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

Abstract

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 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.