# nLab associative 3-form

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

classification of special holonomy manifolds by Berger's theorem:

$\,$G-structure$\,$$\,$special holonomy$\,$$\,$dimension$\,$$\,$preserved differential form$\,$
$\,\mathbb{C}\,$$\,$Kähler manifold$\,$$\,$U(k)$\,$$\,2k\,$$\,$Kähler forms $\omega_2\,$
$\,$Calabi-Yau manifold$\,$$\,$SU(k)$\,$$\,2k\,$
$\,\mathbb{H}\,$$\,$quaternionic Kähler manifold$\,$$\,$Sp(n).Sp(1)$\,$$\,4k\,$$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
$\,$hyper-Kähler manifold$\,$$\,$Sp(k)$\,$$\,4k\,$$\,\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\,$$\,$associative 3-form$\,$

## References

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Last revised on February 19, 2018 at 01:18:17. See the history of this page for a list of all contributions to it.