# nLab G2 manifold

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A $G_2$-structure on a manifold of dimension 7 is a choice of reduction of the structure group of the tangent bundle along the inclusion of G2 into $GL(7)$.

Given that $G_2$ is the subgroup of the general linear group on the Cartesian space $\mathbb{R}^7$ which preserves the associative 3-form on $\mathbb{R}^7$, a $G_2$ structre is a higher analog of an almost symplectic structure under lifting from symplectic geometry to 2-plectic geometry (Ibort).

A $G_2$-manifold is a manifold equipped with an “integrable” or “parallel” $G_2$-structure. This is equivalently a Riemannian manifold of dimension 7 with special holonomy group being the exceptional Lie group G2.

$G_2$-manifolds may be understood as 7-dimensional analogs of real 6-dimensional Calabi-Yau manifolds.

## Definition

### $G_2$-structure

###### Definition

For $X$ a smooth manifold of dimension $7$ a $G_2$-structure on $X$ is a choice of differential 3-form $\omega \in \Omega^3(X)$ such that there is an atlas over which this 3-form locally identifies with the associative 3-form on the Cartesian space $\mathbb{R}^7$.

Equivalently, this is a choice of reduction of the structure group of the tangent bundle along the inclusion

$G_2 \hookrightarrow GL(7) \,.$
###### Remark

A $G_2$-structure in particular implies an orthogonal structure, hence a Riemannian metric.

### $G_2$-holonomy

###### Definition

A manifold equipped with a $G_2$-structure $\omega$, def. 1, is called a $G_2$-manifold if $\omega$ is “parallel” or “integrable” in that

1. $d \omega = 0$

2. $d \star \omega = 0$

(where $d$ is the de Rham differential and $\star$ is the Hodge star operator of the canonical Riemannian metric of remark 1).

For instance (Joyce, p. 4).

The holonomy of the Levi-Civita connection on a $G_2$-manifold is contained in $G_2$.

### Weak $G_2$-holonomy

There is a useful weakened notion of $G_2$-holonomy.

###### Definition

A 7-dimensional manifold is said to be of weak $G_2$-holonomy if it carries a 3-form $\omega$ with the relation of def. 2 generalized to

$d \omega = \lambda \star \omega$

and hence

$d \star \omega = 0$

for $\lambda \in \mathbb{R}$. For $\lambda = 0$ this reduces to strict $G_2$-holonomy, by 2.

(See for instance (Bilal-Derendinger-Sfetsos).)

## Properties

### Existence

###### Proposition

A 7-manifold admits a $G_2$-structure, def. 1, precisely if it admits a spin structure.

### Metric structure

The canonical Riemannian metric $G_2$ manifold is Ricci flat. More generally a manifold of weak $G_2$-holonomy, def. 3, with weakness parameter $\lambda$ is an Einstein manifold with cosmological constant $\lambda$.

## Applications

### In supergravity

In string phenomenology models obtained from compactification of 11-dimensional supergravity/M-theory on $G_2$-manifolds (see for instance Duff) can have attractive phenomenological properties, see for instance the G2-MSSM.

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
Kähler manifoldU(k)$2k$Kähler forms
Calabi-Yau manifoldSU(k)$2k$
hyper-Kähler manifoldSp(k)$4k$
G2 manifoldG2$7$associative 3-form
Spin(7) manifoldSpin(7)8Cayley form

## References

### General

Compact $G_2$-manifolds were first found in

• Dominic Joyce, Compact Riemannian 7-manifolds with holonomy $G_2$, Journal of Differential Geometry vol 43, no 2 (pdf)
• Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000)

Surveys include

• Spiro Karigiannis, What is… a $G_2$-manifold (pdf)

• Spiro Karigiannis, $G_2$-manifolds – Exceptional structures in geometry arising from exceptional algebra (pdf)

The relation to multisymplectic geometry/2-plectic geometry is mentioned explicitly in

(but beware of some mistakes in that article…)

For more see the references at exceptional geometry.

### Application in supergravity

The following references discuss the role of $G_2$-manifolds in M-theory on G2-manifolds:

A survey of the corresponding string phenomenology for M-theory on G2-manifolds (see there for more) is in

• Bobby Acharya, $G_2$-manifolds at the CERN Large Hadron collider and in the Galaxy, talk at $G_2$-days (2012) (pdf)

Weak $G_2$-holonomy is discussed in

• Adel Bilal, J.-P. Derendinger, K. Sfetsos, (Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry, Nucl.Phys. B628 (2002) 112-132 (arXiv:hep-th/0111274)
• Adel Bilal, Steffen Metzger?, Compact weak $G_2$-manifolds with conical singularities (pdf)

Revised on January 4, 2013 08:22:36 by Urs Schreiber (89.204.137.169)