A representation formula for maps on supermanifolds

The theory of supermanifolds, first proposed by Salam and Strathdee [15] as a geometrical framework for understanding the supersymmetry, is now well understood mathematically and can be formulated in roughly two different ways: either by defining a notion of superdifferential structure with ”supernumbers” which generalizes the differential structure of R and by gluing together these local models to build a supermanifold. This is the approach proposed by Dewitt [6] and Rogers [14]. Alternatively one can define supermanifolds as ringed spaces, i.e. objects on which the algebra (or the sheaf) of functions is actually a superalgebra (or a sheaf of superalgebras). This point of view was adopted by Berezin [4], Lĕıtes [12], Manin [13] and was recently further developped by Deligne and Morgan [8], Freed [10] and Varadarajan [16]. The first approach is influenced by differential geometry, whereas the second one is inspired by algebraic geometry. Of course all these points of view are strongly related, but they may lead to some subtle differences (see Batchelor [3], Bartocci, Bruzzo and Hernández-Ruipérez [2] and Bahraini [1]).