The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology

Abstract

Over the $(1,N)$-dimensional supercircle $S^{1|N}$, we classify $\mathfrak{n}(1|N)$-invariant linear differential operators acting on the superspaces of weighted densities on $S^{1|N}$, where $\mathfrak{n}(1|N)$ is the Heisenberg Lie superalgebra. This result allows us to compute the first differential $\mathfrak{n}(1|N)$-relative cohomology of the Lie superalgebra $\mathcal{K}(N)$ of contact vector fields with coefficients in the superspace of weighted densities. For $N=0,1,2,$ we investigate the first $\mathfrak{n}(1|N)$-relative cohomology space associated with the embedding of $\mathcal{K}(N)$ in the superspace of the supercommutative algebra $\mathcal{SP}(N)$ of pseudodifferential symbols on $S^{1|N}$ and in the Lie superalgebra $\mathcal{S}\Psi \mathcal{D}\mathcal{O} (S^{1|N})$ of superpseudodifferential operators with smooth coeffcients. We explicity give 1-cocycles spanning these cohomology spaces.