Pure spinor formalism for Osp(N 4)backgrounds

We start from the Maurer-Cartan (MC) equations of the Osp(N4) superalgebras satisfied by the left-invariant superforms realized on supercoset manifolds of the corresponding supergroups and we derive some new pure spinor constraints. They are obtained by ghostifying the MC forms and extending the differential d to a BRST differential. From the superalgebras G = Osp(N4) we single out different subalgebras H \subset G associated with the different cosets G/H: each choice of leads to a different weakening of the pure spinor constraints. In each case, the number of parameter is counted and we show that in the cases of Osp(6|4)/U(3)×SO(1, 3), Osp(4|4)/SO(3) ×SO(1, 3) and finally Osp(4|4)/U(2) ×SO(1, 3) the bosonic and fermionic degrees of freedom match in order to provide a c = 0 superconformal field theory. We construct both the Green-Schwarz and the pure spinor sigma model for the case Osp(6|4)/U(3) ×SO(1, 3) corresponding to AdS₄ ׳. The pure spinor sigma model can be consistently quantized.