# nLab generalized G₂-manifold

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

# Contents

## 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

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)$

see also: coset space structure on n-spheres

## 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)

Last revised on July 17, 2024 at 12:32:47. See the history of this page for a list of all contributions to it.