Abstract
We develop the generalized Cartan Calculus for the groups \( G=\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{R}}\right)\times {\mathrm{\mathbb{R}}}^{+},\mathrm{S}\mathrm{L}\left(5,\mathrm{\mathbb{R}}\right) \) and SO(5, 5). They are the underlying algebraic structures of d = 9, 7, 6 exceptional field theory, respectively. These algebraic identities are needed for the “tensor hierarchy” structure in exceptional field theory. The validity of Poincaré lemmas in this new differential geometry is also discussed. Finally we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Cremmer and B. Julia, The N = 8 Supergravity Theory. I. The Lagrangian, Phys. Lett. B 80 (1978) 48 [INSPIRE].
E. Cremmer and B. Julia, The SO(8) Supergravity, Nucl. Phys. B 159 (1979) 141 [INSPIRE].
B. de Wit and H. Nicolai, Hidden Symmetry in d = 11 Supergravity, Phys. Lett. B 155 (1985) 47 [INSPIRE].
B. de Wit and H. Nicolai, d = 11 Supergravity With Local SU(8) Invariance, Nucl. Phys. B 274 (1986) 363 [INSPIRE].
B. de Wit and H. Nicolai, Hidden symmetries, central charges and all that, Class. Quant. Grav. 18 (2001) 3095 [hep-th/0011239] [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].
P.C. West, E11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].
P. Henry-Labordere, B. Julia and L. Paulot, Borcherds symmetries in M-theory, JHEP 04 (2002) 049 [hep-th/0203070] [INSPIRE].
T. Damour, M. Henneaux and H. Nicolai, E10 and a ‘small tension expansion’ of M-theory, Phys. Rev. Lett. 89 (2002) 221601 [hep-th/0207267] [INSPIRE].
P.C. West, E11, SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].
C.M. Hull, Generalised Geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
C. Hillmann, Generalized E : 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [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].
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].
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].
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].
G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [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].
H. Samtleben, Lectures on Gauged Supergravity and Flux Compactifications, Class. Quant. Grav. 25 (2008) 214002 [arXiv:0808.4076] [INSPIRE].
M. Cederwall, Non-gravitational exceptional supermultiplets, JHEP 07 (2013) 025 [arXiv:1302.6737] [INSPIRE].
O. Hohm and H. Samtleben, Gauge theory of Kaluza-Klein and winding modes, Phys. Rev. D 88 (2013) 085005 [arXiv:1307.0039] [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 I: E6(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. E7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].
O. Hohm and H. Samtleben, Exceptional field theory. III. E8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].
O. Hohm and Y.-N. Wang, Tensor hierarchy and generalized Cartan calculus in SL(3) × SL(2) exceptional field theory, JHEP 04 (2015) 050 [arXiv:1501.01600] [INSPIRE].
A. Abzalov, I. Bakhmatov and E.T. Musaev, Exceptional field theory: SO(5, 5), JHEP 06 (2015) 088 [arXiv:1504.01523] [INSPIRE].
H. Godazgar, M. Godazgar, O. Hohm, H. Nicolai and H. Samtleben, Supersymmetric E7(7) Exceptional Field Theory, JHEP 09 (2014) 044 [arXiv:1406.3235] [INSPIRE].
E. Musaev and H. Samtleben, Fermions and supersymmetry in E6(6) exceptional field theory, JHEP 03 (2015) 027 [arXiv:1412.7286] [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 S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].
M. Cederwall, J. Edlund and A. Karlsson, Exceptional geometry and tensor fields, JHEP 07 (2013) 028 [arXiv:1302.6736] [INSPIRE].
E. Bergshoeff, T. de Wit, U. Gran, R. Linares and D. Roest, (Non)Abelian gauged supergravities in nine-dimensions, JHEP 10 (2002) 061 [hep-th/0209205] [INSPIRE].
D. Roest, M-theory and gauged supergravities, Fortsch. Phys. 53 (2005) 119 [hep-th/0408175] [INSPIRE].
J.H. Schwarz, The power of M-theory, Phys. Lett. B 367 (1996) 97 [hep-th/9510086] [INSPIRE].
J.H. Schwarz, Lectures on superstring and M-theory dualities: Given at ICTP Spring School and at TASI Summer School, Nucl. Phys. Proc. Suppl. 55B (1997) 1 [hep-th/9607201] [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].
E.T. Musaev, Gauged supergravities in 5 and 6 dimensions from generalised Scherk-Schwarz reductions, JHEP 05 (2013) 161 [arXiv:1301.0467] [INSPIRE].
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
N. Berkovits, ICTP Lectures on Covariant Quantization of the Superstring, hep-th/0209059.
P.A. Grassi, G. Policastro, M. Porrati and P. Van Nieuwenhuizen, Covariant quantization of superstrings without pure spinor constraints, JHEP 10 (2002) 054 [hep-th/0112162] [INSPIRE].
E. Musaev, U-dualities in Type II string theories and M-theory, arXiv:1311.3331.
E. Bergshoeff, H. Samtleben and E. Sezgin, The Gaugings of Maximal D = 6 Supergravity, JHEP 03 (2008) 068 [arXiv:0712.4277] [INSPIRE].
G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, The gauge structure of Exceptional Field Theories and the tensor hierarchy, JHEP 04 (2014) 049 [arXiv:1312.4549] [INSPIRE].
J.-H. Park and Y. Suh, U-gravity: SL(N), JHEP 06 (2014) 102 [arXiv:1402.5027] [INSPIRE].
C. Strickland-Constable, Subsectors, Dynkin Diagrams and New Generalised Geometries, arXiv:1310.4196 [INSPIRE].
O. Hohm and H. Samtleben, Consistent Kaluza-Klein Truncations via Exceptional Field Theory, JHEP 01 (2015) 131 [arXiv:1410.8145] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1504.04780
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wang, YN. Generalized Cartan Calculus in general dimension. J. High Energ. Phys. 2015, 114 (2015). https://doi.org/10.1007/JHEP07(2015)114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2015)114