Currents on Grassmann algebras

TL;DR

This paper introduces the concept of currents on Grassmann algebras, linking them to Hochschild cohomology and cyclic cocycles, and provides an explicit construction for closed currents using Berezin integration.

Contribution

It defines currents on Grassmann algebras and connects them to Hochschild cohomology and cyclic cocycles, offering an explicit construction for closed currents.

Findings

Currents on Grassmann algebras are characterized as distributions on the exterior algebra.

Closed currents are interpreted via cyclic cocycles and multilinear forms.

An explicit Berezin integration method constructs the space of closed currents.

Abstract

We define currents on a Grassmann algebra $Gr(N)$ with $N$ generators as distributions on its exterior algebra (using the symmetric wedge product). We interpret the currents in terms of ${\Z}_2$-graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on $Gr(N)$). An explicit construction of the vector space of closed currents of degree $p$ on $Gr(N)$ is given by using Berezin integration.