nLab generalized Spin(7)-manifold

Idea

A generalized Spin(7)-structure on an 8-manifold $X$ is a reduction of the structure group of the generalized tangent bundle $T X \oplus T^\ast X$ along the inclusion of the direct product group of two copies of Spin(7) into the spin-Narain group, along

$$Spin(7) \times Spin(7) \longrightarrow Spin(8,8) \,.$$

This generalizes the reduction of the plain tangent bundle along the inclusion of Spin(7) into Spin(8)

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

which goes with Spin(7)-manifolds, whence the name.

Properties

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

Related concepts

• generalized Calabi-Yau manifold

• generalized G₂-manifold

References

The concept was maybe first considered in

• Frederik Witt, section 5 of Generalised $G_2$-manifolds, Commun.Math.Phys. 265 (2006) 275-303 (arXiv:math/0411642)

Further discussion includes

• Dimitrios Tsimpis, section 4.1. of M-theory on eight-manifolds revisited: $N=1$ supersymmetry and generalized $Spin(7)$ structures, JHEP 0604 (2006) 027 (arXiv:hep-th/0511047)

• Mariana Graña, Carlos S. Shahbazi, Marco Zambon, Spin(7)-manifolds in compactifications to four dimensions, High Energ. Phys. (2014) 2014: 46 (arXiv:1405.3698)