A SUPERSPACE SURVEY

A survey of superspaces associated with four-dimensional Minkowski spa cetime is given from a superconformal perspective. The discussion begi ns in the complex setting and focuses on a class of supermanifolds-fla g supermanifolds-which includes not only conventional superspaces and chiral superspaces, but also supertwistor spaces and harmonic superspa ces. All of them are homogeneous spaces of superconformal groups. The approach is closely related to twister theory; in particular, the twis torial notion of a double fibration which links together sets of three such homogeneous supermanifolds is exploited to give a geometrical in terpretation of superconformal transformations and is also applied to supersymmetric Yang-Mills theories. In the real setting the complex tw ister space associated to Euclidean space is replaced in the supersymm etric case by supermanifolds which have a CR-structure. Representation s of superconformal groups on fields are also studied; in the case of harmonic superspace this involves representations which differ slightl y from standard induced representations.