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\begin{document}

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\section*{3d quantum gravity}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 ([[Kaluza-Klein mechanism|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 \emph{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 \emph{[[de Sitter gravity]]} and at \emph{\href{http://ncatlab.org/nlab/show/Chern-Simons+gravity}{Chern-Simons Gravity}}.

This means that one can take the [[quantization]] of $Iso(2,1)$-[[Chern-Simons theory]] as the \emph{definition} of 3d [[quantum gravity]]. This has first been noticed and successfully carried out in (\hyperlink{Witten88}{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 (\hyperlink{Witten88}{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 functional]]s 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 \emph{[[Chern-Simons gravity]]} .

\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

One can add additional terms arriving at what is called \emph{massive 3d gravity models} . Very relevant for its study is the [[holographic principle|AdS3/CFT2 correspondence]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[3d TQFT]]


\item [[3-dimensional supergravity]]


\item [[quantum gravity]]


\item [[Chern-Simons gravity]]


\item [[Boulatov model]]


\item [[2d quantum gravity]]


\item [[volume conjecture]]


\item [[piecewise flat spacetime]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A classical article on 3d [[gravity]] is

\begin{itemize}%
\item [[Stanley Deser]], [[Roman Jackiw]], [[Gerard `t Hooft]], \emph{Three-dimensional Einstein gravity: Dynamics of flat space}, Ann. Phys. 152 (1984) 220.

\end{itemize}
The first correct (complete, i.e. non-[[perturbation theory|perturbative]]) quantization of 3-dimensional gravity, on manifolds of the product form $\Sigma \times \mathbb{R}$ appears in

\begin{itemize}%
\item [[Edward Witten]], \emph{(2+1)-Dimensional Gravity as an Exactly Soluble System} Nucl. Phys. B311 (1988) 46. (\href{http://adsabs.harvard.edu/abs/1988NuPhB.311...46W}{web})

\end{itemize}
A textbook account discussing this and a variety of approaches to quantization of 3d gravity is

\begin{itemize}%
\item [[Steven Carlip]], \emph{Quantum Gravity in 2+1 Dimensions}, Cambridge Monographs on Mathematical Physics (2003) (\href{http://www.cambridge.org/de/academic/subjects/physics/cosmology-relativity-and-gravitation/quantum-gravity-21-dimensions}{publisher})

\end{itemize}
see also the further pointers \href{http://www.physics.ucdavis.edu/Text/Carlip.html#2+1}{here on Carlip's webpage}.

More recent developments include

\begin{itemize}%
\item [[Edward Witten]], \emph{Three-dimensional gravity revisited}, (2007) \href{http://arxiv.org/abs/0706.3359}{arxiv/0706.3359}


\item [[Paul Townsend]], \emph{Massive 3d (super)gravity}, slides, (\href{http://superfields.web.cern.ch/Superfields/docs/Seminars/Townsend.pdf}{pdf})


\item Gaston Giribet, \emph{Black hole physics and AdS3/CFT2 correspondence}, lectures at [[Croatian Black Hole School]] 2010


\item Alan Garbarz, Gaston Giribet, Yerko V\'a{}squez, \emph{Asymptotically AdS$_3$ solutions to topologically massive gravity at special values of the coupling constants}, \href{http://arxiv.org/abs/0811.4464}{arxiv/0811.4464}


\item Rudranil Basu, Samir K Paul, \emph{Consistent 3D Quantum Gravity on Lens Spaces} (\href{http://arxiv.org/abs/1109.0793}{arXiv:1109.0793})



\end{itemize}
Reviews include

\begin{itemize}%
\item [[Steven Carlip ]], \emph{Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe} (\href{http://relativity.livingreviews.org/open?pubNo=lrr-2005-1&page=articlesu10.html}{living reviews})


\item Laura Donnay, \emph{Asymptotic dynamics of three-dimensional gravity} (\href{http://arxiv.org/abs/1602.09021}{arXiv:1602.09021})



\end{itemize}
Authors of [[spin foam]] models view them as an approach to quantum gravity.



\end{document}