Given a Sasaki-Einstein manifold, M_7, there is the N=2 supersymmetric AdS_4 x M_7 Freund-Rubin solution of eleven-dimensional supergravity and the corresponding non-supersymmetric solutions: the perturbatively stable skew-whiffed solution, the pertu
rbatively unstable Englert solution, and the Pope-Warner solution, which is known to be perturbatively unstable when M_7 is the seven-sphere or, more generally, a tri-Sasakian manifold. We show that similar perturbative instability of the Pope-Warner solution will arise for any Sasaki-Einstein manifold, M_7, admitting a basic, primitive, transverse (1,1)-eigenform of the Hodge-de Rham Laplacian with the eigenvalue in the range between 2(9-4sqrt 3) and 2(9+4sqrt 3). Existence of such (1,1)-forms on all homogeneous Sasaki-Einstein manifolds can be shown explicitly using the Kahler quotient construction or the standard harmonic expansion. The latter shows that the instability arises from the coupling between the Pope-Warner background and Kaluza-Klein scalar modes that at the supersymmetric point lie in a long Z-vector supermultiplet. We also verify that the instability persists for the orbifolds of homogeneous Sasaki-Einstein manifolds that have been discussed recently.
We study the system of equations derived twenty five years ago by B. de Wit and the first author [Nucl. Phys. B281 (1987) 211] as conditions for the consistent truncation of eleven-dimensional supergravity on AdS_4 x S^7 to gauged N = 8 supergravity
in four dimensions. By exploiting the E_7(7) symmetry, we determine the most general solution to this system at each point on the coset space E_7(7)/SU(8). We show that invariants of the general solution are given by the fluxes in eleven-dimensional supergravity. This allows us to both clarify the explicit non-linear ansatze for the fluxes given previously and to fill a gap in the original proof of the consistent truncation. These results are illustrated with several examples.
