#### Volume 23, issue 7 (2019)

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

### Thomas Bruun Madsen and Andrew Swann

Geometry & Topology 23 (2019) 3459–3500
##### 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 $3×3$ 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.

##### Keywords
exceptional holonomy, multimoment maps, toric geometry, Gibbons–Hawking ansatz
##### Mathematical Subject Classification 2010
Primary: 53C25
Secondary: 53C29, 53D20, 57R45, 70G45