Self-duality inD<~8-dimensional Euclidean gravity

In the context of $D$-dimensional Euclidean gravity, we define the natural generalization to $D$ dimensions of the self-dual Yang-Mills equations as duality conditions on the curvature two-form of a Riemannian manifold. Solutions to these self-duality equations are provided by manifolds of SU(2), SU(3), $G_2$, and Spin(7) holonomy. The equations in eight dimensions are a master set for those in lower dimensions. By considering gauge fields propagating on these self-dual manifolds and embedding the spin connection in the gauge connection, solutions to the $D$-dimensional equations for self-dual Yang-Mills fields are found. We show that the Yang-Mills action on such manifolds is topologically bounded from below, with the bound saturated precisely when the Yang-Mills field is self-dual. These results have a natural interpretation in supersymmetric string theory.