The geometry of supermanifolds and new supersymmetric actions

This is the first of two papers in which we construct the Hodge dual for supermanifolds by means of the Grassmannian Fourier transform of superforms. In this paper we introduce the fundamental concepts and a method for computing Hodge duals in simple cases. We refer to a subsequent publication cite{lavoro3} for a more general approach and the required mathematical details.} In the case of supermanifolds it is known that superforms are not sufficient to construct a consistent integration theory and that {it integral forms} are needed. They are distribution-like forms which can be integrated on supermanifolds as a top form can be integrated on a conventional manifold. In our construction of the Hodge dual of superforms they arise naturally. The compatibility between Hodge duality and supersymmetry is exploited and applied to several examples. We define the irreducible representations of supersymmetry in terms of integral and super forms in a new way which can be easily generalised to several models in different dimensions. The construction of supersymmetric actions based on the Hodge duality is presented and new supersymmetric actions with higher derivative terms are found. These terms are required by the invertibility of the Hodge operator.