# nLab Spin(7) manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

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

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto 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)$

### Relation to J-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)$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$
###### Remark

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

## References

### General

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):

• 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

Last revised on July 15, 2020 at 15:51:34. See the history of this page for a list of all contributions to it.