Currents on Grassmann algebras (Q1842249)

In analogy with De Rham currents on \(\mathbb{R}^N\), currents on a Grassmann algebra \(\text{Gr} (N)\) with \(N\) generators are constructed as distributions on its exterior algebra. The latter is endowed with a symmetric wedge product, and the analogue of De Rham cohomology is trivial. The currents may be interpreted in terms of \(\mathbb{Z}_2\)-graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on \(\text{Gr} (N)\)). The closed currents are constructed explicitly in terms of symmetric tensors over \(\text{Gr} (N)\), using the Berezin integral.