Consistency of Kaluza-Klein Sphere Reductions of Symmetric Potentials

In a recent paper, the complete (non-linear) Kaluza-Klein Ansatz for the consistent embedding of certain scalar plus gravity subsectors of gauged maximal supergravity in D = 4, 5 and 7 was presented, in terms of sphere reductions fromD = 11 or type IIB supergravity. The scalar fields included in the truncations were the diagonal fields in the SL(N,R)/SO(N) scalar submanifolds of the full scalar sectors of the corresponding maximal supergravities, with N = 8, 6 and 5. The embeddings were used for obtaining an interpretation of extremal D = 4, 5 or 7 AdS domain walls in terms of distributed M-branes or D-branes in the higher dimensions. Although strong supporting evidence for the correctness of the embedding Ansätze was presented, a full proof of the consistency was not given. Here, we complete the proof, by showing explicitly that the full set of higher-dimensional equations of motion are satisfied if and only if the lower-dimensional fields satisfy the relevant scalar plus gravity equations. 1 Research supported in part by DOE grant DOE-FG02-95ER40893 2 Research supported in part by DOE grant DOE-FG03-95ER40917