Quantum gravity at D = 3 Quantization of gravity (general relativity) seems to be impossible for spacetime dimension D >= 4. Instead, quantum gravity is described by string theory which is something more than quantization (for example it introduces an infinite number of fields). More direct approaches to quantization of gravity have failed (my opinion; not everyone would agree on this).
However, gravity in dimensions D < 4 is special, because it is topological (carries no dynamic degrees of freedom locally). It is possible to quantize gravity coupled to other fields at D = 2: in fact perturbative string theory is exactly that! What about D = 3? Are there approaches to quantization of gravity (coupled to other fields) which have been more successful at D = 3 than at D = 4?
 A: Two quick comments: 
First, having no perturbative excitations is not the same as being topological, at least with the conventional use of these terms.
Secondly, whether quantization of the metric “works” depends strongly on what your expectations are and what tests you subject your theory to, so you’ll get different stories from different camps. For gravity with negative CC, Witten’s work in http://arxiv.org/abs/0706.3359 and especially some followups strongly suggest that regardless of which trick you might try using to quantize gravity, in the end there is simply no quantum theory with the two properties:


*

*It reproduces the correct spectrum of black hole states. 

*Its classical limit is pure classical gravity. 


This seems a model independent result to me, excluding many previous attempts including Witten’s own celebrated contribution which started the ball rolling. It could also be that with zero or positive CC the result may be different, but I personally don’t see why tricks that give the wrong answer for negative CC (for which we at least know which questions are well-defined) will somehow miraculously work in a much less well-understood context.
