nLab exceptional generalized geometry



Exceptional structures

Duality in string theory



A variant of the idea of generalized complex geometry given by passing from generalization of complex geometry to generalization of exceptional geometry. Instead of by reduction of structure groups along inclusions like O(d)×O(d)O(d,d)O(d)\times O(d) \to O(d,d) it is controled by inclusions into split real forms of exceptional Lie groups.

This serves to neatly encode U-duality groups in supergravity as well as higher supersymmetry of supergravity compactifications. See also at exceptional field theory for more on this.


Higher supersymmetry

Compactification of 11-dimensional supergravity on a manifold of dimension 7 preserves N=1N = 1 supersymmetry precisely if its generalized tangent bundle has G-structure for the inclusion

SU(7)E 7(7) SU(7) \hookrightarrow E_{7(7)}

of the special unitary group in dimension 7 into the split real form of E7. This is shown in (Pacheco-Waldram 08).

One dimension down, compactification of 10-dimensional type II supergravity on a 6-manifold XX preserves N=2N = 2 supersymmetry precisely if the generalized tangent bundle TXT *XT X \otimes T^* X in the NS-NS sector admits G-structure for the inclusion

SU(3)×SU(3)O(6,6). SU(3) \times SU(3) \hookrightarrow O(6,6) \,.

This is reviewed in (GLSW, section 2).



Original articles:


Relation to Borcherds superalgebras is surveyed and discussed in

  • Jakob Palmkvist, Exceptional geometry and Borcherds superalgebras (arXiv:1507.08828)

black branes in the exotic spacetime are discussed in

The string and membrane sigma-models on exceptional spacetime (the “exceptional sigma models”) are discussed in

The generalized-U-duality+diffeomorphism invariance in 11d is discussed in

For the worldvolume theory of the M5-brane this is discussed in

  • Machiko Hatsuda, Kiyoshi Kamimura, M5 algebra and SO(5,5)SO(5,5) duality (arXiv:1305.2258)

Super-exceptional generalized geometry

The combination/unification of exceptional generalized geometry with supergeometry used to be an open problem:

Plausibility arguments that the bosonic body of the superspace underlying the M-theory Lie algebra serves as the unifying exceptional generalized geometry for M-theory for n=11n = 11:

  • Igor Bandos, Exceptional field theories, superparticles in an enlarged 11D superspace and higher spin theories, Nucl. Phys. B925 (2017) 28-62 (arXiv:1612.01321)

Arguments that super-exceptional M-geometry for n=11n = 11 is in fact a further fermionic extension of that (to the “hidden supergroup” of D’Auria-Fre):

A super-exceptional geometry for n=7n = 7:

See also the references on the corresponding super-geometry-enhancement of type II generalized geometry (“doubled geometry”): doubled geometry – References – Doubled supergeometry.

E 6E_6, E 7E_7, E 8E_8

E6,E7, E8-geometry is discussed in

(see also at 3d supergravity – possible gaugings).

E 10E_{10}

The E10-geometry of 11-dimensional supergravity compactified to the line is discussed in

E 11E_{11}

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

Review includes

Original articles include the following:

The observation that E 11E_{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 11E_{11} with its fundamental representation is due to

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

  • Alexander G. Tumanov, Peter West, E 11E_{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 11E_{11} in 11d11d, 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 11E_{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 11E_{11} formulation of M-theory include

Discussion of the semidirect product of E 11E_{11} with its l 1l_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 1l_1 originate in

and was further explored in

Relation to exceptional field theory is discussed in

Relation to Borcherds superalgebras is discussed in

Last revised on July 21, 2022 at 07:34:20. See the history of this page for a list of all contributions to it.