G2 manifold
In differential geometry, a G_{2} manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G_{2}. The group G_2 is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of special orthogonal group SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of the general linear group GL(7) which preserves the non-degenerate 3-form \phi, the associative form. The Hodge dual, \psi=*\phi is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Harvey–Lawson, and thus define special classes of 3- and 4-dimensional submanifolds.
Contents
- Properties 1
- History 2
- Connections to physics 3
- See also 4
- References 5
Properties
If M is a G_2-manifold, then M is:
History
A manifold with holonomy G_2 was firstly introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat. The first complete, but noncompact 7-manifolds with holonomy G_2 were constructed by Robert Bryant and Salamon in 1989. The first compact 7-manifolds with holonomy G_2 were constructed by Dominic Joyce in 1994, and compact G_2 manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.
Connections to physics
These manifolds are important in string theory. They break the original supersymmetry to 1/8 of the original amount. For example, M-theory compactified on a G_2 manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry. The resulting low energy effective supergravity contains a single supergravity supermultiplet, a number of chiral supermultiplets equal to the third Betti number of the G_2 manifold and a number of U(1) vector supermultiplets equal to the second Betti number.
See also
References
- .
- Bryant, R.L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics (Annals of Mathematics) 126 (2): 525–576, .
- Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal 58: 829–850, .
- M. Fernandez; A. Gray (1982), "Riemannian manifolds with structure group G2", Ann. Mat. Pura Appl. 32: 19–845.
- Harvey, R.; Lawson, H.B. (1982), "Calibrated geometries", Acta Mathematica 148: 47–157, .
- Joyce, D.D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, .
- Karigiannis, Spiro (2011), -Manifold?"_{2}G"What Is . . . a (PDF), AMS Notices 58 (04): 580–581.