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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., Comment: 19 pages

In this note, we revisit some well-known examples of instantons on flat space that were originally discovered in the physics literature. In particular, we explain how the basic instanton on $\mathbb{R}^4$, with its flat hyperkaehler structure, has natural generalisations to $\mathbb{R}^7$ and $\mathbb{R}^8$ viewed as flat $G_2$- and Spin(7)-manifolds, respectively. We also provide the details of an arguably less known construction of ASD instantons on $\mathbb{H}^n$, in the sense of quaternionic geometry., Comment: 13 pages

We study $\mathrm{Spin}(7)$-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons-Hawking type ansatz that describes such geometries in terms of a symmetric $4\times4$-matrix of functions. This description leads to the first known $\mathrm{Spin}(7)$-manifolds with a rank $4$ symmetry group and full holonomy. We also show that the multi-moment map exhibits the full orbit space topologically as a smooth four-manifold, containing a trivalent graph in $\mathbb{R}^4$ as the image of the set of the special orbits., Comment: 14 pages. This article builds on arXiv:1803.06646 applying the techniques to a new situation

We consider $G_2$-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of $T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometry on an open dense set in terms a symmetric $3\times 3$-matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to $G_2$. We prove that the multi-moment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples., Comment: 33 pages; improved version of Q operator, typos corrected

We describe the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kaehler metric on the simply-connected 8-manifold G_2/SO(4) that carry a closed fundamental 4-form but are not Einstein., Comment: 20 Pages, dedicated to Nigel Hitchin

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G_2 that arises from SO(3)-structures with invariant intrinsic torsion., Comment: 35 pages. To appear in Complex Manifolds

We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough investigation of the intrinsic torsion of such structures, leading to the construction of explicit Lie group examples with invariant intrinsic torsion., Comment: 24 pages; v2, added two remarks concerning a potential analogous construction related to Spin(7) and the existence of a lattice in the solvable examples. To appear in Transformation Groups

We describe left-invariant half-flat SU(3)-structures on S^3xS^3 using the representation theory of SO(4) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy G_2 obtained on 7-manifolds with equidistant S^3xS^3 hypersurfaces. The generic case is analysed numerically., Comment: 23 pages, 6 figures. To appear in Annals of Global Analysis and Geometry

We introduce and study a notion of `Sasaki with torsion structure' (ST) as an odd-dimensional analogue of K\"ahler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is shown to admit such a structure; in this case the structure is left-invariant and has closed torsion form. We illustrate the relation between ST structures and other generalizations of Sasaki geometry, and explain how some standard constructions in Sasaki geometry can be adapted to this setting. In particular, we relate the ST structure to a KT structure on the space of leaves, and show that both the cylinder and the cone over an ST manifold are KT, although only the cylinder behaves well with respect to closedness of the torsion form. Finally, we introduce a notion of `G-moment map'. We provide criteria based on equivariant cohomology ensuring the existence of these maps, and then apply them as a tool for reducing ST structures., Comment: 34 pages; v2: added a small generalization (Proposition 3.6) of the cone construction; two references added. To appear on Ann. Mat. Pura Appl

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples., Comment: 36 pages

Metrics of exceptional holonomy are vacuum solutions to the Einstein equation. In this paper we describe manifolds with holonomy contained in Spin(7) preserved by a three-torus symmetry in terms of tri-symplectic geometry of four-manifolds. These complement examples that have appeared in the context of domain wall problems in supergravity.

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by a two-torus symmetry in terms of tri-symplectic geometry of four-manifolds., Comment: 27 pages

For geometries with a closed three-form we briefly overview the notion of multi-moment maps. We then give concrete examples of multi-moment maps for homogeneous hypercomplex and nearly Kaehler manifolds. A special role in the theory is played by Lie algebras with second and third Betti numbers equal to zero. These we call (2,3)-trivial. We provide a number of examples of such algebras including a complete list in dimensions up to and including five.

A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable groups in dimension four that admit invariant complex structures but have no invariant SKT structure., Comment: 17 pages