A 4D gravity theory and $G_2$-holonomy manifolds

Bryant and Salamon gave a construction of metrics of $G_2$ 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 $\mathrm{SO}(3)$ bundle (with fibers $\mathbb{R}^3$) 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 $G_2$ 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 $\mathrm{SO}(3)$ connections. Thus, every solution of this $4$-dimensional gravity theory can be lifted to a $G_2$-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 $G_2$ holonomy. We describe several examples that are of cohomogeneity one on the base.

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K.K. and C.S. were supported by ERC Starting Grant 277570-DIGT. Y.S. acknowledges support from the same grant and from the State Fund for Fundamental Research of Ukraine. Y.H. was supported by a grant from ENS Lyon.

Published 15 July 2019