Cohomology and Extensions
Formal Lie groupoids
One of the exceptional Lie groups.
Consider the vector space
of dimension . This is naturally a symplectic vector space with symplectic form given by the natural pairing between linear 2-forms? and bivectors.
In addition, consider on this space the quartic form which sends an element to
Now is the subgroup of the general linear group acting on which preserves both the symplectic form as well as the quartic form . See also below.
This presentation is due to Cartan, for review see Cremmer-Julia 79, appendix B, Pacheco-Waldram 08, B.1. A construction via octonions is due to (Freudenthal 54), one via quaternions is due to (Wilson 2014).
As part of the ADE pattern
– The smallest fundamental representation
The smallest fundamental representation of is the defining one (from the definition above), of dimension . Under the special linear subgroup this decomposes as (e.g. Cacciatori et al. 10, section 4, also Pacheco-Waldram 08, appendix B)
Under the further subgroup inclusion this decomposes further as
where is regarded as the subspace of 2-forms with vanishing 8-components, and where is the Poincaré dual to the complementary subspace of of 2-forms with non-trivial 8-component.
This is due to Cartan, for review see Cremmer-Julia 79, appendix B Pacheco-Waldram 08, B.1.
– The adjoint representation
The adjoint representation of decomposes under as (Pacheco-Waldram 08 (B.7))
In this decomposition the subspace corresponding to the subalgebra is the vector space
where the first summand denotes the skew-symmetric matrices, and the second summand the Hodge anti-self dual 4-forms (Pacheco-Waldram 08 (B.29) (B.30) and below (2.34)).
Under the full adjoint representation decomposes further into (Pacheco-Waldram 08 (B.21))
Here is the -component of and dually, while the -component carries no information by tracelessness; and is the -component of , while is the 7-dimensional Poincaré dual of the complement of the -component (Pacheco-Waldram 08 (B.22)).
Taken together this means that under the subspace is that spanned by
sums of a 3-form with its 8d-Hodge+7d-Poincaré-dual 3-vector;
sums of a 6-form with its dual 6-vector
Hence the tangent space to the coset may be identified as
As U-Duality group of 4d SuGra
is the U-duality group (see there) of 11-dimensional supergravity compactified on a 7-dimensional fiber to 4-dimensional supergravity (e.g. M-theory on G2-manifolds).
Specifically, (Hull 07, section 4.4, Pacheco-Waldram 08, section 2.2) identifies the vector space underlying the -decomposition of the smallest fundamental representation
as the exceptional tangent bundle-structure to the 7-dimensional fiber space which one obtains as discussed at M-theory supersymmetry algebra – As an 11-dimensional boundary condition. Here is the ordinary tangent space itself, is interpreted as the local incarnation of the possible M2-brane charges, the M5-brane charges and as the charges of Kaluza-Klein monopoles.
|supergravity gauge group (split real form)||T-duality group (via toroidal KK-compactification)||U-duality||maximal gauged supergravity|
|1|| S-duality||10d type IIB supergravity|
|SL O(1,1)||9d supergravity|
|SU(3) SU(2)||SL||8d supergravity|
|E9||2d supergravity||E8-equivariant elliptic cohomology|
(Hull-Townsend 94, table 1, table 2)
The description of the defining fundamental -representation of is due to
- Eli Cartan, Thesis, in Oeuvres complètes T1, Part I, Gauthier-Villars, Paris 1952
and recalled for instance in
A construction via the octonions is due to
- Hans Freudenthal, Beziehungen der e7 und e8 zur Oktavenebene, I, II, Indag. Math. 16 (1954), 218–230, 363–368. III, IV, Indag. Math. 17 (1955), 151–157, 277–285. V — IX, Indag. Math. 21 (1959), 165–201, 447–474. X, XI, Indag. Math. 25 (1963) 457–487
A quaternionic construction is given in
In view of U-duality
The hidden E7-U-duality symmetry of the KK-compactification of 11-dimensional supergravity on a 7-dimensional fiber to 4d supergravity was first noticed in (Cremmer-Julia 79) and then expanded on in
The proposal to make this hidden -symmetry manifest via exceptional generalized geometry is due to
Further discussion includes