nLab Cayley form

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.

Definition

(Harvey-Lawson 82, Def. 1.21)

Properties

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).

(Harvey-Lawson 82, Prop. 1.36)

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).

(Harvey-Lawson 82, Def. 1.23)

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):

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

(Harvey-Lawson 82, Theorem 1.38, see also Bryant-Harvey 89, (3.19), Gluck-Mackenzie-Morgan 95, (5.20))

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)).

See also at Spin Grassmannians.

Related concepts

• 8-manifold

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$\,$

References

General

• Reese Harvey, H. Blaine Lawson, Calibrated geometries, Acta Math. Volume 148 (1982), 47-157 (Euclid:1485890157)

• Robert Bryant, Reese Harvey, Submanifolds in Hyper-Kähler Geometry, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 1-31 (jstor:1990911)

• Herman Gluck, Dana Mackenzie, Frank Morgan, Volume-minimizing cycles in Grassmann manifolds, Duke Math. J. Volume 79, Number 2 (1995), 335-404 (euclid:1077285156)

• Liviu Ornea, Paolo Piccinni, Cayley 4-frames and a quaternion-Kähler reduction related to $Spin(7)$, Proceedings of the International Congress of Differential Geometry in the memory of A. Gray, held in Bilbao, Sept. 2000 (arXiv:math/0106116)

• Mikhail Katz, Steven Shnider, Cayley 4-form, comass, and triality isomorphisms (arXiv:0801.0283)

• Kirill Krasnov, Dynamics of Cayley Forms [arXiv:2403.16661]

On a complex and Lorentzian variant:

• Kirill Krasnov, Lorentzian Cayley Form [arXiv:2304.01118]

In string theory/M-theory

In string theory/M-theory:

• Katrin Becker, Melanie Becker, David Morrison, Hirosi Ooguri, Y. Oz, Z. Yin, Supersymmetric Cycles in Exceptional Holonomy Manifolds and Calabi-Yau 4-Folds, Nucl. Phys. B480:225-238, 1996 (arXiv:hep-th/9608116)