Deformations of the $\mathfrak{o}\mathfrak{s}\mathfrak{p}(n\mathrm{\mid }2)$-action on the superspace of symbols of differential operators on ${\mathbb{R}}^{1\mathrm{\mid }n}$

We study formal deformations of the natural $\text{osp}(n\mathrm{\mid }2)$-action, $n\ge 3$, on the superspace ${\text{Sc}}_d^n={⨁}_{k\ge 0}{\text{Fc}}_{d-\frac{k}{2}}^n$ of symbols of linear differential operators on weighted densities over ${\mathbb{R}}^{1\mathrm{\mid }n}$. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product ${\text{Hd}}^1\vee {\text{Hd}}^1\to {\text{Hd}}^2$ which carries the quadratic obstructions. The answer is governed by the $\text{osp}(n\mathrm{\mid }2)$-invariant operators $A_k={\eta }_1\cdots {\eta }_n{\mathrm{\partial }}_x^{k-1}$: the two cocycles $h_k$ and ${\tilde{h}}_k$ spanning the off-diagonal part of ${\text{Hd}}^1$ are exactly the two derivatives of the coboundary of $A_k$ with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each $k$, by a single non-trivial 2-cocycle ${\mathrm{\Omega }}_k$. If $2d\mathrm{\notin }\mathbb{N}$ the space ${\text{Hd}}^1\vee {\text{Hd}}^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\in \mathbb{N}$ we obtain exactly $m$ quadratic integrability conditions, ${\tau }_{2-n-k}(t_k-{\tilde{t}}_k)+{\tau }_k{\tilde{t}}_k=0$, $1\le k\le m$, and we prove that they are also sufficient: no condition of order $\ge 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.