# nLab generalized Spin(7)-manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## 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

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)$

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

Last revised on March 30, 2019 at 10:01:36. See the history of this page for a list of all contributions to it.