Toric geometry of G2–manifolds

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.

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Keywords

exceptional holonomy, multimoment maps, toric geometry, Gibbons–Hawking ansatz

Mathematical Subject Classification 2010

Primary: 53C25

Secondary: 53C29, 53D20, 57R45, 70G45

References

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

Volume 23, issue 7 (2019)

Thomas Bruun Madsen and Andrew Swann

Abstract