Consistent Kaluza-Klein Sphere Reductions

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 S4 and S7 reductions of D = 11 supergravity, and the S5 reduction of type IIB supergravity), the only other possibilities that allow consistent reductions are for p = 2, reduced on S2, and for p = 3, reduced on S3 or SD−3. We construct the fully non-linear Kaluza-Klein Ansätze in all these cases. In particular, we obtain D = 3, N = 8, SO(8) and D = 7, N = 2, SO(4) gauged supergravities from S7 and S3 reductions of N = 1 supergravity in D = 10. 1 Research supported in part by DOE grant DOE-FG02-95ER40893 2 Research supported in part by DOE grant DOE-FG03-95ER40917