Conformal superspaces, projectivity and quantization

Abstract

In this paper we study the complex conformal superspace in D = 4 for N = 1 supersymmetry as a projective superspace. Among the different superspaces that carry a transitive action of the conformal supergroup, the superflag Fl(2|0, 2|1, 4|1) is distinguished as the minimal one that has a reality condition preserved by the real conformal supergroup. As in the non super case, the different flag supermanifolds are related to parabolic subalgebras and imposing the constraint that the reduced manifold must be the Grassmannian Gr(2,4), the complex conformal space, leaves us with five choices, corresponding to the five non isomorphic Borel subalgebras. We identify the ones that are used in physics. As it is well known, not all superflags have a projective embedding, but in the cases of interest one has a super Plucker embedding, so they are projective super varieties. For the superflag Fl(2|0, 2|1, 4|1), this is followed by a super Segre embedding, which then provides a method for its quantization.