Toric geometry of G2-manifolds

Thomas Bruun Madsen, Andrew Francis Swann

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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.

Original languageEnglish
JournalGeometry & Topology
Pages (from-to)3459-3500
Number of pages42
Publication statusPublished - Dec 2019


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