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The gauge structure of generalised diffeomorphisms

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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.

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References

  1. C. Hull and P. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. E. Cremmer, B. Julia, H. Lü and C. Pope, Dualization of dualities. 1., Nucl. Phys. B 523 (1998) 73 [hep-th/9710119] [INSPIRE].

    Article  ADS  Google Scholar 

  3. N. Obers and B. Pioline, U-duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. B. Julia, Group disintegrations, invited paper presented at Nuffield Gravity Workshop, Cambridge England, 22 June - 12 July 1980.

  5. H. Nicolai, The integrability of N = 16 supergravity, Phys. Lett. B 194 (1987) 402 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  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).

  7. T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145 [hep-th/0212256] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. T. Damour and H. Nicolai, Symmetries,Singularities and the De-Emergence of Space, arXiv:0705.2643 [INSPIRE].

  9. T. Damour, M. Henneaux and H. Nicolai, E 10 and a ’small tension expansion’ of M-theory, Phys. Rev. Lett. 89 (2002) 221601 [hep-th/0207267] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. A. Kleinschmidt and H. Nicolai, E 10 and SO(9, 9) invariant supergravity, JHEP 07 (2004) 041 [hep-th/0407101] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  12. P.C. West, E 11 , SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].

    ADS  Google Scholar 

  13. A. Kleinschmidt and P.C. West, Representations of G +++ and the role of space-time, JHEP 02 (2004) 033 [hep-th/0312247] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. P.C. West, E 11 origin of brane charges and U-duality multiplets, JHEP 08 (2004) 052 [hep-th/0406150] [INSPIRE].

    Article  ADS  Google Scholar 

  15. F. Englert, L. Houart, A. Taormina and P.C. West, The symmetry of M theories, JHEP 09 (2003) 020 [hep-th/0304206] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. B. de Wit and H. Nicolai, D = 11 supergravity with local SU(8) invariance, Nucl. Phys. B 274 (1986) 363 [INSPIRE].

    Article  ADS  Google Scholar 

  17. B. de Wit, M theory duality and BPS extended supergravity, Int. J. Mod. Phys. A 16 (2001) 1002 [hep-th/0010292] [INSPIRE].

    ADS  Google Scholar 

  18. P.C. West, Hidden superconformal symmetry in M-theory, JHEP 08 (2000) 007 [hep-th/0005270] [INSPIRE].

    Article  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. B. de Wit and H. Nicolai, Hidden symmetries, central charges and all that, Class. Quant. Grav. 18 (2001) 3095 [hep-th/0011239] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  21. C. Hillmann, Generalized E 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. C. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  23. C. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].

    Article  ADS  Google Scholar 

  24. P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. N. Hitchin, Lectures on generalized geometry, arXiv:1008.0973 [INSPIRE].

  26. C.M. Hull, Doubled Geometry and T-Folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, JHEP 05 (2012) 126 [arXiv:1112.5296].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  34. I. Jeon, K. Lee and J.-H. Park, Stringy differential geometry, beyond Riemann, Phys. Rev. D 84 (2011) 044022 [arXiv:1105.6294] [INSPIRE].

    ADS  Google Scholar 

  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. O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  37. M. Cederwall, M-branes on U-folds, arXiv:0712.4287 [INSPIRE].

  38. D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. V. Bengtsson, M. Cederwall, H. Larsson and B.E. Nilsson, U-duality covariant membranes, JHEP 02 (2005) 020 [hep-th/0406223] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  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].

    MathSciNet  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  45. A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) × R + Generalised Geometry, Connections and M-theory, arXiv:1112.3989 [INSPIRE].

  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].

    Article  ADS  Google Scholar 

  47. O. Hohm, S.K. Kwak and B. Zwiebach, Double field theory of type II strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. A. Coimbra, C. Strickland-Constable and D. Waldram, Generalised geometry and type-II supergravity, Fortsch. Phys. 60 (2012) 982 [arXiv:1202.3170] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  49. I. Jeon, K. Lee and J.-H. Park, Incorporation of fermions into double field theory, JHEP 11 (2011) 025 [arXiv:1109.2035] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. O. Hohm and S.K. Kwak, N = 1 supersymmetric double field theory, JHEP 03 (2012) 080 [arXiv:1111.7293] [INSPIRE].

    Article  ADS  Google Scholar 

  51. B. de Wit, H. Nicolai and H. Samtleben, Gauged supergravities, tensor hierarchies and M-theory, JHEP 02 (2008) 044 [arXiv:0801.1294] [INSPIRE].

    Article  Google Scholar 

  52. B. de Wit and H. Samtleben, The end of the p-form hierarchy, JHEP 08 (2008) 015 [arXiv:0805.4767] [INSPIRE].

    Article  Google Scholar 

  53. A. Kleinschmidt, Counting supersymmetric branes, JHEP 10 (2011) 144 [arXiv:1109.2025] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  54. P. Henry-Labordere, B. Julia and L. Paulot, Borcherds symmetries in M-theory, JHEP 04 (2002) 049 [hep-th/0203070] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  55. M. Henneaux, B.L. Julia and J. Levie, E 11 , Borcherds algebras and maximal supergravity, JHEP 04 (2012) 078 [arXiv:1007.5241] [INSPIRE].

    Article  ADS  Google Scholar 

  56. J. Palmkvist, Tensor hierarchies, Borcherds algebras and E 11, JHEP 02 (2012) 066 [arXiv:1110.4892] [INSPIRE].

    Article  ADS  Google Scholar 

  57. P. West, Generalised BPS conditions, Mod. Phys. Lett. A 27 (2012) 1250202 [arXiv:1208.3397] [INSPIRE].

    ADS  Google Scholar 

  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].

    Article  ADS  Google Scholar 

  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].

    Article  MathSciNet  ADS  Google Scholar 

  60. M. Chesterman, Ghost constraints and the covariant quantization of the superparticle in ten-dimensions, JHEP 02 (2004) 011 [hep-th/0212261] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. N. Berkovits and N. Nekrasov, The character of pure spinors, Lett. Math. Phys. 74 (2005) 75 [hep-th/0503075] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. M. Cederwall, Pure spinor superfields, with application to D = 3 conformal models, arXiv:0906.5490 [INSPIRE].

  63. A.M. Cohen, M. van Leeuwen and B. Lisser, LiE v. 2.2 (1998), http://www-math.univ-poitiers.fr/∼maavl/LiE/.

  64. M. Cederwall, Jordan algebra dynamics, Phys. Lett. B 210 (1988) 169 [INSPIRE].

    MathSciNet  Google Scholar 

  65. M. Günaydin, Generalized conformal and superconformal group actions and Jordan algebras, Mod. Phys. Lett. A 8 (1993) 1407 [hep-th/9301050] [INSPIRE].

    ADS  Google Scholar 

  66. M Günaydin, K. Koepsell and H. Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Commun. Math. Phys. 221 (2001) 57.

    Article  Google Scholar 

  67. B. Pioline and A. Waldron, The automorphic membrane, JHEP 06 (2004) 009 [hep-th/0404018] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  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. T. Damour, A. Kleinschmidt and H. Nicolai, K(E 10 ), supergravity and fermions, JHEP 08 (2006) 046 [hep-th/0606105] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Centre for Research in String Theory, School of Physics, Queen Mary College, University of London, Mile End Road, London, E1 4NS, England

    David S. Berman

  2. Department of Fundamental Physics, Chalmers University of Technology, SE 412 96, Gothenburg, Sweden

    Martin Cederwall

  3. Max-Planck-Institut für Gravitationsphysik (Albert Einstein Institut), Am Mühlenberg 1, 14476, Golm, Germany

    Axel Kleinschmidt

  4. International Solvay Institutes, Campus Plaine C.P. 231, Boulevard du Triomphe, 1050, Bruxelles, Belgium

    Axel Kleinschmidt & Daniel C. Thompson

  5. Theoretische Natuurkunde, Vrije Universiteit Brussel, International Solvay Institutes, Pleinlaan 2, B-1050, Brussels, Belgium

    Daniel C. Thompson

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  1. David S. Berman
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Correspondence to Martin Cederwall.

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ArXiv ePrint: 1208.5884

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Berman, D.S., Cederwall, M., Kleinschmidt, A. et al. The gauge structure of generalised diffeomorphisms. J. High Energ. Phys. 2013, 64 (2013). https://doi.org/10.1007/JHEP01(2013)064

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