# G2 manifold

In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group ${\displaystyle G_{2}}$ 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 ${\displaystyle \phi }$, the associative form. The Hodge dual, ${\displaystyle \psi =*\phi }$ is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Harvey–Lawson,[1] and thus define special classes of 3- and 4-dimensional submanifolds.

## Properties

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

## History

A manifold with holonomy ${\displaystyle G_{2}}$ was first introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat.[2] The first complete, but noncompact 7-manifolds with holonomy ${\displaystyle G_{2}}$ were constructed by Robert Bryant and Salamon in 1989.[3] The first compact 7-manifolds with holonomy ${\displaystyle G_{2}}$ were constructed by Dominic Joyce in 1994, and compact ${\displaystyle G_{2}}$ manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.[4] In 2013, it was shown by M. Firat Arikan, Hyunjoo Cho, and Sema Salur that any manifold with a spin structure, and, hence, a ${\displaystyle G_{2}}$-structure, admits a compatible almost contact metric structure, and an explicit compatible almost contact structure was constructed for manifolds with ${\displaystyle G_{2}}$-structure.[5] In the same paper, it was shown that certain classes of ${\displaystyle G_{2}}$-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 ${\displaystyle G_{2}}$ 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 ${\displaystyle G_{2}}$ manifold and a number of U(1) vector supermultiplets equal to the second Betti number.

5. ^ Arikan, M. Firat; Cho, Hyunjoo; Salur, Sema (2013), "Existence of compatible contact structures on ${\displaystyle G_{2}}$-manifolds", Asian J. Math, International Press of Boston, 17 (2): 321–334, doi:10.4310/AJM.2013.v17.n2.a3.