The α^′ expansion on a compact manifold of exceptional holonomy

In the approximation corresponding to the classical Einstein equations, which is valid at large radius, string theory compactification on a compact manifold M of G₂ or Spin(7) holonomy gives a supersymmetric vacuum in three or two dimensions. Do α^′ corrections to the Einstein equations disturb this statement? Explicitly analyzing the leading correction, we show that the metric of M can be adjusted to maintain supersymmetry. Beyond leading order, a general argument based on low energy effective field theory in spacetime implies that this is true exactly (not just to all finite orders in α^′). A more elaborate field theory argument that includes the massive Kaluza-Klein modes matches the structure found in explicit calculations. In M-theory compactification on a manifold M of G₂ or Spin(7) holonomy, similar results hold to all orders in the inverse radius of M — but not exactly. The classical moduli space of G₂ metrics on a manifold M is known to be locally a Lagrangian submanifold of H³(M,ℝ)⊕H⁴(M,ℝ). We show that this remains valid to all orders in the α^′ or inverse radius expansion