A superspace survey

A survey of superspaces associated with four-dimensional Minkowski spacetime is given from a superconformal perspective. The discussion begins in the complex setting and focuses on a class of supermanifolds---flag supermanifolds---which includes not only conventional superspaces and chiral superspaces, but also supertwistor spaces and harmonic superspaces. All of them are homogeneous spaces of superconformal groups. The approach is closely related to twistor theory; in particular, the twistorial notion of a double fibration which links together sets of three such homogeneous supermanifolds is exploited to give a geometrical interpretation of superconformal transformations and is also applied to supersymmetric Yang--Mills theories. In the real setting the complex twistor space associated to Euclidean space is replaced in the supersymmetric case by supermanifolds which have a CR -structure. Representations of superconformal groups on fields are also studied; in the case of harmonic superspace this involves representations which differ slightly from standard induced representations.