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Toric geometry of
$\mathrm{G}_2$–manifolds
Thomas Bruun Madsen and Andrew Swann |
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Geometry & Topology 23 (2019) 3459–3500
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Abstract |
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We consider –manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of –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 matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to . 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. |
Keywords
exceptional holonomy, multimoment maps, toric geometry,
Gibbons–Hawking ansatz
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Mathematical Subject Classification 2010
Primary: 53C25
Secondary: 53C29, 53D20, 57R45, 70G45
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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
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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 |
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