Self-Duality in D <= 8-dimensional Euclidean Gravity

TL;DR

This work extends self-duality from four dimensions to D \le 8 Euclidean gravity and Yang-Mills theory by formulating a master set of eight-dimensional duality equations based on a Spin(7) invariant 4-form. It shows that eight-dimensional self-duality yields Ricci-flat manifolds with holonomy Spin(7) and, via dimensional reduction, lower-dimensional dualities with holonomies G2, SU(3), and SU(2). By embedding the spin connection into the gauge connection, it constructs self-dual Yang-Mills fields on these backgrounds and proves a universal topological bound on the Yang-Mills action that is saturated by self-dual configurations, with reductions to familiar four-dimensional instanton bounds. The results link these geometric constructions to supersymmetry and heterotic string theory, and underscore octonion-related structures as a unifying theme across dimensions.

Abstract

In the context of D-dimensional Euclidean gravity, we define the natural generalisation to D-dimensions of the self-dual Yang-Mills equations, as duality conditions on the curvature 2-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.