Symetries of modules of differential operators on $S^{1|1}$

TL;DR

This paper investigates the symmetries of differential operator modules on the supercircle $S^{1|1}$, focusing on their $rak{osp}(1|2)$-module structure and the linear maps commuting with this action.

Contribution

It characterizes the space of linear maps commuting with the $rak{osp}(1|2)$-action on modules of differential operators on $S^{1|1}$, revealing their symmetry properties.

Findings

Determined the $rak{osp}(1|2)$-module structure of differential operators.

Identified the space of linear maps commuting with the $rak{osp}(1|2)$-action.

Provided explicit descriptions of symmetries of modules of differential operators.

Abstract

Let $\frak F_{\l}$ be the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|1}$. We consider the space $\mathfrak{D}_{\lambda,\mu}^k$ of k-th order linear differential operators from $\frak F_{\l}$ to $\frak F_{\mu}$ as $\mathfrak{osp}(1|2)$-module and we determine the space $\mathfrak{I}_{\lambda,\mu}^k$ of linear maps on $\mathfrak{D}_{\lambda,\mu}^k$ commuting with the $\mathfrak{osp}(1|2)$-action.