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

Abstract

We study formal deformations of the natural $\osp(n|2)$-action, $n\geq 3$, 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 $\R^{1|n}$. 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 $\widetilde{h}_k$ 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 $2d\notin\N$ the space $\Hd^1\vee\Hd^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\in\N$ we obtain exactly $m$ quadratic integrability conditions, $\tau_{2-n-k}(t_k-\widetilde{t}_k)+\tau_k\widetilde{t}_k=0$, $1\leq k\leq m$, and we prove that they are also sufficient: no condition of order $\geq 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.