On the relation between d = 4 and d = 11 supergravity

We investigate the supersymmetry transformation laws in an arbitrary compactification of d = 11 supergravity to four dimensions. The d = 4 fields of gauged N = 8 supergravity are identified in a class of SO(7)− invariant backgrounds. The two stationary points in these background configurations correspond to the round and parallelized S7. We explicitly demonstrate that the latter coincides with the SO(7)− stationary point of the N = 8 supergravity potential. Chiral SU(8) is found to play a crucial role in establishing these results; we speculate on its possible relevance in the full d = 11 theory