Nonlinear sigma models with AdS supersymmetry in three dimensions

Understand

In three-dimensional anti-de Sitter (AdS) space, there exist several realizations of N-extended supersymmetry, which are traditionally labelled by two non-negative integers p>=q such that p+q=N.

• Different choices of p and q, with N fixed, prove to lead to different restrictions on the target space geometry of supersymmetric nonlinear sigma-models.
• We classify all possible types of hyperkahler target spaces for the cases N=3 and N=4 by making use of two different realizations for the most general (p,q) supersymmetric sigma-models: (i) off-shell formulations in terms of N=3 and N=4 projective supermultiplets; and (ii) on-shell formulations in terms of covariantly chiral scalar superfields in (2,0) AdS superspace.
• Depending on the type of N=3,4 AdS supersymmetry, nonlinear sigma-models can support one of the following target space geometries: (i) hyperkahler cones; (ii) non-compact hyperkahler manifolds with a U(1) isometry group which acts non-trivially on the two-sphere of complex structures; (iii) arbitrary hyperkahler manifolds including compact ones.