laura Donnay, Asymptotic dynamics of three-dimensional gravity (arXiv:1602.09021)
Authors of spin foam models view them as an approach to quantum gravity.