Toric geometry of Spin(7)-manifolds

2018/10/30 by Thomas Bruun Madsen, Andrew Swann, Madsen, Thomas Bruun +1
Mathematics · #53C25 (Primary) 53C29 #53D20 #57R45 #70G45 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C25 #msc:53C29 #msc:53D20 #msc:57R45 #msc:70G45

paper · pdf · doi:10.48550/arxiv.1810.12962

14 pages. This article builds on arXiv:1803.06646 applying the techniques to a new situation

arxiv created 2018/11/01 · arxiv updated 2018/11/02

Abstract

We study 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 4×4-matrix of functions. This description leads to the first known 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 ℝ⁴ as the image of the set of the special orbits.

Related

• Multi-toric geometries with larger compact symmetry
• The 4-Planes of Spin(7) Manifolds
• Connection with torsion, parallel spinors and geometry of Spin(7) manifolds
• A homotopy classification of Spin(7)-structures with applications to exceptional Riemannian holonomy
• Multi-moment maps on nearly Kähler six-manifolds