nLab associative 3-form

Definition

On the Cartesian space $\mathbb{R}^7$ the associative 3-form is the differential 3-form $\omega \in \Omega^3(\mathbb{R}^7)$ which is constant and whose value at the origin on three vectors is

$$\omega(u,v,w) \coloneqq \langle u, v \times w\rangle \,,$$

where

• $\langle -,-\rangle$ is the canonical bilinear form on $\mathbb{R}^7$;

• $(-)\times (-)$ is the cross product in $\mathbb{R}^7$.

This are the structure constants of the unit octonions.

The Hodge dual of the associative 3-form is sometimes called the co-associative 4-form.

Properties

Relation to $G_2$

The group of linear diffeomorphisms of $\mathbb{R}^7$ which preserve this form is the exceptional Lie group G₂.

Related concepts

classification of special holonomy manifolds by Berger's theorem:

	$\,$G-structure$\,$	$\,$special holonomy$\,$	$\,$dimension$\,$	$\,$preserved differential form$\,$
$\,\mathbb{C}\,$	$\,$Kähler manifold$\,$	$\,$U(n)$\,$	$\,2n\,$	$\,$Kähler forms $\omega_2\,$
	$\,$Calabi-Yau manifold$\,$	$\,$SU(n)$\,$	$\,2n\,$
$\,\mathbb{H}\,$	$\,$quaternionic Kähler manifold$\,$	$\,$Sp(n).Sp(1)$\,$	$\,4n\,$	$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
	$\,$hyper-Kähler manifold$\,$	$\,$Sp(n)$\,$	$\,4n\,$	$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$	$\,$Spin(7) manifold$\,$	$\,$Spin(7)$\,$	$\,$8$\,$	$\,$Cayley form$\,$
	$\,$G₂ manifold$\,$	$\,$G₂$\,$	$\,7\,$	$\,$associative 3-form$\,$

• associative submanifold

• octonions

• Hitchin functional

References

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