Multi-toric geometries with larger compact symmetry

TL;DR

This paper investigates special holonomy manifolds with multi-toric structures and larger symmetry groups, classifying possible geometries and providing local examples with singular orbits.

Contribution

It classifies multi-toric special holonomy manifolds with larger symmetry groups, especially for Spin(7), G2, Calabi-Yau, and hyperK"ahler cases, and constructs local models with singular orbits.

Findings

Spin(7) manifolds with cohomogeneity-two $T^3 \times SU(2)$ symmetry

Classification of G2, Calabi-Yau, and hyperK"ahler geometries with multi-toric structures

Construction of local Spin(7) examples with singular orbits

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.