The Binary $\mathfrak{aff}(n|1)$-Invariant Differential Operators On Weighted Densities On The Superspace $\mathbb{R}^{1|n}$ And $\mathfrak{aff}(n|1)$-Relative Cohomology

TL;DR

This paper classifies invariant binary differential operators on weighted densities in superspaces and computes their relative cohomology, advancing understanding of symmetries in supergeometry.

Contribution

It provides a complete classification of $rak{aff}(n|1)$-invariant binary differential operators and calculates their first relative cohomology in the context of supergeometry.

Findings

Classification of invariant binary differential operators.

Calculation of first $rak{aff}(n|1)$-relative cohomology.

Enhanced understanding of symmetries in superspaces.

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.