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E7

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Contents

Idea

One of the exceptional Lie groups.

Definition

Consider the vector space

W 2 8 2( 8) * W \coloneqq \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast

of dimension 5656. This is naturally a symplectic vector space with symplectic form ω\omega given by the natural pairing between linear 2-forms? and bivectors.

In addition, consider on this space the quartic form q:Wq \colon W \to \mathbb{R} which sends an element v=({v ab,w ab})Wv = (\{v^{a b}, w_{a b}\}) \in W to

q(v)v abw bcv cdw da14v abw abv cdw cd+196(ϵ a 1a 2a 3a 4a 5a 6a 7a 8v a 1a 2v a 3a 4v a 5a 6v a 7a 8+ϵ a 1a 2a 3a 4a 5a 6a 7a 8w a 1a 2w a 3a 4w a 5a 6w a 7a 8). q(v) \coloneqq v^{a b } w_{b c} v^{c d} w_{d a} - \tfrac{1}{4} v^{a b} w_{a b} v^{c d} w_{c d} + \tfrac{1}{96} \left( \epsilon_{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} v^{a_1 a_2} v^{a_3 a_4} v^{a_5 a_6} v^{a_7 a_8} + \epsilon^{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} w_{a_1 a_2} w_{a_3 a_4} w_{a_5 a_6} w_{a_7 a_8} \right) \,.

Now E 7(7)GL(56,)E_{7(7)} \subset GL(56,\mathbb{R}) is the subgroup of the general linear group acting on WW which preserves both the symplectic form ω\omega as well as the quartic form qq. 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).

Properties

As part of the ADE pattern

ADE classification

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)simple Lie group
A lA_lcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupE7
E 8E_8dodecahedron/icosahedronicosahedral groupE8

Representation

56\mathbf{56} – The smallest fundamental representation

The smallest fundamental representation of E 7E_7 is the defining one (from the definition above), 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) * 2 8 5( 7) * 6 7 2( 8) *, \mathbf{56} \simeq \underset{\simeq \wedge^2 \mathbb{R}^8}{\underbrace{\mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast}} \oplus \underset{\simeq \wedge^2 (\mathbb{R}^8)^\ast}{\underbrace{\wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7}} \,,

where 2( 7) 2( 8) *\wedge^2 (\mathbb{R}^7) \subset \wedge^2 (\mathbb{R}^8)^\ast is regarded as the subspace of 2-forms with vanishing 8-components, and where 6 7\wedge^6 \mathbb{R}^7 is the Poincaré dual to the complementary subspace of 2( 8*)\wedge^2 (\mathbb{R}^8^\ast) 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.

133\mathbf{133} – The adjoint representation

The adjoint representation 133\mathbf{133} of E 7E_7 decomposes under SL(8,)SL(8,\mathbb{R}) as (Pacheco-Waldram 08 (B.7))

𝔢 7=133( 8( 8) *) traceless 4( 8) *. \mathfrak{e}_7 = \mathbf{133} \simeq (\mathbb{R}^8 \otimes (\mathbb{R}^8)^\ast)_{traceless} \oplus \wedge^4 (\mathbb{R}^8)^\ast \,.

In this decomposition the subspace corresponding to the subalgebra 𝔰𝔲(8)𝔢 8\mathfrak{su}(8) \hookrightarrow \mathfrak{e}_8 is the vector space

𝔰𝔲(8)𝔰𝔬(8)( 4( 8) *) , \mathfrak{su}(8) \simeq \mathfrak{so}(8) \oplus (\wedge^4 (\mathbb{R}^8)^\ast)_- \,,

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 GL(7,)SL(8,)GL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R}) the full adjoint representation decomposes further into (Pacheco-Waldram 08 (B.21))

133( 7( 7) *)( 6 7 6( 7) *)( 3 7 3( 7) *). \mathbf{133} \simeq \left(\mathbb{R}^7 \otimes (\mathbb{R}^7)^\ast\right) \oplus \left(\wedge^6 \mathbb{R}^7 \oplus \wedge^6 (\mathbb{R}^7)^\ast\right) \oplus \left( \wedge^3 \mathbb{R}^7 \oplus \wedge^3 (\mathbb{R}^7)^\ast \right) \,.

Here 6( 7) * 7\wedge^6 (\mathbb{R}^7)^\ast \simeq \mathbb{R}^7 is the (,8)(-,8)-component of 7( 7) *\mathbb{R}^7 \oplus (\mathbb{R}^7)^\ast and dually, while the (8,8)(8,8)-component carries no information by tracelessness; and 3( 7) *\wedge^3 (\mathbb{R}^7)^\ast is the (,,,8)(-,-,-,8)-component of 4( 8) *\wedge^4 (\mathbb{R}^8)^\ast, while 3 7\wedge^3 \mathbb{R}^7 is the 7-dimensional Poincaré dual of the complement of the (,,,8)(-,-,-,8)-component (Pacheco-Waldram 08 (B.22)).

Taken together this means that under GL(7,)GL(7,\mathbb{R}) the subspace 𝕤𝕦(8)𝔢 8\mathbb{su}(8) \hookrightarrow \mathfrak{e}_8 is that spanned by

  1. 𝔰𝔬(7)\mathfrak{so}(7)-elements;

  2. sums of a 3-form with its 8d-Hodge+7d-Poincaré-dual 3-vector;

  3. sums of a 6-form with its dual 6-vector

hence is

𝔰𝔲(8)𝔰𝔬(8) 3 7 6 7. \mathfrak{su}(8) \simeq \mathfrak{so}(8) \oplus \wedge^3 \mathbb{R}^7 \oplus \wedge^6 \mathbb{R}^7 \,.

Hence the tangent space to the coset E 7(7)/(SU(8)/ 2)E_{7(7)}/(SU(8)/\mathbb{Z}_2) may be identified as

𝔢 7/𝔰𝔲(8) 2( 7) * 3( 7) * 6( 7) *. \mathfrak{e}_7/\mathfrak{su}(8) \simeq \odot^2 (\mathbb{R}^7)^\ast \oplus \wedge^3 (\mathbb{R}^7)^\ast \oplus \wedge^6 (\mathbb{R}^7)^\ast \,.

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 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 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

The description of the defining fundamental 56\mathbf{56}-representation of E 7(7)E_{7(7)} 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

reviewed in

A quaternionic construction is given in

See also

  • wikipedia, E7

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 E 7E_7-symmetry manifest via exceptional generalized geometry is due to

Further discussion includes

Revised on August 20, 2015 09:35:03 by Urs Schreiber (131.227.23.37)