New complete noncompact Spin(7) manifolds

Abstract

We construct new explicit metrics on complete noncompact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A 8 , is topologically R 8 and another, which we denote by B 8 , is the bundle of chiral spinors over S 4. Unlike the previously-known complete noncompact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S 4, our new 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. We use the metrics to construct new supersymmetric brane solutions in M-theory and string theory. In particular, we construct resolved fractional M2-branes involving the use of the L 2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A 8 and B 8 Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the radial coordinate to be either positive or negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy. In Appendix A we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B 8 + and B 8 − similar to B 8 .