New cohomogeneity one metrics with Spin(7) holonomy

✍ Scribed by M. Cvetič; G.W. Gibbons; H. Lü; C.N. Pope

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 173 KB
• Volume: 49
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(03)00108-6

✦ Synopsis

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A 8 , is complete and non-singular on R 8 . The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S 4 , and are denoted by B + 8 , B - 8 and B 8 . The metrics on B + 8 and B - 8 occur in families with a non-trivial parameter. The metric on B 8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A 8 . The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L 2 -normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold.