nLab generalized G₂-manifold

Definition

reduction of the structure group of the generalized tangent bundle of a 7-dimensional manifold along the inclusion

$$G_2 \times G_2 \to SO(7,7)$$

of two copies of G₂.

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

• G₂-manifold

• exceptional generalized geometry

• generalized Calabi-Yau manifold, type II geometry

• generalized Spin(7)-manifold

References

• Frederik Witt, Generalised $G_2$-structures, (math.dg/0411642)

• Claus Jeschek, Frederik Witt, Generalised $G_2$-structures and type IIB superstrings (arXiv:hep-th/0412280)

• Generalised $G_2$-manifolds (2005) (pdf)

• Anna Fino, Adriano Tomoassini, Generalized $G_2$-manifolds and $SU(3)$-structure, Int. J. Math. 19, 1147 (2008) (web)

• J. de Boer, P de Medeiros, S El-Showk and A Sinkovics, $G_2$ Hitchin functionals at one loop, Class. Quantum Grav. 25 (web)