Abstract
We address the construction of manifest U-duality invariant generalized diffeomorphisms. The closure of the algebra requires an extension of the tangent space to include a tensor hierarchy indicating the existence of an underlying unifying structure, compatible with E 11 and Borcherds algebras constructions. We begin with four-dimensional gauged maximal supergravity, and build a generalized Lie derivative that encodes all the gauge transformations of the theory. A generalized frame is introduced, which accommodates for all the degrees of freedom, including the tensor hierarchy. The generalized Lie derivative defines generalized field-dependent fluxes containing all the covariant quantities in the theory, and the closure conditions give rise to their corresponding Bianchi Identities. We then move towards the construction of a full generalized Lie derivative defined on an extended space, analyze the closure conditions, and explore the connection with that of maximal gauged supergravity via a generalized Scherk-Schwarz reduction, and with 11-dimensional supergravity.
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References
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].
G. Aldazabal, D. Marques and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].
D.S. Berman and D.C. Thompson, Duality symmetric string and M-theory, arXiv:1306.2643 [INSPIRE].
O. Hohm, D. Lüst and B. Zwiebach, The spacetime of double field theory: review, remarks and outlook, Fortsch. Phys. 61 (2013) 926 [arXiv:1309.2977] [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].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
P. Koerber, Lectures on generalized complex geometry for physicists, Fortsch. Phys. 59 (2011) 169 [arXiv:1006.1536] [INSPIRE].
M. Graña, R. Minasian, M. Petrini and D. Waldram, T-duality, generalized geometry and non-geometric backgrounds, JHEP 04 (2009) 075 [arXiv:0807.4527] [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].
G. Dall’Agata, N. Prezas, H. Samtleben and M. Trigiante, Gauged supergravities from twisted doubled tori and non-geometric string backgrounds, Nucl. Phys. B 799 (2008) 80 [arXiv:0712.1026] [INSPIRE].
C. Hillmann, Generalized E 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [INSPIRE].
G. Aldazabal, E. Andres, P.G. Camara and M. Graña, U-dual fluxes and generalized geometry, JHEP 11 (2010) 083 [arXiv:1007.5509] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) × \( \mathbb{R} \) + 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) × \( \mathbb{R} \) + 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, W. Baron, D. Marques and C. Núñez, The effective action of double field theory, JHEP 11 (2011) 052 [Erratum ibid. 1111 (2011) 109] [arXiv:1109.0290] [INSPIRE].
D. Geissbuhler, Double field theory and N = 4 gauged supergravity, JHEP 11 (2011) 116 [arXiv:1109.4280] [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, M.J. Perry and P. West, Duality invariant actions and generalised geometry, JHEP 02 (2012) 108 [arXiv:1111.0459] [INSPIRE].
D.S. Berman, H. Godazgar, M. Godazgar and M.J. Perry, The local symmetries of M-theory and their formulation in generalised geometry, JHEP 01 (2012) 012 [arXiv:1110.3930] [INSPIRE].
H. Godazgar, M. Godazgar and M.J. Perry, E 8 duality and dual gravity, JHEP 06 (2013) 044 [arXiv:1303.2035] [INSPIRE].
C. Strickland-Constable, Subsectors, Dynkin diagrams and new generalised geometries, arXiv:1310.4196 [INSPIRE].
H. Samtleben and M. Weidner, Gauging hidden symmetries in two dimensions, JHEP 08 (2007) 076 [arXiv:0705.2606] [INSPIRE].
P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].
F. Riccioni and P.C. West, The E 11 origin of all maximal supergravities, JHEP 07 (2007) 063 [arXiv:0705.0752] [INSPIRE].
E.A. Bergshoeff, I. De Baetselier and T.A. Nutma, E 11 and the embedding tensor, JHEP 09 (2007) 047 [arXiv:0705.1304] [INSPIRE].
F. Riccioni and P.C. West, E 11 -extended spacetime and gauged supergravities, JHEP 02 (2008) 039 [arXiv:0712.1795] [INSPIRE].
P. West, Generalised geometry, eleven dimensions and E 11, JHEP 02 (2012) 018 [arXiv:1111.1642] [INSPIRE].
P. West, E 11 , generalised space-time and equations of motion in four dimensions, JHEP 12 (2012) 068 [arXiv:1206.7045] [INSPIRE].
F. Riccioni, D. Steele and P. West, The E 11 origin of all maximal supergravities: the hierarchy of field-strengths, JHEP 09 (2009) 095 [arXiv:0906.1177] [INSPIRE].
J. Palmkvist, Tensor hierarchies, Borcherds algebras and E 11, JHEP 02 (2012) 066 [arXiv:1110.4892] [INSPIRE].
J. Palmkvist, The tensor hierarchy algebra, J. Math. Phys. 55 (2014) 011701 [arXiv:1305.0018] [INSPIRE].
J. Greitz, P. Howe and J. Palmkvist, The tensor hierarchy simplified, arXiv:1308.4972 [INSPIRE].
M. Henneaux, B.L. Julia and J. Levie, E 11 , Borcherds algebras and maximal supergravity, JHEP 04 (2012) 078 [arXiv:1007.5241] [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].
G. Aldazabal, M. Graña, D. Marqués and J. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [INSPIRE].
J.-H. Park and Y. Suh, U-geometry: SL(5), JHEP 04 (2013) 147 [arXiv:1302.1652] [INSPIRE].
M. Cederwall, J. Edlund and A. Karlsson, Exceptional geometry and tensor fields, JHEP 07 (2013) 028 [arXiv:1302.6736] [INSPIRE].
C.D. 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].
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, U-duality covariant gravity, JHEP 09 (2013) 080 [arXiv:1307.0509] [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: E 6(6) covariant form of M-theory and type IIB, 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].
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].
M. Weidner, Gauged supergravities in various spacetime dimensions, Fortsch. Phys. 55 (2007) 843 [hep-th/0702084] [INSPIRE].
B. de Wit, H. Nicolai and H. Samtleben, Gauged supergravities, tensor hierarchies and M-theory, JHEP 02 (2008) 044 [arXiv:0801.1294] [INSPIRE].
B. de Wit and H. Samtleben, The end of the p-form hierarchy, JHEP 08 (2008) 015 [arXiv:0805.4767] [INSPIRE].
B. de Wit and M. van Zalk, Supergravity and M-theory, Gen. Rel. Grav. 41 (2009) 757 [arXiv:0901.4519] [INSPIRE].
E.A. Bergshoeff, J. Hartong, O. Hohm, M. Huebscher and T. Ortín, Gauge theories, duality relations and the tensor hierarchy, JHEP 04 (2009) 123 [arXiv:0901.2054] [INSPIRE].
C.M. Hull, U duality and BPS spectrum of super Yang-Mills theory and M-theory, JHEP 07 (1998) 018 [hep-th/9712075] [INSPIRE].
E.A. Bergshoeff and F. Riccioni, The D-brane U-scan, arXiv:1109.1725 [INSPIRE].
J. de Boer and M. Shigemori, Exotic branes in string theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].
M. Graña and D. Marques, Gauged double field theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].
D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring double field theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].
G. Dibitetto, J. Fernandez-Melgarejo, D. Marques and D. Roest, Duality orbits of non-geometric fluxes, Fortsch. Phys. 60 (2012) 1123 [arXiv:1203.6562] [INSPIRE].
O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].
D.S. Berman and K. Lee, Supersymmetry for gauged double field theory and generalised Scherk-Schwarz reductions, Nucl. Phys. B 881 (2014) 369 [arXiv:1305.2747] [INSPIRE].
M. Graña and F. Orsi, N = 1 vacua in exceptional generalized geometry, JHEP 08 (2011) 109 [arXiv:1105.4855] [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].
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Aldazabal, G., Graña, M., Marqués, D. et al. The gauge structure of exceptional field theories and the tensor hierarchy. J. High Energ. Phys. 2014, 49 (2014). https://doi.org/10.1007/JHEP04(2014)049
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DOI: https://doi.org/10.1007/JHEP04(2014)049