Strings and branes are waves

  • Joel Berkeley
  • David S. Berman
  • Felix J. Rudolph
Open Access
Article

Abstract

We examine the equations of motion of double field theory and the duality manifest form of M-theory. We show the solutions of the equations of motion corresponding to null waves correspond to strings or membranes from the usual spacetime perspective. A Goldstone mode analysis of the null wave solution in double field theory produces the equations of motion of the duality manifest string.

Keywords

p-branes Space-Time Symmetries M-Theory String Duality 

References

  1. [1]
    E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    P.K. Townsend, M theory from its superalgebra, hep-th/9712004 [INSPIRE].
  3. [3]
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].ADSGoogle Scholar
  4. [4]
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].ADSGoogle Scholar
  5. [5]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    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].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    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
  11. [11]
    I. Jeon, K. Lee and J.-H. Park, Incorporation of fermions into double field theory, JHEP 11 (2011) 025 [arXiv:1109.2035] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    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].
  13. [13]
    I. Jeon, K. Lee, J.-H. Park and Y. Suh, Stringy unification of type IIA and IIB supergravities under N = 2 D = 10 supersymmetric double field theory, Phys. Lett. B 723 (2013) 245 [arXiv:1210.5078] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    G. Aldazabal, W. Baron, D. Marqués and C. Núñez, The effective action of double field theory, JHEP 11 (2011) 052 [Erratum ibid. 11 (2011) 109] [arXiv:1109.0290] [INSPIRE].
  15. [15]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    G. Aldazabal, D. Marqués and C. Núñez, Double field theory: a pedagogical review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    D.S. Berman and D.C. Thompson, Duality symmetric string and M-theory, arXiv:1306.2643 [INSPIRE].
  18. [18]
    O. Hohm, D. Lüst and B. Zwiebach, The spacetime of double field theory: review, remarks and outlook, Fortsch. Phys. 61 (2013) 926 [arXiv:1309.2977] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    O. Hohm and B. Zwiebach, Towards an invariant geometry of double field theory, J. Math. Phys. 54 (2013) 032303 [arXiv:1212.1736] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    D.S. Berman, M. Cederwall and M.J. Perry, Global aspects of double geometry, arXiv:1401.1311 [INSPIRE].
  21. [21]
    M. Cederwall, The geometry behind double geometry, arXiv:1402.2513 [INSPIRE].
  22. [22]
    C. Hillmann, Generalized E 7(7) coset dynamics and D = 11 supergravity, JHEP 03 (2009) 135 [arXiv:0901.1581] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    C.M. 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].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) × ℝ+ generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry II: E d(d) × ℝ+ and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  28. [28]
    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].ADSCrossRefMathSciNetGoogle Scholar
  29. [29]
    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].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    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].ADSCrossRefMathSciNetGoogle 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]
    A. Dabholkar, G.W. Gibbons, J.A. Harvey and F. Ruiz Ruiz, Superstrings and solitons, Nucl. Phys. B 340 (1990) 33 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    T. Adawi, M. Cederwall, U. Gran, B.E.W. Nilsson and B. Razaznejad, Goldstone tensor modes, JHEP 02 (1999) 001 [hep-th/9811145] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  34. [34]
    A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  35. [35]
    A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  36. [36]
    M.J. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    M.J. Duff and J.X. Lu, Duality rotations in membrane theory, Nucl. Phys. B 347 (1990) 394 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, The gauge structure of exceptional field theories and the tensor hierarchy, JHEP 04 (2014) 049 [arXiv:1312.4549] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    O. Hohm and H. Samtleben, Exceptional form of D = 11 supergravity, Phys. Rev. Lett. 111 (2013) 231601 [arXiv:1308.1673] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    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
  41. [41]
    O. Hohm and H. Samtleben, Exceptional field theory II: E 7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].ADSGoogle Scholar
  42. [42]
    P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  43. [43]
    F. Englert, L. Houart, A. Taormina and P.C. West, The symmetry of M theories, JHEP 09 (2003) 020 [hep-th/0304206] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  44. [44]
    P.C. West, E 11 , SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    A. Kleinschmidt and P.C. West, Representations of G +++ and the role of space-time, JHEP 02 (2004) 033 [hep-th/0312247] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  46. [46]
    P.C. West, E 11 origin of brane charges and U-duality multiplets, JHEP 08 (2004) 052 [hep-th/0406150] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    P. West, Generalised BPS conditions, Mod. Phys. Lett. A 27 (2012) 1250202 [arXiv:1208.3397] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    D.S. Berman and N.B. Copland, The string partition function in Hulls doubled formalism, Phys. Lett. B 649 (2007) 325 [hep-th/0701080] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    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] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  50. [50]
    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] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  51. [51]
    O. Hohm, W. Siegel and B. Zwiebach, Doubled α -geometry, JHEP 02 (2014) 065 [arXiv:1306.2970] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    A. Betz, R. Blumenhagen, D. Lüst and F. Rennecke, A note on the CFT origin of the strong constraint of DFT, JHEP 05 (2014) 044 [arXiv:1402.1686] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    D.S. Berman, C.D.A. Blair, E. Malek and M.J. Perry, The O D,D geometry of string theory, arXiv:1303.6727 [INSPIRE].
  54. [54]
    C.D.A. Blair, E. Malek and A.J. Routh, An O D,D invariant Hamiltonian action for the superstring, arXiv:1308.4829 [INSPIRE].
  55. [55]
    C.D.A. Blair, E. Malek and J.-H. Park, M-theory and type IIB from a duality manifest action, JHEP 01 (2014) 172 [arXiv:1311.5109] [INSPIRE].CrossRefGoogle Scholar
  56. [56]
    K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, arXiv:1401.3360 [INSPIRE].
  57. [57]
    K. Lee and J.-H. Park, Covariant action for a string indoubled yet gaugedspacetime, Nucl. Phys. B 880 (2014) 134 [arXiv:1307.8377] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    J.-H. Park and Y. Suh, U-gravity: SL(N), arXiv:1402.5027 [INSPIRE].
  59. [59]
    M. Cederwall, J. Edlund and A. Karlsson, Exceptional geometry and tensor fields, JHEP 07 (2013) 028 [arXiv:1302.6736] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  60. [60]
    T. Ortín, Gravity and strings, Cambridge University Press, Cambridge U.K. (2004).CrossRefMATHGoogle Scholar
  61. [61]
    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].ADSCrossRefMathSciNetGoogle Scholar
  62. [62]
    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].ADSCrossRefMathSciNetGoogle Scholar
  63. [63]
    D.S. Berman and K. Lee, Supersymmetry for gauged double field theory and generalised Scherk-Schwarz reductions, Nucl. Phys. B 881 (2014) 369 [arXiv:1305.2747] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  64. [64]
    M. Graña and D. Marqués, Gauged double field theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    D. Geissbuhler, Double field theory and N = 4 gauged supergravity, JHEP 11 (2011) 116 [arXiv:1109.4280] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  66. [66]
    G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    P.C. Aichelburg and R.U. Sexl, On the gravitational field of a massless particle, Gen. Rel. Grav. 2 (1971) 303 [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    D.M. Kaplan and J. Michelson, Zero modes for the D = 11 membrane and five-brane, Phys. Rev. D 53 (1996) 3474 [hep-th/9510053] [INSPIRE].ADSMathSciNetGoogle Scholar
  69. [69]
    E. Malek, U-duality in three and four dimensions, arXiv:1205.6403 [INSPIRE].
  70. [70]
    N. Arkani-Hamed et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605 [INSPIRE].
  71. [71]
    S. Jensen, The KK-monopole/NS5-brane in doubled geometry, JHEP 07 (2011) 088 [arXiv:1106.1174] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Joel Berkeley
    • 1
  • David S. Berman
    • 1
  • Felix J. Rudolph
    • 1
  1. 1.Queen Mary University of London, Centre for Research in String Theory, School of PhysicsLondonU.K.

Personalised recommendations