Abstract
The recent investigation of the gauge structure of extended geometry is generalised to situations when ancillary transformations appear in the commutator of two generalised diffeomorphisms. The relevant underlying algebraic structure turns out to be a tensor hierarchy algebra rather than a Borcherds superalgebra. This tensor hierarchy algebra is a non-contragredient superalgebra, generically infinite-dimensional, which is a double extension of the structure algebra of the extended geometry. We use it to perform a (partial) analysis of the gauge structure in terms of an L∞ algebra for extended geometries based on finite-dimensional structure groups. An invariant pseudo-action is also given in these cases. We comment on the continuation to infinite-dimensional structure groups. An accompanying paper [1] deals with the mathematical construction of the tensor hierarchy algebras.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Cederwall and J. Palmkvist, Tensor hierarchy algebras and extended geometry. Part I. Construction of the algebra, JHEP 02 (2020) 144 [arXiv:1908.08695] [INSPIRE].
M. Cederwall and J. Palmkvist, Extended geometries, JHEP 02 (2018) 071 [arXiv:1711.07694] [INSPIRE].
M. Cederwall and J. Palmkvist, L∞ Algebras for Extended Geometry from Borcherds Superalgebras, Commun. Math. Phys. 369 (2019) 721 [arXiv:1804.04377] [INSPIRE].
J. Palmkvist, Exceptional geometry and Borcherds superalgebras, JHEP 11 (2015) 032 [arXiv:1507.08828] [INSPIRE].
M. Cederwall and J. Palmkvist, Superalgebras, constraints and partition functions, JHEP 08 (2015) 036 [arXiv:1503.06215] [INSPIRE].
J. Palmkvist, The tensor hierarchy algebra, J. Math. Phys. 55 (2014) 011701 [arXiv:1305.0018] [INSPIRE].
L. Carbone, M. Cederwall and J. Palmkvist, Generators and relations for Lie superalgebras of Cartan type, J. Phys. A 52 (2019) 055203 [arXiv:1802.05767] [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, Manifest duality in low-energy superstrings, in International Conference on Strings 93, Berkeley, California, 24–29 May 1993, pp. 353–363 (1993) [hep-th/9308133] [INSPIRE].
N. Hitchin, Lectures on generalized geometry, arXiv:1008.0973 [INSPIRE].
C.M. Hull, A Geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].
C.M. Hull, Doubled Geometry and T-Folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [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].
I. Jeon, K. Lee, J.-H. Park and Y. Suh, Stringy Unification of Type IIA and IIB Supergravities under N = 2 D = 10 Supersymmetric Double Field Theory, Phys. Lett. B 723 (2013) 245 [arXiv:1210.5078] [INSPIRE].
J.-H. Park, Comments on double field theory and diffeomorphisms, JHEP 06 (2013) 098 [arXiv:1304.5946] [INSPIRE].
D.S. Berman, M. Cederwall and M.J. Perry, Global aspects of double geometry, JHEP 09 (2014) 066 [arXiv:1401.1311] [INSPIRE].
M. Cederwall, The geometry behind double geometry, JHEP 09 (2014) 070 [arXiv:1402.2513] [INSPIRE].
M. Cederwall, T-duality and non-geometric solutions from double geometry, Fortsch. Phys. 62 (2014) 942 [arXiv:1409.4463] [INSPIRE].
M. Cederwall, Double supergeometry, JHEP 06 (2016) 155 [arXiv:1603.04684] [INSPIRE].
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, E7(7) and d = 11 supergravity, Ph.D. Thesis, Humboldt-Universität zu Berlin (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, Ed(d) × 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: Ed(d) × 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].
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].
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].
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].
M. Cederwall and J.A. Rosabal, E8 geometry, JHEP 07 (2015) 007 [arXiv:1504.04843] [INSPIRE].
D. Butter, H. Samtleben and E. Sezgin, E7(7) Exceptional Field Theory in Superspace, JHEP 01 (2019) 087 [arXiv:1811.00038] [INSPIRE].
G. Bossard, M. Cederwall, A. Kleinschmidt, J. Palmkvist and H. Samtleben, Generalized diffeomorphisms for E9 , Phys. Rev. D 96 (2017) 106022 [arXiv:1708.08936] [INSPIRE].
G. Bossard, F. Ciceri, G. Inverso, A. Kleinschmidt and H. Samtleben, E9 exceptional field theory. Part I. The potential, JHEP 03 (2019) 089 [arXiv:1811.04088] [INSPIRE].
G. Bossard, A. Kleinschmidt and E. Sezgin, On supersymmetric E11 exceptional field theory, JHEP 10 (2019) 165 [arXiv:1907.02080] [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 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].
O. Hohm and H. Samtleben, U-duality covariant gravity, JHEP 09 (2013) 080 [arXiv:1307.0509] [INSPIRE].
G. Bossard, A. Kleinschmidt, J. Palmkvist, C.N. Pope and E. Sezgin, Beyond E11 , JHEP 05 (2017) 020 [arXiv:1703.01305] [INSPIRE].
T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys. 32 (1993) 1087 [hep-th/9209099] [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].
O. Hohm and B. Zwiebach, L∞ Algebras and Field Theory, Fortsch. Phys. 65 (2017) 1700014 [arXiv:1701.08824] [INSPIRE].
D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, math/9802118.
V.G. Kac, Lie Superalgebras, Adv. Math. 26 (1977) 8 [INSPIRE].
A. Deser and C. Sämann, Extended Riemannian Geometry I: Local Double Field Theory, arXiv:1611.02772 [INSPIRE].
A. Deser and C. Sämann, Derived Brackets and Symmetries in Generalized Geometry and Double Field Theory, PoS(CORFU2017)141 (2018) [arXiv:1803.01659] [INSPIRE].
O. Hohm and H. Samtleben, Leibniz-Chern-Simons Theory and Phases of Exceptional Field Theory, Commun. Math. Phys. 369 (2019) 1055 [arXiv:1805.03220] [INSPIRE].
O. Hohm and H. Samtleben, Higher Gauge Structures in Double and Exceptional Field Theory, Fortsch. Phys. 67 (2019) 1910008 [arXiv:1903.02821] [INSPIRE].
R. Bonezzi and O. Hohm, Leibniz Gauge Theories and Infinity Structures, arXiv:1904.11036 [INSPIRE].
S. Lavau and J. Palmkvist, Infinity-enhancing of Leibniz algebras, arXiv:1907.05752 [INSPIRE].
J. Greitz, P. Howe and J. Palmkvist, The tensor hierarchy simplified, Class. Quant. Grav. 31 (2014) 087001 [arXiv:1308.4972] [INSPIRE].
S. Lavau, Tensor hierarchies and Leibniz algebras, J. Geom. Phys. 144 (2019) 147 [arXiv:1708.07068] [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: 1908.08696
Rights and permissions
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.
About this article
Cite this article
Cederwall, M., Palmkvist, J. Tensor hierarchy algebras and extended geometry. Part II. Gauge structure and dynamics. J. High Energ. Phys. 2020, 145 (2020). https://doi.org/10.1007/JHEP02(2020)145
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2020)145