nLab
E11

Contents

Context

Exceptional structures

Group Theory

Lie theory

∞-Lie theory (higher geometry)

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 0d 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 (West 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.

Review includes

Original articles include the following:

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

A precursor to (West 01) is

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

The derivation of the equations of motion of 11-dimensional supergravity and maximally supersymmetric 5d supergravity from a vielbein with values in the semidirect product E 11E_{11} with its fundamental representation is due to

  • Peter West, Generalised geometry, eleven dimensions and E 11E_{11}, J. High Energ. Phys. (2012) 2012: 18 (arXiv:1111.1642)

  • Alexander G. Tumanov, Peter West, E 11E_{11} must be a symmetry of strings and branes, Physics Letters B Volume 759, 10 August 2016, Pages 663–671 (arXiv:1512.01644)

  • Alexander G. Tumanov, Peter West, E 11E_{11} in 11d11d, Physics Letters B Volume 758, 10 July 2016, Pages 278–285 (arXiv:1601.03974)

This way that elements of cosets of the semidirect product E 11E_{11} with its fundamental representation may encode equations of motion of 11-dimensional supergravity follows previous considerations for Einstein equations in

  • Abdus Salam, J. Strathdee, Nonlinear realizations. 1: The Role of Goldstone bosons, Phys. Rev. 184 (1969) 1750,

  • Chris Isham, Abdus Salam, J. Strathdee, Spontaneous, breakdown of conformal symmetry, Phys. Lett. 31B (1970) 300.

  • A. Borisov, V. Ogievetsky, Theory of dynamical affine and conformal symmetries as the theory of the gravitational field, Theor. Math. Phys. 21 (1973) 1179-1188 (web)

  • V. Ogievetsky, Infinite-dimensional algebra of general covariance group as the closure of the finite dimensional algebras of conformal and linear groups, Nuovo. Cimento, 8 (1973) 988.

Further developments of the proposed E 11E_{11} formulation of M-theory 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

On E11-exceptional field theory:


  1. private communication with Axel Kleinschmidt

Last revised on March 26, 2021 at 00:51:16. See the history of this page for a list of all contributions to it.