On the nonlinear KK reductions on spheres of supergravity theories

We address some issues related to the construction of general Kaluza-Klein (KK) ansaetze for the compactification of a supergravity (sugra) theory on a sphere Sm. We first reproduce various ansaetze for compactification to 7D from the ansatz for the full nonlinear KK reduction of 11D sugra on AdS7xS4. As a side result, we obtain a lagrangian formulation of 7D N=2 gauged sugra, which so far had only a on-shell formulation, through field equations and constraints. The AdS7xS4 ansatz generalizes therefore all previous sphere compactifications to 7D. Then we consider the case when the scalars in the lower dimensional theory are in a coset Sl(m+1)/SO(m+1), and we keep the maximal gauge group SO(m+1). The 11-dimensional sugra truncated on S4 fits precisely the case under consideration, and serves as a model for our construction. We find that the metric ansatz has a universal expression, with the internal space deformed by the scalar fluctuations to a conformally rescaled ellipsoid. We also find the ansatz for the dependence of the antisymmetric tensor on the scalars. We comment on the fermionic ansatz, which will contain a matrix U interpolating between the spinorial SO(m+1) indices of the spherical harmonics and the R-symmetry indices of the fermionic fields in the lower dimensional sugra theory. We derive general conditions which the matrix U has to satisfy and we give a formula for the vielbein in terms of U. As an application of our methods we obtain the full ansatz for the metric and vielbein for 10D sugra on AdS5xS5 (with no restriction on any fields)