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 an “integrable” or “parallel” -structure. 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.
For a smooth manifold of dimension a -structure on is a choice of differential 3-form such that there is an atlas over which this 3-form locally identifies with the associative 3-form on the Cartesian space .
G_2 \hookrightarrow GL(7) \,.
A manifold equipped with a -structure , def. 1, is called a -manifold if is “parallel” or “integrable” in that
For instance (Joyce, p. 4).
There is a useful weakened notion of -holonomy.
A 7-dimensional manifold is said to be of weak -holonomy if it carries a 3-form with the relation of def. 2 generalized to
d \omega = \lambda \star \omega
d \star \omega = 0
for . For this reduces to strict -holonomy, by 2.
(See for instance (Bilal-Derendinger-Sfetsos).)
In string phenomenology models obtained from compactification of 11-dimensional supergravity/M-theory on -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|
|G2 manifold||G2||associative 3-form|
|Spin(7) manifold||Spin(7)||8||Cayley form|
Compact -manifolds were first found in
Spiro Karigiannis, What is… a -manifold (pdf)
Spiro Karigiannis, -manifolds – Exceptional structures in geometry arising from exceptional algebra (pdf)
(but beware of some mistakes in that article…)
For more see the references at exceptional geometry.
The following references discuss the role of -manifolds in M-theory on G2-manifolds:
Weak -holonomy is discussed in