nLab exceptional generalized geometry

Idea

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 $O(d)\times O(d) \to O(d,d)$ 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.

Examples

Higher supersymmetry

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

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

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 $X$ preserves $N = 2$ supersymmetry precisely if the generalized tangent bundle $T X \otimes T^* X$ in the NS-NS sector admits G-structure for the inclusion

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

This is reviewed in (GLSW, section 2).

Related concepts

• non-geometric string theory vacua

• generalized complex geometry, exceptional geometry

• generalized tangent bundle, exceptional tangent bundle

• generalized Calabi-Yau manifold, generalized G₂-manifold

• Bn-geometry, differential T-duality

• universal exceptionalism

References

General

Original articles:

• K. Koepsell, Hermann Nicolai, Henning Samtleben, An exceptional geometry for d=11 supergravity?, Class. Quant. Grav. 17 (2000) 3689-3702 [arXiv:hep-th/0006034, doi:10.1088/0264-9381/17/18/308]

• Chris Hull, Generalised Geometry for M-Theory, JHEP 0707:079 (2007) [arXiv:hep-th/0701203, doi:10.1088/1126-6708/2007/07/079]

• Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 0809:123 (2008) [arXiv:0804.1362]

• Mariana Graña, Jan Louis, Aaron Sim, Daniel Waldram, $E_{7(7)}$ formulation of $N=2$ backgrounds (arXiv:0904.2333)

• G. Aldazabala, E. Andrésb, P. Cámarac, Mariana Graña, U-dual fluxes and generalized geometry, JHEP 1011:083,2010 (arXiv:1007.5509)

• Mariana Graña, Francesco Orsi, $N=1$ vacua in Exceptional Generalized Geometry (arXiv:1105.4855)

• Mariana Graña, Francesco Orsi, N=2 vacua in Generalized Geometry, (arXiv:1207.3004)

• André Coimbra, Charles Strickland-Constable, Daniel Waldram, $E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory (arXiv:1112.3989)

• David Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, Journal of Geometry and Physics 62 (2012), pp. 903-934 (arXiv:1101.0856)

• David Baraglia, Exceptional generalized geometry and $N = 2$ backgrounds (pdf)

Review:

• Daniel Persson, Arithmetic and Hyperbolic Structures in String Theory (arXiv:1001.3154)

• Nassiba Tabti, Kac-Moody algebraic structures in supergravity theories (arXiv:0910.1444)

• Daniel Waldram, Fluxes, holography and uses of Exceptional Generalized Geometry, talk at Strings 2022 [indico:4940880, slides, video ]

  on generalized G-structures and exceptional generalized geometry of flux compactifications and in AdS/CFT

Review of U-duality and exceptional generalized geometry in KK-compactification of D=11 supergravity:

• Henning Samtleben, 11D Supergravity and Hidden Symmetries, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2303.12682]

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

• Ilya Bakhmatov, David Berman, Axel Kleinschmidt, Edvard Musaev, Ray Otsuki, arXiv:1710.09740

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

• Yuho Sakatani, Shozo Uehara, Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry, Phys. Rev. Lett. 117 191601 (2016) [arXiv:1607.04265]

• Alex Arvanitakis, Chris D. A. Blair, Type II strings are Exceptional (arXiv:1712.07115)

• Alex Arvanitakis, Chris Blair, The Exceptional Sigma Model (arXiv:1802.00442)

Generalized U-duality+diffeomorphism invariance in 11d:

• David Berman, Martin Cederwall, Axel Kleinschmidt, Daniel C. Thompson, The gauge structure of generalised diffeomorphisms, High Energ. Phys. 2013, 64 (2013) [arXiv:1208.5884]

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

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

More on generalized exceptional sigma-models:

• David Osten, On the universal exceptional structure of world-volume theories in string and M-theory [arXiv:2402.10269]

Super-exceptional generalized geometry

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

• Martin Cederwall, p. 39 of Fundamental issues in extended geometry, 8th Mathematical Physics Meeting, Aug 2014 Belgrade, Serbia (spire:1477275)

• Martin Cederwall, Joakim Edlund, Anna Karlsson, p. 4, 7 of Exceptional geometry and tensor fields, J. High Energ. Phys. (2013) 2013: 28 (arXiv:1302.6736)

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 $n = 11$:

• 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 $n = 11$ is in fact a further fermionic extension of that (to the “hidden supergroup” of D’Auria-Fre):

• Silvia Vaula, On the underlying $E_{11}$ symmetry of the $D= 11$ Free Differential Algebra, JHEP 0703:010, 2007 (arXiv:hep-th/0612130)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Higher T-duality of super M-branes (arXiv:1803.05634)

• Hisham Sati, Urs Schreiber, Higher T-duality in M-theory via local supersymmetry, Physics Letters B 781 (2018) [arXiv:1805.00233]

• Daniel Butter, Henning Samtleben, Ergin Sezgin, $E_{7(7)}$ Exceptional Field Theory in Superspace, JHEP 01 (2019) 087 [arXiv:1811.00038, doi:10.1007/JHEP01(2019)087]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5, Journal of High Energy Physics 2020 107 (2020) [arXiv:1908.00042, doi:10.1007/JHEP02(2020)107]

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

$E_6$, $E_7$, $E_8$

E6,E7, E8-geometry is discussed in

• Christian Hillmann, Generalized E(7(7)) coset dynamics and D=11 supergravity, JHEP 0903 (2009) 135 (arXiv:0901.1581)

• Hadi Godazgar, Mahdi Godazgar, Hermann Nicolai, Generalised geometry from the ground up (arXiv:1307.8295)

• Olaf Hohm, Henning Samtleben, Exceptional Form of $D=11$ Supergravity, Phys. Rev. Lett. 111, 231601 (2013) (arXiv:1308.1673)

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

$E_{10}$

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

• Thibault Damour, Hermann Nicolai, Higher order M theory corrections and the Kac-Moody algebra E10 (arXiv:hep-th/0504153)

• Hermann Nicolai, Wonders of $E_{10}$ and $K(E_{10})$ (2008) (pdf)

• Axel Kleinschmidt, Hermann Nicolai, Standard model fermions and $K(E_{10})$ (arXiv:1504.01586)

$E_{11}$

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

• Peter West, $E_{11}$ and M theory, Class. Quant. Grav., 18:4443–4460, 2001. (arXiv:hep-th/0104081)

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 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 $E_{11}$, J. High Energ. Phys. (2012) 2012: 18 (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)

• 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 $E_{11}$ (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.B575:333-

  342,2003 (arXiv:hep-th/0307098)

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

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

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)