Journal of High Energy Physics

, 2019:19 | Cite as

Adjoint orbits, generalised parallelisable spaces and consistent truncations

  • Louise AndersonEmail author
Open Access
Regular Article - Theoretical Physics


The aim of this note is to present some new explicit examples of O(d, d)generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits \( \mathcal{O} \) of some semi-simple Lie group G. Using this construction, an explicit expression for a generalised frame is given in the case when the orbits are regular, but subtleties arise when they become degenerate. In the case of regular orbits, the resulting space is a globally flat fiber bundle over \( \mathcal{O} \) which can be made compact, allowing for a generalised Scherk-Schwartz reduction. This means these spaces should admit consistent supergravity truncations. For degenerate orbits, the procedure hinges on the existence of a suitable metric, allowing for a consistent normalisation of the generalised frame.


Differential and Algebraic Geometry Supergravity Models 


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.


  1. [1]
    J. Scherk and J.H. Schwarz, Spontaneous Breaking of Supersymmetry Through Dimensional Reduction, Phys. Lett. 82B (1979) 60 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    J. Scherk and J.H. Schwarz, How to Get Masses from Extra Dimensions, Nucl. Phys. B 153 (1979) 61 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    M.J. Duff, B.E.W. Nilsson and C.N. Pope, Kaluza-Klein Supergravity, Phys. Rept. 130 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, Consistent group and coset reductions of the bosonic string, Class. Quant. Grav. 20 (2003) 5161 [hep-th/0306043] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    B. de Wit and H. Nicolai, The Consistency of the S 7 Truncation in D = 11 Supergravity, Nucl. Phys. B 281 (1987) 211 [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11-D supergravity on AdS 7 × S 4 and selfduality in odd dimensions, Phys. Lett. B 469 (1999) 96 [hep-th/9905075] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS 7 × S 4 reduction and the origin of selfduality in odd dimensions, Nucl. Phys. B 581 (2000) 179 [hep-th/9911238] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Cvetič, H. Lü, C.N. Pope, A. Sadrzadeh and T.A. Tran, Consistent SO(6) reduction of type IIB supergravity on S 5, Nucl. Phys. B 586 (2000) 275 [hep-th/0003103] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Cvetič, H. Lü, C.N. Pope, A. Sadrzadeh and T.A. Tran, S 3 and S 4 reductions of type IIA supergravity, Nucl. Phys. B 590 (2000) 233 [hep-th/0005137] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E.T. Musaev, Gauged supergravities in 5 and 6 dimensions from generalised Scherk-Schwarz reductions, JHEP 05 (2013) 161 [arXiv:1301.0467] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Baguet, O. Hohm and H. Samtleben, Consistent Type IIB Reductions to Maximal 5D Supergravity, Phys. Rev. D 92 (2015) 065004 [arXiv:1506.01385] [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    B. de Wit and H. Nicolai, Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions, JHEP 05 (2013) 077 [arXiv:1302.6219] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. Godazgar, M. Godazgar and H. Nicolai, Embedding tensor of Scherk-Schwarz flux compactifications from eleven dimensions, Phys. Rev. D 89 (2014) 045009 [arXiv:1312.1061] [INSPIRE].ADSzbMATHGoogle Scholar
  16. [16]
    H. Godazgar, M. Godazgar and H. Nicolai, Testing the non-linear flux ansatz for maximal supergravity, Phys. Rev. D 87 (2013) 085038 [arXiv:1303.1013] [INSPIRE].ADSGoogle Scholar
  17. [17]
    H. Godazgar, M. Godazgar and H. Nicolai, Generalised geometry from the ground up, JHEP 02 (2014) 075 [arXiv:1307.8295] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    H. Godazgar, M. Godazgar and H. Nicolai, Nonlinear Kaluza-Klein theory for dual fields, Phys. Rev. D 88 (2013) 125002 [arXiv:1309.0266] [INSPIRE].ADSGoogle Scholar
  19. [19]
    K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017) 1700048 [arXiv:1401.3360] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    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].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    O. Hohm and H. Samtleben, Consistent Kaluza-Klein Truncations via Exceptional Field Theory, JHEP 01 (2015) 131 [arXiv:1410.8145] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Malek and H. Samtleben, Dualising consistent IIA/IIB truncations, JHEP 12 (2015) 029 [arXiv:1510.03433] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  23. [23]
    A. Baguet, C.N. Pope and H. Samtleben, Consistent Pauli reduction on group manifolds, Phys. Lett. B 752 (2016) 278 [arXiv:1510.08926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    G. Inverso, H. Samtleben and M. Trigiante, Type II supergravity origin of dyonic gaugings, Phys. Rev. D 95 (2017) 066020 [arXiv:1612.05123] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    E. Malek and H. Samtleben, Ten-dimensional origin of Minkowski vacua in N = 8 supergravity, Phys. Lett. B 776 (2018) 64 [arXiv:1710.02163] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    E. Malek, 7-dimensional \( \mathcal{N} \) = 2 Consistent Truncations using SL(5) Exceptional Field Theory, JHEP 06 (2017) 026 [arXiv:1612.01692] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    E. Malek, Half-Maximal Supersymmetry from Exceptional Field Theory, Fortsch. Phys. 65 (2017) 1700061 [arXiv:1707.00714] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  28. [28]
    P. du Bosque, F. Hassler and D. Lüst, Generalized parallelizable spaces from exceptional field theory, JHEP 01 (2018) 117 [arXiv:1705.09304] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    D.S. Berman and M.J. Perry, Generalized Geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    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].ADSCrossRefGoogle Scholar
  32. [32]
    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].ADSGoogle Scholar
  33. [33]
    O. Hohm and H. Samtleben, Exceptional field theory. II. E 7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].
  34. [34]
    O. Hohm and H. Samtleben, Exceptional field theory. III. E 8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].
  35. [35]
    A. Abzalov, I. Bakhmatov and E.T. Musaev, Exceptional field theory: SO(5, 5), JHEP 06 (2015) 088 [arXiv:1504.01523] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    E.T. Musaev, Exceptional field theory: SL(5), JHEP 02 (2016) 012 [arXiv:1512.02163] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
  38. [38]
    M. Gualtieri, Generalized complex geometry, math/0401221.
  39. [39]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry I: Type II Theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].ADSMathSciNetGoogle Scholar
  42. [42]
    O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  43. [43]
    I. Jeon, K. Lee and J.-H. Park, Differential geometry with a projection: Application to double field theory, JHEP 04 (2011) 014 [arXiv:1011.1324] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    I. Jeon, K. Lee and J.-H. Park, Stringy differential geometry, beyond Riemann, Phys. Rev. D 84 (2011) 044022 [arXiv:1105.6294] [INSPIRE].ADSGoogle Scholar
  45. [45]
    C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    F. Hassler, The Topology of Double Field Theory, JHEP 04 (2018) 128 [arXiv:1611.07978] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    G. Inverso, Generalised Scherk-Schwarz reductions from gauged supergravity, JHEP 12 (2017) 124 [arXiv:1708.02589] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    A. Besse, Einstein Manifolds, Classics in mathematics. Springer Berlin Heidelberg (1987).Google Scholar
  52. [52]
    P. Crooks, Complex adjoint orbits in lie theory and geometry, Expo. Math. (2018), in press.Google Scholar
  53. [53]
    C.-L. Terng, Isoparametric submanifolds and their coxeter groups, J. Diff. Geom. 21 (1985) 79.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    A.V. Bolsinov and B. Jovanovic, Magnetic Flows on Homogeneous Spaces, math-ph/0609005.
  55. [55]
    G. Rudolph and M. Schmidt, Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics, Springer Netherlands (2017).Google Scholar
  56. [56]
    J. Bernatska and P. Holod, Geometry and topology of coadjoint orbits of semisimple lie groups, in Proceedings of the Ninth International Conference on Geometry, Integrability and Quantization, Sofia, Bulgaria, pp. 146–166, Softex (2008) [].
  57. [57]
    O. de Felice, Generalised Geometry, Parallelisations and Consistent Truncations, Imperial College London, Department of Physics, September (2014).Google Scholar
  58. [58]
    L.J. Boya, A.M. Perelomov and M. Santander, Berry phase in homogeneous Kähler manifolds with linear Hamiltonians, J. Math. Phys. 42 (2001) 5130 [math-ph/0111022].
  59. [59]
    E. Inonu and E.P. Wigner, On the Contraction of Groups and Their Representations, Proc. Natl. Acad. Sci. USA 36 (1953) 510.ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.The Blackett LaboratoryImperial College LondonLondonUnited Kingdom

Personalised recommendations