Journal of High Energy Physics

, 2012:108 | Cite as

Duality invariant actions and generalised geometry

  • David S. Berman
  • Hadi Godazgar
  • Malcolm J. Perry
  • Peter West
Article

Abstract

We construct the non-linear realisation of the semi-direct product of E11 and its first fundamental representation at lowest order and appropriate to spacetime dimensions four to seven. This leads to a non-linear realisation of the duality groups and introduces fields that depend on a generalised space which possess a generalised vielbein. We focus on the part of the generalised space on which the duality groups alone act and construct an invariant action.

Keywords

M-Theory String Duality Supergravity Models 

References

  1. [1]
    P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  2. [2]
    P.C. West, E 11 , SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].ADSGoogle Scholar
  3. [3]
    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
  4. [4]
    P.C. West, E 11 origin of brane charges and U-duality multiplets, JHEP 08 (2004) 052 [hep-th/0406150] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    P.P. Cook and P.C. West, Charge multiplets and masses for E 11, JHEP 11 (2008) 091 [arXiv:0805.4451] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    F. Riccioni and P.C. West, Dual fields and E 11, Phys. Lett. B 645 (2007) 286 [hep-th/0612001] [INSPIRE].MathSciNetADSGoogle Scholar
  7. [7]
    S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett. 18 (1967) 188 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    J.S. Schwinger, Chiral dynamics, Phys. Lett. B 24 (1967) 473 [INSPIRE].ADSGoogle Scholar
  9. [9]
    J.S. Schwinger, A theory of the fundamental interactions, Annals Phys. 2 (1957) 407 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev. 166 (1968) 1568 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    M. Gell-Mann and M. Levy, The axial vector current in beta decay, Nuovo Cim. 16 (1960) 705 [INSPIRE].MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    J. Wess and B. Zumino, Lagrangian method for chiral symmetries, Phys. Rev. 163 (1967) 1727 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1, Phys. Rev. 177 (1969) 2239 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2, Phys. Rev. 177 (1969) 2247 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    C. Isham, A. Salam and J. Strathdee, Nonlinear realizations of space-time symmetries. Scalar and tensor gravity, Annals Phys. 62 (1971) 98 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  16. [16]
    A. Borisov and V. Ogievetsky, Theory of dynamical affine and conformal symmetries as gravity theory, Theor. Math. Phys. 21 (1975) 1179 [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    D. Volkov and V. Akulov, Is the neutrino a goldstone particle?, Phys. Lett. B 46 (1973) 109 [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]
    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
  20. [20]
    E. Cremmer, B. Julia, H. Lü and C. Pope, Dualization of dualities. 2. Twisted selfduality of doubled fields and superdualities, Nucl. Phys. B 535 (1998) 242 [hep-th/9806106] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    E. Cremmer, B. Julia, H. Lü and C. Pope, Higher dimensional origin of D = 3 coset symmetries, hep-th/9909099 [INSPIRE].
  22. [22]
    I. Schnakenburg and P.C. West, Kac-Moody symmetries of 2B supergravity, Phys. Lett. B 517 (2001) 421 [hep-th/0107181] [INSPIRE].MathSciNetADSGoogle Scholar
  23. [23]
    F. Riccioni and P.C. West, The E 11 origin of all maximal supergravities, JHEP 07 (2007) 063 [arXiv:0705.0752] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    F. Riccioni and P.C. West, E 11 -extended spacetime and gauged supergravities, JHEP 02 (2008) 039 [arXiv:0712.1795] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    F. Riccioni and P. West, Local E 11, JHEP 04 (2009) 051 [arXiv:0902.4678] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    F. Riccioni, D. Steele and P. West, The E 11 origin of all maximal supergravities: the hierarchy of field-strengths, JHEP 09 (2009) 095 [arXiv:0906.1177] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    E. Cremmer and B. Julia, The SO(8) supergravity, Nucl. Phys. B 159 (1979) 141 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    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
  29. [29]
    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
  30. [30]
    E.A. Bergshoeff, I. De Baetselier and T.A. Nutma, E 11 and the embedding tensor, JHEP 09 (2007) 047 [arXiv:0705.1304] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    P.C. West, Brane dynamics, central charges and E 11, JHEP 03 (2005) 077 [hep-th/0412336] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    M. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    M. Duff and J. Lu, Duality rotations in membrane theory, Nucl. Phys. B 347 (1990) 394 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    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
  38. [38]
    C. Hillmann, Generalized E 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939) 149.MathSciNetCrossRefGoogle Scholar
  40. [40]
    G.W. Mackey, On induced representations of groups, Amer. J. Math. 73 (1951) 576.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    G.W. Mackey, Induced representations of locally compact groups. I, Ann. Math. 55 (1952) 101.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    G.W. Mackey, The theory of group representations: lecture notes in three volumes, University of Chicago, Chicago U.S.A. (1955).Google Scholar
  43. [43]
    G.W. Mackey, The mathematical foundations of quantum mechanics: a lecture-note volume, W.A. Benjamin Inc., U.S.A. (1963).MATHGoogle Scholar
  44. [44]
    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].ADSCrossRefGoogle Scholar
  45. [45]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].MathSciNetADSCrossRefGoogle 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].MathSciNetADSCrossRefGoogle 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].MathSciNetADSCrossRefGoogle 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].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations, Cambridge University Press, Cambridge U.K. (1997).MATHGoogle Scholar
  50. [50]
    C. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    I. Schnakenburg and P.C. West, Massive IIA supergravity as a nonlinear realization, Phys. Lett. B 540 (2002) 137 [hep-th/0204207] [INSPIRE].MathSciNetADSGoogle Scholar
  53. [53]
    M.R. Gaberdiel, D.I. Olive and P.C. West, A class of lorentzian Kac-Moody algebras, Nucl. Phys. B 645 (2002) 403 [hep-th/0205068] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    P.C. West, Very extended E 8 and A 8 at low levels, gravity and supergravity, Class. Quant. Grav. 20 (2003) 2393 [hep-th/0212291] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  55. [55]
    P. West, E 11 , generalised space-time and IIA string theory, Phys. Lett. B 696 (2011) 403 [arXiv:1009.2624] [INSPIRE].ADSGoogle Scholar
  56. [56]
    A. Rocen and P. West, E 11 , generalised space-time and IIA string theory: the RR sector, arXiv:1012.2744 [INSPIRE].
  57. [57]
    P. West, Introduction to strings and branes, to be published, Cambridge University Press, Cambridge U.K. (2012).Google Scholar
  58. [58]
    S. Elitzur, A. Giveon, D. Kutasov and E. Rabinovici, Algebraic aspects of matrix theory on T d, Nucl. Phys. B 509 (1998) 122 [hep-th/9707217] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  59. [59]
    N. Obers, B. Pioline and E. Rabinovici, M theory and U duality on T d with gauge backgrounds, Nucl. Phys. B 525 (1998) 163 [hep-th/9712084] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  60. [60]
    N. Obers and B. Pioline, U duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    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
  62. [62]
    T. Damour, M. Henneaux and H. Nicolai, Cosmological billiards, Class. Quant. Grav. 20 (2003) R145 [hep-th/0212256] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  63. [63]
    T. Damour and H. Nicolai, Eleven dimensional supergravity and the E 10 /E E10 σ-model at low A 9 levels, hep-th/0410245 [INSPIRE].
  64. [64]
    H. Nicolai and A. Kleinschmidt, E 10 : Eine fundamentale Symmetrie der Physik?, Phys. Unserer Zeit 3N41 (2010) 134.ADSCrossRefGoogle Scholar
  65. [65]
    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
  66. [66]
    É. Cartan, Sur la structure des groupes de transformations finis et continus, Thése, Paris, France (1984).Google Scholar
  67. [67]
    É. Cartan, Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. 31 (1914) 263.MathSciNetGoogle Scholar
  68. [68]
    H. Freudenthal, Sur le groupe exceptionnel E 7, Nederl. Akad. Wetensch. Proc. Ser. A 56 (1953) 81.MathSciNetMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • David S. Berman
    • 1
  • Hadi Godazgar
    • 2
  • Malcolm J. Perry
    • 2
  • Peter West
    • 3
  1. 1.Department of PhysicsQueen Mary University of LondonLondonU.K.
  2. 2.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeU.K.
  3. 3.Department of MathematicsKing’s CollegeLondonU.K.

Personalised recommendations