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I was briefly going through the idea of 3d quantum gravity on nLab, where it is stated that:

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

So what do we mean by quantizing gravity if there are no propagating fields for gravity in (2+1) dim spacetime (i.e. no conformal curvature)? I might be wrong, but to me it sounds like quantizing an electromagnetic field with a vanishing Maxwell tensor everywhere(which is just absurd), this is because Maxwell tensor plays the same role in electromagnetism as Weyl tensor in Einstein's gravity.

So, by the term "gravity", do we mean arbitrary spacetime geometry in any dimension, or need for a propagating gravitational field (i.e. nonvanishing Weyl curvature) is also an absolutely necessary ingredient?

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  • $\begingroup$ "quantizing an electromagnetic field with a vanishing Maxwell tensor everywhere(which is just absurd)" not absurd at all – gauge theories with vanishing curvature tensors still have topological degrees of freedom and give curious mathematical results, e.g. the Chern-Simons theory $\endgroup$ Jun 12, 2021 at 21:15
  • $\begingroup$ Thanks for the clarification :) $\endgroup$
    – KP99
    Jun 13, 2021 at 3:09

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Theoretically, gravity in $n$ dimensions is described by the Einstein-Hilbert action:

$$ S\sim\int\mathrm d^nx\sqrt{-g}\ R $$

However in pure 3D gravity, there are no propagating gravitational degrees of freedom. This is because the Weyl tensor is identically zero, and since it encodes the behaviour of gravitational waves, it is clear that gravitational information cannot propagate across the manifold. The Weyl tensor also contains the information about the Newtonian potential and higher-order potentials, so surprisingly in 3D gravity, masses do not attract! As a result it was actually thought for a long time that the theory was trivial - that is, until the discovery of the BTZ black hole in 1992. Furthermore, 3D gravity with $\Lambda<0$ also shows features like spinning particles, which appear as local conical defects, as well asymptotic symmetries of spacetime.

However, recall that the independent components of the Riemann tensor (the "field strength") can be decomposed into to the Weyl tensor and the Ricci tensor, the latter describing the curvature due to matter distribution ("sources"), which is non-zero in general. So 3D gravity is not trivial as you seem to think.

Whether or not one wishes to deem a theory with features like this as "gravitational" is obviously a purely semantic choice, but theorists like 3D quantum gravity for multiple reasons:

The famous AdS/CFT correspondence is a duality between a gravitational theory in $n$ dimensions and a specific quantum theory in $n-1$ dimensions, and provides non-perturbative results in string theory. The original discovery was apropos AdS5/CFT4 by Maldacena, however analogues have been found in other dimensions. An AdS spacetime is one with a constant negative cosmological constant, and the first benefit in 3D is that any spacetime with negative curvature is locally equivalent to the AdS spacetime. One might think to simplify further to AdS2, but this is in some regards too trivial: AdS3 provides the right balance between complexity and tractability. As I mentioned, 3D gravity has rotating black hole solutions that asymptote to AdS3, and its partner CFT2 is sufficiently well-understood that one can translate results between the two in order to provide insights on solving the information loss paradox, aid in microscopic black hole entropy computations, etc. AdS3 is also amenable to serve as a background for WZW-models of string propagation. As such, 3D quantum gravity is heavily entwined with string theory - in fact, it is not even known whether there us a well-defined description of it without embedding it in string theory.

An important theoretical endeavour is to describe the gravitational force as a quantum theory and incorporate it into our current descriptions of the other forces to form a theory of everything. This is incredibly difficult, so analysing the more tractable 3D quantum gravity is a good way to seek hints, extrapolate results and develop the requisite tools. Note that pure 3D gravity is not the only option here: one is free to couple Chern-Simons/topological terms in order to produce gravitons.

While 3D quantum gravity is naïvely non-renormalisable and looks doomed as a quantum theory, the techniques mentioned above enable us to gain insights about non-perturbative formulations and features of such a quantum theory. A good deal of progress in this regard came from the Chern-Simons formulation of 3D gravity, a construction due to Achúcarro, Townsend and Witten: I highly recommend Witten's paper for a good review of the modern methods and results.

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  • $\begingroup$ Thanks :) Is it possible to extend 3d gravity construction to a complex 3 dimensional Riemannian Manifold and then later restrict attention to a real 4 dim Lorentzian sub manifold? Seems like one can then obtain some form of a 4D quantum gravity which might have direct physical relevance $\endgroup$
    – KP99
    Jun 11, 2021 at 11:01
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    $\begingroup$ This is a nice answer. +1. I have a question about this part: "surprisingly in 3D gravity, masses do not attract". Do you know a reference that discusses this in detail? I like to see this result by a direct computation in path integral formalism. I'm curious that how a direct computation by zero-loop approximation of quantum gravity in path integral formalism gives the same result (I can imagine it now but not sure yet). $\endgroup$
    – SG8
    Jun 11, 2021 at 11:14
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The answer by Nihar Karve is really nice and relevant, although I would have some additional comments.

It does make sense to talk about gravity in any dimensions. If one would say that the discussion of gravity in 3d does not make sense then what about 11 or 26? :).

Gravity can be discussed in terms of geometry, and using the Einstei-Hilbert action one can formulate the lattice version of quantum gravity based on Regge calculus and the path integral formalism. This project is running under the name Dynamical Triangulations (DT) . Depending on the manifold constraints you can have EDT or CDT. EDT uses wick rotation but keeps the d-dimensional manifolds as the element of the study, but CDT introduces also a foliation, thus 3-dimensional submanifolds are connected causally to each other.

Its theorised, but not yet proven, that every time slice of CDT is actually described by a lower dimensional EDT. So the 4d CDT is actually 3+1 d quantum gravity, but the spatial submanifolds may be described by 3d EDT. This is one of the reasons why it could be potentially interesting to study 3d quantum gravity!

(if you are interested I can extend my answer)

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  • $\begingroup$ Can 4d CDT incorporate different matter field Lagrangians or is it just for vacuum solutions? I was wondering if one could define analogue of Dirac fields or arbitrary spinor fields (as in 4d space-time) in 3d space-time. $\endgroup$
    – KP99
    Jun 12, 2021 at 7:28
  • $\begingroup$ Definitely! We are just publishing the first results when matter fields (scalar) are added to 4d CDT. You can find the results in arxiv: arxiv.org/abs/2105.10086 . It turns out that it has a serious effect on the geometry! It initiates an effective change of topology, so there is a matter driven phase transition observed. We also have some results related to particles (massline), but it is not yet out. $\endgroup$
    – Kregnach
    Jun 12, 2021 at 7:33
  • $\begingroup$ @KP99 if you are interested, look for "cosmic voids and filaments from quantum gravity" :) ( or matter driven phase transition in lattice quantumgravity ) $\endgroup$
    – Kregnach
    Aug 15, 2021 at 21:07
  • $\begingroup$ Thank you. These ideas are quite interesting and I will go through them soon (although I could hardly scratch the surface) . Since past few months I am thinking about QG along the lines of R. Penrose's Twistor theory (keeping in mind that it is lagging behind other popular approaches and it is sophisticated). It is really fascinating that quantization of twistor space is necessary for vacuum Einstein's solutions to exist. I can recommend this as a fun reading :) (I shouldn't pretend that I understand twistors entirely, I'm just a grad student, & mostly these are my self studies) $\endgroup$
    – KP99
    Aug 16, 2021 at 14:43
  • $\begingroup$ I am familiar with twistor theory, but my knowledge is shallow, as it is on loop quantum gravity too. I didn't have time to delve into it yet. Althou it is "just a gauge theory", just certain things are called differently. Maybe next year I'll look into it deeper :) $\endgroup$
    – Kregnach
    Aug 16, 2021 at 14:50

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