The binary aff(n|1)-invariant differential operators on weighted densities on the superspace R1|n and aff(n|1)-relative cohomology

Abstract

We consider the $\mathfrak{aff}(n|1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{aff}(n|1)-$invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first $\mathfrak{aff}(n|1)-$relative differential cohomology of $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities.