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5) and used the fact that its spatial component ~a I = 0, if one does not incorporate ordinary translations. 86) are obtained as follows : define f o r the s u p e r f i e l d s F = ~, D ~, D ~ , . . 88) yields the susy transformations of the space-time fields. 9o) 43 The projection method also allows a straightforward superfields. g. 89) of the ehiral superfield kinetic energy terms from e=o ~ we obtain the familiar 44 (iii) Rigid supersynlnetry on non-trivial base manifolds It is possible to formulate rigid supersyrmnetric theories on more general spaces than ordinary Minkowski space (upon removing the degrees of freedom associated with gravity).

De Witt who used a very different approach (ref. 39)). These remarks essentially close our account of the geometry of rigid superspace. Before adding a few related comments we will compare the previously des- cribed standard approach with the Riemannian one whichmight be more familiar from the study of other physical theories. 39). e. on a manifold (pseudo-) Riemannian metric g. M which is equipped with some Thus it seems natural to attempt a generalization of this framework to supersymmetric theories and to introduce some supermetric on rigid superspace body manifold SP/L ~4.

Thus our Riemannian superspace does not represent a flat space, but a curved "extension" of ordinary Minkowski space (iii) (~,~)~). Relation between Killing vector fields and susy transformations Before discussing the Killing vector fields associated to the previously defined supermetric, we recall a few general facts about these fields and consider the Minkowski metric as a simple example. On a (pseudo-) Riemannian manifold (M,g) there is a particularly interesting class of vector fields namely those which generate diffeomorphisms of leave the metric g associated to and they can be characterized by the condition g invariant.

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