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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 ⊕ 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 φ on R⁷. If we identify R⁷ with the imaginary octonions, this form is given is terms of octonion multiplication by

$\varphi (x,y,z)=\langle xy,z\rangle .$

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 φ on M by taking φ to be φ₀ in any local trivialization. This is well-defined since the form φ₀ 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.

References

• Bryant, R.L. (1987), "Metrics with exceptional holonomy", Annals of Mathematics, 126 (2): 525–576.
• Bryant, R.L.; Salamon, S.M. (1989), "On the construction of some complete metrics with exceptional holonomy", Duke Mathematical Journal, 58: 829–850.
• Harvey, R.; Lawson, H.B. (1982), "Calibrated geometries", Acta Mathematica, 148: 47–157.
• Joyce, D.D. (2000), Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-850601-5.
• McLean, R.C. (1998), "Deformations of calibrated submanifolds", Communications in Analysis and Geometry, 6: 705–747.