Connection with torsion, parallel spinors and geometry of Spin(7) manifolds

Abstract

We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior differential. We show the vanishing of the (\hat)-A genus and obtain a linear relation between Betti numbers of a compact Spin(7) manifolds which are locally but not globally conformally equivalent to a space with closed fundamental 4-form. A general solution to the Killing spinor equations is presented

LaTeX, 15 pages, references added, some typos corrected, final version to appear in Math. Res. Lett

References in corpus (3)

• Fivebranes Wrapped on SLAG Three-Cycles and Related Geometry
• The supergravity dual of a theory with dynamical supersymmetry breaking
• On types of non-integrable geometries

Cited by in corpus (6)

• The Geometry of D=11 Killing Spinors
• The spinorial geometry of supersymmetric heterotic string backgrounds
• The spinorial geometry of supersymmetric IIB backgrounds
• Geometry of all supersymmetric type I backgrounds
• On $\mathcal{M}$-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to $\mathcal{O}(R^4)$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries
• Special Holonomy and Two-Dimensional Supersymmetric Sigma-Models