Three-dimensional (p, q) AdS superspaces and matter couplings

We introduce N-extended (p,q) AdS superspaces in three space-time dimensions, with p + q = N and p ≥ q, and analyse their geometry. We show that all (p, q) AdS superspaces with X = 0 are conformally flat. Nonlinear σ-models with (p,q) AdS supersymmetry exist for p + q ≤ 4 (for N > 4 the target space geometries are highly restricted). Here we concentrate on studying off-shell N = 3 supersymmetric σ-models in AdS . For each of the cases (3,0) and (2,1), we give three different realisations of the supersymmetric action. We show that (3,0) AdS supersymmetry requires the σ-model to be superconformal, and hence the corresponding target space is a hyperkähler cone. In the case of (2,1) AdS supersymmetry, the σ-model target space must be a non-compact hyperkähler manifold endowed with a Killing vector field which generates an SO(2) group of rotations of the two-sphere of complex structures