M-Theory Vacua from Warped Compactifications on Spin(7) Manifolds
  At the leading order, M-theory admits minimal supersymmetric compactifications if the internal manifold has exceptional holonomy. Once we take into account higher order quantum correction terms in the low energy effective action, the supergravity vacua have to be deformed away from the exceptional holonomy if we want to preserve the supersymmetry of the solution. In this paper we discuss the Spin(7) holonomy case. We derive a perturbative set of solutions which emerges from a warped compactification with non-vanishing flux for the M-theory field strength and we identify the supersymmetric vacua out of this general set of solutions. For this purpose we have to evaluate the value of the quartic polynomial J0 on a Spin(7) holonomy manifold as well as its first variation with respect to the internal metric. We show that in general the Ricci flatness of the internal manifold is lost. 
Dragos Constantin