New complete noncompact Spin(7) manifolds
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We construct new explicit metrics on complete noncompact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by double-struck A8, is topologically 8 and another, which we denote by double-struck 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 3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L2-normalisable harmonic 4-form for the double-struck A 8 manifold, and two such 4-forms (of opposite dualities) for the double-struck B8 manifold. We use the metrics to construct new supersyrhmetric 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 double-struck A8 and double-struck 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 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 double-struck B8+ and double-struck B8- similar to double-struck B8. 2002 Published by Elsevier Science B.V.