The $\alpha'$ 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 $G_2$ or $\mathrm{Spin}(7)$ holonomy gives a supersymmetric vacuum in three or two dimensions. Do $\alpha'$ 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 $\alpha'$). 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 $G_2$ or Spin(7) holonomy, similar results hold to all orders in the inverse radius of $M$ -- but not exactly. The classical moduli space of $G_2$ metrics on a manifold $M$ is known to be locally a Lagrangian submanifold of $H^3(M,{\mathbf R})\oplus H^4(M,\mathbf{R})$. We show that this remains valid to all orders in the $\alpha'$ or inverse radius expansion.