There exists two distinct off-shell N = 2 supergravities in three dimensions. They are also referred to as N = (1, 1) and N = (2, 0) supergravities, and they arise from the coupling of the Weyl multiplet to a compensating scalar or vector multiplet, respectively, followed by fixing of conformal symmetries. The N = (p, q) terminology refers to the underlying anti-de Sitter superalgebras OSp(2, p)⊕OSp(2, q) with R-symmetry group SO(p) × SO(q). We construct off-shell invariants of these theories up to fourth order in derivatives. As an application of these results, we determine the special combinations of the N = (1, 1) invariants that admit anti-de Sitter vacuum solution about which there is a ghost-free massive spin-2 multiplet of propagating modes. We also show that the N = (2, 0) invariants do not allow such possibility.

We revisit the problem of the bulk-boundary unitarity clash in 2 + 1 dimensional gravity theories, which has been an obstacle in providing a viable dual two-dimensional conformal field theory for bulk gravity in anti-de Sitter (AdS) spacetime. Chiral gravity, which is a particular limit of cosmological topologically massive gravity (TMG), suffers from perturbative log-modes with negative energies inducing a non-unitary logarithmic boundary field theory. We show here that any f (R) extension of TMG does not improve the situation. We also study the perturbative modes in the metric formulation of minimal massive gravityoriginally constructed in a first-order formulation-and find that the massive mode has again negative energy except in the chiral limit. We comment on this issue and also discuss a possible solution to the problem of negative energy modes. In any of these theories, the infinitesimal dangerous deformations might not be integrable to full solutions; this suggests a linearization instability of AdS spacetime in the direction of the perturbative log-modes.

Using an off-shell Killing spinor analysis we perform a systematic investigation of the supersymmetric background and black hole solutions of the N = (1, 1) Cosmological New Massive Gravity model. The solutions with a null Killing vector are the same pp-wave solutions that one finds in the N = 1 model but we find new solutions with a time-like Killing vector that are absent in the N = 1 case. An example of such a solution is a Lifshitz spacetime. We also consider the supersymmetry properties of the so-called rotating hairy BTZ black holes and logarithmic black holes in an AdS 3 background. Furthermore, we show that under certain assumptions there is no supersymmetric Lifshitz black hole solution.

In three dimensions, there exist modifications of Einstein's gravity akin to the topologically massive gravity that describe massive gravitons about maximally symmetric backgrounds. These theories are built on the three-dimensional version of the Bach tensor (a curl of the Cotton-York tensor) and its higher derivative generalizations; and they are on-shell consistent without a Lagrangian description based on the metric tensor alone. We give a generic construction of these models, find the spectra and compute the conserved quantities for the Banados-Teitelboim-Zanelli black hole.

The requirement of the existence of a holographic c-function for higher derivative theories is a very restrictive one, and hence most theories do not possess this property. Here, we show that, when some of the parameters are fixed, the D ≥ 3 Born-Infeld gravity theories admit a holographic c-function. We work out the details of the D ¼ 3 theory with no free parameters, which is a nonminimal Born-Infeld type extension of new massive gravity. Moreover, we show that these theories generate an infinite number of higher derivative models admitting a c-function in a suitable expansion, and therefore they can be studied at any truncated order.