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

TL;DR

This paper explores self-duality equations in D-dimensional Euclidean gravity, linking special holonomy manifolds to solutions of Yang-Mills equations and their implications in supersymmetric string theory.

Contribution

It generalizes self-duality conditions to D dimensions, identifies solutions via special holonomy manifolds, and connects these to supersymmetric string theory.

Findings

Solutions involve manifolds with SU(2), SU(3), G_2, Spin(7) holonomy.

Yang-Mills action is topologically bounded and saturated by self-dual fields.

Provides a framework linking geometry, gauge fields, and string theory.

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.