#### 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
##### Publication
Received: 22 March 2018
Revised: 19 November 2018
Accepted: 25 May 2019
Published: 30 December 2019
Proposed: Gang Tian
Seconded: Simon Donaldson, Yasha Eliashberg
##### Authors
 Thomas Bruun Madsen School of Computing University of Buckingham Buckingham United Kingdom Centre for Quantum Geometry of Moduli Spaces Aarhus University Aarhus Denmark Andrew Swann Department of Mathematics, Centre for Quantum Geometry of Moduli Spaces and Aarhus University Centre for Digitalisation, Big Data and Data Analytics University of Aarhus Aarhus Denmark