nLab E₉

Idea

One of the affine Kac-Moody groups in the E-series.

• E7, E6, E8, E9, E10, E11, $\cdots$

Properties

Representations of maximal compact subalgebra

In contrast to the full affine Lie algebra $\mathfrak{e}_{9(9)}$, the “maximal compact subalgebra” $\mathfrak{k}_{9(9)} \,\coloneqq\, Lie\big(K(E_{9(9)})\big)$ has non-trivial finite-dimensional representations [Nicolai & Samtleben 2005, KKLN22].

Specifically, under the restriction (“branching”) of the basic representation of $\mathfrak{e}_{9(9)}$ along the inclusion $\mathfrak{k}_{9(9)} \hookrightarrow \mathfrak{e}_{9(9)}$ the first (namely: parabolic level zero) $\mathfrak{k}$-irrep summand has dimension $2^{rk(\mathfrak{e}_8)} = 256$. [König 2024, top of p 38 and pp 41-42].

As U-duality group of 2d supergravity

$E_9$ is the U-duality group (see there) of 11-dimensional supergravity compactified to 2 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)

Related concepts

• exceptional Lie group

• exceptional geometry, exceptional generalized geometry

• U-duality group of 11-dimensional supergravity compactified to 2 dimensions.

References

General

(…)

As U-duality symmetry

Discussion of $E_9$ in view of U-duality-symmetry of D=11 supergravity reduced to $D=2$:

On finite-dimensional representations of the maximal compact subalgebra $\mathfrak{k} = Lie\big(K(E_9)\big)$:

• Hermann Nicolai, Henning Samtleben: On $K(E_9)$, Pure and Applied Mathematics Quarterly 1 1 (2005) 180–204 [arXiv:hep-th/0407055, doi:10.4310/PAMQ.2005.v1.n1.a8]

• Axel Kleinschmidt, Ralf Köhl, Robin Lautenbacher, Hermann Nicolai: Representations of involutory subalgebras of affine Kac-Moody algebras, Commun. Math. Phys. 392 (2022) 89–123 [arXiv:2102.00870, doi:10.1007/s00220-022-04342-9]

• Benedikt König: $\mathfrak{k}$-structure of basic representation of affine algebras [arXiv:2407.12748]

On BPS states of D=11 supergravity via the representation theory of $E_9$:

• François Englert, Laurent Houart, Axel Kleinschmidt, Hermann Nicolai, Nassiba Tabti, An E9 multiplet of BPS states, JHEP 0705:065 (2007) [arXiv:hep-th/0703285, doi:10.1088/1126-6708/2007/05/065]

On $E_9$-exceptional field theory-formulation of D=11 supergravity:

• Guillaume Bossard, Franz Ciceri, Gianluca Inverso, Axel Kleinschmidt, Henning Samtleben: $E_9$ exceptional field theory I. The potential J. High Energ. Phys. 2019 89 (2019) [arXiv:1811.04088, doi:10.1007/JHEP03(2019)089]

• Guillaume Bossard, Franz Ciceri, Gianluca Inverso, Axel Kleinschmidt, Henning Samtleben: $E_9$ exceptional field theory II. The complete dynamics, J. High Energ. Phys. 2021 107 (2021). [arXiv:2103.12118, doi:10.1007/JHEP05(2021)107]

Maximal compact subalgebra

On the maximal compact subalgebras of E9 (and E10) and their finite-dimensional linear representations:

• Hermann Nicolai, Henning Samtleben: On $K(E_9)$, Q. J. Pure Appl. Math. 1 (2005) 180-204 [arXiv:hep-th/0407055, doi:10.4310/PAMQ.2005.v1.n1.a8]

• Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist: $K(E_9)$ from $K(E_{10})$, Journal of High Energy Physics 2007 JHEP06 (2007) [arXiv:hep-th/0611314, doi:10.1088/1126-6708/2007/06/051]

• Axel Kleinschmidt, Hermann Nicolai: Generalised holonomies and $K(E_9)$, J. High Energ. Phys. 2021 54 (2021) [arXiv:2107.02445, doi:10.1007/JHEP09(2021)054]