Toric geometry of $G_2$-manifolds

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.

arXiv comment33 pages; improved version of Q operator, typos corrected

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Cited by in corpus (2)

• Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds
• Special Holonomy Manifolds, Domain Walls, Intersecting Branes and T-folds