nLab E₁₀

Idea

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

• E₆, E₇, E₈, E₉, E₁₀, E₁₁, $\cdots$

Properties

Maximal compact subalgebra

In contrast to $\mathfrak{e}_{10(10)}$ itself, its “maximal compact subalgebra” $\mathfrak{k}_{10(10)}$ has non-trivial finite dimensional representations (Kleinschmidt, Nicolai & Vigano 2020, KKLN22).

Among these is in particular a spinor representation $\mathbf{32}$ and a vector-spinor representation $\mathbf{320}$

$$\mathbf{32} ,\, \mathbf{320} \;\in\; Rep^{fdim}_{\mathbb{R}}\big(\mathfrak{k}_{10(10)}\big)$$

akin to the familiar reps of $\mathfrak{so}(10)$ of the same name/dimension [deBuyl, Henneaux & Paulot 2005 §8, Kleinschmidt & Nicolai 2006].

Remarkably, the symmetrized tensor product of this spinorial $\mathbf{32}$ with itself decomposes as a 1-dimensional trivial representation with a 527-dimensional irrep:

$$\mathbf{32} \otimes_{sym} \mathbf{32} \;\; \simeq \;\; \mathbf{1} \oplus \mathbf{527} \;\;\;\;\; \in \;\; Rep^{fdim}_{\mathbb{R}}\big( \mathfrak{k}_{10(10)} \big) \,.$$

(Damour, Kleinschmidt & Nicolai 2006 p 37).

As U-duality group of 1d M-theory

$E_{10}$ is conjectured (e.g. Nicolai 08) to be the U-duality group (see there) of M-theory compactified to 1 dimension (see also F/M-theory on elliptically fibered Calabi-Yau 5-folds).

	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)

References

General

Lecture notes:

• Hermann Nicolai (notes by Oliver Schlotterer): Infinite dimensional symmetries, lecture at Saalburg summer school in Wolfersdorf, Thuringia (2009) [pdf]

The fact that every simply laced hyperbolic Kac-Moody algebra is a sub Lie algebra of $E_{10}$:

• Sankaran Viswanath: Embeddings of hyperbolic Kac-Moody algebras into $E_{10}$ [arXiv:0801.2586, doi:10.1007/s11005-007-0214-7, doi:10.1007/s11005-007-0214-7]

See also:

• 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]

• Thomas Fischbacher: The structure of $E_{10}$ at higher $A_9$ levels – a first algorithmic approach, JHEP 0508:012 (2005) [arXiv:hep-th/0504230, doi:10.1088/1126-6708/2005/08/012]

Relation to supergravity

Discussion of a 1d sigma-model on $E_{10}/K(E_{10})$ as U-duality-covariant formuation of 11D supergravity/M-theory:

For bosonic degrees of freedom:

• Thibault Damour, Marc Henneaux, Hermann Nicolai: $E(10)$ and a “small tension expansion” of M theory, Phys. Rev. Lett. 89 221601 (2002) [arXiv:hep-th/0207267, doi:10.1103/PhysRevLett.89.221601]

• Axel Kleinschmidt, Hermann Nicolai: $E(10)$ and $SO(9,9)$ invariant supergravity, JHEP 0407,

  041 (2004) [arXiv:hep-th/0407101, doi:10.1088/1126-6708/2004/07/041]

• Thibault Damour, Hermann Nicolai: Eleven dimensional supergravity and the $E_{10}/K(E_{10})$ sigma-model at low $A_9$ levels, in Proceedings of the XXV International Colloquium on Group Theoretical Methods in Physics, 2-6 August 2004, Cocoyoc, Mexico, Routledge (2005) 93-111 [arXiv:hep-th/0410245, ISBN:9780750310086]

• Axel Kleinschmidt, Hermann Nicolai: $E_{10}$ Cosmology, JHEP 01 (2006) 137 [arXiv:hep-th/0511290, doi:10.1088/1126-6708/2006/01/137]

  (cosmological solutions)

• Marc Henneaux, Mauricio Leston, Daniel Persson, Philippe Spindel: Geometric Configurations, Regular Subalgebras of E10 and M-Theory Cosmology, JHEP 0610:021 (2006) [arXiv:hep-th/0606123, doi:10.1088/1126-6708/2006/10/021]

  (cosmological solutions)

• Marc Henneaux, Ella Jamsin, Axel Kleinschmidt, Daniel Persson: On the $E_{10}$/Massive Type IIA Supergravity Correspondence, Phys. Rev. D 79 (2009) 045008 [arXiv:0811.4358, doi:10.1103/PhysRevD.79.045008]

  (relation to massive type IIA string theory)

• Thibault Damour, Hermann Nicolai: Higher order M theory corrections and the Kac-Moody algebra $E_{10}$, Class. Quant. Grav. 22 (2005) 2849-2880 [arXiv:hep-th/0504153, doi:10.1088/0264-9381/22/14/003]

  (relating to higher curvature corrections)

• Thibault Damour, Axel Kleinschmidt, Hermann Nicolai: Constraints and the $E_{10}$ Coset Model, Class. Quant. Grav. 24 (2007) 6097-6120 [arXiv:0709.2691, doi:10.1088/0264-9381/24/23/025]

• Eric A. Bergshoeff, Olaf Hohm, Axel Kleinschmidt, Hermann Nicolai, Teake A. Nutma, Jakob Palmkvist: $E_{10}$ and Gauged Maximal Supergravity, JHEP 0901:020 (2009) [doi:10.1088/1126-6708/2009/01/020, arXiv:0810.5767]

  (relation to D=3 gauged supergravity)

• Thibault Damour, Axel Kleinschmidt, Hermann Nicolai: Sugawara-type constraints in hyperbolic coset models, Commun. Math. Phys. 302 (2011) 755-788 [arXiv:0912.3491, doi:10.1007/s00220-011-1188-y]

• Axel Kleinschmidt, Hermann Nicolai: The $E_{10}$ Wheeler-DeWitt operator at low levels, J. High Energ. Phys. 2022 92 (2022) [arXiv:2202.12676, doi:10.1007/JHEP04(2022)092]

  (relation to the Wheeler-DeWitt equation for 11D supergravity)

and for fermionic degrees of freedom

• S. de Buyl, Marc Henneaux, L. Paulot: Extended $E_8$ Invariance of 11-Dimensional Supergravity, Journal of High Energy Physics 2006 JHEP02 (2006) [arXiv:hep-th/0512292, doi:10.1088/1126-6708/2006/02/056]

• 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: $K(E_{10})$, Supergravity and Fermions, JHEP 0608:046 (2006) [arXiv:hep-th/0606105, doi:10.1088/1126-6708/2006/08/046]

• Axel Kleinschmidt: Unifying R-symmetry in M-theory, in New Trends in Mathematical Physics, Springer (2009) 389-401 [arXiv:hep-th/0703262, doi:10.1007/978-90-481-2810-5_26]

and application to supersymmetric quantum cosmology:

• Thibault Damour, Quantum supersymmetric Bianchi IX Cosmology and its hidden Kac-Moody Structure, talk at Gravitation, Solitons and Symmetries [pdf]

Review:

• Luca Carlevaro, part III of: Three approaches to M-theory, PhD thesis (2006) [hdl:123456789/16186, pdf, spire:1253257]

• Axel Kleinschmidt, Hermann Nicolai: Maximal supergravities and the $E_{10}$ model, International Journal of Modern Physics D 15 10 (2006) 1619-1642 [doi:10.1142/S0218271806009005]

• Hermann Nicolai: Wonders of $E_{10}$ and $K(E_{10})$, talk at Wonders of Gauge Theory and Supergravity, IHP Paris (2008) [pdf]

• Hermann Nicolai, On Exceptional Geometry and Supergravity, talk at Gravitation, Solitons and Symmetries [pdf]

• Hermann Nicolai: $\mathcal{N}=8$ Supergravity, and beyond [arXiv:2409.18656]

Discussion of phenomenology:

• Axel Kleinschmidt, Hermann Nicolai: Standard model fermions and $K(E_{10})$, Physics Letters B

  747 (2015) 251-254 [arXiv:1504.01586, doi:10.1016/j.physletb.2015.06.005]

• Krzysztof A. Meissner, Hermann Nicolai, Standard Model Fermions and Infinite-Dimensional R-Symmetries, Phys. Rev. Lett. 121 091601 (2018) [arXiv:1804.09606, doi:10.1103/PhysRevLett.121.091601]

• Krzysztof A. Meissner, Hermann Nicolai: Planck Mass Charged Gravitino Dark Matter, Phys. Rev. D 100 035001 (2019) [arXiv:1809.01441]

See also:

• Jeffrey Brown, Surya Ganguli, Ori J. Ganor, Craig Helfgott: $E_{10}$ Orbifolds, Journal of High Energy Physics 2005 JHEP06 (2005) [arXiv:hep-th/0409037, doi:10.1088/1126-6708/2005/06/057]

Maximal compact subalgebra

More on the maximal compact subalgebras of E9 and E10, respectively, and their finite-dimensional linear representations:

• Axel Kleinschmidt, Hermann Nicolai, Adriano Viganò: On spinorial representations of involutory subalgebras of Kac-Moody algebras, In: Partition Functions and Automorphic Forms, Moscow Lectures 5, Springer (2020) [arXiv:1811.11659, doi:10.1007/978-3-030-42400-8_4]

• Sophie de Buyl, Marc Henneaux, Louis Paulot: Hidden Symmetries and Dirac Fermions, Class. Quant. Grav. 22 (2005) 3595-3622 [arXiv:hep-th/0506009, doi:10.1088/0264-9381/22/17/018]

• 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]

• 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: On higher spin realizations of $K(E_{10})$, J. High Energ. Phys. 2013 41 (2013) [arXiv:1307.0413, doi:10.1007/JHEP08(2013)041]