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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 ${\displaystyle G_{2}}$ 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

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.[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. Compact ${\displaystyle G_{2}}$  manifolds are therefore sometimes known as "Joyce manifolds", especially in the physics literature.[5]

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.[6]

## 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, 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.
• 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.