New complete noncompact Spin(7) manifolds Original Research Article

We construct new explicit metrics on complete noncompact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A8, is topologically R8 and another, which we denote by B8, is the bundle of chiral spinors over S4. Unlike the previously-known complete noncompact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S4, 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 CP3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L2-normalisable harmonic 4-form for the A8 manifold, and two such 4-forms (of opposite dualities) for the B8 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 L2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A8 and B8 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 we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B8+ and B8− similar to B8.