A -structure on a manifold of dimension 7 is a choice of G-structure on , for the exceptional Lie group G2. Hence it is a reduction of the structure group of the frame bundle of along the canonical (the defining) inclusion into the general linear group.
Given that is the subgroup of the general linear group on the Cartesian space which preserves the associative 3-form on , a structre is a higher analog of an almost symplectic structure under lifting from symplectic geometry to 2-plectic geometry (Ibort).
A -manifold is a manifold equipped with -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.
-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.
(e.g. Bryant 05, definition 2)
Often it is useful to exhibit prop. 1 in the following way.
For a smooth manifold of dimension 7, write for its frame bundle. By the discussion at vielbein – in terms of basic forms on the frame bundle there is a universal -valued differential form on the total space of the frame bundle
is the corresponding local vielbein field. Hence one obtains a universal 3-form on the frame bundle by setting
with the canonical components of the associative 3-form and with summation over repeated indices understood.
Conversely, given a 3-form such that on an atlas over which the frame bundle trvializes it is of this form
then the -valued transition functions of the given local trivialization must factor through and hence exhibit a -structure: because we have and hence
But by the nature of the universal vielbein, its local pullbacks are related by
and hence (1) says that
which is precisely the defining condition for to take values in .
Viewed this way, the definite 3-forms characterizing -structures are an example of a more general kind of differential forms obtained from a constant form on some linear model space 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
for 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 -manifold structure as in def. 4.
The following is important for the analysis:
By definition of as the stabilizer group of the associative 3-form, the orbit it generates under the -action is the coset . The dimension of this as a smooth manifold is 49-14 = 35. This is however already the full dimension of the space of 3-forms in 7d that the orbit sits in. Therefore (since does not have a boundary) the orbit must be an open subset.
(e.g. Bryant 05, (4.31))
such that the globally defined 3-form is locally gauge equivalent to the canonical associative 3-form
via a 2-form on .
(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 with vanishing at that point such that at that point. But since the -action on is open, there is an open neighbourhood around that point where this is still the case.
where 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 as a map to this moduli stack, which when restricted to the cover becomes gauge equivalent to the pullback of the associative 3-form , similarly regarded as a map, to the cover, where the gauge equivalence is exhibited by a homotopy (of maps of formal smooth -groupoids) which is the 2-form on .
A manifold equipped with a -structure, def. 1, is called a -manifold if the following equivalent conditions hold
has special holonomy .
The higher torsion invariants of -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 -manifold, def. 4, does not, in general admit an atlas be adapted coordinate charts equal to .
The space of second order torsion invariants of -structures is for instance in (Bryant 05 (4.7)).
A 7-dimensional manifold is said to be of weak -holonomy if it carries a 3-form with the relation of def. 4 generalized to
for . For this reduces to strict -holonomy, by 4.
When used as KK-compactification-fibers for M-theory on G2-manifolds, then for realistic phenomenology one needs to consider ADE orbifolds with “-manifold” structure, i.e. -orbifolds, also called Joyce orbifolds. Moreover, for F-theory purposes this -orbifold is to be a fibration by a K3 surface .
For instance the Cartesian product admits a -manifold structure. There is a canonical SO(3)-action on the tangent spaces of , given on by rotation of the hyper-Kähler manifold-structure of and on by the standard rotation. For a finite subgroup of , hence a finite group in the ADE classification, then is a -orbifold. (Acharya 98, p.3). (For not a cyclic group then this has precisely one parallel spinor.)
Codimension-4 ADE singularities in -manifolds are discussed in (Barrett 06).
That orientability and spinnability is necessary follows directly from the fact that is connected and simply connected. That these conditions are already sufficient is due to (Gray 69), see also (Bryant 05, remark 3).
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.
|G-structure||special holonomy||dimension||preserved differential form|
|Kähler manifold||U(k)||Kähler forms|
|quaternionic Kähler manifold|
|G2 manifold||G2||associative 3-form|
|Spin(7) manifold||Spin(7)||8||Cayley form|
The concept goes back to
Non-compact -manifolds were first constructed in
Compact -manifolds were first found in
Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000)
The sufficiency of spin structure for -structure is due to
Discussion of -orbifolds includes
Adam B. Barrett, M-Theory on Manifolds with Holonomy, 2006 (arXiv:hep-th/0612096)
(but beware of some mistakes in that article…)
For more see the references at exceptional geometry.
Discussion of the moduli space of -structures includes
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
Disucssion of the more general concept of Riemannian manifolds equipped with covariantly constant 3-forms is in
Discussion of -structures in view of the existence of Killing spinors includes
The following references discuss the role of -manifolds in M-theory on G2-manifolds:
Weak -holonomy is discussed in
Thomas House, Andrei Micu, M-theory Compactifications on Manifolds with Structure (arXiv:hep-th/0412006)
For more on this see at M-theory on G2-manifolds