Toric geometry of Spin(7)-manifolds

TY - JOUR

T1 - Toric geometry of Spin(7)-manifolds

AU - Madsen, Thomas Bruun

AU - Swann, Andrew Francis

PY - 2021/11/1

Y1 - 2021/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85122211420&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz279

DO - 10.1093/imrn/rnz279

M3 - Journal article

SN - 1073-7928

VL - 2021

SP - 16511

EP - 16529

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 21

ER -