# nLab G2 manifold

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A $G_2$-structure on a manifold $X$ of dimension 7 is a choice of G-structure on $X$, for $G$ the exceptional Lie group G2. Hence it is a reduction of the structure group of the frame bundle of $X$ along the canonical (the defining) inclusion $G_2 \hookrightarrow GL(\mathbb{R}^7)$ into the general linear group.

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 $G_2$-structure that is integrable to first order, i.e. torsion-free (prop. 3 below). 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. Accordingly the relation between Calabi-Yau manifolds and supersymmetry lifts from string theory to M-theory on G2-manifolds.

## Definition

### $G_2$-structure

###### Definition

For $X$ a smooth manifold of dimension $7$ a $G_2$-structure on $X$ is a G-structure for $G =$ G2 $\hookrightarrow GL(7)$.

###### Remark

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

Given the definition of G2 as the stabilizer group of the associative 3-form on $\mathbb{R}^7$, there is accordingly an equivalent formulation of def. 1 in terms of differential forms:

###### Definition

Write $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast$ for the orbit of the associative 3-form $\phi$ under the canonical $GL(7)$-action. Similarly for $X$ a smooth manifold of dimension 7, write

$\Omega^3_+(X) \hookrightarrow \Omega^3(X)$

for the subset of the set of differential 3-forms on those that, as sections to the exterior power of the cotangent bundle, are pointwise in $\Lambda^3_+(\mathbb{R}^7)^\ast$.

These are also called the positive forms (Joyce 00, p. 243) or the definite differential forms (Bryant 05, section 3.1.1) on $X$.

(e.g. Bryant 05, definition 2)

###### Proposition

A $G_2$-structure on $X$, def. 1, is equivalently a choice of definite 3-form $\sigma$ on $X$, def. 2.

Often it is useful to exhibit prop. 1 in the following way.

###### Example

For $X$ a smooth manifold of dimension 7, write $Fr(X) \to X$ for its frame bundle. By the discussion at vielbein – in terms of basic forms on the frame bundle there is a universal $\mathbb{R}^7$-valued differential form on the total space of the frame bundle

$E_u \in \Omega^1(Fr(X), \mathbb{R}^7)$

(whose components we write $(E_u^a)_{a = 1}^7$) such that given an orthogonal structure $i \colon Fr_O(X)\hookrightarrow Fr(X)$ and a local section $\sigma_i \colon (U_i \subset X) \to Fr_O(X)$ of orthogonal frames, then the pullback of differential forms

$E_i \coloneqq \sigma_i^\ast i^\ast E_u$

is the corresponding local vielbein field. Hence one obtains a universal 3-form $\phi_u \in \Omega^3(Fr(x))$ on the frame bundle by setting

$\phi_u \coloneqq \phi_{a b c} E_u^a \wedge E_u^b \wedge E_u^c$

with $(\phi_{a b c})$ the canonical components of the associative 3-form and with summation over repeated indices understood.

By construction this is such that on a chart $(U_i \subset X)$ any definite 3-form, def. 2, restricts to the pullback of $\phi_u$ via a section $\sigma_i \colon U_i \to Fr(X)$ and hence is of the form

$\phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c \,.$

Conversely, given a 3-form $\sigma \in \Omega^3(X)$ such that on an atlas $(U_i \to X)$ over which the frame bundle trvializes it is of this form

$\sigma|_{U_i} = \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c$

then the $GL(7)$-valued transition functions $g_{i j}$ of the given local trivialization must factor through $G_2\hookrightarrow SO(7) \hookrightarrow GL(7)$ and hence exhibit a $G_2$-structure: because we have $\sigma|_{U_i} = \sigma|_{U_j} \;\;\; on \;U_i \cap U_j$ and hence

(1)$\phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c = \phi_{a b c} E_j^a \wedge E_j^b \wedge E_j^c \;\;\; on \; U_i \cap U_j \,.$

But by the nature of the universal vielbein, its local pullbacks are related by

$E_j = g_{i j} E_i$

i.e.

$E_j^a = (g_{i j})^a{}_b E_i^b$

and hence (1) says that

$\phi_{a b c} = \phi_{a' b' c'} (g_{i j})^{a'}{}_a (g_{i j})^{b'}{}_b (g_{i j})^{c'}{}_c \;\;\; on \; U_i \cap U_j$

which is precisely the defining condition for $g_{i j}$ to take values in $G_2$.

Viewed this way, the definite 3-forms characterizing $G_2$-structures are an example of a more general kind of differential forms obtained from a constant form on some linear model space $V$ by locally contracting with a vielbein field. For instance on a super-spacetime solving the equations of motion of 11-dimensional supergravity there is a super-4-form part of the field strength of the supergravity C-field which is cnstrained to be locally of the form

$\Gamma_{a b \alpha \beta} E_i^a \wedge E_i^b \wedge E_i^\alpha \wedge E_i^\beta$

for $(E^A)= (E^a, E^\alpha) = (E^a, \Psi^\alpha)$ the super-vielbein. See at Green-Schwarz action functional – Membrane in 11d SuGra Background. Indeed, by the discussion there this 4-form is required to be covariantly constant, which is precisely the analog of $G_2$-manifold structure as in def. 4.

References that write definite 3-forms in this form locally as $\phi_{a b c}E^a \wedge E^b \wedge E^c$ include (BGGG 01 (2.9), …).

The following is important for the analysis:

###### Remark

The subset $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast$ in def. 2 is an open subset, hence $\phi$ is a stable form (e.g. Hitchin, def. 1.1).

###### Proof

By definition of $G_2$ as the stabilizer group of the associative 3-form, the orbit it generates under the $GL_+(7)$-action is the coset $GL_+(7)/G_2$. The dimension of this as a smooth manifold is 49-14 = 35. This is however already the full dimension $\left(7 \atop 3\right) = 35$ of the space of 3-forms in 7d that the orbit sits in. Therefore (since $G_+(7)/G_2$ does not have a boundary) the orbit must be an open subset.

### Closed $G_2$-structure

###### Definition

A $G_2$-structure, def. 1, is called closed if the definite 3-form $\sigma$ corresponding to it via prop. 1 is a closed differential form, $\mathbf{d}\sigma = 0$.

(e.g. Bryant 05, (4.31))

###### Proposition

For a closed $G_2$-structure, def. 3, on a manifold $X$ there exists an atlas by open subsets

$\mathbb{R}^7 \underoverset{et}{f}{\leftarrow} U \underset{et}{\rightarrow} X$

such that the globally defined 3-form $\sigma \in \Omega^3_+(X)$ is locally gauge equivalent to the canonical associative 3-form $\phi$

$\sigma|_U = f^\ast \phi + \mathbf{d}\beta$

via a 2-form $\beta$ on $U$.

(e.g. Bryant 05, p. 21)

This follows from the fact, remark 2, that the definite 3-forms are an open subset inside all 3-forms: given a chart centered around any point then there is $\beta$ with $\mathbf{d}\beta$ vanishing at that point such that $\sigma|_U \simeq f^\ast \phi + \mathbf{d}\beta$ at that point. But since the $GL(7)$-action on $\phi$ is open, there is an open neighbourhood around that point where this is still the case.

###### Remark

When regarding smooth manifolds in the wider context of higher differential geometry, then the situation of prop. 2 corresponds to a diagram of formal smooth infinity-groupoids of the following form:

$\array{ && U \\ & {}^{\mathllap{f}}\swarrow && \searrow \\ \mathbb{R}^7 && \swArrow_{\mathrlap{\beta}} && X \\ & {}_{\mathllap{\phi}}\searrow && \swarrow_{\mathrlap{\sigma}} \\ && \flat_{dR}\mathbf{B}^3\mathbb{R} } \,,$

where $\flat_{dR}\mathbf{B}^3\mathbb{R}$ is the higher moduli stack of flat 3-forms with 2-form gauge transformations between them (and 1-form gauge transformation between these). The diagram expresses the 3-form $\sigma$ as a map to this moduli stack, which when restricted to the cover $U$ becomes gauge equivalent to the pullback of the associative 3-form $\phi$, similarly regarded as a map, to the cover, where the gauge equivalence is exhibited by a homotopy (of maps of formal smooth $\infty$-groupoids) which is the 2-form $\beta$ on $U$.

### $G_2$-holonomy / $G_2$-manifold

###### Definition

A manifold $X$ equipped with a $G_2$-structure, def. 1, is called a $G_2$-manifold if the following equivalent conditions hold

1. we have

1. $\mathbf{d} \omega = 0$;

2. $\mathbf{d} \star_g \omega = 0$;

2. $\nabla^g \omega = 0$;

3. $(X,g)$ has special holonomy $Hol(g) \subset G_2$.

Here

• $d$ is the de Rham differential;

• $\omega$ is the 3-form $\omega$ corresponding to the given $G_2$-structure via prop. 1;

• $g$ is the induced Riemannian metric of remark 1);

• $\star_g$ is the Hodge star operator of this metric;

• $\nabla^g$ is the covariant derivative of this metric;

For instance (Joyce, p. 4, Joyce 00, prop. 10.1.3).

###### Proposition

The conditions on a $G_2$-structure in def. 1 are equivalent to the torsion of the $G_2$-structure to vanish.

###### Remark

The higher torsion invariants of $G_2$-structures do not necessarily vanish (contrary to the case for instance of symplectic structure and complex structure, see at integrability of G-structures – Examples). Therefore, even in view of prop. 3, a $G_2$-manifold, def. 4, does not, in general admit an atlas be adapted coordinate charts equal to $(\mathbb{R}^7, \phi)$.

The space of second order torsion invariants of $G_2$-structures is for instance in (Bryant 05 (4.7)).

### Variants and weakenings

There are several variants of the definition of $G_2$-manifolds, def.4, given by imposing other constraints on the torsion.

#### With skew-symmetric torsion

Discussion for totally skew symmetric torsion of a Cartan connection includes (Friedrich-Ivanov 01, theorem 4.7, theorem 4.8)

#### Weak $G_2$-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. 4 generalized to

$\mathbf{d} \omega = \lambda \star \omega$

and hence

$\mathbf{d} \star \omega = 0$

for $\lambda \in \mathbb{R}$. For $\lambda = 0$ this reduces to strict $G_2$-holonomy, by 4.

(See for instance (Bilal-Derendinger-Sfetsos 02, Bilal-Metzger 03).)

## Properties

### Existence

###### Proposition

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

That orientability and spinnability is necessary follows directly from the fact that $G_2 \hookrightarrow GL(7)$ is connected and simply connected. That these conditions are already sufficient is due to (Gray 69), see also (Bryant 05, remark 3).

### Metric structure

The canonical Riemannian metric $G_2$ manifold is Ricci flat. More generally a manifold of weak $G_2$-holonomy, def. 5, 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 G2-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 $\omega_2$
Calabi-Yau manifoldSU(k)$2k$
hyper-Kähler manifoldSp(k)$4k$$\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$)
quaternionic Kähler manifold$4k$$\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$
G2 manifoldG2$7$associative 3-form
Spin(7) manifoldSpin(7)8Cayley form

## References

### General

The concept goes back to

• E. Bonan, (1966), Sur les variétés riemanniennes à groupe d’holonomie G2 ou Spin(7), C. R. Acad. Sci. Paris 262: 127–129.

Non-compact $G_2$-manifolds were first constructed in

• Robert Bryant, ; S.M. Salamon, (1989), On the construction of some complete metrics with exceptional holonomy, Duke Mathematical Journal 58: 829–850.

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 (Euclid)

• Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000)

The sufficiency of spin structure for $G_2$-structure is due to

• A. Gray, Vector cross products on manifolds, Trans. Amer. Math. Soc. 141 (1969), 465–504.

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)

• Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)

• Robert Bryant, Some remarks on $G_2$-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 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.

### Variants and generalizations

Disucssion of the more general concept of Riemannian manifolds equipped with covariantly constant 3-forms is in

• Hong Van Le , Geometric structures associated with a simple Cartan 3-form, Journal of Geometry and Physics (2013) (arXiv:1103.1201)

### Relation to Killing spinors

Discussion of $G_2$-structures in view of the existence of Killing spinors includes

### 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)

• Andreas Brandhuber, Jaume Gomis, Steven S. Gubser, Sergei Gukov, Gauge Theory at Large N and New G2 Holonomy Metrics_, Nucl.Phys. B611 (2001) 179-204 (arXiv:hep-th/0106034)

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 (arXiv:hep-th/0302021)

• Thomas House, Andrei Micu, M-theory Compactifications on Manifolds with $G_2$ Structure (arXiv:hep-th/0412006)

For more on this see at M-theory on G2-manifolds

Revised on January 22, 2015 21:10:14 by Urs Schreiber (88.100.66.95)