The gauge structure of generalised diffeomorphisms

  • David S. Berman
  • Martin Cederwall
  • Axel Kleinschmidt
  • Daniel C. Thompson
Article

Abstract

We investigate the generalised diffeomorphisms in M-theory, which are gauge transformations unifying diffeomorphisms and tensor gauge transformations. After giving an E n(n)-covariant description of the gauge transformations and their commutators, we show that the gauge algebra is infinitely reducible, i.e., the tower of ghosts for ghosts is infinite. The Jacobiator of generalised diffeomorphisms gives such a reducibility transformation. We give a concrete description of the ghost structure, and demonstrate that the infinite sums give the correct (regularised) number of degrees of freedom. The ghost towers belong to the sequences of representations previously observed appearing in tensor hierarchies and Borcherds algebras. All calculations rely on the section condition, which we reformulate as a linear condition on the cotangent directions. The analysis holds for n < 8. At n = 8, where the dual gravity field becomes relevant, the natural guess for the gauge parameter and its reducibility still yields the correct counting of gauge parameters.

Keywords

Space-Time Symmetries M-Theory 

References

  1. [1]
    C. Hull and P. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    E. Cremmer, B. Julia, H. Lü and C. Pope, Dualization of dualities. 1., Nucl. Phys. B 523 (1998) 73 [hep-th/9710119] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    N. Obers and B. Pioline, U-duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    B. Julia, Group disintegrations, invited paper presented at Nuffield Gravity Workshop, Cambridge England, 22 June - 12 July 1980.Google Scholar
  5. [5]
    H. Nicolai, The integrability of N = 16 supergravity, Phys. Lett. B 194 (1987) 402 [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    B. Julia, Kac-Moody symmetry of gravitation and supergravity theories, in Lectures in Applied Mathematics. Vol. 21, AMS Publishing, New York U.S.A. (1985).Google Scholar
  7. [7]
    T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145 [hep-th/0212256] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    T. Damour and H. Nicolai, Symmetries,Singularities and the De-Emergence of Space, arXiv:0705.2643 [INSPIRE].
  9. [9]
    T. Damour, M. Henneaux and H. Nicolai, E 10 and asmall tension expansionof M-theory, Phys. Rev. Lett. 89 (2002) 221601 [hep-th/0207267] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    A. Kleinschmidt and H. Nicolai, E 10 and SO(9, 9) invariant supergravity, JHEP 07 (2004) 041 [hep-th/0407101] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  12. [12]
    P.C. West, E 11 , SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].ADSGoogle Scholar
  13. [13]
    A. Kleinschmidt and P.C. West, Representations of G +++ and the role of space-time, JHEP 02 (2004) 033 [hep-th/0312247] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    P.C. West, E 11 origin of brane charges and U-duality multiplets, JHEP 08 (2004) 052 [hep-th/0406150] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    F. Englert, L. Houart, A. Taormina and P.C. West, The symmetry of M theories, JHEP 09 (2003) 020 [hep-th/0304206] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    B. de Wit and H. Nicolai, D = 11 supergravity with local SU(8) invariance, Nucl. Phys. B 274 (1986) 363 [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    B. de Wit, M theory duality and BPS extended supergravity, Int. J. Mod. Phys. A 16 (2001) 1002 [hep-th/0010292] [INSPIRE].ADSGoogle Scholar
  18. [18]
    P.C. West, Hidden superconformal symmetry in M-theory, JHEP 08 (2000) 007 [hep-th/0005270] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    K. Koepsell, H. Nicolai and H. Samtleben, An exceptional geometry for D = 11 supergravity?, Class. Quant. Grav. 17 (2000) 3689 [hep-th/0006034] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  20. [20]
    B. de Wit and H. Nicolai, Hidden symmetries, central charges and all that, Class. Quant. Grav. 18 (2001) 3095 [hep-th/0011239] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  21. [21]
    C. Hillmann, Generalized E 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    C. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    C. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    N. Hitchin, Lectures on generalized geometry, arXiv:1008.0973 [INSPIRE].
  26. [26]
    C.M. Hull, Doubled Geometry and T-Folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, JHEP 05 (2012) 126 [arXiv:1112.5296].ADSCrossRefGoogle Scholar
  30. [30]
    D. Andriot, M. Larfors, D. Lüst and P. Patalong, A ten-dimensional action for non-geometric fluxes, JHEP 09 (2011) 134 [arXiv:1106.4015] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, A geometric action for non-geometric fluxes, Phys. Rev. Lett. 108 (2012) 261602 [arXiv:1202.3060] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, Non-geometric fluxes in supergravity and double field theory, Fortsch. Phys. 60 (2012) 1150 [arXiv:1204.1979] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    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].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    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
  35. [35]
    I. Jeon, K. Lee and J.-H. Park, Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity, Phys. Rev. D 85 (2012) 081501 [Erratum ibid. D 86 (2012) 089903] [arXiv:1112.0069] [INSPIRE].
  36. [36]
    O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    M. Cederwall, M-branes on U-folds, arXiv:0712.4287 [INSPIRE].
  38. [38]
    D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    V. Bengtsson, M. Cederwall, H. Larsson and B.E. Nilsson, U-duality covariant membranes, JHEP 02 (2005) 020 [hep-th/0406223] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    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].ADSCrossRefGoogle Scholar
  41. [41]
    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].MathSciNetADSGoogle Scholar
  42. [42]
    D.S. Berman, E.T. Musaev and M.J. Perry, Boundary terms in generalized geometry and doubled field theory, Phys. Lett. B 706 (2011) 228 [arXiv:1110.3097] [INSPIRE].MathSciNetADSGoogle Scholar
  43. [43]
    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].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    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].ADSCrossRefGoogle Scholar
  45. [45]
    A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) × R + Generalised Geometry, Connections and M-theory, arXiv:1112.3989 [INSPIRE].
  46. [46]
    O. Hohm, S.K. Kwak and B. Zwiebach, Unification of Type II Strings and T-duality, Phys. Rev. Lett. 107 (2011) 171603 [arXiv:1106.5452] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    O. Hohm, S.K. Kwak and B. Zwiebach, Double field theory of type II strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Generalised geometry and type-II supergravity, Fortsch. Phys. 60 (2012) 982 [arXiv:1202.3170] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  49. [49]
    I. Jeon, K. Lee and J.-H. Park, Incorporation of fermions into double field theory, JHEP 11 (2011) 025 [arXiv:1109.2035] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    O. Hohm and S.K. Kwak, N = 1 supersymmetric double field theory, JHEP 03 (2012) 080 [arXiv:1111.7293] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    B. de Wit, H. Nicolai and H. Samtleben, Gauged supergravities, tensor hierarchies and M-theory, JHEP 02 (2008) 044 [arXiv:0801.1294] [INSPIRE].CrossRefGoogle Scholar
  52. [52]
    B. de Wit and H. Samtleben, The end of the p-form hierarchy, JHEP 08 (2008) 015 [arXiv:0805.4767] [INSPIRE].CrossRefGoogle Scholar
  53. [53]
    A. Kleinschmidt, Counting supersymmetric branes, JHEP 10 (2011) 144 [arXiv:1109.2025] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    P. Henry-Labordere, B. Julia and L. Paulot, Borcherds symmetries in M-theory, JHEP 04 (2002) 049 [hep-th/0203070] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    M. Henneaux, B.L. Julia and J. Levie, E 11 , Borcherds algebras and maximal supergravity, JHEP 04 (2012) 078 [arXiv:1007.5241] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    J. Palmkvist, Tensor hierarchies, Borcherds algebras and E 11, JHEP 02 (2012) 066 [arXiv:1110.4892] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    P. West, Generalised BPS conditions, Mod. Phys. Lett. A 27 (2012) 1250202 [arXiv:1208.3397] [INSPIRE].ADSGoogle Scholar
  58. [58]
    E. Cremmer, H. Lü, C. Pope and K. Stelle, Spectrum generating symmetries for BPS solitons, Nucl. Phys. B 520 (1998) 132 [hep-th/9707207] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    K. Koepsell, H. Nicolai and H. Samtleben, On the Yangian [Y(E 8 )] quantum symmetry of maximal supergravity in two-dimensions, JHEP 04 (1999) 023 [hep-th/9903111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    M. Chesterman, Ghost constraints and the covariant quantization of the superparticle in ten-dimensions, JHEP 02 (2004) 011 [hep-th/0212261] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    N. Berkovits and N. Nekrasov, The character of pure spinors, Lett. Math. Phys. 74 (2005) 75 [hep-th/0503075] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  62. [62]
    M. Cederwall, Pure spinor superfields, with application to D = 3 conformal models, arXiv:0906.5490 [INSPIRE].
  63. [63]
    A.M. Cohen, M. van Leeuwen and B. Lisser, LiE v. 2.2 (1998), http://www-math.univ-poitiers.fr/∼maavl/LiE/.
  64. [64]
    M. Cederwall, Jordan algebra dynamics, Phys. Lett. B 210 (1988) 169 [INSPIRE].MathSciNetGoogle Scholar
  65. [65]
    M. Günaydin, Generalized conformal and superconformal group actions and Jordan algebras, Mod. Phys. Lett. A 8 (1993) 1407 [hep-th/9301050] [INSPIRE].ADSGoogle Scholar
  66. [66]
    M Günaydin, K. Koepsell and H. Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Commun. Math. Phys. 221 (2001) 57.CrossRefGoogle Scholar
  67. [67]
    B. Pioline and A. Waldron, The automorphic membrane, JHEP 06 (2004) 009 [hep-th/0404018] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  68. [68]
    M.B. Green, S.D. Miller and P. Vanhove, Small representations, string instantons and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa), arXiv:1111.2983 [INSPIRE].
  69. [69]
    T. Damour, A. Kleinschmidt and H. Nicolai, K(E 10 ), supergravity and fermions, JHEP 08 (2006) 046 [hep-th/0606105] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • David S. Berman
    • 1
  • Martin Cederwall
    • 2
  • Axel Kleinschmidt
    • 3
    • 4
  • Daniel C. Thompson
    • 4
    • 5
  1. 1.Centre for Research in String Theory, School of Physics, Queen Mary CollegeUniversity of LondonLondonEngland
  2. 2.Department of Fundamental PhysicsChalmers University of TechnologyGothenburgSweden
  3. 3.Max-Planck-Institut für Gravitationsphysik (Albert Einstein Institut)GolmGermany
  4. 4.International Solvay InstitutesBruxellesBelgium
  5. 5.Theoretische Natuurkunde, Vrije Universiteit Brussel, International Solvay InstitutesBrusselsBelgium

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