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

Abstract

We start from the Maurer–Cartan (MC) equations of the [Formula: see text] 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 [Formula: see text] we single out different subalgebras [Formula: see text] associated with the different cosets [Formula: see text]: 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|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 AdS 4 ×ℙ3. The pure spinor sigma model can be consistently quantized.