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
This generalizes the reduction of the plain tangent bundle along the inclusion of Spin(7) into Spin(8)
which goes with Spin(7)-manifolds, whence the name.
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
The concept was maybe first considered in
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)
Last revised on July 18, 2024 at 11:46:01. See the history of this page for a list of all contributions to it.