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-IJKL = 0 are conformally flat. Nonlinear sigma-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 sigma-models in AdS 3. 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 sigma-model to be superconformal, and hence the corresponding target space is a hyperkahler cone. In the case of (2,1) AdS supersymmetry, the sigma-model target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO(2) group of rotations of the two-sphere of complex structures.