Abstract
We give the supersymmetric extension of exceptional field theory for E7(7), which is based on a (4 + 56)-dimensional generalized spacetime subject to a covariant constraint. The fermions are tensors under the local Lorentz group SO(1, 3) × SU(8) and transform as scalar densities under the E7(7) (internal) generalized diffeomorphisms. The supersymmetry transformations are manifestly covariant under these symmetries and close, in particular, into the generalized diffeomorphisms of the 56-dimensional space. We give the fermionic field equations and prove supersymmetric invariance. We establish the consistency of these results with the recently constructed generalized geometric formulation of D = 11 supergravity.
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References
E. Cremmer and B. Julia, The N = 8 Supergravity Theory. 1. The Lagrangian, Phys. Lett. B 80 (1978) 48 [INSPIRE].
E. Cremmer, B. Julia and J. Scherk, Supergravity Theory in Eleven-Dimensions, Phys. Lett. B 76 (1978) 409 [INSPIRE].
B. de Wit and H. Nicolai, d = 11 Supergravity With Local SU(8) Invariance, Nucl. Phys. B 274 (1986) 363 [INSPIRE].
H. Nicolai, D = 11 Supergravity With Local SO(16) Invariance, Phys. Lett. B 187 (1987) 316 [INSPIRE].
K. Koepsell, H. Nicolai and H. Samtleben, An exceptional geometry for D = 11 supergravity?, Class. Quant. Grav. 17 (2000) 3689 [hep-th/0006034] [INSPIRE].
B. de Wit and H. Nicolai, Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions, JHEP 05 (2013) 077 [arXiv:1302.6219] [INSPIRE].
H. Godazgar, M. Godazgar and H. Nicolai, Generalised geometry from the ground up, JHEP 02 (2014) 075 [arXiv:1307.8295] [INSPIRE].
H. Godazgar, M. Godazgar and H. Nicolai, Einstein-Cartan Calculus for Exceptional Geometry, JHEP 06 (2014) 021 [arXiv:1401.5984] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
O. Hohm and H. Samtleben, Exceptional Form of D = 11 Supergravity, Phys. Rev. Lett. 111 (2013) 231601 [arXiv:1308.1673] [INSPIRE].
O. Hohm and H. Samtleben, Exceptional Field Theory II: E 7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].
C.M. Hull, Generalised Geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, \( {E}_{d(d)}\times {\mathrm{\mathbb{R}}}^{+} \) generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry II: \( {E}_{d(d)}\times {\mathrm{\mathbb{R}}}^{+} \) and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281 [math/0209099] [INSPIRE].
M. Gualtieri, Generalized complex geometry, math/0401221 [INSPIRE].
P.C. West, E 11 , SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].
C. Hillmann, Generalized E(7(7)) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [INSPIRE].
D.S. Berman, H. Godazgar, M.J. Perry and P. West, Duality Invariant Actions and Generalised Geometry, JHEP 02 (2012) 108 [arXiv:1111.0459] [INSPIRE].
O. Hohm, S.K. Kwak and B. Zwiebach, Unification of Type II Strings and T-duality, Phys. Rev. Lett. 107 (2011) 171603 [arXiv:1106.5452] [INSPIRE].
O. Hohm, S.K. Kwak and B. Zwiebach, Double Field Theory of Type II Strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].
O. Hohm and H. Samtleben, U-duality covariant gravity, JHEP 09 (2013) 080 [arXiv:1307.0509] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry I: Type II Theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
O. Hohm and S.K. Kwak, N = 1 Supersymmetric Double Field Theory, JHEP 03 (2012) 080 [arXiv:1111.7293] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity, Phys. Rev. D 85 (2012) 081501 [Erratum ibid. D 86 (2012) 089903] [arXiv:1112.0069] [INSPIRE].
O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].
O. Hohm and B. Zwiebach, On the Riemann Tensor in Double Field Theory, JHEP 05 (2012) 126 [arXiv:1112.5296] [INSPIRE].
O. Hohm and B. Zwiebach, Towards an invariant geometry of double field theory, J. Math. Phys. 54 (2013) 032303 [arXiv:1212.1736] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Stringy differential geometry, beyond Riemann, Phys. Rev. D 84 (2011) 044022 [arXiv:1105.6294] [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].
M. Cederwall, J. Edlund and A. Karlsson, Exceptional geometry and tensor fields, JHEP 07 (2013) 028 [arXiv:1302.6736] [INSPIRE].
V.I. Ogievetsky, Infinite-dimensional algebra of general covariance group as the closure of finite-dimensional algebras of conformal and linear groups, Lett. Nuovo Cim. 8 (1973) 988 [INSPIRE].
A.B. Borisov and V.I. Ogievetsky, Theory of Dynamical Affine and Conformal Symmetries as Gravity Theory, Theor. Math. Phys. 21 (1975) 1179 [INSPIRE].
P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].
B. de Wit and H. Nicolai, The Consistency of the S 7 Truncation in D = 11 Supergravity, Nucl. Phys. B 281 (1987) 211 [INSPIRE].
H. Nicolai and K. Pilch, Consistent Truncation of D = 11 Supergravity on AdS 4 × S 7, JHEP 03 (2012) 099 [arXiv:1112.6131] [INSPIRE].
H. Godazgar, M. Godazgar and H. Nicolai, Nonlinear Kaluza-Klein theory for dual fields, Phys. Rev. D 88 (2013) 125002 [arXiv:1309.0266] [INSPIRE].
B. de Wit, H. Nicolai and N.P. Warner, The Embedding of Gauged N = 8 Supergravity Into d = 11 Supergravity, Nucl. Phys. B 255 (1985) 29 [INSPIRE].
H. Godazgar, M. Godazgar and H. Nicolai, Testing the non-linear flux ansatz for maximal supergravity, Phys. Rev. D 87 (2013) 085038 [arXiv:1303.1013] [INSPIRE].
H. Godazgar, M. Godazgar and H. Nicolai, The 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, arXiv:1401.3360 [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].
B. de Wit, H. Samtleben and M. Trigiante, The Maximal D = 4 supergravities, JHEP 06 (2007) 049 [arXiv:0705.2101] [INSPIRE].
G. Dall’Agata, G. Inverso and M. Trigiante, Evidence for a family of SO(8) gauged supergravity theories, Phys. Rev. Lett. 109 (2012) 201301 [arXiv:1209.0760] [INSPIRE].
G. Dall’Agata, G. Inverso and A. Marrani, Symplectic Deformations of Gauged Maximal Supergravity, JHEP 07 (2014) 133 [arXiv:1405.2437] [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].
E. Bergshoeff, R. Kallosh, T. Ortín, D. Roest and A. Van Proeyen, New formulations of D = 10 supersymmetry and D8 - O8 domain walls,Class. Quant. Grav. 18 (2001) 3359 [hep-th/0103233] [INSPIRE].
B. de Wit, H. Samtleben and M. Trigiante, On Lagrangians and gaugings of maximal supergravities, Nucl. Phys. B 655 (2003) 93 [hep-th/0212239] [INSPIRE].
O. Hohm and H. Samtleben, Gauge theory of Kaluza-Klein and winding modes, Phys. Rev. D 88 (2013) 085005 [arXiv:1307.0039] [INSPIRE].
B. de Wit and H. Nicolai, N=8 Supergravity, Nucl. Phys. B 208 (1982) 323 [INSPIRE].
O. Hohm and H. Samtleben, Exceptional Field Theory III: E 8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].
F.W. Hehl, P. Von Der Heyde, G.D. Kerlick and J.M. Nester, General Relativity with Spin and Torsion: Foundations and Prospects, Rev. Mod. Phys. 48 (1976) 393 [INSPIRE].
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Godazgar, H., Godazgar, M., Hohm, O. et al. Supersymmetric E7(7) exceptional field theory. J. High Energ. Phys. 2014, 44 (2014). https://doi.org/10.1007/JHEP09(2014)044
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DOI: https://doi.org/10.1007/JHEP09(2014)044