The linear $$\mathfrak {osp}(n|2)$$ osp ( n | 2 ) -invariant differential operators and cohomology

Abstract

Over the \((1, n)\)-dimensional real superspace, we consider the Lie superalgebra \(\mathcal {K}(n)\) of contact vector fields on \({\mathbb {R}}^{1|n}\). We classify \(\mathfrak {osp}(n|2)\)-invariant linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the first differential cohomology of \(\mathcal {K}(n)\) with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra \(\mathfrak {osp}(n|2)\). We explicitly give 1-cocycles spanning these cohomology spaces.