User:Fropuff/Drafts/G2 manifold

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In differential geometry, a G2-manifold is a seven-dimensional Riemannian manifold with holonomy group G2. The group G2 is one of two exceptional cases appearing in Berger's list of possible holonomy groups of (irreducible, nonsymmetric) Riemannian manifolds, the other being Spin(7). These two groups are therefore referred to as the exceptional holonomy groups.

G2-manifolds and their 8-dimensional cousins, Spin(7)-manifolds, are sometimes called Joyce manifolds after Dominic Joyce who constructed the first compact examples.

The group G2
The group G2 is one of the five exceptional Lie groups. It is a compact, simply-connected, simple Lie group of dimension 14. The smallest nontrivial irreducible representation of G2 is 7-dimensional. It is this representation which is important in Riemannian geometry. There are numerous concrete ways to describe the group G2. We list three below which are important for an understanding of G2-manifolds.

The group G2 can be described as the automorphism group of the octonions. The octonions form a 8-dimensional normed division algebra over the reals, so G2 is naturally a subgroup of GL(8,R). Moreover, G2 must preserve the (positive-definite) norm on the octonions so G2 lies in SO(8). This representation of G2 is reducible, however, since it leaves invariant the decomposition O = R &oplus; Im(O). The representation of G2 on the purely imaginary octonions is faithful and irreducible and so gives G2 as a subgroup of SO(7). This is the 7-dimensional fundamental representation of G2.

Secondly, G2 can be given as the subgroup of GL(7,R) which preserves a certain 3-form &phi; on R7. If we identify R7 with the imaginary octonions, this form is given is terms of octonion multiplication by
 * $$\varphi(x,y,z) = \lang xy, z\rang.$$

Here x, y, and z are imaginary octonions and the bracket is the inner product on O. This form is closely related to the seven-dimensional cross product. The stabilizer group of this form necessarily preserves the standard inner product and orientation on R7 so that  G2 is, again, a subgroup of SO(7).

Finally, G2 can be described as a the subgroup of Spin(7) that preserves a spinor in the eight-dimensional spin representation of Spin(7).

G2-structures
Let M be a 7-dimensional smooth manifold. A G2-structure on M is a G-structure on M for G = G2. That is, it is a given reduction of the structure group of the tangent frame bundle of M from GL(7,R) to G2. Since G2 lies in SO(7), a G2-structure on M naturally determines a Riemannian metric and orientation on M. Moreover, since G2 is simply-connected any G2-structure determines a natural spin structure on M.

A smooth 7-manifold admits a G2-structure if and only if it is spin. For a given spin structure, the compatible G2-structures are in one-to-one correspondence with the global spinor fields on M with unit norm.

A G2-structure on M is equivalent to a choice of a certain "nondegenerate" 3-form on M. If M is equipped with a G2-structure then we can construct a 3-form &phi; on M by taking &phi; to be &phi;0 in any local trivialization. This is well-defined since the form &phi;0 is invariant under the action of G2.

G2-manifolds
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.

$$G_2$$ manifolds are Ricci-flat, see Bryant. The first complete, but noncompact 7-manifold with holonomy $$G_2$$ were constructed by Bryant and Salamon.

Applications in string theory and M-theory
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.