Toric geometry of $\mathrm{G}_2$–manifolds

We consider [math] –manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of [math] –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 [math] matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to [math] . We prove that the multimoment 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.