The Binary Invariant Differential Operators on Weighted Densities on the superspace $\mathbb{R}^{1|n}$ and Cohomology

TL;DR

This paper classifies invariant binary differential operators on weighted densities over superspace and computes the first cohomology of the contact vector fields Lie superalgebra, extending classical results to the super setting.

Contribution

It provides a classification of invariant operators and explicit cohomology calculations for the superalgebra of contact vector fields on superspace.

Findings

Classification of invariant binary differential operators.

Explicit 1-cocycles for first cohomology.

Extension of classical results to superspace.

Abstract

Over the $(1,n)$-dimensional real superspace, $n>1$, we classify $\mathcal{K}(n)$-invariant binary differential operators acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of %the Lie superalgebra $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities--a superisation of a result by Feigin and Fuchs. We explicitly give 1-cocycles spanning these cohomology spaces.