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 $G_2$ structure 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. 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.
The definition of $G_2$-manifold stuctures proceeds in stages:
and then there are
For $X$ a smooth manifold of dimension $7$ a $G_2$-structure on $X$ is a G-structure for $G =$ G2 $\hookrightarrow GL(7)$.
Since the inclusion of $G_2$ into $GL(7)$ factors through $SO(6)$ (see here), a $G_2$-structure induces 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. in terms of differential forms:
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
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)
A $G_2$-structure on $X$, def. , is equivalently a choice of definite 3-form $\sigma$ on $X$, def. .
(e.g. Joyce 00, p. 243, Bryant 05, section 3.1.1)
Often it is useful to exhibit prop. in the following way.
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
(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
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
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. , restricts to the pullback of $\phi_u$ via a section $\sigma_i \colon U_i \to Fr(X)$ and hence is of the form
Conversely, given a 3-form $\sigma \in \Omega^3(X)$ such that on an atlas $(U_i \to X)$ over which the frame bundle trivializes it is of this form
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
But by the nature of the universal vielbein, its local pullbacks are related by
i.e.
and hence (1) says that
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 constrained to be locally of the form
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. .
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:
The subset $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast$ in def. is an open subset, hence $\phi$ is a stable form (e.g. Hitchin, def. 1.1).
(e.g. Joyce 00, p. 243, Bryant 05, 2.8)
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.
A $G_2$-structure, def. , is called closed if the definite 3-form $\sigma$ corresponding to it via prop. is a closed differential form, $\mathbf{d}\sigma = 0$.
(e.g. Bryant 05, (4.31))
For a closed $G_2$-structure, def. , on a manifold $X$ there exists an atlas by open subsets
such that the globally defined 3-form $\sigma \in \Omega^3_+(X)$ is locally gauge equivalent to the canonical associative 3-form $\phi$
via a 2-form $\beta$ on $U$.
(e.g. Bryant 05, p. 21)
This follows from the fact, remark , 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.
When regarding smooth manifolds in the wider context of higher differential geometry, then the situation of prop. corresponds to a diagram of formal smooth infinity-groupoids of the following form:
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 $U$.
A manifold $X$ equipped with a $G_2$-structure, def. , is called a $G_2$-manifold if the following equivalent conditions hold
we have
$\mathbf{d} \omega = 0$ (closed)
$\mathbf{d} \star_g \omega = 0$ (co-closed);
$\nabla^g \omega = 0$;
$(X,g)$ has special holonomy $Hol(g) \subset G_2$;
$Ric(g) = 0$ (vanishing Ricci curvature);
$R(g) = 0$ (vanishing scalar curvature);
$\tau = 0$ (vanishing torsion of the G2-structure).
Here
$d$ is the de Rham differential;
$\omega$ is the 3-form $\omega$ corresponding to the given $G_2$-structure via prop. ;
$g$ is the induced Riemannian metric, via remark ;
$\star_g$ is the Hodge star operator of this metric;
$\nabla^g$ is the covariant derivative of this metric;
For the equivalence of the first items see for instance Joyce 96, p. 294 (4 of 38), Joyce 00, prop. 10.1.3. For the equivalence to the vanishing curvature invariant see also Bryant 05, corollary 1, and for the equivalence to the vanishing torsion of a G-structure see Bryant 05, prop. 2.
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. , a $G_2$-manifold, def. , does not, in general admit an atlas by adapted coordinate charts equal to $(\mathbb{R}^7, \phi)$.
The space of second order torsion invariants of $G_2$-structures is discussed by Bryant 05 (4.7).
There are several variants of the definition of $G_2$-manifolds, def., given by imposing other constraints on the torsion.
On a kind of instantons on $G_2$-manifolds:
Discussion for totally skew symmetric torsion of a Cartan connection includes Friedrich & Ivanov 01, theorem 4.7, theorem 4.8.
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. generalized to
and hence
for $\lambda \in \mathbb{R}$. For $\lambda = 0$ this reduces to strict $G_2$-holonomy, by .
(See for instance Bilal & Derendinger-Sfetsos 02Bilal & Metzger 03](#BilalMetzger03).)
When used as KK-compactification-fibers for M-theory on G2-manifolds, then for quadi-realistic phenomenology one needs to consider ADE orbifolds with “$G_2$-manifold” structure, i.e. G2-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 G2-orbifold (Acharya 1098, 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 G2-orbifolds are discussed in some detail by Atiyah & Witten 2001. Families of examples are constructed in Reidegeld 2015.
Codimension-4 ADE singularities in $G_2$-manifolds are discussed by Acharya & Gukov 2004, section 5.1 and Barrett 2006.
A 7-manifold admits a $G_2$-structure, def. , 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).
The canonical Riemannian metric $G_2$ manifold is Ricci flat. More generally a manifold of weak $G_2$-holonomy, def. , with weakness parameter $\lambda$ is an Einstein manifold with cosmological constant $\lambda$.
classification of special holonomy manifolds by Berger's theorem:
$\,$G-structure$\,$ | $\,$special holonomy$\,$ | $\,$dimension$\,$ | $\,$preserved differential form$\,$ | |
---|---|---|---|---|
$\,\mathbb{C}\,$ | $\,$Kähler manifold$\,$ | $\,$U(n)$\,$ | $\,2n\,$ | $\,$Kähler forms $\omega_2\,$ |
$\,$Calabi-Yau manifold$\,$ | $\,$SU(n)$\,$ | $\,2n\,$ | ||
$\,\mathbb{H}\,$ | $\,$quaternionic Kähler manifold$\,$ | $\,$Sp(n).Sp(1)$\,$ | $\,4n\,$ | $\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$ |
$\,$hyper-Kähler manifold$\,$ | $\,$Sp(n)$\,$ | $\,4n\,$ | $\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$) | |
$\,\mathbb{O}\,$ | $\,$Spin(7) manifold$\,$ | $\,$Spin(7)$\,$ | $\,$8$\,$ | $\,$Cayley form$\,$ |
$\,$G2 manifold$\,$ | $\,$G2$\,$ | $\,7\,$ | $\,$associative 3-form$\,$ |
$\;$normed division algebra$\;$ | $\;\mathbb{A}\;$ | $\;$Riemannian $\mathbb{A}$-manifolds$\;$ | $\;$special Riemannian $\mathbb{A}$-manifolds$\;$ |
---|---|---|---|
$\;$real numbers$\;$ | $\;\mathbb{R}\;$ | $\;$Riemannian manifold$\;$ | $\;$oriented Riemannian manifold$\;$ |
$\;$complex numbers$\;$ | $\;\mathbb{C}\;$ | $\;$Kähler manifold$\;$ | $\;$Calabi-Yau manifold$\;$ |
$\;$quaternions$\;$ | $\;\mathbb{H}\;$ | $\;$quaternion-Kähler manifold$\;$ | $\;$hyperkähler manifold$\;$ |
$\;$octonions$\;$ | $\;\mathbb{O}\;$ | $\;$Spin(7)-manifold$\;$ | $\;$G2-manifold$\;$ |
(Leung 02)
Spin(8)-subgroups and reductions to exceptional geometry
reduction | from spin group | to maximal subgroup |
---|---|---|
Spin(7)-structure | Spin(8) | Spin(7) |
G2-structure | Spin(7) | G2 |
CY3-structure | Spin(6) | SU(3) |
SU(2)-structure | Spin(5) | SU(2) |
generalized reduction | from Narain group | to direct product group |
generalized Spin(7)-structure | $Spin(8,8)$ | $Spin(7) \times Spin(7)$ |
generalized G2-structure | $Spin(7,7)$ | $G_2 \times G_2$ |
generalized CY3 | $Spin(6,6)$ | $SU(3) \times SU(3)$ |
see also: coset space structure on n-spheres
The metric cone over complex projective 3-space carries the structure of a G2-manifold whose Riemannian metric is invariant under the canonical Sp(2) action by left-matrix multiplication on homomogeneous coordinates in $\mathbb{H}^2 \simeq_{\mathbb{R}} \mathbb{C}^4 \to \mathbb{C}P^3$ (Byant-Salamon 89, see also Acharya-Bryant-Salamon 20).
by resolution of singularities in compact flat orbifolds
(…)
by twisted connected sum-constructions
graphics grabbed from Klemm 16
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.
General survey:
Spiro Karigiannis, What is… a $G_2$-manifold (pdf)
Spiro Karigiannis, $G_2$-manifolds – Exceptional structures in geometry arising from exceptional algebra (pdf)
Spiro Karigiannis, $G_2$-Conifolds: A survey, 2014 (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 (arXiv:math/0305124, webpage, pdf)
The concept of $G_2$-manifolds goes back to
Non-compact $G_2$-manifolds were first constructed in
See also:
Compact$\;G_2$-manifolds were first found in:
Dominic Joyce, Compact Riemannian 7-manifolds with holonomy $G_2$, Journal of Differential Geometry 43 2 (1996) [Euclid:jdg/1214458109]
Dominic Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs (2000) [ISBN:9780198506010]
Review in:
Dominic Joyce, Compact Riemannian manifolds with exceptional holonomy, Surveys in Differential Geometry
Volume 6 (2001) (doi:10.4310/SDG.2001.v6.n1.a3)
The sufficiency of spin structure for $G_2$-structure is due to:
and the compact twisted connected sum G2-manifolds due to:
Review includes
and from the point of view of M-theory on G2-manifolds:
More compact examples are constructed in
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.
On G2-conifolds (G2-manifolds with conical singularities):
Survey:
Spiro Karigiannis, $G_2$-conifolds: A survey, 2014 (pdf)
Mark Haskins, Exotic Einstein metrics on $S^6$ and $S^3 \times S^3$ nearly Kähler 6-manifolds and $G_2$-holonomy cones, 2016 (pdf)
Three simply connected $G_2$-cones are known: the
metric cone on $S^3 \times S^3$
metric cone on $SU(3) / (U(1) \times U(1))$
Robert Bryant, Simon Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. Volume 58, Number 3 (1989), 829-850 (euclid:euclid.dmj/1077307681)
Gary Gibbons, Don Page, Christopher Pope, Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 (euclid:cmp/1104180218)
More on the metric cone over complex projective 3-space as a G2-manifold:
Discussion of G2-orbifolds includes
Bobby Acharya, M theory, Joyce Orbifolds and Super Yang-Mills, Adv.Theor.Math.Phys. 3 (1999) 227-248 (arXiv:hep-th/9812205)
Michael Atiyah, Edward Witten $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)
Bobby Acharya, Sergei Gukov, M theory and Singularities of Exceptional Holonomy Manifolds, Phys.Rept.392:121-189,2004 (arXiv:hep-th/0409191)
Adam B. Barrett, M-Theory on Manifolds with $G_2$ Holonomy, 2006 (arXiv:hep-th/0612096)
Frank Reidegeld, $G_2$-orbifolds from K3 surfaces with ADE-singularities (arXiv:1512.05114, spire:1409963, doi:10.17877/DE290R-18940)
Frank Reidegeld, K3 surfaces with a pair of commuting non-symplectic involutions (arXiv:1809.07501)
Bobby Acharya, Andreas Braun, Eirik Eik Svanes, Roberto Valandro, Counting Associatives in Compact $G_2$ Orbifolds (arXiv:1812.04008)
Daniel Platt, Existence of torsion-free $G_2$-structures on resolutions of $G_2$-orbifolds using weighted Hölder norms (arXiv:2011.00482)
Discussion of the moduli space of $G_2$-structures:
Sergey Grigorian, Shing-Tung Yau, Local geometry of the $G_2$ moduli space, Commun.Math.Phys.287:459-488,2009 (arXiv:0802.0723)
Spiro Karigiannis, Naichung Conan Leung, Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps, Proceedings of the London Mathematical Society (3) 99, 297-325 (2009) [arXiv:0709.2987, talk slides pdf]
Relating to moduli of flat connections on tori:
Discussion of the more general concept of Riemannian manifolds equipped with covariantly constant 3-forms is in
Discussion of $G_2$-structures in view of the existence of Killing spinors includes
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
See also
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
An analysis of the de Rham complex of $G_2$ manifolds, and an analogue of Dolbeault cohomology is in
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