New Complete Non-compact Spin(7) Manifolds

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  • 21 March 2001
Abstract
With a view to future applications in M-Theory, we construct new explicit metrics on two complete non-compact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A_8, is topologically R^8 and the other, which we denote by B_8, is the bundle of positive (or negative) chirality spinors over S^4. Unlike the previously-known complete non-compact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S^4, both 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 L^2-normalisable harmonic 4-forms (of opposite dualities) for the B_8 manifold. The new metrics allow the construction of various supersymmetric brane solutions in M-theory and string theory. In particular, we obtain resolved fractional M2-brane solutions 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 two new Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the original radial coordinate to be either positive or negative. We make a comparison with the Ricci-flat Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy.

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