Multi-toric geometries with larger compact symmetry

Thomas Bruun Madsen; Andrew Swann

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arXiv · 2407.03693

Multi-toric geometries with larger compact symmetry

Thomas Bruun Madsen·Andrew Swann

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Abstract

We study complete, simply-connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For $\mathrm{Spin}(7)$-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $T^{3} \times \mathrm{SU}(2)$. We then specialise the analysis to holonomy $G_{2}$, to Calabi-Yau geometries in real dimension six and to hyperK\"ahler four-manifolds. Finally, we consider weakly coherent triples on $\mathbb{R} \times \mathrm{SU}(2)$, and their extensions over singular orbits, to give local examples in the $\mathrm{Spin}(7)$-case that have singular orbits where the stabiliser is of rank one.

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BibTeX RIS

Thomas Bruun Madsen, Andrew Swann. 2024-07-04. Multi-toric geometries with larger compact symmetry. https://arxiv.org/abs/2407.03693

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