# Quantum gravity at D = 3

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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?

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Yes, this question has many layers and I wouldn't be able to cover all of them. But 3D quantum gravity has been attacked in many ways: by AdS3/CFT2 which has some special features because of the Virasoro symmetry of CFT2 (Witten's CFT dual for pure 3D gravity involves monster symmetry); by equivalences with other topological theories i.e. Chern-Simons theory (which only holds perturbatively etc.), and others. The dynamics of pure gravity in 3D is seemingly trivial.

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But I think that what we're learned is that when we study the problem nonperturbatively and accurately, the simplicity really goes away, the simplified formulae and equivalences to field theory don't work, and there's still a whole emerging structure that requires a theory of the "same degree of complexity" as string theory, if I want to avoid questions whether e.g. the CFT duals of pure 3D gravity are a part of string theory or not.

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Classically, GR can be written as a BF-theory, with a simplicity constraint on B. Quantum mechanically this can be made to work in 3D (i.e. canonical quantization as a constrained system). Google terms: "Turaev-Viro theory". This gives the expected TQFT.

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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:

1. It reproduces the correct spectrum of black hole states.
2. 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.

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answered Dec 18, 2011 by (2,405 points)
Quick comment before I understand the rest of the answer: I think that having no excitations and being diffeomorphism invariant _is_ the same as being topological.

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The qualifier "perturbative" in my statement is important.

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If you're interested in the state of affairs before Witten's reconsideration above, you can check out Carlip's book, "Quantum Gravity in 2+1 Dimensions". It's been claimed that Witten's old quantization (Moshe's second link) via CS theory with a noncompact gauge group is equivalent to various LQG techniques, but the actual reference where that is done seems to be hard to track down.

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But @Moshe, the abstract suggests "The monster theory may be the first in a discrete series of CFT's that are dual to three-dimensional gravity" so apparently Witten thinks 3D quantum gravity _does_ make sense. What am I missing?

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The classical limit is the large k limit, Witten found the k=1 theory, subject to constraint 1 above. Subsequent papers suggest k is bounded by a relatively small number, so theories that produce the right black hole spectrum have no classical limit. This essentially says there are no large enough unitary representations of the asymptotic symmetry group, whose modular invariant partition functions fits the spectrum of black hole degeneracy.

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Or, you can start with a classical theory you hope to obtain as a limit of some quantum theory, and "quantize" it, which means do an $\hbar$ (in this context 1/k) expansion starting with that classical theory. None of the methods to obtain such quantization, all of which involve quite a bit of guesswork, obtain the right black hole spectrum for negative CC. The failed search for a quantum theory with the right properties, something which does not single out a method of "quantization", is likely the underlying reason.

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