Clifford systems, Clifford structures, and their canonical differential forms

Kai Brynne M. Boydon; Paolo Piccinni

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arXiv · 2008.10288

Clifford systems, Clifford structures, and their canonical differential forms

Kai Brynne M. Boydon·Paolo Piccinni

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Abstract

A comparison among different constructions of the quaternionic $4$-form $\Phi_{Sp(2)Sp(1)}$ and of the Cayley calibration $\Phi_{Spin(7)}$ shows that one can start for them from the same collections of "K\"ahler 2-forms", entering in dimension 8 both in quaternion K\"ahler and in $Spin(7)$ geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension $16$, similar constructions allow to write explicit formulas for the canonical $4$-forms $\Phi_{Spin(8)}$ and $\Phi_{Spin(7)U(1)}$, associated with Clifford systems related with the subgroups $Spin(8)$ and $Spin(7)U(1)$ of $SO(16)$. We characterize the calibrated $4$-planes of the $4$-forms $\Phi_{Spin(8)}$ and $\Phi_{Spin(7)U(1)}$, extending in two different ways the notion of Cayley $4$-plane to dimension $16$.

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Kai Brynne M. Boydon, Paolo Piccinni. 2020-08-24. Clifford systems, Clifford structures, and their canonical differential forms. https://arxiv.org/abs/2008.10288

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