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Cayley form

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Context

Differential cohomology

Algebra

Contents

Idea

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

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

which constitutes an exceptional calibration of 4\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 8\mathbb{R}^8 (Harvey-Lawson 82)

Grassmannian of Cayley 4-planes

A calibrated submanifold for Φ\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 CAYCAY (Bryant-Harvey 89, (2.19)) or CAYLEYCAYLEY (Gluck-Mackenzie-Morgan 95, (5.20)), is hence a topological subspace of the Grassmannian of all 4-planes in 8-dimensions:

CAYGr(4,8) 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):

CAYSpin(7)/(Spin(4)Spin(3)) 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 sLSpin(6)/(Spin(3)Spin(3))SU(4)/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.

classification of special holonomy manifolds by Berger's theorem:

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

References

General

In string theory/M-theory

In string theory/M-theory:

Last revised on April 5, 2019 at 11:33:05. See the history of this page for a list of all contributions to it.