Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds

Abstract

We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor.

arXiv comment25 pages; typos and misprints corrected; to appear in Journal of Geometry and Physics

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• A new example of a compact ERP G2-structure
• Einstein warped G2 and Spin(7) manifolds
• Killing and twistor spinors with torsion
• Deformations of Asymptotically Conical $G_2$-Instantons
• The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor
• Deformations of ${\mathrm{G}}_2$-instantons on nearly ${\mathrm{G}}_2$ manifolds
• The heterotic G${}_2$ system with reducible characteristic holonomy
• Half-flat structures inducing Einstein metrics on homogeneous spaces
• A new $\frac{1}{2}$-Ricci type formula on the spinor bundle and applications
• Topological field theories of 2- and 3-forms in six dimensions
• Moduli spaces of oriented Type A manifolds of dimension at least 3
• Multiplet Classification of Reducible Verma Modules over the $G_2$ Algebra
• Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint
• Moduli spaces of Type~$\mathcal{B}$ surfaces with torsion
• Harmonic $SU(3)$- and $G_2$-structures via spinors
• Parallel spinors for ${\mathrm{G}}_2^{\ast }$ and isotropic structures