Toric geometry of Spin(7)-manifolds

We study $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-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 $4\times 4$-matrix of functions. This description leads to the first known $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-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.