The G₂ invariant compactifications in eleven-dimensional supergravity

In this article we study the most general compactifications of eleven-dimensional supergravity to four-dimensional anti-de Sitter space, using a seven-dimensional internal compact manifold whose local isometry group contains the exceptional group, G₂. We find that the only such compactifications are the well known ones involving the round seven-sphere. We discuss the implications of this result for conjectured relationship between the eleven-dimensional theory and gauged N = 8 supergravity.