The ${\alpha }^{\mathrm{\prime }}$ 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 $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ holonomy gives a supersymmetric vacuum in three or two dimensions. Do ${\alpha }^{\mathrm{\prime }}$ 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 ${\alpha }^{\mathrm{\prime }}$). 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}$)$\mathrm{.}Weshowthatthisremainsvalidtoallordersinthe$\alpha’$ or inverse radius expansion.