G. Russo - A. Swann

Nearly Parallel G_2-Structures with Torus Symmetry

Published Online

Inserted: 1 sep 2025
Last Updated: 19 jun 2026

Journal: Math. Z.
Volume: 313
Number: 44
Year: 2026
Doi: https://link.springer.com/article/10.1007/s00209-026-04062-z

Abstract:

We study nearly parallel G₂-structures with a three-torus symmetry via multi-moment map techniques. An effective three-torus action on a nearly parallel G₂-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 G₂-structures from three-dimensional data. We observe that locally this may produce examples with four-torus symmetry.