Abstract
In this work we revisit the \( {E}_8\times {\mathrm{\mathbb{R}}}^{+} \) generalised Lie derivative encoding the algebra of diffeomorphisms and gauge transformations of compactifications of M-theory on eight-dimensional manifolds, by extending certain features of the \( {E}_7\times {\mathrm{\mathbb{R}}}^{+} \) one. Compared to its \( {E}_d\times {\mathrm{\mathbb{R}}}^{+} \), d ≤ 7 counterparts, a new term is needed for consistency. However, we find that no compensating parameters need to be introduced, but rather that the new term can be written in terms of the ordinary generalised gauge parameters by means of a connection. This implies that no further degrees of freedom, beyond those of the field content of the E 8 group, are needed to have a well defined theory. We discuss the implications of the structure of the \( {E}_8\times {\mathrm{\mathbb{R}}}^{+} \) generalised transformation on the construction of the d = 8 generalised geometry. Finally, we suggest how to lift the generalised Lie derivative to eleven dimensions.
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References
C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [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].
C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
M.J. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].
A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].
A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math.DG/0209099] [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].
M. Graña and D. Marques, Gauged double field theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].
G. Dibitetto, J.J. Fernandez-Melgarejo, D. Marques and D. Roest, Duality orbits of non-geometric fluxes, Fortschr. Phys. 60 (2012) 1123 [arXiv:1203.6562] [INSPIRE].
G. Aldazabal, D. Marques and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].
R. Blumenhagen, F. Hassler and D. Lüst, Double field theory on group manifolds, JHEP 02 (2015) 001 [arXiv:1410.6374] [INSPIRE].
A. Betz, R. Blumenhagen, D. Lüst and F. Rennecke, A note on the CFT origin of the strong constraint of DFT, JHEP 05 (2014) 044 [arXiv:1402.1686] [INSPIRE].
M. Cederwall, The geometry behind double geometry, JHEP 09 (2014) 070 [arXiv:1402.2513] [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].
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].
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, H. Samtleben and M. Trigiante, The maximal D = 4 supergravities, JHEP 06 (2007) 049 [arXiv:0705.2101] [INSPIRE].
A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, arXiv:1401.3360 [INSPIRE].
W.H. Baron, Gaugings from E 7(7) extended geometries, Phys. Rev. D 91 (2015) 024008 [arXiv:1404.7750] [INSPIRE].
J. Berkeley, D.S. Berman and F.J. Rudolph, Strings and branes are waves, JHEP 06 (2014) 006 [arXiv:1403.7198] [INSPIRE].
O.A. Bedoya, D. Marques and C. Núñez, Heterotic α′-corrections in double field theory, JHEP 12 (2014) 074 [arXiv:1407.0365] [INSPIRE].
D.S. Berman and F.J. Rudolph, Branes are waves and monopoles, JHEP 05 (2015) 015 [arXiv:1409.6314] [INSPIRE].
H. Godazgar, M. Godazgar and M.J. Perry, E 8 duality and dual gravity, JHEP 06 (2013) 044 [arXiv:1303.2035] [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].
B. de Wit and H. Samtleben, Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems, Fortschr. Phys. 53 (2005) 442 [hep-th/0501243] [INSPIRE].
B. de Wit, H. Samtleben and M. Trigiante, Magnetic charges in local field theory, JHEP 09 (2005) 016 [hep-th/0507289] [INSPIRE].
B. de Wit, H. Nicolai and H. Samtleben, Gauged supergravities, tensor hierarchies and M-theory, JHEP 02 (2008) 044 [arXiv:0801.1294] [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].
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].
A. Le Diffon and H. Samtleben, Supergravities without an action: gauging the trombone, Nucl. Phys. B 811 (2009) 1 [arXiv:0809.5180] [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].
F. Riccioni, Local E 11 and the gauging of the trombone symmetry, Class. Quant. Grav. 27 (2010) 125009 [arXiv:1001.1316] [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].
A. Le Diffon, H. Samtleben and M. Trigiante, N = 8 supergravity with local scaling symmetry, JHEP 04 (2011) 079 [arXiv:1103.2785] [INSPIRE].
O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, JHEP 05 (2012) 126 [arXiv:1112.5296] [INSPIRE].
É. Cartan, Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. 31 (1914) 263.
C. Strickland-Constable, Subsectors, Dynkin diagrams and new generalised geometries, arXiv:1310.4196 [INSPIRE].
T. Curtright, Generalized gauge fields, Phys. Lett. B 165 (1985) 304 [INSPIRE].
C.S. Aulakh, I.G. Koh and S. Ouvry, Higher spin fields with mixed symmetry, Phys. Lett. B 173 (1986) 284 [INSPIRE].
B. Zwiebach, Closed string field theory: quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [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 and B. Zwiebach, Double field theory at order α′, JHEP 11 (2014) 075 [arXiv:1407.3803] [INSPIRE].
A.G. Tumanov and P. West, Generalised vielbeins and non-linear realisations, JHEP 10 (2014) 009 [arXiv:1405.7894] [INSPIRE].
P. West, Generalised space-time and gauge transformations, JHEP 08 (2014) 050 [arXiv:1403.6395] [INSPIRE].
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Rosabal, J.A. On the exceptional generalised Lie derivative for d ≥ 7. J. High Energ. Phys. 2015, 153 (2015). https://doi.org/10.1007/JHEP09(2015)153
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DOI: https://doi.org/10.1007/JHEP09(2015)153