Multi-toric geometries with larger compact symmetry.

The article investigates complete, simply connected eight-dimensional manifolds with holonomy group |$\mathrm{Spin}(7)$| that admit toric symmetry via multi-moment maps and possess a connected non-Abelian symmetry group containing the torus |$T^4$|. It establishes that such manifolds cannot have cohomogeneity-one actions with enhanced symmetry and identifies the unique possibility as a cohomogeneity-two action by |$T^3 \times \mathrm{SU}(2)$|. The study further specializes these results to related Ricci-flat geometries with holonomy groups |$G_2$|, |$\mathrm{SU}(3)$| (Calabi–Yau), and hyperKähler structures, providing classifications of possible symmetry groups and orbit types. Finally, it constructs local examples of |$\mathrm{Spin}(7)$|-manifolds with |$T^3 \times \mathrm{SU}(2)$|-symmetry using weakly coherent triples on four-manifolds of the form |$\mathbb{R} \times \mathrm{SU}(2)$|, including cases with singular orbits where stabilizers have rank one.