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,^[1] 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 first introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat.^[2] The first complete, but noncompact 7-manifolds with holonomy $G_{2}$ were constructed by Robert Bryant and Salamon in 1989.^[3] 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.^[4] In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a $G_{2}$-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with $G_{2}$-structure.^[5] In the same paper, it was shown that certain classes of $G_{2}$-manifolds admit a contact structure.

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

1. Harvey, R.; Lawson, H.B. (1982), "Calibrated geometries", Acta Mathematica, 148: 47–157, doi:10.1007/BF02392726.
2. E. Bonan, (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris, 262: 127–129.
3. Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal, 58: 829–850, doi:10.1215/s0012-7094-89-05839-0.
4. Joyce, D.D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
5. Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on $G_{2}$-manifolds", Asian J. Math, International Press of Boston, 17 (2): 321–334, doi:10.4310/AJM.2013.v17.n2.a3.

• Bryant, R.L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, Annals of Mathematics, 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360.
• M. Fernandez; A. Gray (1982), "Riemannian manifolds with structure group G2", Ann. Mat. Pura Appl., 32: 19–845.
• Karigiannis, Spiro (2011), "What Is . . . a G₂-Manifold?" (PDF), AMS Notices, 58 (04): 580–581.