PURE SPINOR FORMALISM FOR ${\rm Osp}({\mathcal N}\vert 4)$ BACKGROUNDS

We start from the Maurer-Cartan (MC) equations of the Osp(N vertical bar 4) 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) over cap = Osp(N vertical bar 4) we single out different subalgebras H subset of (G) over cap associated with the different cosets (C) over cap /H: each choice of H 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 vertical bar 4)/U(3) x SO(1, 3), Osp(4 vertical bar 4)/SO(3) x SO(1, 3) and finally Osp(4 vertical bar 4)/U(2) x 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 vertical bar 4)/U(3) x SO(1, 3) corresponding to AdS(4) x P-3. The pure spinor sigma model can be consistently quantized