G2 manifold

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The correct title of this article is G₂ manifold. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

A G₂ manifold is a seven-dimensional Riemannian manifold with holonomy group G₂. 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 SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of GL(7) which preserves a positive, nondegenerate 3-form, $φ 0$ . The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in $\Lambda^3(\Bbb R^7)$ . The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature $(3,4)$ . Thus, $G 2$ is a subgroup of $S O (7)$ . By covariant transport, a manifold with holonomy $G 2$ has a Riemannian metric and a parallel (covariant constant) 3-form, $φ$ , the associative form. The Hodge dual, $ψ = * φ$ 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, respectively. The deformation theory of such submanifolds was studied by McLean.

1 Properties
2 History
3 Connections to physics
4 References

[edit] Properties

If M is a $G 2$ -manifold, then M is:

[edit] History

The first complete, but noncompact 7-manifolds with holonomy $G 2$ were constructed by Bryant and Salamon in 1989. The first compact 7-manifolds with holonomy $G 2$ were constructed by Dominic Joyce in 1994.

[edit] 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.

[edit] References

Bryant, R.L. (1987), “Metrics with exceptional holonomy”, 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 .
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, ISBN 0-19-850601-5 .
McLean, R.C. (1998), “Deformations of calibrated submanifolds”, Communications in Analysis and Geometry 6: 705–747 .