Consistent Kaluza-Klein sphere reductions - art. no. 064028

We study the circumstances under which a Kaluza-Klein reduction on an n-sph ere, with a massless truncation that includes all the Yang-Mills fields of SO(n + 1), can be consistent at the full non-linear level. We take as the s tarting point a theory comprising a p-form field strength and (possibly) a dilaton, coupled to gravity in the higher dimension D. We show that aside f rom the previously studied cases with (D, p) = (11,4) and (10,5) (associate d with the S-4 and S-7 reductions of D = 11 supergravity, and the S-5 reduc tion of type IIB supergravity), the only other possibilities that allow con sistent reductions are for p = 2, reduced on S-2 and for p = 3, reduced on S-3 or SD - 3. We construct the fully non-linear Kaluza-Klein Ansatze in al l these cases. In particular, we obtain D = 3, N = 8, SO(8) and D = 7, N = 2, SO(4) gauged supergravities from S-7 and S-3 reductions of N = 1 supergr avity in D = 10.