Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry

Giovanni Russo; Andrew Swann

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arXiv · 2508.21703

Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry

Giovanni Russo·Andrew Swann

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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.

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BibTeX RIS

Giovanni Russo, Andrew Swann. 2025-08-29. Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry. https://arxiv.org/abs/2508.21703

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