Independently of the formal development of massive gravity in four dimensions described above, there has been interest in constructing a purely three dimensional theory of massive gravity. Three dimensions are special for the following reason: for a massless graviton in three dimensions there are no propagating degrees of freedom. This follows simply by counting, a symmetric tensor in three dimensions has six components. A massless graviton must admit a diffeomorphism symmetry which renders three of the degrees of freedom pure gauge, and the remaining three are non-dynamical due to the associated first class constraints. On the contrary, a massive graviton in three dimensions has the same number of degrees of freedom as a massless graviton in four dimensions, namely two. Combining these two facts together, in three dimensions it should be possible to construct a diffeomorphism invariant theory of massive gravity. The usual massless graviton implied by diffeomorphism invariance is absent and only the massive degree of freedom remains.
A diffeomorphism and parity invariant theory in three dimensions was given in  and referred to as ‘new massive gravity’ (NMG). In its original formulation the action is taken to be.
The auxiliary field formulation of new massive gravity is also useful for understanding the absence of the BD ghost [141*]. Setting as imposed previously and working with the formulation (13.2*), we can introduce new vector and scalar degrees of freedom as follows
Introducing new fields in this way also introduced new symmetries. Specifically there is a symmetry[141*] it is shown that the resulting equations of motion of all fields are second order due to these special Fierz–Pauli combinations.
As a result of the introduction of the new gauge symmetries, we straightforwardly count the number of non-perturbative degrees of freedom. The total number of fields are 16: six from , six from , three from and one from . The total number of gauge symmetries are 7: three from diffeomorphisms, three from linear diffeomorphisms and one from the . Thus, the total number of degrees of freedom are which agrees with the linearized analysis. An independent argument leading to the same result is given in  where NMG including its topologically massive extension (see below) are presented in Hamiltonian form using Einstein–Cartan language (see also ).
The formalism of Section 13.2 is also useful for deriving the decoupling limit of NMG which as in the higher dimensional case, determines the leading interactions for the helicity-0 mode. The decoupling limit  is defined as the limit
The decoupling limit clarifies one crucial aspect of NMG. It has been suggested that NMG could be power counting renormalizable following previous arguments for topological massive gravity  due to the softer nature of divergences in three-dimensional and the existence of a dimensionless combination of the Planck mass and the graviton mass. This is in fact clearly not the case since the above cubic interaction is a non-renormalizable operator and dominates the Feynman diagrams leading to perturbative unitarity violation at the strong coupling scale (see Section 10.5 for further discussion on the distinction between the breakdown of perturbative unitarity and the breakdown of the theory).
The existence of the NMG theory at first sight appears to be something of an anomaly that cannot be reproduced in higher dimensions. There also does not at first sight seem to be any obvious connection with the diffeomorphism breaking ghost-free massive gravity model (or dRGT) and multi-gravity extensions. However, in [425*] it was shown that NMG, and certain extensions to it, could all be obtained as scaling limits of the same 3-dimensional bi-gravity models that are consistent with ghost-free massive gravity in a different decoupling limit. As we already mentioned, the key to seeing this is the auxiliary formulation where the tensor is related to the missing extra metric of the bi-gravity theory.
Starting with the 3-dimensional version of bi-gravity  in the form6.4*) 6.7*) in terms of the two dynamical metrics and . The scale is defined as . The idea is to define a scaling limit [425*] as follows
The generic form of the auxiliary field formulation of NMG derived above  and Born–Infeld extension NMG [279*]. Although additional higher derivative corrections have been proposed based on consistency with the holographic c-theorem , the above connection suggests that Eq. (13.23*) is the most general set of interactions allowed in NMG which are free from the BD ghost.
In the specific case of the Born–Infeld extension  the action is4 holographic renormalization group . The significance of this relation is unclear at present.
In four dimensions, the massive spin-2 representations of the Poincaré group must come in positive and negative helicity pairs. By contrast, in three dimensions the positive and negative helicity states are completely independent. Thus, while a parity preserving theory of massive gravity in three dimensions will contain two propagating degrees of freedom, it seems possible in principle for there to exist an interacting theory for one of the helicity modes alone. What is certainly possible is that one can give different interactions to the two helicity modes. Such a theory necessarily breaks parity, and was found in [180, 179]. This theory is known as ‘topologically massive gravity’ (TMG) and is described by the Einstein–Hilbert action, with cosmological constant, supplemented by a term constructed entirely out of the connection (hence the name topological)
The equations of motion for topologically massive gravity take the form to map the theory of gravity on an asymptotically AdS3 space to a 2D CFT living at the boundary.
As with any gravitational theory, it is natural to ask whether extensions exist which exhibit local supersymmetry, i.e., supergravity. A supersymmetric extension to topologically massive gravity was given in . An supergravity extension of NMG including the topologically massive gravity terms was given in [21*] and further generalized in [67*]. The construction requires the introduction of an ‘auxiliary’ bosonic scalar field so that the form of the action is[21, 67*]. The extensions of this supergravity theory to larger numbers of supersymmetries is considered at the linearized level in .
Finally, let us comment on a special case of three dimensional gravity known as log gravity  or critical gravity in analogy with the general dimension case [384, 183, 16]. For a special choice of parameters of the theory, there is a degeneracy in the equations of motion for the two degrees of freedom leading to the fact that one of the modes of the theory becomes a ‘logarithmic’ mode.
Indeed, at the special point , (where is the AdS length scale, ), known as the ‘chiral point’ the left-moving (in the language of the boundary CFT) excitations of the theory become pure gauge and it has been argued that the theory then becomes purely an interacting theory for the right moving graviton [93*]. In Ref.  it was earlier argued that there was no massive graviton excitations at the critical point , however Ref.  found one massive graviton excitation for every finite and non-zero value of , including at the critical point .
This case was further analyzed in [273*], see also Ref. [274*] for a recent review. It was shown that the degeneration of the massive graviton mode with the left moving boundary graviton leads to logarithmic excitations.
To be more precise, starting with the auxiliary formulation of NMG with a cosmological constant
As usual, it is apparent that this theory describes one massless graviton (with no propagating degrees of freedom) and one massive one whose mass is given by . However, by choosing the massive mode becomes degenerate with the existing massless one.
In this case, the action is13.45*). These are so-called because they behave logarithmically in asymptotically when the AdS metric is put in the form . The presence of these log modes was shown to remain beyond the linear regime, see Ref. .
Based on this result as well as on the finiteness and conservation of the stress tensor and on the emergence of a Jordan cell structure in the Hamiltonian, the correspondence to a logarithmic CFT was conjectured in Ref. , where the to be dual log CFTs representations have degeneracies in the spectrum of scaling dimensions.
Strong indications for this correspondence appeared in many different ways. First, consistent boundary conditions which allow the log modes were provided in Ref. , were it was shown that in addition to the Brown–Henneaux boundary conditions one could also consider more general ones. These boundary conditions were further explored in [305, 397], where it was shown that the stress-energy tensor for these boundary conditions are finite and not chiral, giving another indication that the theory could be dual to a logarithmic CFT.
Then specific correlator functions were computed and compared. Ref.  checked the 2-point correlators and Ref.  the 3-point ones. A similar analysis was also performed within the context of NMG in Ref.  where the 2-point correlators were computed at the chiral point and shown to behave as those of a logarithmic CFT.
It has been shown, however, that ultimately these theories are non-unitary due to the fact that there is a non-zero inner product between the log modes and the normal models and the inability to construct a positive definite norm on the Hilbert space .
A great deal of physics can be learned from studying exact solutions, in particular those corresponding to black hole geometries. Black holes are also important probes of the non-perturbative aspects of gravitational theories. We briefly review here the types of exact solutions obtained in the literature.
In the case of topologically massive gravity, a one-parameter family of extensions to the BTZ black hole have been obtained in . In the case of NMG as well as the usual BTZ black holes obtained in the presence of a negative cosmological constant there are in a addition a class of warped AdS3 black holes  whose metric takes the form. Further work on extensions to black hole solutions, including charged black hole solutions can be found in [418, 101, 253, 5, 6, 372, 427, 250]. We note in particular the existence of a class of Lifshitz black holes [32*] that exhibit the Lifshitz anisotropic scale symmetry   and an exact, non-stationary solution of TMG and NMG with the asymptotic charges of a BTZ black hole was find in . This exact solution was shown to admit a timelike singularity. Other exact asymptotically AdS-like solutions were found in Ref. .
One of the most interesting avenues of exploration for NMG has been in the context of Maldacena’s AdS/CFT correspondence . According to this correspondence, NMG with a cosmological constant chosen so that there are asymptotically anti-de Sitter solutions is dual to a conformal field theory (CFT). This has been considered in [67*, 381, 380] where it was found that the requirements of bulk unitarity actually lead to a negative central charge.
The argument for this proceeds from the identification of the central charge of the dual two dimensional field theory with the entropy of a black hole in the bulk using Cardy’s formula. The entropy of the black hole is given by 
A universal formula for this central charge has been obtained as is given by for a higher derivative gravity theory and identifying this with the central change through the Cardy formula. Applying this argument for new massive gravity we obtain 
As we have seen, there is a conflict in NMG between unitarity in the bulk, i.e., the requirement that the massive gravitons are not ghosts, and unitarity in dual CFT as required by the positivity of the central charge. This conflict may be resolved, however, by replacing NMG with the 3-dimensional bi-gravity extension of ghost-free massive gravity that we have already discussed. In particular, if we work in the Einstein–Cartan formulation in three dimensions, then the metric is replaced by a ‘dreibein’ and since this is a bi-gravity model, we need two ‘dreibeins’. This gives us the Zwei-dreibein gravity [63*]. it is shown that there is an open set of solutions which are close to the special case , , and . This result is not in contradiction with the scaling limit that reproduces NMG, because this scaling limit requires the choice which is in contradiction with positive central charge.
These results potentially have an impact on the higher dimensional case. We see that in three dimensions we potentially have a diffeomorphism invariant theory of massive gravity (i.e., bi-gravity) which at least for AdS solutions exhibits unitarity both in the bulk and in the boundary CFT for a finite range of parameters in the theory. However, these bi-gravity models are easily extended into all dimensions as we have already discussed and it is similarly easy to find AdS solutions which exhibit bulk unitarity. It would be extremely interesting to see if the associated dual CFTs are also unitary thus providing a potential holographic description of generalized theories of massive gravity.