G2-structures and octonion bundles

Abstract

We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. We also interpret the choice of a particular G2-structure within the same metric class as a choice of gauge and show that under a change of this gauge, the torsion does transform as an octonion-valued connection 1-form. Finally, we also show an explicit relationship between the octonion bundle and the spinor bundle, define an octonionic Dirac operator and explore an energy functional for octonion sections. We then prove that critical points correspond to divergence-free torsion, which is shown to be an octonionic analog of the Coulomb gauge.

48 pages (40 plus appendix), corrected typos, expanded some proofs

References in corpus (6)

• The Srni lectures on non-integrable geometries with torsion
• Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
• Curvature decomposition of G_2 manifolds
• The spinorial energy functional on surfaces
• Local geometry of the G2 moduli space

Cited by in corpus (18)

• Harmonic flow of geometric structures
• A gradient flow of isometric ${\mathrm{G}}_2$ structures
• Estimates and monotonicity for a heat flow of isometric G2-structures
• A flow of isometric $G_2$-structures. Short-time existence
• Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a $Spin(9)$-structure
• Nonassociative analogs of Lie groupoids
• Non-associative Categories of Octonionic Bimodules
• G2-structures for N=1 supersymmetric AdS4 solutions of M-theory
• Harmonic flow of $\mathrm{Spin}(7)$-structures
• Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere
• Slice Dirac operator over octonions
• Para-linearity as the nonassociative counterpart of linearity
• Flows of SU(2)-structures
• Additional fermionic fields onto parallelizable 7-spheres
• An energy functional on the universal spinor bundle
• Octonionic isometric isomorphisms and partial isometry
• Cohomogeneity one solitons for the isometric flow of $G_2$-structures
• Slice regular functions of several octonionic variables