Abstract
We formalise the teleparallel version of extended geometry (including gravity) by the introduction of a complex, the differential of which provides the linearised dynamics. The main point is the natural replacement of the two-derivative equations of motion by a differential which only contains terms of order 0 and 1 in derivatives. Second derivatives arise from homotopy transfer (elimination of fields with algebraic equations of motion). The formalism has the advantage of providing a clear consistency relation for the algebraic part of the differential, the “dualisation”, which then defines the dynamics of physical fields. It remains unmodified in the interacting BV theory, and the full non-linear models arise from covariantisation. A consequence of the use of the complex is that symmetry under local rotations becomes as good as manifest, instead of arising for a specific combination of tensorial terms, for less obvious reasons. We illustrate with a derivation of teleparallel Ehlers geometry, where the extended coordinate module is the adjoint module of a finite-dimensional simple Lie group.
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References
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].
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, Tensor hierarchy algebras and extended geometry. Part II. Gauge structure and dynamics, JHEP 02 (2020) 145 [arXiv:1908.08696] [INSPIRE].
M. Cederwall and J. Palmkvist, Teleparallelism in the algebraic approach to extended geometry, JHEP 04 (2022) 164 [arXiv:2112.08403] [INSPIRE].
G. Bossard et al., Extended geometry of magical supergravities, arXiv:2301.10974 [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 the proceedings of the International Conference on Strings 93, Berkeley, California, 24–29 May 1993, p. 353–363 [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 et al., Generalized diffeomorphisms for E9, Phys. Rev. D 96 (2017) 106022 [arXiv:1708.08936] [INSPIRE].
G. Bossard et al., 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].
G. Bossard et al., E9 exceptional field theory. Part II. The complete dynamics, JHEP 05 (2021) 107 [arXiv:2103.12118] [INSPIRE].
G. Bossard, A. Kleinschmidt and E. Sezgin, A master exceptional field theory, JHEP 06 (2021) 185 [arXiv:2103.13411] [INSPIRE].
J. Palmkvist, Exceptional geometry and Borcherds superalgebras, JHEP 11 (2015) 032 [arXiv:1507.08828] [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].
M. Cederwall and J. Palmkvist, Tensor Hierarchy Algebra Extensions of Over-Extended Kac-Moody Algebras, Commun. Math. Phys. 389 (2022) 571 [arXiv:2103.02476] [INSPIRE].
M. Cederwall and J. Palmkvist, Tensor hierarchy algebras and restricted associativity, arXiv:2207.12417 [INSPIRE].
A.M. Zeitlin, Conformal Field Theory and Algebraic Structure of Gauge Theory, JHEP 03 (2010) 056 [arXiv:0812.1840] [INSPIRE].
M. Rocek and A.M. Zeitlin, Homotopy algebras of differential (super)forms in three and four dimensions, Lett. Math. Phys. 108 (2018) 2669 [arXiv:1702.03565] [INSPIRE].
M. Reiterer, A homotopy BV algebra for Yang-Mills and color-kinematics, arXiv:1912.03110 [INSPIRE].
S.V. Lapin, Differential perturbations and d∞-differential modules, Sbornik Math. 192 (2001) 1639.
M. Cederwall and A. Karlsson, Pure spinor superfields and Born-Infeld theory, JHEP 11 (2011) 134 [arXiv:1109.0809] [INSPIRE].
M. Cederwall, Towards a manifestly supersymmetric action for 11-dimensional supergravity, JHEP 01 (2010) 117 [arXiv:0912.1814] [INSPIRE].
M. Cederwall, D = 11 supergravity with manifest supersymmetry, Mod. Phys. Lett. A 25 (2010) 3201 [arXiv:1001.0112] [INSPIRE].
M. Cederwall, Pure spinor superfields — an overview, Springer Proc. Phys. 153 (2014) 61 [arXiv:1307.1762] [INSPIRE].
M. Cederwall, Pure spinors in classical and quantum supergravity, arXiv:2210.06141 [INSPIRE].
R. Eager, F. Hahner, I. Saberi and B.R. Williams, Perspectives on the pure spinor superfield formalism, J. Geom. Phys. 180 (2022) 104626 [arXiv:2111.01162] [INSPIRE].
V.C. De Andrade, L.C.T. Guillen and J.G. Pereira, Teleparallel gravity: An Overview, in the proceedings of the 9th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 9), (2000) [gr-qc/0011087] [INSPIRE].
A. Golovnev, The geometrical meaning of the Weitzenböck connection, arXiv:2302.13599 [INSPIRE].
A. Kleinschmidt, H. Nicolai and J. Palmkvist, K(E9) from K(E10), JHEP 06 (2007) 051 [hep-th/0611314] [INSPIRE].
A. Kleinschmidt, H. Nicolai and A. Viganò, On spinorial representations of involutory subalgebras of Kac-Moody algebras, arXiv:1811.11659 [INSPIRE].
A. Kleinschmidt, R. Köhl, R. Lautenbacher and H. Nicolai, Representations of Involutory Subalgebras of Affine Kac-Moody Algebras, Commun. Math. Phys. 392 (2022) 89 [arXiv:2102.00870] [INSPIRE].
Acknowledgments
The authors would like to thank Michael Reiterer, Olaf Hohm, Ingmar Saberi and Guillaume Bossard for stimulating discussions and input. Part of this work was done during the workshop on Higher Structures, Gravity and Fields at the Mainz Institute for Theoretical Physics of the DFG Cluster of Excellence PRISMA+ (Project ID 39083149). We thank the institute for its hospitality.
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Cederwall, M., Palmkvist, J. The teleparallel complex. J. High Energ. Phys. 2023, 68 (2023). https://doi.org/10.1007/JHEP05(2023)068
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DOI: https://doi.org/10.1007/JHEP05(2023)068