nLab Spin(7) manifold

Idea

An 8-manifold of special holonomy Spin(7).

Equivalently: an 8-manifold equipped with a globalization of the Cayley 4-form.

Properties

As part of the Berger classification

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$\,$

As $\mathbb{O}$-Riemannian manifolds

$\;$normed division algebra$\;$	$\;\mathbb{A}\;$	$\;$Riemannian $\mathbb{A}$-manifolds$\;$	$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$	$\;\mathbb{R}\;$	$\;$Riemannian manifold$\;$	$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$	$\;\mathbb{C}\;$	$\;$Kähler manifold$\;$	$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$	$\;\mathbb{H}\;$	$\;$quaternion-Kähler manifold$\;$	$\;$hyperkähler manifold$\;$
$\;$octonions$\;$	$\;\mathbb{O}\;$	$\;$Spin(7)-manifold$\;$	$\;$G₂-manifold$\;$

(Leung 02)

As exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

see also: coset space structure on n-spheres

Relation to tangentially twisted Cohomotopy

On a spin-manifold of dimension 8 a choice of topological Spin(7)-structure is equivalently a choice of cocycle in J-twisted Cohomotopy cohomology theory. This follows (FSS 19, 3.4) from

1. the standard coset space-structures on the 7-sphere (see here)

2. the fact that coset spaces $G/H$ are the homotopy fibers of the maps $B H \to B G$ of the corresponding classifying spaces (see here)

Characteristic classes

Proposition

Let $X$ be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

$$G = Spin(7) \hookrightarrow Spin(8)$$

then the Euler class $\chi$, the second Pontryagin class $p_2$ and the cup product-square $(p_1)^2$ of the first Pontryagin class (the combination proportional to the I8-term) of the frame bundle/tangent bundle are related by

(1)$$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$$

(Isham-Pope 88 (3.36))

Remark

The same conclusion (1) also holds for Spin(5).Spin(3)-structure, see there.

See also at C-field tadpole cancellation.

Related concepts

• Spin(7)-instanton

• M-theory on Spin(7)-manifolds

• F-theory on Spin(7)-manifolds

• 8-manifold

References

General

The concept goes back to:

• E. Bonan, (1966), Sur les variétés riemanniennes à groupe d’holonomie G2 ou Spin(7), C. R. Acad. Sci. Paris 262: 127–129.

Characterization in terms of the Euler 8-class and the I8-invariant polynomial of the tangent bundle:

• H. Blaine Lawson, Marie-Louise Michelsohn, Thm. 10.7 in: Spin geometry, Princeton University Press (1989) [ISBN 9780691085425]

Construction of compact Spin(7)-manifolds:

• Christine Taylor, Compact Manifolds with Holonomy Spin(7) (1996) (pdf)

• Dominic Joyce, A new construction of compact 8-manifolds with holonomy $Spin(7)$, J. Differential Geom. Volume 53, Number 1 (1999), 89-130 (euclid:jdg/1214425448)

• Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000) (ISBN:9780198506010)

In terms of G-structure:

• Robert Bryant, Metrics with Exceptional Holonomy, Annals of Mathematics Second Series, Vol. 126, No. 3 (Nov., 1987), pp. 525-576 (jstor:1971360)

and motivated from special supersymmetry (such as in M-theory on Spin(7)-manifolds):

• Chris Isham, Christopher Pope, Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings, Class. Quant. Grav. 5 (1988) 257 (spire:251240, doi:10.1088/0264-9381/5/2/006)

• Chris Isham, Christopher Pope, Nicholas Warner, Nowhere-vanishing spinors and triality rotations in 8-manifolds, Classical and Quantum Gravity, Volume 5, Number 10, 1988 (cds:185144, doi:10.1088/0264-9381/5/10/009)

  (focus on Spin(7)-structure)

On Spin(7)-orbifolds:

• Ya. V. Bazaikin, On the new examples of complete noncompact Spin(7)-holonomy metrics, Sib Math J 48, 8–25 (2007) (doi:10.1007/s11202-007-0003-7)

Relation to Higgs bundles

Relating M-theory on Spin(7)-manifolds with F-theory on Spin(7)-manifolds via Higgs bundles:

• Mirjam Cvetic, Jonathan Heckman, Thomas B. Rochais, Ethan Torres, Gianluca Zoccarato, Geometric Unification of Higgs Bundle Vacua (arXiv:2003.13682)