Roughly speaking, De Rham currents are distributions that can be pictured as possibly singular functions with several variables on a manifold. More precisely they extend the concept of distributions to the case of exterior forms. One can generalize currents to graded-commutative calculus, i.e., in the simplest case, to Grassmann algebras. The analogy with the classical situation is useful but can be, at times, confusing, since the space of exterior forms over a Grassmann algebra is endowed with a symmetric wedge product in contradistinction with the usual exterior algebra; in particular the dimension of this space extends to infinity. With this warning in mind one notices that Grassmannian currents, like their classical counterpart in commutative geometry, i.e. the De Rham currents, and also like submanifolds may have boundary or not. The space of closed (i.e. without boundary) Grassmannian currents of degree p over a Grassmann algebra with N generators is a finite dimensional -graded vector space.
In this paper, we show that these closed currents are in correspondence with -graded cyclic cocycles over the Grassmann algebra: This is therefore a -graded generalization of what is already known in the case of manifolds [1, 2]. The situation is actually quite simple because, in the case of Grassmann algebras, the super-commutative analogue of De Rham cohomology is trivial. In simple terms we shall determine the dimension of the space of closed currents and express them explicitly in terms of symmetric tensor (over the Grassmann algebra) whose Grassmann divergence is zero. The dimension of the space of cyclic cocycles (for general p) was already known after the work [5] and an explicit expression in the case p=1 was given in [3].
Our purpose here is
1) To interpret cyclic cocycles in terms of closed Grassmannian
currents
2) Build explicitly these objects in terms of symmetric
tensors over the Grassmann algebra, using the Berezin integral.