String solitons and T-duality

  • Eric A. Bergshoeff
  • Fabio Riccioni
Open Access


We construct for arbitrary dimensions a universal T-duality covariant expression for the Wess-Zumino terms of supersymmetric String Solitons in toroidally compactified string theories with 32 supercharges. The worldvolume fields occurring in the effective action of these String Solitons form either a vector or a tensor multiplet with 16 supercharges. We determine the dimensions of the conjugacy classes under T-duality to which these String Solitons belong. We do this in two steps. First, we determine the T-duality representations of the p-forms of maximal supergravities that contain the potentials that couple to these String Solitons. We find that these are p-forms, with D − 4 ≤ p ≤ 6 if D ≥ 6 and with D − 4 ≤ p ≤ D if D < 6, transforming in the antisymmetric representation of rank m = p + 4 − D ≤ 4 of the T-duality symmetry SO(10 − D, 10 − D). All branes support vector multiplets except when m = 10 − D. In that case the T-duality representation splits, for D < 10, into a selfdual and anti-selfdual part, corresponding to 5-branes supporting either a vector or a tensor multiplet. In a second step we show that only certain well-defined lightlike directions in the anti-symmetric tensor representations of the T-duality group correspond to supersymmetric String Solitons. These lightlike directions define the conjugacy classes. As a by-product we show how by a straightforward procedure all solitonic fields of maximal supergravity are derived using the Kac-Moody algebra E11.


p-branes D-branes String Duality 


  1. 1.
    E. Bergshoeff, M. de Roo, M.B. Green, G. Papadopoulos and P.K. Townsend, Duality of Type II 7-branes and 8-branes, Nucl. Phys. B 470 (1996) 113 [hep-th/9601150] [SPIRES].ADSCrossRefGoogle Scholar
  2. 2.
    E.A. Bergshoeff, M. de Roo, S.F. Kerstan and F. Riccioni, IIB Supergravity Revisited, JHEP 08 (2005) 098 [hep-th/0506013] [SPIRES].ADSCrossRefGoogle Scholar
  3. 3.
    E.A. Bergshoeff, M. de Roo, S.F. Kerstan, T. Ortín and F. Riccioni, IIA ten-forms and the gauge algebras of maximal supergravity theories, JHEP 07 (2006) 018 [hep-th/0602280] [SPIRES].ADSCrossRefGoogle Scholar
  4. 4.
    E.A. Bergshoeff, J. Hartong, P.S. Howe, T. Ortín and F. Riccioni, IIA/ IIB Supergravity and Ten-forms, JHEP 05 (2010) 061 [arXiv:1004.1348] [SPIRES].ADSCrossRefGoogle Scholar
  5. 5.
    L.J. Romans, Massive N = 2a Supergravity in Ten-Dimensions, Phys. Lett. B 169 (1986) 374 [SPIRES].MathSciNetADSGoogle Scholar
  6. 6.
    B. de Wit, H. Nicolai and H. Samtleben, Gauged Supergravities, Tensor Hierarchies and M-theory, JHEP 02 (2008) 044 [arXiv:0801.1294] [SPIRES].CrossRefGoogle Scholar
  7. 7.
    F. Riccioni and P.C. West, The E 11 origin of all maximal supergravities, JHEP 07 (2007) 063 [arXiv:0705.0752] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    E.A. Bergshoeff, I. De Baetselier and T.A. Nutma, E 11 and the embedding tensor, JHEP 09 (2007) 047 [arXiv:0705.1304] [SPIRES].ADSCrossRefGoogle Scholar
  9. 9.
    P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    E.A. Bergshoeff and F. Riccioni, D-Brane Wess-Zumino Terms and U-duality, JHEP 11 (2010) 139 [arXiv:1009.4657] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    P.P. Cook and P.C. West, Charge multiplets and masses for E 11, JHEP 11 (2008) 091 [arXiv:0805.4451] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    P.P. Cook, Exotic E 11 branes as composite gravitational solutions, Class. Quant. Grav. 26 (2009) 235023 [arXiv:0908.0485] [SPIRES].ADSCrossRefGoogle Scholar
  13. 13.
    J.H. Schwarz, An SL(2, Z) multiplet of type IIB superstrings, Phys. Lett. B 360 (1995) 13 [hep-th/9508143] [SPIRES].ADSGoogle Scholar
  14. 14.
    M. Aganagic, J. Park, C. Popescu and J.H. Schwarz, Dual D-brane actions, Nucl. Phys. B 496 (1997) 215 [hep-th/9702133] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    P.K. Townsend, Membrane tension and manifest IIB S-duality, Phys. Lett. B 409 (1997) 131 [hep-th/9705160] [SPIRES].MathSciNetADSGoogle Scholar
  16. 16.
    M. Cederwall and A. Westerberg, World-volume fields, SL(2, Z) and duality: The type IIB 3-brane, JHEP 02 (1998) 004 [hep-th/9710007] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    A. Westerberg and N. Wyllard, Towards a manifestly SL(2, Z)-covariant action for the type IIB (p,q) super-five-branes, JHEP 06 (1999) 006 [hep-th/9905019] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    E.A. Bergshoeff, M. de Roo, S.F. Kerstan, T. Ortín and F. Riccioni, SL(2,R)-invariant IIB brane actions, JHEP 02 (2007) 007 [hep-th/0611036] [SPIRES].ADSCrossRefGoogle Scholar
  19. 19.
    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    C.M. Hull, Doubled geometry and T-folds, JHEP 07 (2007) 080 [hep-th/0605149] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    C.M. Hull and R.A. Reid-Edwards, Gauge Symmetry, T-duality and Doubled Geometry, JHEP 08 (2008) 043 [arXiv:0711.4818] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    P. Meessen and T. Ortín, An SL(2, Z) multiplet of nine-dimensional type-II supergravity theories, Nucl. Phys. B 541 (1999) 195 [hep-th/9806120] [SPIRES].ADSCrossRefGoogle Scholar
  23. 23.
    G. Dall’Agata, K. Lechner and M. Tonin, D = 10, N = IIB supergravity: Lorentz-invariant actions and duality, JHEP 07 (1998) 017 [hep-th/9806140] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    E.A. Bergshoeff, J. Hartong, T. Ortín and D. Roest, Seven-branes and supersymmetry, JHEP 02 (2007) 003 [hep-th/0612072] [SPIRES].ADSCrossRefGoogle Scholar
  25. 25.
    E.A. Bergshoeff, M. de Roo, S.F. Kerstan, T. Ortín and F. Riccioni, IIB nine-branes, JHEP 06 (2006) 006 [hep-th/0601128] [SPIRES].ADSCrossRefGoogle Scholar
  26. 26.
    R. Slansky, Group Theory for Unified Model Building, Phys. Rept. 79 (1981) 1 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    P.K. Townsend, M-theory from its superalgebra, hep-th/9712004 [SPIRES].
  28. 28.
    E. Bergshoeff, B. Janssen and T. Ortín, Kaluza-Klein monopoles and gauged σ-models, Phys. Lett. B 410 (1997) 131 [hep-th/9706117] [SPIRES].ADSGoogle Scholar
  29. 29.
    I. Schnakenburg and P.C. West, Kac-Moody symmetries of IIB supergravity, Phys. Lett. B 517 (2001) 421 [hep-th/0107181] [SPIRES].MathSciNetADSGoogle Scholar
  30. 30.
    A. Kleinschmidt, I. Schnakenburg and P.C. West, Very-extended Kac-Moody algebras and their interpretation at low levels, Class. Quant. Grav. 21 (2004) 2493 [hep-th/0309198] [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    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
  33. 33.
    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] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  34. 34.
    F. Riccioni and P.C. West, Dual fields and E 11, Phys. Lett. B 645 (2007) 286 [hep-th/0612001] [SPIRES].MathSciNetADSGoogle Scholar
  35. 35.
    V.G. Kac, Infinite dimensional Lie algebras, Cambridge Univ. Pr., Cambridge U.K. (1990) [SPIRES].zbMATHCrossRefGoogle Scholar
  36. 36.
    P.C. West, The IIA, IIB and eleven dimensional theories and their common E 11 origin, Nucl. Phys. B 693 (2004) 76 [hep-th/0402140] [SPIRES].ADSCrossRefGoogle Scholar
  37. 37.
    E.A. Bergshoeff, M. de Roo, S.F. Kerstan, A. Kleinschmidt and F. Riccioni, Dual Gravity and Matter, Gen. Rel. Grav. 41 (2009) 39 [arXiv:0803.1963] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    A. Van Proeyen, Tools for supersymmetry, hep-th/9910030 [SPIRES].
  39. 39.
    F. Riccioni, Low-energy structure of six-dimensional open-string vacua, hep-th/0203157 [SPIRES].

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© The Author(s) 2011

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Centre for Theoretical PhysicsUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of MathematicsKings College LondonStrand LondonU.K.

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