Bilinear differential operators and $\mathfrak {osp}(1|2)$-relative cohomology on $\mathbb{R}^{1|1}$

Abstract

We consider the $1|1$-dimensional real superspace $\mathbb{R}^{1|1}$ endowed with its standard contact structure defined by the 1-form $\alpha$. The conformal Lie superalgebra $\mathcal{K}(1)$ acts on $\mathbb{R}^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra $\mathfrak{osp}(1|2)$. We classify $\mathfrak{osp}(1|2)$-invariant superskew symmetric binary differential operators from $\mathcal{K}(1)\wedge\mathcal{K}(1)$ to $\mathfrak{D}_{\lambda,\mu;\nu}$ vanishing on $\mathfrak{osp}(1|2)$, where $\mathfrak{D}_{\lambda,\mu;\nu}$ is the superspace of bilinear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential $\mathfrak{osp}(1|2)$-relative cohomology of $\mathcal{K}(1)$ with coefficients in $\mathfrak{D}_{\lambda,\mu;\nu}$.