In differential geometry, a G₂ 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 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.^[6] 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.^[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 ${\displaystyle G_{2}}$ 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.)^[6] play an important role in ${\displaystyle G_{2}}$ geometry".^[8]

See also

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

References

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.