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

E6, E7, E8, E9, E10, E 11E_{11}, …


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



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} (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

  1. private communication with Axel Kleinschmidt

Revised on March 7, 2017 15:18:27 by Urs Schreiber (