On the nonlinear KK reductions on spheres of supergravity theories Original Research Article

We address some issues related to the construction of general Kaluza–Klein (KK) ansätze for the compactification of a supergravity (sugra) theory on a sphere Sm. We first reproduce various ansätze for compactification to 7D from the ansatz for the full nonlinear KK reduction of 11D sugra on AdS7×S4. 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 AdS7×S4 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 AdS5×S5 (with no restriction on any fields).