Generalized geometry and M theory

Article

Abstract

We reformulate the Hamiltonian form of bosonic eleven dimensional supergravity in terms of an object that unifies the three-form and the metric. For the case of four spatial dimensions, the duality group is manifest and the metric and C-field are on an equal footing even though no dimensional reduction is required for our results to hold. One may also describe our results using the generalized geometry that emerges from membrane duality. The relationship between the twisted Courant algebra and the gauge symmetries of eleven dimensional supergravity are described in detail.

Keywords

M-Theory Space-Time Symmetries p-branes 

References

  1. [1]
    B. Julia, Group disintegrations, in Superspace and supergravity: proceedings of the Nuffield Workshop, Cambridge 1980, S.W. Hawking and M. Rocek eds., Cambridge University Press, Cambridge U.K. (1981).Google Scholar
  2. [2]
    B. Julia, Gravity, supergravities and integrable systems, in Group Theoretical Methods in Physics: Proceedings, Istanbul, Turkey 1982, M. Serdaroglu and E. Inonu eds., Spinger, U.S.A. (1983).Google Scholar
  3. [3]
    J. Thierry-Mieg and B. Morel, Superalgebras in exceptional gravity, in Superspace and supergravity: proceedings of the Nuffield Workshop, Cambridge 1980, S.W. Hawking and M. Rocek eds., Cambridge University Press, Cambridge U.K. (1981).Google Scholar
  4. [4]
    E. Cremmer, Supergravities in 5 dimensions, in Supergravities in diverse dimensions, volume 1, A. Salam and E. Sezgin, World Scientific, Singapore (1989).Google Scholar
  5. [5]
    G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries, Commun. Math. Phys. 66 (1979) 291 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    B. de Wit and H. Nicolai, D = 11 supergravity with local SU(8) invariance, Nucl. Phys. B 274 (1986) 363 [SPIRES].ADSCrossRefGoogle Scholar
  7. [7]
    H. Nicolai, D = 11 supergravity with local SO(16) invariance, Phys. Lett. B 187 (1987) 316 [SPIRES].MathSciNetADSGoogle Scholar
  8. [8]
    K. Koepsell, H. Nicolai and H. Samtleben, An exceptional geometry for D = 11 supergravity?, Class. Quant. Grav. 17 (2000) 3689 [hep-th/0006034] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  9. [9]
    B. de Wit and H. Nicolai, Hidden symmetries, central charges and all that, Class. Quant. Grav. 18 (2001) 3095 [hep-th/0011239] [SPIRES].ADSMATHCrossRefGoogle Scholar
  10. [10]
    P. West, Generalised space-time and duality, Phys. Lett. B 693 (2010) 373 [arXiv:1006.0893] [SPIRES].ADSGoogle Scholar
  11. [11]
    P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [SPIRES].ADSMATHCrossRefGoogle Scholar
  12. [12]
    P.C. West, E 11 , SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [SPIRES].ADSGoogle Scholar
  13. [13]
    P.C. West, E 11 origin of brane charges and U-duality multiplets, JHEP 08 (2004) 052 [hep-th/0406150] [SPIRES].ADSCrossRefGoogle Scholar
  14. [14]
    A. Kleinschmidt and P.C. West, Representations of G+++ and the role of space-time, JHEP 02 (2004) 033 [hep-th/0312247] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    P.C. West, Brane dynamics, central charges and E 11, JHEP 03 (2005) 077 [hep-th/0412336] [SPIRES].ADSCrossRefGoogle Scholar
  16. [16]
    N.A. Obers and B. Pioline, U-duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    F. Riccioni and P.C. West, E 11 -extended spacetime and gauged supergravities, JHEP 02 (2008) 039 [arXiv:0712.1795] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    H. Nicolai and A. Kleinschmidt, E 10 : eine fundamentale Symmetrie der Physik?, Phys. Unserer Zeit 3 N41 (2010) 134.ADSCrossRefGoogle Scholar
  19. [19]
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145 [hep-th/0212256] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    D. Persson, Arithmetic and hyperbolic structures in string theory, arXiv:1001.3154 [SPIRES].
  22. [22]
    C. Hillmann, Generalized E 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281 [math/0209099]. = MATH/0209099;MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    N. Hitchin, Brackets, forms and invariant functionals, math/0508618.
  25. [25]
    M. Gualtieri, Generalized complex geometry, math/0401221.
  26. [26]
    C.M. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [SPIRES].ADSCrossRefGoogle Scholar
  27. [27]
    P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    P.A.M. Dirac, The theory of gravitation in Hamiltonian form, Proc. Roy. Soc. Lond. A 246 (1958) 333 [SPIRES].MathSciNetADSGoogle Scholar
  29. [29]
    P.A.M. Dirac, Fixation of coordinates in the Hamiltonian theory of gravitation, Phys. Rev. 114 (1959) 924 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  30. [30]
    R.L. Arnowitt, S. Deser and C.W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. 116 (1959) 1322 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  31. [31]
    S. Deser, R. Arnowitt and C.W. Misner, Consistency of canonical reduction of general relativity, J. Math Phys. 1 (1960) 434 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  32. [32]
    R.L. Arnowitt, S. Deser and C.W. Misner, Canonical variables for general relativity, Phys. Rev. 117 (1960) 1595 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  33. [33]
    R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, in Gravitation. An introduction to current research, L. Witten ed., John Wiley & Sons, U.S.A. (1962).Google Scholar
  34. [34]
    B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory, Phys. Rev. 160 (1967) 1113 [SPIRES].ADSMATHCrossRefGoogle Scholar
  35. [35]
    C.M. Hull, Duality and the signature of space-time, JHEP 11 (1998) 017 [hep-th/9807127] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    M.J. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [SPIRES].MathSciNetADSGoogle Scholar
  38. [38]
    A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    C.M. Hull, Global aspects of T-duality, gauged σ-models and T-folds, JHEP 10 (2007) 057 [hep-th/0604178] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    C.M. Hull, Doubled geometry and T-folds, JHEP 07 (2007) 080 [hep-th/0605149] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    M.J. Duff and J.X. Lu, Duality rotations in membrane theory, Nucl. Phys. B 347 (1990) 394 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  46. [46]
    D.S. Berman and N.B. Copland, The string partition function in Hull’s doubled formalism, Phys. Lett. B 649 (2007) 325 [hep-th/0701080] [SPIRES].MathSciNetADSGoogle Scholar
  47. [47]
    D.S. Berman, N.B. Copland and D.C. Thompson, Background field equations for the duality symmetric string, Nucl. Phys. B 791 (2008) 175 [arXiv:0708.2267] [SPIRES].MathSciNetADSGoogle Scholar
  48. [48]
    D.S. Berman and D.C. Thompson, Duality symmetric strings, dilatons and O(d, d) effective actions, Phys. Lett. B 662 (2008) 279 [arXiv:0712.1121] [SPIRES].MathSciNetADSGoogle Scholar
  49. [49]
    S.D. Avramis, J.-P. Derendinger and N. Prezas, Conformal chiral boson models on twisted doubled tori and non-geometric string vacua, Nucl. Phys. B 827 (2010) 281 [arXiv:0910.0431] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    G. Bonelli and M. Zabzine, From current algebras for p-branes to topological M-theory, JHEP 09 (2005) 015 [hep-th/0507051] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    G. Bonelli, A. Tanzini and M. Zabzine, On topological M-theory, Adv. Theor. Math. Phys. 10 (2006) 239 [hep-th/0509175] [SPIRES].MathSciNetMATHGoogle Scholar
  52. [52]
    G. Bonelli, A. Tanzini and M. Zabzine, Topological branes, p-algebras and generalized Nahm equations, Phys. Lett. B 672 (2009) 390 [arXiv:0807.5113] [SPIRES].MathSciNetADSGoogle Scholar
  53. [53]
    G. Aldazabal, E. Andres, P.G. Camara and M. Graña, U-dual fluxes and generalized geometry, JHEP 11 (2010) 083 [arXiv:1007.5509] [SPIRES].ADSCrossRefGoogle Scholar
  54. [54]
    M. Graña, R. Minasian, M. Petrini and D. Waldram, T-duality, generalized geometry and non-geometric backgrounds, JHEP 04 (2009) 075 [arXiv:0807.4527] [SPIRES].ADSCrossRefGoogle Scholar
  55. [55]
    E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  57. [57]
    R.A. Reid-Edwards, Bi-algebras, generalised geometry and T-duality, arXiv:1001.2479 [SPIRES].
  58. [58]
    N. Halmagyi, Non-geometric backgrounds and the first order string σ-model, arXiv:0906.2891 [SPIRES].
  59. [59]
    J. McOrist, D.R. Morrison and S. Sethi, Geometries, non-geometries and fluxes, arXiv:1004.5447 [SPIRES].
  60. [60]
    J. de Boer and M. Shigemori, Exotic branes and non-geometric backgrounds, Phys. Rev. Lett. 104 (2010) 251603 [arXiv:1004.2521] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    V. Moncrief and C. Teitelboim, Momentum constraints as integrability conditions for the hamiltonian constraint in general relativity, Phys. Rev. D 6 (1972) 966 [SPIRES].MathSciNetADSGoogle Scholar
  62. [62]
    G.W. Gibbons, S.W. Hawking and M.J. Perry, Path integrals and the indefiniteness of the gravitational action, Nucl. Phys. B 138 (1978) 141 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    H.A. Buchdahl, Reciprocal static solutions of the equations of the gravitational field, Austral. J. Phys. 9 (1956) 13.MathSciNetADSMATHGoogle Scholar
  64. [64]
    J. Ehlers, Konstruktionen und Charakterisierung von Losungen der Einsteinschen Gravitationsfeldgleichungen, Ph.D. thesis, University of Hamburg, Hamburg, Germany (1957).Google Scholar
  65. [65]
    R.P. Geroch, A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971) 918 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  66. [66]
    R.P. Geroch, A Method for generating new solutions of Einstein’s equation. 2, J. Math. Phys. 13 (1972) 394 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  67. [67]
    J. Thierry-Mieg, BRS structure of the antisymmetric tensor gauge theories, Nucl. Phys. B 335 (1990) 334 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  68. [68]
    L. Baulieu and M. Henneaux, P forms and diffeomorphisms: hamiltonian formulation, Phys. Lett. B 194 (1987) 81 [SPIRES].MathSciNetADSGoogle Scholar
  69. [69]
    T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990) 631.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    E. Bergshoeff, E. Sezgin and P.K. Townsend, Properties of the eleven-dimensional super membrane theory, Ann. Phys. 185 (1988) 330 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  71. [71]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  72. [72]
    C.M. Hull and B. Julia, Duality and moduli spaces for time-like reductions, Nucl. Phys. B 534 (1998) 250 [hep-th/9803239] [SPIRES].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Department of PhysicsQueen MaryLondonEngland U.K.
  2. 2.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridgeEngland U.K.
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada

Personalised recommendations