Abstract
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 specialize the analysis to holonomy G2, to Calabi-Yau geometries in real dimension six and to hyperKähler 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 stabilizer is of rank one.
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