Some rigidity results for supergravity backgrounds in 11 dimensions

This paper is a contribution to the supersymmetry gap problem for supergravity backgrounds $(M,g,F)$ in $11$ dimensions. We study restrictions on the curvature of $(M,g,F)$ and, using the bijective correspondence between the space of certain filtered deformations of Lie superalgebras and the space of highly supersymmetric supergravity backgrounds, we establish the following general rigidity result: if the $4$-form $F$ has rank $\mathrm{rk}(F)\le 6$, Euclidean support, and the space ${\mathfrak{k}}_{\bar{1}}$ of Killing spinors has dimension $\dim {\mathfrak{k}}_{\bar{1}}>26$ then $(M,g,F)$ is locally isometric to the maximally supersymmetric Minkowski spacetime or Freund Rubin background ${\mathrm{A}\mathrm{d}\mathrm{S}}_7\times {\mathrm{S}}^4$. The same rigidity result but with finer estimates on $\dim {\mathfrak{k}}_{\bar{1}}$ is provided for certain types of ${\mathfrak{k}}_{\bar{1}}$ and specific orbits of the $4$-form under the action of the Lorentz group.