nLab
E11

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

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

E6, E7, E8, E9, E10, E 11E_{11}, …

Properties

As U-duality group of 1d supergravity

E 11E_{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,)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)

Fundamental representation and brane charges

The first fundamental representation of E 11E_{11}, usually denoted l 1l_1, is argued, since (West 04), to contain in its decomposition into representations of GL(11)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 (Wess 11, (2.17)), the level decomposition of l 1l_1 under GL(11)GL(11) starts out as so:

l 1 10,1level0 2( 10,1) *level1 5( 10,1) *level2 7( 10,1) * s( 10,1) * 8( 10,1) *level3 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 level2level \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

General

Relation to supergravity

Literature discussing E 11E_{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 11E_{11} seems to neatly organite the structures in 11-dimensional supergravity/M-theory is due to

  • Peter West, E 11E_{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 11E_{11} with its l 1l_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 1l_1 originate in

and was further explored in

Relation to exceptional field theory is discussed in

Relation to Borcherds superalgebras is discussed in


  1. private communication with Axel Kleinschmidt

Revised on August 3, 2015 17:02:16 by Urs Schreiber (147.231.89.193)