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Spin(7) manifold

Contents

Contents

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\,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\,

As 𝕆\mathbb{O}-Riemannian manifolds

\;normed division algebra\;𝔸\;\mathbb{A}\;\;Riemannian 𝔸\mathbb{A}-manifolds\;\;special Riemannian 𝔸\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\;𝕆\;\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)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

see also: coset space structure on n-spheres

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/HG/H are the homotopy fibers of the maps BHBGB H \to B G of the corresponding classifying spaces (see here)

Characteristic classes

Proposition

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

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

then the Euler class χ\chi, the second Pontryagin class p 2p_2 and the cup product-square (p 1) 2(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)8χ=4p 2(p 1) 2. 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.


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)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):

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:

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