Nearly Parallel ${\mathrm{G}}_2$-Structures with Torus Symmetry

Abstract

We study nearly parallel ${\mathrm{G}}_2$-structures with a three-torus symmetry via multi-moment map techniques. An effective three-torus action on a nearly parallel ${\mathrm{G}}_2$-manifold yields a multi-moment map. The torus acts freely on its regular level sets, so they are torus bundles over smooth three-dimensional manifolds. We show that the geometry of the base spaces is specified by two triples of closed two-forms related by a Riemannian metric. We then describe an inverse construction producing invariant nearly parallel ${\mathrm{G}}_2$-structures from three-dimensional data. We observe that locally this may produce examples with four-torus symmetry.

18 pages. A typo corrected in Proposition 1.3. Accepted in Math. Z