Some rigidity results for supergravity backgrounds in 11 dimensions

TL;DR

This paper establishes rigidity results for 11-dimensional supergravity backgrounds with high supersymmetry, showing they are locally isometric to known maximally supersymmetric solutions under certain conditions.

Contribution

It provides new rigidity theorems linking the rank of the 4-form, the dimension of Killing spinors, and the geometry of supergravity backgrounds in 11 dimensions.

Findings

High supersymmetry implies the background is locally Minkowski or AdS4×S7.

Restrictions on the 4-form's rank lead to classification results.

Finer estimates are given for specific orbit types of the 4-form.

Abstract

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 $\operatorname{rk}(F)\leq 6$, 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 $\mathrm{AdS}_7\times\mathrm{S}^4$. 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.