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In differential geometry, a G₂ manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G₂. The group G2{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 ϕ{displaystyle phi }, the associative form. The Hodge dual, ψ=∗ϕ{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

If M is a G2{displaystyle G_{2}}-manifold, then M is:

• 
  Ricci-flat,


• 
  orientable,


• a spin manifold.

History

The fact that G2{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 then made an interesting 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 G2{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 G2{displaystyle G_{2}} were constructed by Bryant and Simon Salamon in 1989.^[4] The first compact 7-manifolds with holonomy G2{displaystyle G_{2}} were constructed by Dominic Joyce in 1994, and compact G2{displaystyle G_{2}} manifolds are 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 G2{displaystyle G_{2}}-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with G2{displaystyle G_{2}}-structure.^[6] In the same paper, it was shown that certain classes of G2{displaystyle G_{2}}-manifolds admit a contact structure.

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 G2{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 G2{displaystyle G_{2}} manifold and a number of U(1) vector supermultiplets equal to the second Betti number.

See also

• Spin(7)-manifold


• Calabi–Yau manifold

References

• 
  Bryant, R.L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, Annals of Mathematics, 126 (2): 525&ndash, 576, doi:10.2307/1971360, JSTOR 1971360.


• 
  M. Fernandez; A. Gray (1982), "Riemannian manifolds with structure group G2", Ann. Mat. Pura Appl., 32: 19&ndash, 845.


• 
  Karigiannis, Spiro (2011), "What Is . . . a G₂-Manifold?" (PDF), AMS Notices, 58 (04): 580–581.