Abstract
We construct the scalar potential for the exceptional field theory based on the affine symmetry group E9. The fields appearing in this potential live formally on an infinite-dimensional extended spacetime and transform under E9 generalised diffeomorphisms. In addition to the scalar fields expected from D = 2 maximal supergravity, the invariance of the potential requires the introduction of new constrained scalar fields. Other essential ingredients in the construction include the Virasoro algebra and indecomposable representations of E9. Upon solving the section constraint, the potential reproduces the dynamics of either eleven-dimensional or type IIB supergravity in the presence of two isometries.
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References
C.M. Hull, Generalised Geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
P. Pires Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
C. Hillmann, E 7(7) and d = 11 supergravity, Ph.D. thesis, Humboldt University Berlin, Germany, (2008), arXiv:0902.1509 [INSPIRE].
D.S. Berman and M.J. Perry, Generalized Geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].
D.S. Berman, H. Godazgar and M.J. Perry, SO(5, 5) duality in M-theory and generalized geometry, Phys. Lett. B 700 (2011) 65 [arXiv:1103.5733] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) × ℝ+ generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
D.S. Berman, M. Cederwall, A. Kleinschmidt and D.C. Thompson, The gauge structure of generalised diffeomorphisms, JHEP 01 (2013) 064 [arXiv:1208.5884] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry II: E d(d) × ℝ+ and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
J.-H. Park and Y. Suh, U-geometry: SL(5), JHEP 04 (2013) 147 [Erratum ibid. 11 (2013) 210] [arXiv:1302.1652] [INSPIRE].
M. Cederwall, J. Edlund and A. Karlsson, Exceptional geometry and tensor fields, JHEP 07 (2013) 028 [arXiv:1302.6736] [INSPIRE].
M. Cederwall, Non-gravitational exceptional supermultiplets, JHEP 07 (2013) 025 [arXiv:1302.6737] [INSPIRE].
G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [INSPIRE].
G. Bossard, M. Cederwall, A. Kleinschmidt, J. Palmkvist and H. Samtleben, Generalized diffeomorphisms for E 9, Phys. Rev. D 96 (2017) 106022 [arXiv:1708.08936] [INSPIRE].
M. Cederwall and J. Palmkvist, Extended geometries, JHEP 02 (2018) 071 [arXiv:1711.07694] [INSPIRE].
E. Cremmer and B. Julia, The N = 8 Supergravity Theory. 1. The Lagrangian, Phys. Lett. B 80 (1978) 48 [INSPIRE].
B. Julia, Kac-Moody symmetry of gravitation and supergravity theories, Lectures Appl. Math. 21 (1985) 335.
H. Nicolai, The Integrability of N = 16 Supergravity, Phys. Lett. B 194 (1987) 402 [INSPIRE].
E. Cremmer, B. Julia, H. Lü and C.N. Pope, Dualization of dualities. 1., Nucl. Phys. B 523 (1998) 73 [hep-th/9710119] [INSPIRE].
E. Cremmer, B. Julia, H. Lü and C.N. Pope, Dualization of dualities. 2. Twisted self-duality of doubled fields and superdualities, Nucl. Phys. B 535 (1998) 242 [hep-th/9806106] [INSPIRE].
B. de Wit and H. Samtleben, Gauged maximal supergravities and hierarchies of nonAbelian vector-tensor systems, Fortsch. Phys. 53 (2005) 442 [hep-th/0501243] [INSPIRE].
B. de Wit, H. Nicolai and H. Samtleben, Gauged Supergravities, Tensor Hierarchies and M-theory, JHEP 02 (2008) 044 [arXiv:0801.1294] [INSPIRE].
O. Hohm and H. Samtleben, Exceptional Field Theory I: E 6(6) covariant Form of M-theory and Type IIB, Phys. Rev. D 89 (2014) 066016 [arXiv:1312.0614] [INSPIRE].
O. Hohm and H. Samtleben, Exceptional field theory. II. E 7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].
O. Hohm and H. Samtleben, Exceptional field theory. III. E 8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].
D.S. Berman, E.T. Musaev, D.C. Thompson and D.C. Thompson, Duality Invariant M-theory: Gauged supergravities and Scherk-Schwarz reductions, JHEP 10 (2012) 174 [arXiv:1208.0020] [INSPIRE].
E.T. Musaev, Gauged supergravities in 5 and 6 dimensions from generalised Scherk-Schwarz reductions, JHEP 05 (2013) 161 [arXiv:1301.0467] [INSPIRE].
H. Godazgar, M. Godazgar and H. Nicolai, Embedding tensor of Scherk-Schwarz flux compactifications from eleven dimensions, Phys. Rev. D 89 (2014) 045009 [arXiv:1312.1061] [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017) 1700048 [arXiv:1401.3360] [INSPIRE].
O. Hohm and H. Samtleben, Consistent Kaluza-Klein Truncations via Exceptional Field Theory, JHEP 01 (2015) 131 [arXiv:1410.8145] [INSPIRE].
B. Julia, Infinite Lie algebras in physics, proceedings of Johns Hopkins Workshop on Current Problems in Particle Theory, LPTENS-81-14 [INSPIRE].
H. Samtleben and M. Weidner, Gauging hidden symmetries in two dimensions, JHEP 08 (2007) 076 [arXiv:0705.2606] [INSPIRE].
J. de Boer and M. Shigemori, Exotic branes and non-geometric backgrounds, Phys. Rev. Lett. 104 (2010) 251603 [arXiv:1004.2521] [INSPIRE].
J. de Boer and M. Shigemori, Exotic Branes in String Theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].
I. Bakhmatov, D. Berman, A. Kleinschmidt, E. Musaev and R. Otsuki, Exotic branes in Exceptional Field Theory: the SL(5) duality group, JHEP 08 (2018) 021 [arXiv:1710.09740] [INSPIRE].
J.J. Fernández-Melgarejo, T. Kimura and Y. Sakatani, Weaving the Exotic Web, JHEP 09 (2018) 072 [arXiv:1805.12117] [INSPIRE].
D.S. Berman, E.T. Musaev and R. Otsuki, Exotic Branes in Exceptional Field Theory: E 7(7) and Beyond, JHEP 12 (2018) 053 [arXiv:1806.00430] [INSPIRE].
P. Goddard and D.I. Olive, Kac-Moody and Virasoro Algebras in Relation to Quantum Physics, Int. J. Mod. Phys. A 1 (1986) 303 [INSPIRE].
H. Sugawara, A field theory of currents, Phys. Rev. 170 (1968) 1659 [INSPIRE].
B. Julia and H. Nicolai, Conformal internal symmetry of 2-D σ-models coupled to gravity and a dilaton, Nucl. Phys. B 482 (1996) 431 [hep-th/9608082] [INSPIRE].
R.P. Geroch, A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971) 918 [INSPIRE].
P. Breitenlohner and D. Maison, On the Geroch group, Ann. Inst. H. Poincaré Phys. Theor. 46 (1987) 215.
P. Breitenlohner, D. Maison and G.W. Gibbons, Four-Dimensional Black Holes from Kaluza-Klein Theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].
H. Nicolai and H. Samtleben, On K(E 9 ), Q. J. Pure Appl. Math. 1 (2005) 180 [hep-th/0407055] [INSPIRE].
E. Cremmer, H. Lü, C.N. Pope and K.S. Stelle, Spectrum generating symmetries for BPS solitons, Nucl. Phys. B 520 (1998) 132 [hep-th/9707207] [INSPIRE].
V.G. Kac and D.H. Peterson, Defining relations of certain infinite dimensional groups, Astérisque Hors-Série, (1984), pp. 165-208.
J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987) 542.
L. Carbone and H. Garland, Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math. 5 (2003) 813.
T. De Medts, R. Gramlich, M. Horn Iwasawa decompositions of split Kac-Moody groups, J. Lie Theory 19 (2009) 311.
O. Hohm and H. Samtleben, Exceptional Form of D = 11 Supergravity, Phys. Rev. Lett. 111 (2013) 231601 [arXiv:1308.1673] [INSPIRE].
C.D.A. Blair, E. Malek and J.-H. Park, M-theory and Type IIB from a Duality Manifest Action, JHEP 01 (2014) 172 [arXiv:1311.5109] [INSPIRE].
G. Bossard and A. Kleinschmidt, Loops in exceptional field theory, JHEP 01 (2016) 164 [arXiv:1510.07859] [INSPIRE].
H. Godazgar, M. Godazgar and M.J. Perry, E8 duality and dual gravity, JHEP 06 (2013) 044 [arXiv:1303.2035] [INSPIRE].
V.A. Belinsky and V.E. Zakharov, Integration of the Einstein Equations by the Inverse Scattering Problem Technique and the Calculation of the Exact Soliton Solutions, Sov. Phys. JETP 48 (1978) 985 [Zh. Eksp. Teor. Fiz. 75 (1978) 1953] [INSPIRE].
D. Maison, Are the stationary, axially symmetric Einstein equations completely integrable?, Phys. Rev. Lett. 41 (1978) 521 [INSPIRE].
D. Bernard and B. Julia, Twisted self-duality of dimensionally reduced gravity and vertex operators, Nucl. Phys. B 547 (1999) 427 [hep-th/9712254] [INSPIRE].
M. Cederwall and J.A. Rosabal, E 8 geometry, JHEP 07 (2015) 007 [arXiv:1504.04843] [INSPIRE].
A. Baguet and H. Samtleben, E 8(8) Exceptional Field Theory: Geometry, Fermions and Supersymmetry, JHEP 09 (2016) 168 [arXiv:1607.03119] [INSPIRE].
P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].
P. West, Generalised Space-time and Gauge Transformations, JHEP 08 (2014) 050 [arXiv:1403.6395] [INSPIRE].
A.G. Tumanov and P. West, E11 in 11D, Phys. Lett. B 758 (2016) 278 [arXiv:1601.03974] [INSPIRE].
T. Damour, M. Henneaux and H. Nicolai, E 10 and a ‘small tension expansion’ of M-theory, Phys. Rev. Lett. 89 (2002) 221601 [hep-th/0207267] [INSPIRE].
G. Bossard, A. Kleinschmidt, J. Palmkvist, C.N. Pope and E. Sezgin, Beyond E 11, JHEP 05 (2017) 020 [arXiv:1703.01305] [INSPIRE].
F. Ciceri, G. Dibitetto, J.J. Fernández-Melgarejo, A. Guarino and G. Inverso, Double Field Theory at SL(2) angles, JHEP 05 (2017) 028 [arXiv:1612.05230] [INSPIRE].
O. Hohm, E.T. Musaev and H. Samtleben, O(d + 1, d + 1) enhanced double field theory, JHEP 10 (2017) 086 [arXiv:1707.06693] [INSPIRE].
O. Hohm and S.K. Kwak, Double Field Theory Formulation of Heterotic Strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].
F. Ciceri, A. Guarino and G. Inverso, The exceptional story of massive IIA supergravity, JHEP 08 (2016) 154 [arXiv:1604.08602] [INSPIRE].
O. Hohm and B. Zwiebach, L ∞ Algebras and Field Theory, Fortsch. Phys. 65 (2017) 1700014 [arXiv:1701.08824] [INSPIRE].
M. Cederwall and J. Palmkvist, L ∞ algebras for extended geometry from Borcherds superalgebras, arXiv:1804.04377 [INSPIRE].
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Bossard, G., Ciceri, F., Inverso, G. et al. E9 exceptional field theory. Part I. The potential. J. High Energ. Phys. 2019, 89 (2019). https://doi.org/10.1007/JHEP03(2019)089
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DOI: https://doi.org/10.1007/JHEP03(2019)089