∞-Lie theory (higher geometry)
A hyperbolic Kac-Moody Lie algebra in the E-series
… E6, E7, E8, E9, E10, $E_{11}$, …
$E_{11}$ 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)
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:
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})
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)
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
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 $E_{11}$ with its fundamental representation is due to
Peter West, Generalised geometry, eleven dimensions and $E_{11}$ (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)
E. Bergshoeff, I. De Baetselier, T. Nutma, E(11) and the Embedding Tensor (arXiv:0705.1304, poster)
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
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
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)
private communication with Axel Kleinschmidt ↩