G2 manifold explained

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.) play an important role in G _{2} geometry".

See also

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

Further reading

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Notes and References

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7. Corti. Alessio. Alessio Corti. Haskins. Mark. Nordström. Johannes. Pacini. Tommaso . 2015. -manifolds and associative submanifolds via semi-Fano 3-folds. . 164. 10. 1971–2092. 10.1215/00127094-3120743. 119141666.