In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group equal to G2. The group is one of the five exceptional simple Lie groups, and can be described as the automorphism group of the octonions. Equivalently, as a proper subgroup of the double cover Spin(7) of special orthogonal group SO(7), G2 is the subgroup that preserves some non-zero spinor. Finally, G2 is also the subgroup of the general linear group GL(7) which preserves z non-degenerate 3-form , called the associative form; the Hodge dual, is then an invariant 4-form, called the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson, and thus define special classes of 3- and 4-dimensional submanifolds of a G2 manifold.
If M is a -manifold, then M is:
If, in addition, M is compact, then:
- the third and fourth Betti numbers of M are non-zero, while
- the first and sixth Betti numbers of M both vanish.
The fact that might possibly arise as the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, and this possibility remained consistent with the simplified proof of Berger's theorem given by Jim Simons in 1962. Although not a single example of such a manifold had yet been discovered, Edmond Bonan showed in 1966 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. A local existence result for 7-manifolds of holonomy was finally given around 1984 by Robert Bryant, and the full details of his proof, which makes heavy use of the theory of exterior differential systems, appeared in the Annals in 1987. Next, complete and explicit (but still noncompact) 7-manifolds with holonomy were constructed by Bryant and Simon Salamon in 1989. The first compact 7-manifolds with holonomy were constructed by Dominic Joyce in 1994, and compact manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature. A more recent construction due to Corti, Haskins, Nordström, and Pacini, following a suggestion by Simon Donaldson and building on earlier work by Alexei Kovalev, shows that thousands of compact 7-manifolds actually admit metrics of holonomy .
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 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 manifold and a number of U(1) vector supermultiplets equal to the second Betti number.
- Harvey, Reese; Lawson, H. Blaine (1982), "Calibrated geometries", Acta Mathematica, 148: 47–157, doi:10.1007/BF02392726, MR 0666108.
- 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.
- Bryant, Robert L. (1987) "Metrics with exceptional holonomy," Annals of Mathematics (2)126, 525–576.
- 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.
- Joyce, Dominic D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
- Corti, Alessio; Haskins, Mark; Nordström, Johannes; and Pacini, Tommaso (2015) "-manifolds and associative submanifolds via semi-Fano 3-folds," Duke Mathematical Journal 164, 1971–2092
- Bryant, Robert L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, 2, 126 (3): 525–576, doi:10.2307/1971360, JSTOR 1971360, MR 0916718.
- Fernández, Marisa; Gray, Alfred (1982), "Riemannian manifolds with structure group ", Annali di Matematica Pura ed Applicata, 4, 32: 19–45, doi:10.1007/BF01760975, MR 0696037.
- Karigiannis, Spiro (2011), "What Is . . . a G2-Manifold?" (PDF), AMS Notices, 58 (04): 580–581.