Clifford systems, Clifford structures, and their canonical differential forms

TL;DR

This paper explores the relationships between various canonical 4-forms in high-dimensional geometry, linking quaternionic, Spin(7), and Clifford structures, and extends Cayley plane concepts to 16 dimensions.

Contribution

It provides explicit formulas for canonical 4-forms in 8 and 16 dimensions and characterizes their calibrated 4-planes, connecting Clifford systems with special holonomy geometries.

Findings

Explicit formulas for $ ext{Spin}(8)$ and $ ext{Spin}(7)U(1)$ 4-forms in dimension 16.

Characterization of calibrated 4-planes for these forms.

Unified perspective on quaternionic and Spin(7) geometries via Clifford structures.

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 $16$.