Constrained supermanifolds for AdS M-theory backgrounds

A long standing problem is the supergauge completion of AdS ₄ × ({/H}) ₇ or AdS ₅ × ({/H}) ₅ backgrounds which preserve less then maximal supersymmetry. In parallel with the supersolvable realization of the AdS ₄ × ⁷ background based on κ-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 using κ-symmetry, we impose an additional constraint on them reproducing the simplifications of the supersolvable representations. The constraints are quadratic, homogeneous, Sp(4,)-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.