group theory

∞-Lie theory

Contents

Idea

One of the exceptional Lie groups.

Properties

Representation

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

The smallest fundamental representation of $E_7$ is 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

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

As U-Duality group of 4d SuGra

$E_{7(7)}$ 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 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 $\mathbb{R}^7$ 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

• 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

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