Constrained Supermanifolds for AdS M-Theory Backgrounds

A long standing problem is the supergauge completion of AdS_4 x (G/H)_7 or AdS_5 x (G/H)_5 backgrounds which preserve less then maximal supersymmetry. In parallel with the supersolvable realization of the AdS_4 x S^7 background based on Kappa-symmetry, we develop a technique which amounts to solving the above-mentioned problem in a way useful for pure spinor quantization for supermembranes and superstrings. Instead of gauge fixing some of the superspace coordinates to zero, we impose an additional constraint on them reproducing the simplifications of the supersolvable representations. The constraints are quadratic, homogeneous, Sp(4,R)-covariant, and consistent from the quantum point of view in the pure spinor approach. Here we provide the geometrical solution which, in a subsequent work, will be applied to the membrane and the superstring sigma models.