G2 manifold

A "G"₂ manifold is a seven-dimensional Riemannian manifold with holonomy group "G"₂. The group $$G_2 is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of GL(7) which preserves a "positive, nondegenerate" 3-form, $$phi_0. The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in $$Lambda^3(Bbb R^7). The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature $$(3,4). Thus, $$G_2 is a subgroup of $$SO(7). By covariant transport, a manifold with holonomy $$G_2 has a Riemannian metric and a parallel (covariant constant) 3-form, $$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 Harvey-Lawson, and thus define special classes of 3 and 4 dimensional submanifolds, respectively. The deformation theory of such submanifolds was studied by McLean.

Properties

If "M" is a $$G_2-manifold, then "M" is:
* Ricci-flat
* orientable
* a spin manifold

History

The first complete, but noncompact 7-manifolds with holonomy $$G_2 were constructed by Bryant and Salamon in 1989. The first compact 7-manifolds with holonomy $$G_2 were constructed by Dominic Joyce in 1994, and compact $$G_2 manifolds are sometimes known as "Joyce manifolds", espcially in the physics literature.

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.

"See also": Calabi-Yau manifold, Spin(7) manifold

References

*citation | last = Bryant | first = R.L. | title = Metrics with exceptional holonomy | journal = Annals of Mathematics | issue = 2 | volume = 126 | year = 1987 | pages = 525–576.
*citation | last = Bryant | first = R.L. | first2 = S.M. | last2 = Salamon | title = On the construction of some complete metrics with exceptional holonomy | journal = Duke Mathematical Journal | volume = 58 | year = 1989 | pages = 829–850.
*citation | first = R. | last = Harvey | first2 = H.B. | last2 = Lawson | title = Calibrated geometries | journal = Acta Mathematica | volume = 148 | year = 1982 | pages = 47–157.
*citation | first = D.D. | last = Joyce | title = Compact Manifolds with Special Holonomy | series = Oxford Mathematical Monographs | publisher = Oxford University Press | isbn = 0-19-850601-5 | year = 2000.
*citation | first = R.C. | last = McLean | title = Deformations of calibrated submanifolds | journal = Communications in Analysis and Geometry | volume = 6 | year = 1998 | pages = 705–747.