Differential Operators on the Weighted Densities on the Supercircle $S^{1|n}$

Abstract

Over the $(1,n)$-dimensional real supercircle, we consider the $\mathcal{K}(n)$-modules of linear differential operators, $\frak{D}^n_{λ,μ}$, acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules for $n=1$. We also prove that $\frak{D}^{n}_{λ,μ}$ and $\frak{D}_{ρ,ν}^{n}$ are isomorphic for $ρ=\frac{2-n}{2}-μ$ and $ν=\frac{2-n}{2}-λ$. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of Differential Operators on the Real Line, Functional Analysis and Its Applications, Vol. 35, No. 1, pp. 13--18, 2001.]