Deformations of the $\mathfrak{osp}(n|2)$-action on the superspace of symbols of differential operators on ${\mathbb{R}}^{1|n}$

Abstract

We study formal deformations of the natural $\osp(n|2)$-action, $n\ge 3$, on the superspace $\Sc^n_d=\bigoplus_{k\geq 0}\Fc^n_{d-\frac{k}{2}}$ of symbols of linear differential operators on weighted densities over ${\mathbb{R}}^{1|n}$. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product $\Hd^1\vee \Hd^1\to \Hd^2$ which carries the quadratic obstructions. The answer is governed by the $\osp(n|2)$-invariant operators $A_k={\eta }_1\cdots {\eta }_n{\partial }_x^{k-1}$: the two cocycles $h_k$ and ${\tilde{h}}_k$ spanning the off-diagonal part of $\Hd^1$ 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 ${\Omega }_k$. If $2d\notin \mathbb{N}$ the space $\Hd^1\vee\Hd^1$ 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.

14 pages