A 4D gravity theory and G2-holonomy manifolds

TL;DR

This paper generalizes Bryant and Salamon's G2 holonomy metrics by linking them to solutions of a 4D gravity theory, enabling the construction of many new non-topological G2 metrics via SO(3) bundle connections.

Contribution

It introduces a novel method to generate G2 holonomy metrics from 4D gravity solutions using SO(3) bundles, expanding the class of known G2 manifolds.

Findings

G2 metrics correspond to solutions of a 4D gravity PDE

The construction produces non-topological G2 metrics

Several explicit cohomogeneity-one examples are provided

Abstract

Bryant and Salamon gave a construction of metrics of G2 holonomy on the total space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional self-dual Einstein manifold. We generalise it by considering the total space of an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection on this bundle. We make essentially the same ansatz for the calibrating 3-form, but use the curvature 2-forms instead of the ASD ones. We show that the resulting 3-form defines a metric of G2 holonomy if the connection satisfies a certain second-order PDE. This is exactly the same PDE that arises as the field equation of a certain 4-dimensional gravity theory formulated as a diffeomorphism-invariant theory of SO(3) connections. Thus, every solution of this 4-dimensional gravity theory can be lifted to a G2-holonomy metric. Unlike all previously known constructions, the theory that we lift to 7 dimensions is not topological. Thus, our construction should give rise to many new metrics of G2 holonomy. We describe several examples that are of cohomogeneity one on the base.