nLab E11

Idea

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

… E6, E7, E8, E9, E10, $E_{11}$, …

Properties

As U-duality group of 0d 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-duality	maximal gauged supergravity
	$SL(2,\mathbb{R})$	1	$SL(2,\mathbb{Z})$ S-duality	10d 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 supergravity	E8-equivariant elliptic cohomology
E10	$E_{10(10)}$	$O(9,9;\mathbb{Z})$	$E_{10(10)}(\mathbb{Z})$
E11	$E_{11(11)}$	$O(10,10;\mathbb{Z})$	$E_{11(11)}(\mathbb{Z})$

(Hull-Townsend 94, table 1, table 2)

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

• level 0: momentum

• level 1: M2-brane charge

• level 2: M5-brane charge

• level 3: dual graviton charge (West 11, section N) (has two components ¹)

References

General

• Peter West, section 16.7 of Introduction to Strings and Branes

• Hermann Nicolai, Thomas Fischbacher, Low Level Representations for E10 and E11 (arXiv:hep-th/0301017)

• H. Mkrtchyan, R. Mkrtchyan, $E_{11},K_{11}$ and $EE_{11}$ (arXiv:hep-th/0407148)

Relation to supergravity

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

Review includes

• Peter West, section 17.5 of Introduction to Strings and Branes

• Fabio Riccioni, $E_{11}$ and M-theory, talk at Strings07 (pdf slides)

• Fabio Riccioni, Peter West, The $E_{11}$ origin of all maximal supergravities, JHEP 0707:063,2007 (arXiv:0705.0752, spire)

• Paul Cook, Connections between Kac-Moody algebras and M-theory PhD thesis (arXiv:0711.3498)

• Peter West, A brief review of E theory (arXiv:1609.06863)

Original articles include the following:

The observation that $E_{11}$ seems to neatly organize 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. (arXiv:hep-th/0104081)

A precursor to (West 01) is

• Bernard Julia, Dualities in the classical supergravity limits (arXiv:hep-th/9805083)

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 $E_{11}$ with its fundamental representation is due to

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

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

This way that elements of cosets of the semidirect product $E_{11}$ 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 $E_{11}$ formulation of M-theory include

• Peter West, $E_{11}$, ten forms and supergravity, JHEP0603:072,2006 (arXiv:hep-th/0511153)

• Fabio Riccioni, Peter West, Dual fields and $E_{11}$, Phys.Lett.B645:286-292,2007 (arXiv:hep-th/0612001)

• Fabio Riccioni, Peter West, E(11)-extended spacetime and gauged supergravities, JHEP 0802:039,2008 (arXiv:0712.1795)

• Fabio Riccioni, Duncan Steele, Peter West, The E(11) origin of all maximal supergravities - the hierarchy of field-strengths, JHEP 0909:095 (2009) (arXiv:0906.1177)

• Eric Bergshoeff, I. De Baetselier, T. Nutma, E(11) and the Embedding Tensor (arXiv:0705.1304, poster)

• Guillaume Bossard, Axel Kleinschmidt, Jakob Palmkvist, Christopher Pope, Ergin Sezgin, Beyond $E_{11}$ (arXiv:1703.01305)

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

• Axel Kleinschmidt, Peter West, Representations of $G^{+++}$ and the role of space-time, JHEP 0402 (2004) 033 (arXiv:hep-th/0312247)

• Paul Cook, Peter West, Charge multiplets and masses for $E(11)$, JHEP 11 (2008) 091 (arXiv:0805.4451)

• Peter West, $E_{11}$ origin of Brane charges and U-duality multiplets, JHEP 0408 (2004) 052 (arXiv:hep-th/0406150)

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

• Pierre Henry-Labordere, Bernard Julia, Louis Paulot, Borcherds symmetries in M-theory, JHEP 0204 (2002) 049 (arXiv:hep-th/0203070)

• Marc Henneaux, Bernard Julia, Jérôme Levie, $E_{11}$, Borcherds algebras and maximal supergravity (arxiv:1007.5241)

• Jakob Palmkvist, Tensor hierarchies, Borcherds algebras and $E_{11}$, JHEP 1202 (2012) 066 (arXiv:1110.4892)

On E11-exceptional field theory:

• Guillaume Bossard, Axel Kleinschmidt, Ergin Sezgin, On supersymmetric E11 exceptional field theory (arXiv:1907.02080)

• Guillaume Bossard, Axel Kleinschmidt, Ergin Sezgin, A master exceptional field theory (arXiv:2103.13411)

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1. private communication with Axel Kleinschmidt ↩