Toric geometry of Spin(7)-manifolds

Thomas Bruun Madsen, Andrew Francis Swann

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We study $ \operatorname{Spin}(7) $-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons-Hawking type ansatz that describes such geometries in terms of a symmetric $ 4\times 4 $-matrix of functions. This description leads to the 1st known $ \operatorname{Spin}(7) $-manifolds with a rank $ 4 $ symmetry group and full holonomy. We also show that the multi-moment map exhibits the full orbit space topologically as a smooth four-manifold, containing a trivalent graph in $ \mathbb{R}^4 $ as the image of the set of the special orbits.

Original languageEnglish
JournalInternational Mathematics Research Notices
Pages (from-to)16511-16529
Number of pages19
Publication statusPublished - 2021


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