G2 manifold

In differential geometry, 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, $\phi_0$. 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 $SO(7)$. By covariant transport, a manifold with holonomy $G_2$ has a Riemannian metric and a parallel (covariant constant) 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, respectively. The deformation theory of such submanifolds was studied by McLean.

Properties

If M is a $G_2$-manifold, then M is:

• Ricci-flat
• orientable
• a spin manifold

History

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: Calabi-Yau manifold, Spin(7) manifold

References

• E. Bonan, (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris 262: 127–129 .
• Bryant, R.L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics (Annals of Mathematics) 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360 .
• 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, doi:10.1007/BF02392726 .
• 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 .
• Karigiannis, Spiro (2011), "What Is . . . a G₂-Manifold?", AMS Notices 58 (04): 580–581, http://www.ams.org/notices/201104/rtx110400580p.pdf .