nLab
G2

Idea

The Lie group $G_2$ is one (or rather: three) of the exceptional Lie groups. One way to characterize it is as the automorphism group of the octonions. Another way to characterize it is as the subgroup of the general linear group $\mathrm{GL}(7)$ of those elements that preserve the canonical differential 3-form $⟨,(-)\times (-)⟩$ on the Cartesian space ${ℝ}^7$. As such, the group $G_2$ is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any ${ℝ}^{2n}$), obtained by passing from symplectic geometry to 2-plectic geometry.

Definition

Definition

On the Cartesian space ${ℝ}^7$ consider the associative 3-form, the constant differential 3-form $\omega \in {\Omega }^3({ℝ}^7)$ given on tangent vectors $u,v,w\in {ℝ}^7$ by

$$\omega (u,v,w)≔⟨u,v\times w⟩,$$\omega(u,v,w) \coloneqq \langle u , v \times w\rangle \,,

where

• $⟨-,-⟩$ is the canonical bilinear form

• $(-)\times (-)$ is the cross product of vectors.

Then the group $G_2\hookrightarrow \mathrm{GL}(7)$ is the subgroup of the general linear group acting on ${ℝ}^7$ which preserves the canonical orientation and preserves this 3-form $\omega$. Equivalently, it is the subgroup preserving the orientation and the Hodge dual differential 4-form $\star \omega$.

See for instance the introduction of (Joyce).

Properts

General

The inclusion $G_2\hookrightarrow \mathrm{GL}(7)$ of def. 1 factors through the special orthogonal group

$$G_2\hookrightarrow \mathrm{SL}(7)\hookrightarrow \mathrm{GL}(7).$$G_2 \hookrightarrow SL(7) \hookrightarrow GL(7) \,.

Relation to higher prequantum geometry

The 3-form $\omega$ from def. 1 we may regard as equipping ${ℝ}^7$ with 2-plectic structure. From this point of view $G_2$ is the linear subgroup of the 2-plectomorphism group, hence (up to the translations) the image of the Heisenberg group of $({ℝ}^7,\omega )$ in the symplectomorphism group.

Or, dually, we may regard the 4-form $\star \omega$ of def. 1 as being a 3-plectic structure and $G_2$ correspondingly as the linear part in the 3-plectomorphism group of ${ℝ}^7$.

Related concepts

• G2 manifold, generalized G2-manifold

• M-theory on G2-manifolds, G2-MSSM

• G2, F4,

  E6, E7, E8, E9, E10, E11, $\cdots$

References

General

Surveys are in

• Spiro Karigiannis, What is… a $G_2$-manifold (pdf)

• Wikipedia, G2 .

The definitions are reviewed for instance in

• Dominic Joyce, Compact Riemannian 7-manifolds with holonomy $G_2$, Journal of Differential Geometry vol 43, no 2 (pdf)

• The octonions and $G_2$ (pdf)

Discussion in terms of the Heisenberg group in 2-plectic geometry is in

• Alberto Ibort, Multisymplectic geometry: generic and exceptional, Proceedings of the IX Fall workshop on geometry and physics (pdf)

Cohomological properties are discussed in

• Younggi Choi, Homology of the gauge group of exceptional Lie group $G_2$, J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709

Applications in physics

Discussion of Yang-Mills theory with $G_2$ as gauge group is in

• Ernst-Michael Ilgenfritz, Axel Maas, Topological aspects of G2 Yang-Mills theory (arXiv:1210.5963)