nLab E₁₁

Contents

Context

Exceptional structures

Group Theory

Lie theory

Idea
Properties

As U-duality group of 0d supergravity
Fundamental representation and brane charges
Temporal involutory subalgebra

References

General
Relation to supergravity

Idea

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

E₆, E₇, E₈, E₉, E₁₀, E₁₁, $\ldots$

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	D=10 type IIB supergravity
	SL $(2,\mathbb{R}) \times$ O(1,1)	$\mathbb{Z}_2$	$SL(2,\mathbb{Z})$ $\times \mathbb{Z}_2$	D=9 supergravity
SU(3) $\times$ SU(2)	SL $(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$O(2,2;\mathbb{Z})$	$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$	D=8 supergravity
SU(5)	$SL(5,\mathbb{R})$	$O(3,3;\mathbb{Z})$	$SL(5,\mathbb{Z})$	D=7 supergravity
Spin(10)	$Spin(5,5)$	$O(4,4;\mathbb{Z})$	$O(5,5,\mathbb{Z})$	D=6 supergravity
E₆	$E_{6(6)}$	$O(5,5;\mathbb{Z})$	$E_{6(6)}(\mathbb{Z})$	D=5 supergravity
E₇	$E_{7(7)}$	$O(6,6;\mathbb{Z})$	$E_{7(7)}(\mathbb{Z})$	D=4 supergravity
E₈	$E_{8(8)}$	$O(7,7;\mathbb{Z})$	$E_{8(8)}(\mathbb{Z})$	D=3 supergravity
E₉	$E_{9(9)}$	$O(8,8;\mathbb{Z})$	$E_{9(9)}(\mathbb{Z})$	D=2 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)

Temporal involutory subalgebra

In contrast to $\mathfrak{e}_{11(11)}$ itself, its temporal “involutory subalgebra” $\mathfrak{k}_{1,10}$ (Englert & Houart 2004 §2.2, the $H_{11}$ of Keurentjes 2004) has non-trivial finite-dimensional representations (KKLN22),

In particular there is:

an irrep $\mathbf{32} \,\in\, Rep_{\mathbb{R}}( \mathfrak{k}_{1,10} )$ which when restricted to the sub Lie algebra $\mathfrak{so}_{1,10} \hookrightarrow \mathfrak{k}_{11(11)}$ is the usual Majorana spinor representation $\mathbf{32}$ of D=11 supergravity [Bossard, Kleinschmidt & Sezgin 2019 p 42]
an irrep $\mathbf{352} \,\in\, Rep_{\mathbb{R}}( \mathfrak{k}_{1,10} )$ which when restricted to $\mathfrak{k}_{10(10)}$ branches to $\mathbf{320} \oplus \mathbf{32}$ , the first summand now corresponding to the $\Gamma$ -traceless tensor-spinor irrep of $\mathfrak{so}_{1,10}$ [BKS19, p 42]
an irrep $\mathbf{528}$ of $\mathfrak{k}_{1,10}$ [Gomis, Kleinschmidt & Palmkvist 2019 p 29, BKS19, appendix D] and this is the symmetric power of the spinor rep [Bossard, Kleinschmidt & Sezgin 2019 Apdx D]

$\mathbf{32} \otimes_{sym} \mathbf{32} \;\simeq\; \mathbf{528} \;\; \in \; Rep_{\mathbb{R}}(\mathfrak{k}_{1,10}) \,.$

References

General

Peter West, section 16.7 of Introduction to Strings and Branes
Hermann Nicolai, Thomas Fischbacher: Low Level Representations for $E_{10}$ and $E_{11}$ , in Kac-Moody Lie Algebras and Related Topics [arXiv:hep-th/0301017, doi:10.1090/conm/343]
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 and in view of M-theory.

Peter West: $E_11$ and M Theory, Class. Quant. Grav. 18 (2001) 4443-4460 [arXiv:hep-th/0104081, doi:10.1088/0264-9381/18/21/305]

Review:

Peter West, section 17.5 of Introduction to Strings and Branes, Cambridge University Press (2012) [doi:10.1017/CBO9781139045926]
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, King’s College London (2007) [arXiv:0711.3498, webpage]
Peter West: A brief review of E theory, in: Memorial Volume on Abdus Salam’s 90th Birthday, World Scientific (2017) 135-176 [arXiv:1609.06863, doi:10.1142/9789813144873_0009]
Keith Glennon: An Overview of the $E_{11}$ Program, talk at OIST, Okinawa (2024) [pdf, pdf]

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)

On the encoding of spacetime signature(s) in $\mathfrak{e}_{11}$ :

Arjan Keurentjes: $E_{11}$ : Sign of the times, Nucl. Phys. B 697 (2004) 302-318 [arXiv:hep-th/0402090, doi:10.1016/j.nuclphysb.2004.06.058]

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 temporal modification of the Cartan involution needed to define $\mathfrak{k}_{1,10}$ :

François Englert, Laurent Houart: $\mathcal{G}^{+++}$ Invariant Formulation of Gravity and M-Theories: Exact BPS Solutions, JHEP01(2004)002 [arXiv:hep-th/0311255, doi:10.1088/1126-6708/2004/01/002]

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:

Peter West: Generalised geometry, eleven dimensions and $E_{11}$ , J. High Energ. Phys. 2012 (2012) 18 [arXiv:1111.1642, doi:10.1007/JHEP02(2012)018]
Alexander G. Tumanov, Peter West: $E_{11}$ must be a symmetry of strings and branes, Physics Letters B 759 10 (2016) 663-671 [arXiv:1512.01644, doi:10.1016/j.physletb.2016.06.011]
Alexander G. Tumanov, Peter West: $E_{11}$ in $11d$ , Physics Letters B 758 (2016) 278285 [arXiv:1601.03974, doi:10.1016/j.physletb.2016.04.058]

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 $E_{11}$ , JHEP 05 (2017) 020 [doi:10.1007/JHEP05(2017)020, 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. B 575 (2003) 333-342 [arXiv:hep-th/0307098, doi:10.1016/j.physletb.2003.09.059]

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:

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

Relation to Borcherds superalgebras:

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 $E_{11}$ exceptional field theory, J. High Energ. Phys. 2019 165 (2019) [arXiv:1907.02080, doi:10.1007/JHEP10(2019)165]
Guillaume Bossard, Axel Kleinschmidt, Ergin Sezgin, A master exceptional field theory (arXiv:2103.13411)

On spinor representatons of the maximal compact subalgebra $\mathfrak{k}_{11(11)}$ :

Axel Kleinschmidt, Hermann Nicolai: IIA and IIB spinors from $K(E_{10})$ , Phys. Lett. B 637 (2006) 107-112 [arXiv:hep-th/0603205, doi:10.1016/j.physletb.2006.04.007]
Thibault Damour, Axel Kleinschmidt, Hermann Nicolai: Hidden symmetries and the fermionic sector of eleven-dimensional supergravity, Phys. Lett. B 634 (2006) 319-324 [arXiv:hep-th/0512163, doi:10.1016/j.physletb.2006.01.015]
Guillaume Bossard, Axel Kleinschmidt, Ergin Sezgin: On supersymmetric $E_{11}$ exceptional field theory, J. High Energ. Phys. 2019 165 (2019) [arXiv:1907.02080, doi:10.1007/JHEP10(2019)165]

Relation to free super Lie algebras:

Joaquim Gomis, Axel Kleinschmidt, Jakob Palmkvist: Symmetries of M-theory and free Lie superalgebras, J. High Energ. Phys. 2019 160 (2019) [arXiv:1809.09171, doi:10.1007/JHEP03(2019)160]