nLab
E7

Context

Group Theory

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

One of the exceptional Lie groups.

Properties

Representation

56\mathbf{56} – The smallest fundamental representation

The smallest fundamental representation of E 7E_7 is of dimension 5656. Under the special linear subgroup SL(8,)E 7SL(8,\mathbb{R}) \hookrightarrow E_7 this decomposes as (e.g. Cacciatori et al. 10, section 4, also Pacheco-Waldram 08, appendix B)

562828 * 2 8 2( 8) *. \mathbf{56} \simeq \mathbf{28} \oplus \mathbf{28}^\ast \simeq \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast \,.

Under the further subgroup inclusion SL(7,)SL(8,)E 7SL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R}) \hookrightarrow E_7 this decomposes further as

56 7 2( 7) * 5( 7) * 6 7. \mathbf{56} \simeq \mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast \oplus \wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7 \,.

As U-Duality group of 4d SuGra

E 7(7)E_{7(7)} 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 SL(7,)SL(7,\mathbb{R})-decomposition of the smallest fundamental representation

56 7 2( 7) * 5( 7) * 6 7. \mathbf{56} \simeq \mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast \oplus \wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7 \,.

as the generalized 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 7\mathbb{R}^7 is the ordinary tangent space itself, 2( *) 7\wedge^2 (\mathbb{R}^\ast)^7 is interpreted as the local incarnation of the possible M2-brane charges, 5( *) 7\wedge^5 (\mathbb{R}^\ast)^7 the M5-brane charges and 6 7\wedge^6 \mathbb{R}^7 as the charges of Kaluza-Klein monopoles.

supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
SL(2,)SL(2,\mathbb{R})1SL(2,)SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2SL(2,)× 2SL(2,\mathbb{Z}) \times \mathbb{Z}_29d supergravity
SU(3)×\times SU(2)SL(3,)×SL(2,)(3,\mathbb{R}) \times SL(2,\mathbb{R})O(2,2;)O(2,2;\mathbb{Z})SL(3,)×SL(2,)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})8d supergravity
SU(5)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
Spin(10)Spin(5,5)Spin(5,5)O(4,4;)O(4,4;\mathbb{Z})O(5,5,)O(5,5,\mathbb{Z})6d supergravity
E6E 6(6)E_{6(6)}O(5,5;)O(5,5;\mathbb{Z})E 6(6)()E_{6(6)}(\mathbb{Z})5d supergravity
E7E 7(7)E_{7(7)}O(6,6;)O(6,6;\mathbb{Z})E 7(7)()E_{7(7)}(\mathbb{Z})4d supergravity
E8E 8(8)E_{8(8)}O(7,7;)O(7,7;\mathbb{Z})E 8(8)()E_{8(8)}(\mathbb{Z})3d supergravity
E9E 9(9)E_{9(9)}O(8,8;)O(8,8;\mathbb{Z})E 9(9)()E_{9(9)}(\mathbb{Z})2d supergravityE8-equivariant elliptic cohomology
E10E 10(10)E_{10(10)}O(9,9;)O(9,9;\mathbb{Z})E 10(10)()E_{10(10)}(\mathbb{Z})
E11E 11(11)E_{11(11)}O(10,10;)O(10,10;\mathbb{Z})E 11(11)()E_{11(11)}(\mathbb{Z})

(Hull-Townsend 94, table 1, table 2)

References

General

  • wikipedia, E7

  • Sergio L. Cacciatori, Francesco Dalla Piazza, Antonio Scotti, E7 groups from octonionic magic square (arXiv:1007.4758)

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

and more generally in

The proposal to make this hidden E 7E_7-symmetry manifest via exceptional generalized geometry is due to

Further discussion includes

Revised on July 30, 2015 09:22:10 by Urs Schreiber (88.128.80.186)