Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere

Spontaneous compactification of $d=11\mathrm{}$ supergravity on the squashed seven-sphere ($S^7$) yields a spontaneously broken phase of the $N=8\mathrm{}$ theory on the round $S^7$ with SO(8) broken to SO(5) ⊗ SU(2) and with $N=1\mathrm{}$ or $N=0\mathrm{}$ supersymmetry according as the squashing is left or right handed. A feature unique to Kaluza-Klein supergravity is that the massless gravitino of the $N=1\mathrm{}$ phase comes from a massive $N=8\mathrm{}$ supermultiplet, as do the Higgs scalars. A solution is found with squashing and torsion but only in the right-squashed ($N=0\mathrm{}$) case.