G₂-structures and octonion bundles

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• 作者：Sergey Grigorian ^{sergey.grigorian@utrgv.edu}
• 关键词：G2-structures ; Octonions ; Torsion ; Spinors
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：21 February 2017
• 年：2017
• 卷：308
• 期：Complete
• 页码：142-207
• 全文大小：832 K
• 卷排序：308

文摘

We use a G2G2-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 octonionic covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2G2-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 G2G2. We also interpret the choice of a particular G2G2-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.