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).)

When used as KK-compactification-fibers for M-theory on G2-manifolds, then for realistic phenomenology one needs to consider ADE orbifolds with “$G_2$-manifold” structure, i.e. $G_2$-orbifolds, also called Joyce orbifolds. Moreover, for F-theory purposes this $G_2$-orbifold is to be a fibration by a K3 surface $X_{K3}$.

For instance the Cartesian product $X_{K3} \times T^3$ admits a $G_2$-manifold structure. There is a canonical SO(3)-action on the tangent spaces of $X_{K3} \times T^3$, given on $X_{K3}$ by rotation of the hyper-Kähler manifold-structure of $X_{K_3}$ and on $T^3$ by the standard rotation. For $K_{ADE}$ a finite subgroup of $SO(3)$, hence a finite group in the ADE classification, then $(X_{K3}\times T^3)/K_{ADE}$ is a $G_2$-orbifold. (Acharya 98, p.3). (For $K_{ADE}$ not a cyclic group then this has precisely one parallel spinor.)

In a local coordinate chart of $X_{K3}$ by $\mathbb{C}^2$ the orbifold $X_{K3}/K_{ADE}$ locally looks like $\mathbb{C}^2/{G_{ADE}}$, where now $G_{ADE}$ is a finite subgroup of SU(2). Such local $G_2$-orbifolds are discussed in some detail in (Atiyah-Witten 01).

Codimension-4 ADE singularities in $G_2$-manifolds are discussed in (Barrett 06).

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

Discussion of $G_2$-orbifolds includes

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.

Moduli

Discussion of the moduli space of $G_2$-structures includes

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)

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)