Consistent Kaluza-Klein Sphere Reductions

Abstract

We study the circumstances under which a Kaluza-Klein reduction on an n-sphere, 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 starting 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 from the previously-studied cases with (D,p)=(11,4) and (10,5) (associated with the S^4 and S^7 reductions of D=11 supergravity, and the S^5 reduction of type IIB supergravity), the only other possibilities that allow consistent reductions are for p=2, reduced on S^2, and for p=3, reduced on S^3 or S^{D-3}. We construct the fully non-linear Kaluza-Klein Ansatze in all 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 supergravity in D=10.