Supersymmetric vacua in N=2 supergravity
We use the embedding tensor formalism to analyse maximally symmetric backgrounds of N = 2 gauged supergravities which have the full N = 2 supersymmetry. We state the condition for N = 2 vacua and discuss some of their general properties. We show that if the gauged isometries leave the SU(2) R-symmetry invariant, then the N = 2 vacuum must be Minkowski. This implies that there are no AdS backgrounds with eight unbroken supercharges in the effective N = 2 supergravity of six-dimensional SU(3) x SU(3) structure compactifications of type II string theory and M-theory. Combined with previous results on N = 1 vacua, we show that there exist N = 2 supergravities with a given set of gauged Abelian isometries that have both N = 2 and N = 1 vacua. We also argue that an analogue of our analysis holds in five and six spacetime dimensions.