nLab
G2 manifold

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 $\mathrm{GL}(7)$.

Given that $G_2$ is the subgroup of the general linear group on the Cartesian space ${ℝ}^7$ which preserves the associative 3-form on ${ℝ}^7$, 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 ${ℝ}^7$.

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

$$G_2\hookrightarrow \mathrm{GL}(7).$$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

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

and hence

$$d\star \omega =0$$d \star \omega = 0

for $\lambda \in ℝ$. 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.

Related concepts

classification of special holonomy manifolds by Berger's theorem:

G-structure	special holonomy	dimension	preserved differential form
Kähler manifold	U(k)	$2k$	Kähler forms
Calabi-Yau manifold	SU(k)	$2k$
hyper-Kähler manifold	Sp(k)	$4k$
G2 manifold	G2	$7$	associative 3-form
Spin(7) manifold	Spin(7)	8	Cayley form

• Hitchin functional

• M-theory on G2-manifolds, G2-MSSM

• topological M-theory, topological membrane

• generalized G2-manifold

• Calabi-Yau manifold

• exceptional generalized geometry

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

• Alberto Ibort, Multisymplectic geometry: generic and exceptional, Proceedings of the IX Fall workshop on geometry and physics (pdf)

(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:

• Mike Duff, M-theory on manifolds of G2 holonomy: the first twenty years (arXiv:hep-th/0201062)

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)