nLab
G2 manifold

Contents

Idea

A G 2G_2-structure on a manifold XX of dimension 7 is a choice of G-structure on XX, for GG the exceptional Lie group G2. Hence it is a reduction of the structure group of the frame bundle of XX along the canonical (the defining) inclusion G 2GL( 7)G_2 \hookrightarrow GL(\mathbb{R}^7) into the general linear group.

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

A G 2G_2-manifold is a manifold equipped with G 2G_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 2G_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 2G_2-structure

Definition

For XX a smooth manifold of dimension 77 a G 2G_2-structure on XX is a G-structure for G=G = G2 GL(7)\hookrightarrow GL(7).

Remark

A G 2G_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 7\mathbb{R}^7, there is accordingly an equivalent formulation of def. 1 in terms of differential forms:

Definition

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

Ω + 3(X)Ω 3(X) \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 Λ + 3( 7) *\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 XX.

(e.g. Bryant 05, definition 2)

Proposition

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

(e.g. Joyce 00, p. 243, Bryant 05, section 3.1.1)

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

Example

For XX a smooth manifold of dimension 7, write Fr(X)XFr(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 7\mathbb{R}^7-valued differential form on the total space of the frame bundle

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

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

E iσ i *i *E u E_i \coloneqq \sigma_i^\ast i^\ast E_u

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

ϕ uϕ abcE u aE u bE u c \phi_u \coloneqq \phi_{a b c} E_u^a \wedge E_u^b \wedge E_u^c

with (ϕ abc)(\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 iX)(U_i \subset X) any definite 3-form, def. 2, restricts to the pullback of ϕ u\phi_u via a section σ i:U iFr(X)\sigma_i \colon U_i \to Fr(X) and hence is of the form

ϕ abcE i aE i bE i c. \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c \,.

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

σ| U i=ϕ abcE i aE i bE i c \sigma|_{U_i} = \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c

then the GL(7)GL(7)-valued transition functions g ijg_{i j} of the given local trivialization must factor through G 2SO(7)GL(7)G_2\hookrightarrow SO(7) \hookrightarrow GL(7) and hence exhibit a G 2G_2-structure: because we have σ| U i=σ| U jonU iU j\sigma|_{U_i} = \sigma|_{U_j} \;\;\; on \;U_i \cap U_j and hence

(1)ϕ abcE i aE i bE i c=ϕ abcE j aE j bE j conU iU j. \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 ijE i E_j = g_{i j} E_i

i.e.

E j a=(g ij) a bE i b E_j^a = (g_{i j})^a{}_b E_i^b

and hence (1) says that

ϕ abc=ϕ abc(g ij) a a(g ij) b b(g ij) c conU iU j \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 ijg_{i j} to take values in G 2G_2.

Viewed this way, the definite 3-forms characterizing G 2G_2-structures are an example of a more general kind of differential forms obtained from a constant form on some linear model space VV 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

Γ abαβE i aE i bE i αE i β \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 α)=(E a,Ψ α)(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 2G_2-manifold structure as in def. 4.

References that write definite 3-forms in this form locally as ϕ abcE aE bE c\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 Λ + 3( 7) *Λ 3( 7) *\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast in def. 2 is an open subset, hence ϕ\phi is a stable form (e.g. Hitchin, def. 1.1).

(e.g. Joyce 00, p. 243, Bryant 05, 2.8)

Proof

By definition of G 2G_2 as the stabilizer group of the associative 3-form, the orbit it generates under the GL +(7)GL_+(7)-action is the coset GL +(7)/G 2GL_+(7)/G_2. The dimension of this as a smooth manifold is 49-14 = 35. This is however already the full dimension (73)=35\left(7 \atop 3\right) = 35 of the space of 3-forms in 7d that the orbit sits in. Therefore (since G +(7)/G 2G_+(7)/G_2 does not have a boundary) the orbit must be an open subset.

Closed G 2G_2-structure

Definition

A G 2G_2-structure, def. 1, is called closed if the definite 3-form σ\sigma corresponding to it via prop. 1 is a closed differential form, dσ=0\mathbf{d}\sigma = 0.

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

Proposition

For a closed G 2G_2-structure, def. 3, on a manifold XX there exists an atlas by open subsets

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

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

σ| U=f *ϕ+dβ \sigma|_U = f^\ast \phi + \mathbf{d}\beta

via a 2-form β\beta on UU.

(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 β\beta with dβ\mathbf{d}\beta vanishing at that point such that σ| Uf *ϕ+dβ\sigma|_U \simeq f^\ast \phi + \mathbf{d}\beta at that point. But since the GL(7)GL(7)-action on ϕ\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:

U f 7 β X ϕ σ dRB 3, \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 dRB 3\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 σ\sigma as a map to this moduli stack, which when restricted to the cover UU becomes gauge equivalent to the pullback of the associative 3-form ϕ\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 β\beta on UU.

G 2G_2-holonomy / G 2G_2-manifold

Definition

A manifold XX equipped with a G 2G_2-structure, def. 1, is called a G 2G_2-manifold if the following equivalent conditions hold

  1. we have

    1. dω=0\mathbf{d} \omega = 0;

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

  2. gω=0\nabla^g \omega = 0;

  3. (X,g)(X,g) has special holonomy Hol(g)G 2Hol(g) \subset G_2.

Here

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

Proposition

The conditions on a G 2G_2-structure in def. 1 are equivalent to the torsion of the G 2G_2-structure to vanish.

(Bryant 05, prop. 2).

Remark

The higher torsion invariants of G 2G_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 2G_2-manifold, def. 4, does not, in general admit an atlas be adapted coordinate charts equal to ( 7,ϕ)(\mathbb{R}^7, \phi).

The space of second order torsion invariants of G 2G_2-structures is for instance in (Bryant 05 (4.7)).

Variants and weakenings

There are several variants of the definition of G 2G_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 2G_2-holonomy

Definition

A 7-dimensional manifold is said to be of weak G 2G_2-holonomy if it carries a 3-form ω\omega with the relation of def. 4 generalized to

dω=λω \mathbf{d} \omega = \lambda \star \omega

and hence

dω=0 \mathbf{d} \star \omega = 0

for λ\lambda \in \mathbb{R}. For λ=0\lambda = 0 this reduces to strict G 2G_2-holonomy, by 4.

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

Properties

Existence

Proposition

A 7-manifold admits a G 2G_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 2GL(7)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 2G_2 manifold is Ricci flat. More generally a manifold of weak G 2G_2-holonomy, def. 5, with weakness parameter λ\lambda is an Einstein manifold with cosmological constant λ\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)2k2kKähler forms ω 2\omega_2
Calabi-Yau manifoldSU(k)2k2k
hyper-Kähler manifoldSp(k)4k4kω=aω 2 (1)+bω 2 (2)+cω 2 (3)\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2 (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
quaternionic Kähler manifold4k4kω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3
G2 manifoldG277associative 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 2G_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 2G_2-manifolds were first found in

  • Dominic Joyce, Compact Riemannian 7-manifolds with holonomy G 2G_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 2G_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 2G_2-manifold (pdf)

  • Spiro Karigiannis, G 2G_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 2G_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 2G_2-structures in view of the existence of Killing spinors includes

Application in supergravity

The following references discuss the role of G 2G_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 2G_2-manifolds at the CERN Large Hadron collider and in the Galaxy, talk at G 2G_2-days (2012) (pdf)

See also

  • 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 2G_2-holonomy is discussed in

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)