# nLab Cayley form

Contents

### Context

#### Differential cohomology

differential cohomology

## Application to gauge theory

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The Cayley 4-form $\Phi$ (Harvey-Lawson 82) is a certain differential 4-form on the real 8-dimensional space $\mathbb{R}^8$.

$\Phi \;\in\; \Omega^4(\mathbb{R}^n)$

which constitutes an exceptional calibration of $\mathbb{R}^4$ with its Euclidean geometry.

More generally, a Spin(7)-manifold carries a globalization of this 4-form calibration, then also called a Cayley-4-form.

## Invariance

The stabilizer subgroup inside GL(8) of the Cayley 4-from under the action given by pullback of differential forms is the subgroup Spin(7) inside SO(8).

### As a calibration

The Cayley 4-form constitutes a calibration of the Euclidean space $\mathbb{R}^8$ (Harvey-Lawson 82)

### Grassmannian of Cayley 4-planes

A calibrated submanifold for $\Phi$ is also called a Cayley 4-plane (not to be confused with the Cayley plane).

The space (moduli space) of Cayley 4-planes, denoted $CAY$ (Bryant-Harvey 89, (2.19)) or $CAYLEY$ (Gluck-Mackenzie-Morgan 95, (5.20)), is hence a topological subspace of the Grassmannian of all 4-planes in 8-dimensions:

$CAY \subset Gr(4,8)$

This is of codimension 4 (Harvey-Lawson 82, below (5)). In fact, this space is homeomorphic to the coset space of Spin(7) by Spin(4).Spin(3) = Spin(3).Spin(3).Spin(3) = Sp(1).Sp(1).Sp(1):

$CAY \;\simeq\; Spin(7)/\big( Spin(4) \cdot Spin(3)\big)$

Moreover, the coset space of Spin(6) by Spin(3).Spin(3) $\simeq$ SO(4)

$CAY_{sL} \;\simeq\; Spin(6)/\big( Spin(3) \cdot Spin(3)\big) \;\simeq\; SU(4)/SO(4)$

is the Grassmannian of those Cayley 4-planes which are also special Lagrangian submanifolds (BBMOOY 96, p. 7 (8 of 17)).

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\,$$\,$associative 3-form$\,$