G2 manifold

In differential geometry, a G₂ manifold is a seven-dimensional Riemannian manifold with holonomy group contained in 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 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.

Properties

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

• Ricci-flat,
• orientable,
• a spin manifold.

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

• Spin(7)-manifold
• Calabi–Yau manifold

References

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