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Given a Sasaki-Einstein manifold, M_7, there is the N=2 supersymmetric AdS_4x M_7 Freund-Rubin solution of eleven-dimensional supergravity and thecorresponding non-supersymmetric solutions: the perturbatively stableskew-whiffed solution, the perturbatively unstable Englert solution, and thePope-Warner solution, which is known to be perturbatively unstable when M_7 isthe seven-sphere or, more generally, a tri-Sasakian manifold. We show thatsimilar perturbative instability of the Pope-Warner solution will arise for anySasaki-Einstein manifold, M_7, admitting a basic, primitive, transverse(1,1)-eigenform of the Hodge-de Rham Laplacian with the eigenvalue in the rangebetween 2(9-4\sqrt 3) and 2(9+4\sqrt 3). Existence of such (1,1)-forms on allhomogeneous Sasaki-Einstein manifolds can be shown explicitly using the Kahlerquotient construction or the standard harmonic expansion. The latter shows thatthe instability arises from the coupling between the Pope-Warner background andKaluza-Klein scalar modes that at the supersymmetric point lie in a longZ-vector supermultiplet. We also verify that the instability persists for theorbifolds of homogeneous Sasaki-Einstein manifolds that have been discussedrecently.

On perturbative instability of Pope-Warner solutions on Sasaki-Einstein manifolds