Cohomology and Extensions
Formal Lie groupoids
The Lie group 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 of those elements that preserve the canonical differential 3-form on the Cartesian space . As such, the group is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any ), obtained by passing from symplectic geometry to 2-plectic geometry.
On the Cartesian space consider the associative 3-form, the constant differential 3-form given on tangent vectors by
\omega(u,v,w) \coloneqq \langle u , v \times w\rangle
Then the group is the subgroup of the general linear group acting on which preserves the canonical orientation and preserves this 3-form . Equivalently, it is the subgroup preserving the orientation and the Hodge dual differential 4-form .
See for instance the introduction of (Joyce).
The inclusion of def. 1 factors through the special orthogonal group
G_2 \hookrightarrow SL(7) \hookrightarrow GL(7)
Relation to higher prequantum geometry
The 3-form from def. 1 we may regard as equipping with 2-plectic structure. From this point of view is the linear subgroup of the 2-plectomorphism group, hence (up to the translations) the image of the Heisenberg group of in the symplectomorphism group.
Or, dually, we may regard the 4-form of def. 1 as being a 3-plectic structure and correspondingly as the linear part in the 3-plectomorphism group of .
G2 manifold, generalized G2-manifold
M-theory on G2-manifolds, G2-MSSM
E6, E7, E8, E9, E10, E11,
Surveys are in
The definitions are reviewed for instance in
- Dominic Joyce, Compact Riemannian 7-manifolds with holonomy , Journal of Differential Geometry vol 43, no 2 (pdf)
- The octonions and (pdf)
Discussion in terms of the Heisenberg group in 2-plectic geometry is in
Cohomological properties are discussed in
- Younggi Choi, Homology of the gauge group of exceptional Lie group , J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709
Applications in physics
Discussion of Yang-Mills theory with as gauge group is in
- Ernst-Michael Ilgenfritz, Axel Maas, Topological aspects of G2 Yang-Mills theory (arXiv:1210.5963)