Supergravity on seven-dimensional homotopy spheres

Abstract

We discuss a spontaneous compactification scheme of eleven-dimensional supergravity on the exotic spheres Σ 7 and on the ordinary S 7 with the Hopf-fibering. These spheres are the nontrivial fibration of S 4 × SO(4)S 3 with the SO(4)-instantons. In particular two classical solutions are explicitly obtained for the Hopf-fibering of S 7 in the eleven-dimensional supergravity. The ratios of the curvatures in S AdS 4 × S 4′ × S 3 are (−1 : 1 : 1 4 ) and ( −9 5 : 1 : 5 4 ), and the corresponding super-symmetries are N = 8 and N = 1, respectively. They are gauge equivalent to the round and the squashed S 7.