Cohomology of $\mathfrak {osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}

TL;DR

This paper calculates the first cohomology spaces of the Lie superalgebra osp(1|2) acting on differential operators on the supercircle, revealing a more complex structure than previously conjectured.

Contribution

It provides a detailed computation of the first cohomology spaces, correcting and extending earlier conjectures about their structure.

Findings

First cohomology spaces are more complex than previously thought.

Explicit structure of cohomology spaces is determined.

Results refine understanding of osp(1|2) actions on supercircle operators.

Abstract

We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{\lambda,\mu})$ ($\lambda, \mu\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{\lambda,\mu}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)