group theory

∞-Lie theory

# Contents

## Idea

A hyperbolic Kac-Moody Lie algebra in the E-series

E6, E7, E8, E9, E10, $E_{11}$, …

## Properties

### As U-duality group of 1d supergravity

$E_{11}$ is conjectured (West 01) to be the U-duality group (see there) of 11-dimensional supergravity compactified to 0 dimensions.

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})$

### Fundamental representation and brane charges

The first fundamental representation of $E_{11}$, usually denoted $l_1$, is argued, since (West 04), to contain in its decomposition into representations of $GL(11)$ the representations in which the charges of the M-branes and other BPS states transform.

According to (Nicolai-Fischbacher 03, first three rows of table 2 on p. 72, West 04, Kleinschmidt-West 04) and concisely stated for instance in (West 11, (2.17)), the level decomposition of $l_1$ under $GL(11)$ starts out as so:

$l_1 \simeq \underset{level\,0}{ \underbrace{ \mathbb{R}^{10,1} }} \oplus \underset{level\,1}{ \underbrace{ \wedge^2 (\mathbb{R}^{10,1})^\ast }} \oplus \underset{level\,2}{ \underbrace{ \wedge^5 (\mathbb{R}^{10,1})^\ast }} \oplus \underset{level\,3}{\underbrace{ \wedge^7 (\mathbb{R}^{10,1})^\ast \otimes_s (\mathbb{R}^{10,1})^\ast \oplus \wedge^8 (\mathbb{R}^{10,1})^\ast }} \oplus \cdots$

Here the $level \leq 2$-truncation happens to coincide with the bosonic body underlying the M-theory super Lie algebra and via the relation of that to BPS charges in 11-dimensional supergravity/M-theory, the direct summands here have been argued to naturally correspond to

## References

### Relation to supergravity

Literature discussing $E_{11}$ U-duality and in the context of exceptional generalized geometry of 11-dimensional supergravity.

Reviews include

Original articles include the following:

The observation that $E_{11}$ seems to neatly organite the structures in 11-dimensional supergravity/M-theory is due to

• Peter West, $E_{11}$ and M theory, Class. Quant. Grav., 18:4443–4460, 2001.

A precursor to West 01 is

as explained in (Henneaux-Julia-Levie 10).

Further developments include

Discussion of the semidirect product of $E_{11}$ with its $l_1$-representation, and arguments that the charges of the M-theory super Lie algebra and in fact further brane charges may be identified inside $l_1$ originate in

• Peter West, $E_{11}$, $SL(32)$ and Central Charges, Phys.Lett.B575:333- 342,2003 (arXiv:hep-th/0307098)

and was further explored in

Relation to exceptional field theory is discussed in

• Alexander G. Tumanov, Peter West, $E_{11}$ and exceptional field theory (arXiv:1507.08912)

Relation to Borcherds superalgebras is discussed in

1. private communication with Axel Kleinschmidt

Revised on August 24, 2015 02:58:51 by David Corfield (46.208.196.195)