Supergravity, the seven-sphere, and spontaneous symmetry breaking

N=1 supergravity in d=11 dimensions spontaneously compactifies on S7 to an N=8 supergravity d=4 with a local SO(8) x SO(8) invariance, probably enlargeable to SO(8) x SU(8). Apart from group manifolds, S7 is the only compact manifold to admit an absolute parallelism. This permits (a) a ''squashing'' of S7 which gives expectation values to the scalar fields and (b) a parallelizing ''torsion'' which gives expectation values to the pseudoscalars. This correspondence between extrema of the d=4 effective potential and solutions of the d=11 field equations provides a Kaluza-Klein origin for the spontaneous breakdown of gauge symmetries, discrete symmetries, and supersymmetries. It also puts a new perspective on the puzzle of the cosmological constant. (orig.)