Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 8

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

Pages: 2001 – 2034



Yannick Herfray (Départment de Mathématique, Université Libre de Bruxelles, Belgium)

Kirill Krasnov (School of Mathematical Sciences, University of Nottingham, United Kingdom)

Carlos Scarinci (Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, South Korea)

Yuri Shtanov (Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine)


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.

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