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In differential geometry, a G₂ manifold or Joyce 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, and thus define special classes of 3- and 4-dimensional submanifolds.

Properties

All ${\displaystyle G_{2}}$-manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to ${\displaystyle G_{2}}$ has finite fundamental group, non-zero first Pontryagin class, and non-zero third and fourth Betti numbers.

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 nonetheless made a useful 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.

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. Next, complete (but still noncompact) 7-manifolds with holonomy ${\displaystyle G_{2}}$ were constructed by Bryant and Simon Salamon in 1989. The first compact 7-manifolds with holonomy ${\displaystyle G_{2}}$ were constructed by Dominic Joyce in 1994. Compact ${\displaystyle G_{2}}$ manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature. 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. In the same paper, it was shown that certain classes of ${\displaystyle G_{2}}$-manifolds admit a contact structure.

In 2015, a new construction of compact ${\displaystyle G_{2}}$ manifolds, due to Alessio Corti, Mark Haskins, Johannes Nordstrőm, and Tommaso Pacini, combined a gluing idea suggested by Simon Donaldson with new algebro-geometric and analytic techniques for constructing Calabi–Yau manifolds with cylindrical ends, resulting in tens of thousands of diffeomorphism types of new examples.

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. Recently it was shown that almost contact structures (constructed by Sema Salur et al.) play an important role in ${\displaystyle G_{2}}$ geometry".

See also

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

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, Robert L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360.
4. Bryant, Robert L.; Salamon, Simon M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal, 58 (3): 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 Journal of Mathematics, 17 (2): 321–334, arXiv:1112.2951, doi:10.4310/AJM.2013.v17.n2.a3, S2CID 54942812.
7. Corti, Alessio; Haskins, Mark; Nordström, Johannes; Pacini, Tommaso (2015), "G₂-manifolds and associative submanifolds via semi-Fano 3-folds" (PDF), Duke Mathematical Journal, 164 (10): 1971–2092, doi:10.1215/00127094-3120743, S2CID 119141666
8. de la Ossa, Xenia; Larfors, Magdalena; Magill, Matthew (2022), "Almost contact structures on manifolds with a G₂ structure", Advances in Theoretical and Mathematical Physics, 26 (1): 143–215, arXiv:2101.12605, doi:10.4310/atmp.2022.v26.n1.a3, MR 4504848

Further reading

• Becker, Katrin; Becker, Melanie; Schwarz, John H. (2007), "Manifolds with G₂ and Spin(7) holonomy", String Theory and M-Theory : A Modern Introduction, Cambridge University Press, pp. 433–455, ISBN 978-0-521-86069-7.
• Fernandez, M.; Gray, A. (1982), "Riemannian manifolds with structure group G₂", Ann. Mat. Pura Appl., 32: 19–845, doi:10.1007/BF01760975, S2CID 123137620.
• Karigiannis, Spiro (2011), "What Is . . . a G₂-Manifold?" (PDF), AMS Notices, 58 (4): 580–581.

• Differential geometry
• Riemannian geometry
• Structures on manifolds
• Octonions