group theory

∞-Lie theory

# Contents

## Idea

One of the exceptional Lie groups.

## Definition

Consider the vector space

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

of dimension $56$. This is naturally a symplectic vector space with symplectic form $\omega$ given by the natural pairing between linear 2-forms? and bivectors.

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

$q(v) \coloneqq v^{a b } w_{b c} v^{c d} w_{d a} - \tfrac{1}{4} v^{a b} w_{a b} v^{c d} w_{c d} + \tfrac{1}{96} \left( \epsilon_{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} v^{a_1 a_2} v^{a_3 a_4} v^{a_5 a_6} v^{a_7 a_8} + \epsilon^{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8} w_{a_1 a_2} w_{a_3 a_4} w_{a_5 a_6} w_{a_7 a_8} \right) \,.$

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

Dynkin diagramPlatonic solidfinite subgroup of $SO(3)$simple Lie group
$A_l$cyclic groupspecial unitary group
$D_l$dihedron/hosohedrondihedral groupspecial orthogonal group
$E_6$tetrahedrontetrahedral groupE6
$E_7$cube/octahedronoctahedral groupE7
$E_8$dodecahedron/icosahedronicosahedral groupE8

### Representation

#### $\mathbf{56}$ – The smallest fundamental representation

The smallest fundamental representation of $E_7$ is the defining one (from the definition above), of dimension $56$. Under the special linear subgroup $SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes as (e.g. Cacciatori et al. 10, section 4, also Pacheco-Waldram 08, appendix B)

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

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

$\mathbf{56} \simeq \underset{\simeq \wedge^2 \mathbb{R}^8}{\underbrace{\mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast}} \oplus \underset{\simeq \wedge^2 (\mathbb{R}^8)^\ast}{\underbrace{\wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7}} \,,$

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

#### $\mathbf{133}$ – The adjoint representation

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

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

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

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

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

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

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

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

1. $\mathfrak{so}(7)$-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

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

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

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

### As U-Duality group of 4d SuGra

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

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

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 $\mathbb{R}^7$ is the ordinary tangent space itself, $\wedge^2 (\mathbb{R}^\ast)^7$ is interpreted as the local incarnation of the possible M2-brane charges, $\wedge^5 (\mathbb{R}^\ast)^7$ the M5-brane charges and $\wedge^6 \mathbb{R}^7$ 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,\mathbb{R})$1$SL(2,\mathbb{Z})$ S-duality10d type IIB supergravity
SL$(2,\mathbb{R}) \times$ O(1,1)$\mathbb{Z}_2$$SL(2,\mathbb{Z}) \times \mathbb{Z}_2$9d supergravity
SU(3)$\times$ SU(2)SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$$O(2,2;\mathbb{Z})$$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$8d supergravity
SU(5)$SL(5,\mathbb{R})$$O(3,3;\mathbb{Z})$$SL(5,\mathbb{Z})$7d supergravity
Spin(10)$Spin(5,5)$$O(4,4;\mathbb{Z})$$O(5,5,\mathbb{Z})$6d supergravity
E6$E_{6(6)}$$O(5,5;\mathbb{Z})$$E_{6(6)}(\mathbb{Z})$5d supergravity
E7$E_{7(7)}$$O(6,6;\mathbb{Z})$$E_{7(7)}(\mathbb{Z})$4d supergravity
E8$E_{8(8)}$$O(7,7;\mathbb{Z})$$E_{8(8)}(\mathbb{Z})$3d supergravity
E9$E_{9(9)}$$O(8,8;\mathbb{Z})$$E_{9(9)}(\mathbb{Z})$2d supergravityE8-equivariant elliptic cohomology
E10$E_{10(10)}$$O(9,9;\mathbb{Z})$$E_{10(10)}(\mathbb{Z})$
E11$E_{11(11)}$$O(10,10;\mathbb{Z})$$E_{11(11)}(\mathbb{Z})$
• G2, F4,

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

## References

### General

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

• 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

• Bernard de Wit, Hermann Nicolai, D = 11 Supergravity With Local SU(8) Invariance, Nucl. Phys. B 274, 363 (1986) (spire), Local SU(8) invariance in $d = 11$ supergravity (spire)

The proposal to make this hidden $E_7$-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)