Skip to main content

Canonical approach to Courant brackets for D-branes

Abstract

We present an extension of the Courant bracket to the ones for Dp-branes by analyzing Hamiltonians and local superalgebras. Contrast to the basis of the bracket for a fundamental string which consists of the momentum and the winding modes, the ones for Dp-branes contain higher rank R-R coupling tensors. We show that the R-R gauge transformation rules are obtained by these Courant brackets for Dp-branes where the Dirac-Born-Infeld gauge field and the “two-vierbein field” play an essential role. Canonical analysis of the worldvolume theories naturally gives the basis of the brackets and the target space backgrounds keeping T-duality manifest at least for NS-NS sector. In a D3-brane analysis S-duality is manifest as a symmetry of interchanging the NS-NS coupling and the R-R coupling.

This is a preview of subscription content, access via your institution.

References

  1. T. Buscher, A Symmetry of the String Background Field Equations, Phys. Lett. B 194 (1987) 59 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  2. T. Buscher, Path Integral Derivation of Quantum Duality in Nonlinear σ-models, Phys. Lett. B 201 (1988) 466 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  3. A. Giveon, E. Rabinovici and G. Veneziano, Duality in String Background Space, Nucl. Phys. B 322 (1989) 167 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  4. A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  5. M. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  6. A. Giveon and M. Roček, Generalized duality in curved string backgrounds, Nucl. Phys. B 380 (1992) 128 [hep-th/9112070] [INSPIRE].

    ADS  Article  Google Scholar 

  7. J. Maharana and J.H. Schwarz, Noncompact symmetries in string theory, Nucl. Phys. B 390 (1993) 3 [hep-th/9207016] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  8. W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].

    ADS  Google Scholar 

  9. W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].

    ADS  Google Scholar 

  10. W. Siegel, Manifest duality in low-energy superstrings, hep-th/9308133 [INSPIRE].

  11. N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281 [math/0209099] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  12. M. Gualtieri, Generalized complex geometry, math/0401221 [INSPIRE].

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

    MathSciNet  ADS  Article  Google Scholar 

  14. C. Hull and R. Reid-Edwards, Flux compactifications of string theory on twisted tori, Fortsch. Phys. 57 (2009) 862 [hep-th/0503114] [INSPIRE].

    MathSciNet  ADS  Article  MATH  Google Scholar 

  15. C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  16. C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  17. B. Zwiebach, Double Field Theory, T-duality and Courant Brackets, Lect. Notes Phys. 851 (2012)265 [arXiv:1109.1782] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

  22. 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  Article  Google Scholar 

  23. 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].

    MathSciNet  ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

  25. M. Graña, J. Louis, A. Sim and D. Waldram, E 7(7) formulation of \(\mathcal{N} = 2\) backgrounds backgrounds, JHEP 07 (2009) 104 [arXiv:0904.2333] [INSPIRE].

    ADS  Article  Google Scholar 

  26. A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry I: Type II Theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

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

  28. A. Coimbra, C. Strickland-Constable and D. Waldram, Generalised Geometry and type-II Supergravity, arXiv:1202.3170 [INSPIRE].

  29. A. Alekseev and T. Strobl, Current algebras and differential geometry, JHEP 03 (2005) 035 [hep-th/0410183] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  30. M. Hatsuda and K. Kamimura, Covariant quantization of the super D string, Nucl. Phys. B 520 (1998) 493 [hep-th/9708001] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  31. K. Kamimura and M. Hatsuda, Canonical formulation of IIB D-branes, Nucl. Phys. B 527 (1998) 381 [hep-th/9712068] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  32. M. Abe, M. Hatsuda, K. Kamimura and T. Tokunaga, SO(2, 1) covariant IIB superalgebra, Nucl. Phys. B 553 (1999) 305 [hep-th/9903234] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  33. M. Hatsuda and K. Kamimura, Wess-Zumino actions for IIA D-branes and their supersymmetries, Nucl. Phys. B 535 (1998) 499 [hep-th/9804087] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  34. G. Bonelli and M. Zabzine, From current algebras for p-branes to topological M-theory, JHEP 09 (2005) 015 [hep-th/0507051] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  35. J. Ekstrand and M. Zabzine, Courant-like brackets and loop spaces, JHEP 03 (2011) 074 [arXiv:0903.3215] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  36. M. Graña and D. Marques, Gauged Double Field Theory, JHEP 04 (2012) 020 [arXiv:1201.2924] [INSPIRE].

    ADS  Article  Google Scholar 

  37. M. Graña, Flux compactifications in string theory: A Comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].

    ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

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

    MathSciNet  ADS  Article  Google Scholar 

  40. C. Albertsson, T. Kimura and R.A. Reid-Edwards, D-branes and doubled geometry, JHEP 04 (2009) 113 [arXiv:0806.1783] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  41. C. Albertsson, S.-H. Dai, P.-W. Kao and F.-L. Lin, Double Field Theory for Double D-branes, JHEP 09 (2011) 025 [arXiv:1107.0876] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  42. M. Hatsuda and K. Kamimura, Classical AdS superstring mechanics, Nucl. Phys. B 611 (2001) 77 [hep-th/0106202] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  43. I. Bakhmatov, Fermionic T-duality and U-duality in type-II supergravity, arXiv:1112.1983 [INSPIRE].

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Machiko Hatsuda.

Additional information

ArXiv ePrint: 1203.5499

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hatsuda, M., Kimura, T. Canonical approach to Courant brackets for D-branes. J. High Energ. Phys. 2012, 34 (2012). https://doi.org/10.1007/JHEP06(2012)034

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP06(2012)034

Keywords

  • String Duality
  • p-branes
  • D-branes
  • Space-Time Symmetries