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

Thomas Bruun Madsen and Andrew Swann

Geometry & Topology 23 (2019) 3459–3500

We consider G2–manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of T3–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 G2. 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.

exceptional holonomy, multimoment maps, toric geometry, Gibbons–Hawking ansatz
Mathematical Subject Classification 2010
Primary: 53C25
Secondary: 53C29, 53D20, 57R45, 70G45
Received: 22 March 2018
Revised: 19 November 2018
Accepted: 25 May 2019
Published: 30 December 2019
Proposed: Gang Tian
Seconded: Simon Donaldson, Yasha Eliashberg
Thomas Bruun Madsen
School of Computing
University of Buckingham
United Kingdom
Centre for Quantum Geometry of Moduli Spaces
Aarhus University
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