Supertranslation distribution

Definition

Let $\mathbb M^{V|S}$ be super-Minkowski space determined by a supertranslation bracket $\Gamma:\operatorname{Sym}^2S\to V$. Its supertranslation distribution $\mathcal D$ is the rank $0|{\dim S}$ distribution spanned by the odd left-invariant vector fields corresponding to $S$.

The Levi bracket of $\mathcal D$ is induced by $\Gamma$:

Thus $\mathcal D$ is nonintegrable whenever $\Gamma\ne0$. If $\Gamma$ is surjective, brackets of its odd sections generate every even translation direction, so $\mathcal D$ is bracket-generating.

This distribution is additional structure on the underlying affine supermanifold. It records the supertranslation algebra geometrically.