Massive N=2 Supergravity in Three Dimensions

There exists two distinct off-shell $\mathcal{N}=2$ supergravities in three dimensions. They are also referred to as $\mathcal{N}=(1,1)$ and $\mathcal{N}=(2,0)$ 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 $\mathcal{N}=(p,q)$ terminology refers to the underlying anti-de Sitter superalgebras $OSp(2,p) \oplus OSp(2,q)$with$R$-symmetrygroup$SO(p) \times SO(q)$. 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 $\mathcal{N}=(1,1)$ 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 $\mathcal{N}=(2,0)$ invariants do not allow such possibility.