Toric geometry of $G_2$-manifolds

TL;DR

This paper explores $G_2$-manifolds with torus symmetries, deriving explicit metrics and analyzing orbit space topology, especially for $T^3$-actions, leading to new examples of $G_2$ holonomy manifolds.

Contribution

It introduces a Gibbons-Hawking type ansatz for $G_2$-manifolds with torus symmetry, providing explicit metrics and detailed orbit space descriptions.

Findings

Derived explicit $G_2$ metrics with torus symmetry.

Analyzed the topology of orbit spaces and special orbits.

Constructed new examples with $G_2$ holonomy.

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 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.