nLab
generalized Spin(7)-manifold

Contents

Contents

Idea

A generalized Spin(7)-structure on an 8-manifold XX is a reduction of the structure group of the generalized tangent bundle TXT *XT 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)×Spin(7)Spin(8,8). 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)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)-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

References

The concept was maybe first considered in

  • Frederik Witt, section 5 of Generalised G 2G_2-manifolds, Commun.Math.Phys. 265 (2006) 275-303 (arXiv:math/0411642)

Further discussion includes

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