nLab exceptional field theory

1. Idea

In the context of supergravity and string theory, the term exceptional field theory has come to be used for formulations of 11-dimensional supergravity which make the (exceptional, whence the name) U-duality symmetry group structure manifest. This is in generalization of the “double field theory” formulation of 10d type II supergravity which makes (only) the T-duality symmetry manifest.

Accordingly, exceptional field theory is related to exceptional generalized geometry as double field theory is related to generalized complex geometry.

A spacetime in exceptional field theory is locally modeled on the Cartesian product

$$\mathbb{R}^{1,d-1} \times R$$

of a $d$-dimensional Minkowski space with the fundamental representation vector space $R$ of a form of the exceptional Lie group $E_{11-d}$ (the U-duality group). In the literature the former is called external space and the latter internal space. Fields on this spacetime are subject to satisfy a certain differential equation derived from an invariant form of the representation and one considers a generalized isometry algebra on this space which fails the Jacobi identity by a term proportional to this contraint (e.g. Baguet-Hohm-Samtleben 15, section 2).

2. Related concepts

• U-duality, exceptional Lie group

• exceptional generalized geometry, exceptional tangent bundle

• non-geometry, exotic branes

• double field theory

• generalized Scherk-Schwarz reduction, gauged supergravity, tensor hierarchy, embedding tensor

• exceptional geometry

• universal exceptionalism

3. References

General

Precursors on U-duality covariant formulations of D=11 supergravity:

• Bernard de Wit, Hermann Nicolai: D = 11 Supergravity With Local $SU(8)$ Invariance, Nucl. Phys. B 274, 363 (1986) [doi:10.1016/0550-3213(86)90290-7, spire:227409]

• Bernard de Wit, Hermann Nicolai: Local SU(8) invariance in $d = 11$ supergravity, talk at Nuffield Workshop on Supersymmetry and its Applications, 0357 [spire:218601]

• Hermann Nicolai: On M-Theory, J Astrophys Astron 20, (1999) 149–164 [arXiv:hep-th/9801090, doi:10.1007/BF02702349]

  (the term “exceptional geometry” appears here)

• Olaf Hohm, Henning Samtleben: U-duality covariant gravity, J. High Energ. Phys. 2013 80 (2013) [arXiv:1307.0509, doi:10.1007/JHEP09(2013)080]

And precursor geometry enriched by brane winding charges in higher analogy to double field theory:

• David S. Berman, Malcolm J. Perry: Generalized Geometry and M theory, J. High Energ. Phys. 2011 74 (2011) [arXiv:1008.1763v4, doi:10.1007/JHEP06(2011)074]

The original articles introducing the terminology of “exceptional field theory”:

• Olaf Hohm, Henning Samtleben: Exceptional Form of $D=11$ Supergravity, Phys. Rev. Lett. 111 231601 (2013) [arXiv:1308.1673, doi:10.1103/PhysRevLett.111.231601]

• Olaf Hohm, Henning Samtleben: Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB, Phys. Rev. D 89 066016 (2014) [arXiv:1312.0614, doi:10.1103/PhysRevD.89.066016]

• Olaf Hohm, Henning Samtleben: Exceptional Field Theory II: $E_{7(7)}$, Phys. Rev. D 89 066017 (2014) [arXiv:1312.4542, doi:10.1103/PhysRevD.89.066017]

• Olaf Hohm, Henning Samtleben: Exceptional Field Theory III: $E_{8(8)}$, Phys. Rev. D 90 066002 (2014) [arXiv:1406.3348, doi:10.1103/PhysRevD.90.066002]

• 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]

Review:

• David Berman, Chris D. A. Blair, The Geometry, Branes and Applications of Exceptional Field Theory, International Journal of Modern Physics A 35 30 (2020) 2030014 [arXiv:2006.09777, doi:10.1142/S0217751X20300148]

• Henning Samtleben: Exceptional field theories, in Half a century of Supergravity, Cambridge University Press [arXiv:2503.16947]

Review for KK-compactification to 5d supergravity, hence for E6-U-duality, includes

• Arnaud Baguet, Olaf Hohm, Henning Samtleben, $E_{6(6)}$ Exceptional Field Theory: Review and Embedding of Type IIB (arXiv:1506.01065)

Discussion of solitonic black brane (and exotic brane) solutions in terms of EFT includes

• David Berman, Felix Rudolph, Strings, Branes and the Self-dual Solutions of Exceptional Field Theory (arXiv.1412.2768)

Discussion for E11:

• Alexander G. Tumanov, Peter West: $E_{11}$ and exceptional field theory, International Journal of Modern Physics A 31 12 (2016) 1650066 [arXiv:1507.08912, doi:10.1142/S0217751X16500664]

• Guillaume Bossard, Axel Kleinschmidt, Jakob Palmkvist, Christopher Pope, Ergin Sezgin: Beyond $E_{11}$, JHEP 05 (2017) 020 [doi:10.1007/JHEP05(2017)020, arXiv:1703.01305]

• Guillaume Bossard, Axel Kleinschmidt, Ergin Sezgin: On supersymmetric $E_{11}$ exceptional field theory, J. High Energ. Phys. 2019 165 (2019) [arXiv:1907.02080, doi:10.1007/JHEP10(2019)165]

• Guillaume Bossard, Axel Kleinschmidt, Ergin Sezgin: A master exceptional field theory, J. High Energ. Phys. 2021 185 (2021) [arXiv:2103.13411, doi:10.1007/JHEP06(2021)185]

Discussion for E9:

• Guillaume Bossard, Franz Ciceri, Gianluca Inverso, Axel Kleinschmidt, Henning Samtleben, $E_9$ exceptional field theory II. The complete dynamics (arXiv:2103.12118)

Generalization to exceptional super-spacetimes:

• Lars Brink, Paul Howe: The $N = 8$ supergravity in superspace, Physics Letters B 88 3–4 (1979) 268-272 [doi:10.1016/0370-2693(79)90464-7]

• Daniel Butter, Henning Samtleben, Ergin Sezgin: $E_{7(7)}$-Exceptional Field Theory in Superspace, J. High Energ. Phys. 2019 87 (2019) [arXiv:1811.00038, doi:10.1007/JHEP01(2019)087]

• Sudarshan Ananth, Nipun Bhave: Exceptional symmetries in light-cone superspace [arXiv:2410.19463]

On AdS4/CFT3 duality via exceptional field theory and super Chern-Simons theory:

• Oscar Varela: Super-Chern-Simons spectra from Exceptional Field Theory, J. High Energ. Phys. 2021 283 (2021) [arXiv:2010.09743, doi:10.1007/JHEP04(2021)283]

See also:

On application to KK-reduction of D=10 supergravity and D=11 supergravity on squashed 7-spheres:

• Bastien Duboeuf, Emanuel Malek, Henning Samtleben: Kaluza-Klein Spectrometry beyond Consistent Truncations: The Squashed $S^7$, J. High Energ. Phys. 2023 62 (2023) [arXiv:2212.01135, doi:10.1007/JHEP04(2023)062]

Exceptional-geometric brane sigma-models

On U-duality-covariant exceptional geometric super $p$-brane sigma-models (worldvolume exceptional field theory):

• Yuho Sakatani, Shozo Uehara, Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry, Phys. Rev. Lett. 117 191601 (2016) [doi:10.1103/PhysRevLett.117.191601, arXiv:1607.04265, talk slides]

• Yuho Sakatani, Shozo Uehara, Exceptional M-brane sigma models and $\eta$-symbols, Progress of Theoretical and Experimental Physics 2018 3 (2018) 033B05, [doi:10.1093/ptep/pty021, arXiv:1712.10316]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: Super-exceptional geometry: Super-exceptional embedding construction of M5, J. High Energy Physics 2020 107 (2020) [doi:10.1007/JHEP02(2020)107, arXiv:1908.00042]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: Super-exceptional M5-brane model – Emergence of SU(2)-flavor sector, J. Geometry and Physics 170 (2021) 104349 [doi:10.1016/j.geomphys.2021.104349, arXiv:2006.00012]

• David Osten: Currents, charges and algebras in exceptional generalised geometry, J. High Energ. Phys. 2021 70 (2021) [doi:10.1007/JHEP06(2021)070, arXiv:2103.03267]

• Machiko Hatsuda, Ondřej Hulík, William D. Linch, Warren D. Siegel, Di Wang, Yu-Ping Wang: $\mathcal{A}$-theory: A brane world-volume theory with manifest U-duality, J. High Energ. Phys. 2023 87 (2023) [doi:10.1007/JHEP10(2023)087, arXiv:2307.04934]

• David Osten: On exceptional QP-manifolds, J. High Energ. Phys. 2024 28 (2024) [doi:10.1007/JHEP01(2024)028, arXiv:2306.11093]

• David Osten: On the universal exceptional structure of world-volume theories in string and M-theory, Physics Letters B 855 (2024) 138814 [doi:10.1016/j.physletb.2024.138814, arXiv:2402.10269]

• Machiko Hatsuda, Ondřej Hulík, William D. Linch, Warren D. Siegel, Di Wang, Yu-Ping Wang: Strings and membranes from $\mathcal{A}$-theory five brane [arXiv:2410.11197]