G₂ manifold

In differential geometry, a G₂ manifold or Joyce 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 Reese Harvey and H. Blaine Lawson,^[1] and thus define special classes of 3- and 4-dimensional submanifolds.

Properties

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

History

The fact that $G_{2}$ 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.^[2]

The first local examples of 7-manifolds with holonomy $G_{2}$ 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 $G_{2}$ were constructed by Bryant and Simon Salamon in 1989.^[4] The first compact 7-manifolds with holonomy $G_{2}$ were constructed by Dominic Joyce in 1994. Compact $G_{2}$ manifolds are therefore 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 $G_{2}$ -structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with $G_{2}$ -structure.^[6] In the same paper, it was shown that certain classes of $G_{2}$ -manifolds admit a contact structure.

In 2015, a new construction of compact $G_{2}$ 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.^[7]

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. Recently it was shown that almost contact structures (constructed by Sema Salur et al.)^[6] play an important role in $G_{2}$ geometry".^[8]

References

^ 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, 126 (2): 525–576, doi:10.2307/1971360, JSTOR 1971360.
^ 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.
^ Joyce, Dominic D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
^ ^a ^b Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on $G_{2}$ -manifolds", Asian Journal of Mathematics, 17 (2): 321–334, arXiv:1112.2951, doi:10.4310/AJM.2013.v17.n2.a3, S2CID 54942812.
^ Corti, Alessio; Haskins, Mark; Nordström, Johannes; Pacini, Tommaso (2015), " $G 2$ -manifolds and associative submanifolds via semi-Fano 3-folds" (PDF), Duke Mathematical Journal, 164 (10): 1971–2092, doi:10.1215/00127094-3120743, S2CID 119141666
^ de la Ossa, Xenia; Larfors, Magdalena; Magill, Matthew (2022), "Almost contact structures on manifolds with a $G 2$ structure", Advances in Theoretical and Mathematical Physics, 26 (1): 143–215, arXiv:2101.12605, doi:10.4310/atmp.2022.v26.n1.a3, MR 4504848

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G₂ manifold

Properties

History

Connections to physics

See also

References

Further reading