Instability of some Riemannian manifolds with real Killing spinors

Abstract

We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces $N_{k,l}=\mathrm{SU}(3)/i_{k,l}(S^1)$ (which are all nearly ${\mathrm{G}}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected non-symmetric compact homogeneous Einstein spaces of dimensions $5,6,$ and $7$, including the strict nearly Kähler ones (except ${\mathrm{G}}_2/\mathrm{SU}(3)$).

25 pages, typos corrected