G2 manifold

In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group ${\displaystyle G_{2}}$ 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 ${\displaystyle \phi }$, the associative form. The Hodge dual, ${\displaystyle \psi =*\phi }$ is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson,[1] and thus define special classes of 3- and 4-dimensional submanifolds.

Properties

If M is a ${\displaystyle G_{2}}$ -manifold, then M is:

History

The fact that ${\displaystyle G_{2}}$  might possibly be the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this remained consistent with the simplified proof later given by Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered, Edmond Bonan then made an interesting contribution by showing that, if such a manifold did in fact exist, it would carry both a parallel 3-form and a parallel 4-form, and that it would necessarily be Ricci-flat.[2] The first local examples of 7-manifolds with holonomy ${\displaystyle G_{2}}$  were finally constructed around 1984 by Robert Bryant, and his full proof of their existence appeared in the Annals in 1987 .[3] Next, complete (but still noncompact) 7-manifolds with holonomy ${\displaystyle G_{2}}$  were constructed by Bryant and Simon Salamon in 1989.[4] The first compact 7-manifolds with holonomy ${\displaystyle G_{2}}$  were constructed by Dominic Joyce in 1994, and compact ${\displaystyle G_{2}}$  manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.[5] In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a ${\displaystyle G_{2}}$ -structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with ${\displaystyle G_{2}}$ -structure.[6] In the same paper, it was shown that certain classes of ${\displaystyle G_{2}}$ -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 ${\displaystyle G_{2}}$  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 ${\displaystyle G_{2}}$  manifold and a number of U(1) vector supermultiplets equal to the second Betti number.

References

1. ^ Harvey, Reese; Lawson, H. Blaine (1982), "Calibrated geometries", Acta Mathematica, 148: 47–157, doi:10.1007/BF02392726, MR 0666108.
2. ^ Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", Comptes Rendus de l'Académie des Sciences, 262: 127–129.
3. ^ {Bryant, Rober L. (1987) Metrics with exceptional holonomy, Annals of Mathematics (2)126, 525–576.
4. ^ Bryant, Rober L.; Salamon, Simon 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, MR 1016448.
5. ^ Joyce, Dominic D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
6. ^ Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on ${\displaystyle G_{2}}$ -manifolds", Asian J. Math, International Press of Boston, 17 (2): 321–334, arXiv:1112.2951, 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.