Supergravity instabilities of non-supersymmetric quantum critical points

Motivated by the recent use of certain consistent truncations of M-theory to study condensed matter physics using holographic techniques, we study the SU(3)-invariant sector of four-dimensional, N = 8 gauged supergravity and compute the complete scalar spectrum at each of the five non-trivial critical points. We demonstrate that the smaller SU(4)-sector is equivalent to a consistent truncation studied recently by various authors and find that the critical point in this sector, which has been proposed as the ground state of a holographic superconductor, is unstable due to a family of scalars that violate the Breitenlohner-Freedman bound. We also derive the origin of this instability in 11 dimensions and comment on the generalization to other embeddings of this critical point which involve arbitrary Sasaki-Einstein seven manifolds. In the spirit of a resurging interest in consistent truncations, we present a formal treatment of the SU(3)-invariant sector as a U(1) x U(1) gauged N = 2 supergravity theory coupled to one hypermultiplet.