Toric geometry of G2–manifolds

Madsen, Thomas¹ ; Swann, Andrew²

¹ School of Computing, University of Buckingham, Buckingham, United Kingdom, Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Aarhus, Denmark
² 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

Geometry & topology, Tome 23 (2019) no. 7, pp. 3459-3500.

Résumé

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 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.

Détail

DOI : 10.2140/gt.2019.23.3459

Classification : 53C25, 53C29, 53D20, 57R45, 70G45
Keywords: exceptional holonomy, multimoment maps, toric geometry, Gibbons–Hawking ansatz

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