G2 manifold

Summary

In differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group 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 , the associative form. The Hodge dual, 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

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

History

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The fact that   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   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   were constructed by Bryant and Simon Salamon in 1989.[4] The first compact 7-manifolds with holonomy   were constructed by Dominic Joyce in 1994. Compact   manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature.[5]

In 2015, a new construction of compact   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.[6]

Connections to physics

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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.

See also

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References

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  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. ^ 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.

Further reading

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  • 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.