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

We introduce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document}-extended (p, q) AdS superspaces in three space-time dimensions, with p + q = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} and p ≥ q, and analyse their geometry. We show that all (p, q) AdS superspaces with XIJ KL = 0 are conformally flat. Nonlinear σ-models with (p, q) AdS supersymmetry exist for p + q ≤ 4 (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} > {4} $\end{document} the target space geometries are highly restricted). Here we concentrate on studying off-shell \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = {3} $\end{document} supersymmetric σ-models in AdS3. 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.