Advertisement

Gauged double field theory

  • Mariana Graña
  • Diego Marques
Open Access
Article

Abstract

We find necessary and sufficient conditions for gauge invariance of the action of Double Field Theory (DFT) as well as closure of the algebra of gauge symmetries. The so-called weak and strong constraints are sufficient to satisfy them, but not necessary. We then analyze compactifications of DFT on twisted double tori satisfying the consistency conditions. The effective theory is a Gauged DFT where the gaugings come from the duality twists. The action, bracket, global symmetries, gauge symmetries and their closure are computed by twisting their analogs in the higher dimensional DFT. The non-Abelian heterotic string and lower dimensional gauged supergravities are particular examples of Gauged DFT.

Keywords

Flux compactifications String Duality 

References

  1. [1]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    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
  3. [3]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].ADSGoogle Scholar
  6. [6]
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].ADSGoogle Scholar
  7. [7]
    A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    A.A. Tseytlin, Duality symmetric formulation of string world sheet dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].MathSciNetADSGoogle Scholar
  9. [9]
    M. Duff, Duality rotations in string theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    M. Duff and J. Lu, Duality rotations in membrane theory, Nucl. Phys. B 347 (1990) 394 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    O. Hohm and S.K. Kwak, N = 1 supersymmetric double field theory, JHEP 03 (2012) 080 [arXiv:1111.7293] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    I. Jeon, K. Lee and J.-H. Park, Supersymmetric double field theory: stringy reformulation of supergravity, Phys. Rev. D Rapid Communications 85, 081501 (R) (2012) [arXiv:1112.0069] [INSPIRE].
  13. [13]
    N.B. Copland, A double σ-model for double field theory, arXiv:1111.1828 [INSPIRE].
  14. [14]
    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
  15. [15]
    I. Jeon, K. Lee and J.-H. Park, Incorporation of fermions into double field theory, JHEP 11 (2011) 025 [arXiv:1109.2035] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    N. Kan, K. Kobayashi and K. Shiraishi, Equations of motion in double field theory: from particles to scale factors, Phys. Rev. D 84 (2011) 124049 [arXiv:1108.5795] [INSPIRE].ADSGoogle Scholar
  17. [17]
    O. Hohm, S.K. Kwak and B. Zwiebach, Double field theory of type II strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    O. Hohm, S.K. Kwak and B. Zwiebach, Unification of type II strings and T-duality, Phys. Rev. Lett. 107 (2011) 171603 [arXiv:1106.5452] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    N.B. Copland, Connecting T-duality invariant theories, Nucl. Phys. B 854 (2012) 575 [arXiv:1106.1888] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    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].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    D.C. Thompson, Duality invariance: from M-theory to double field theory, JHEP 08 (2011) 125 [arXiv:1106.4036] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    D. Andriot, M. Larfors, D. Lüst and P. Patalong, A ten-dimensional action for non-geometric fluxes, JHEP 09 (2011) 134 [arXiv:1106.4015] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    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
  24. [24]
    I. Jeon, K. Lee and J.-H. Park, Double field formulation of Yang-Mills theory, Phys. Lett. B 701 (2011) 260 [arXiv:1102.0419] [INSPIRE].MathSciNetADSGoogle Scholar
  25. [25]
    O. Hohm and S.K. Kwak, Frame-like geometry of double field theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    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].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    S.K. Kwak, Invariances and equations of motion in double field theory, JHEP 10 (2010) 047 [arXiv:1008.2746] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    O. Hohm and S.K. Kwak, Massive type II in double field theory, JHEP 11 (2011) 086 [arXiv:1108.4937] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    O. Hohm and S.K. Kwak, Double field theory formulation of heterotic strings, JHEP 06 (2011) 096 [arXiv:1103.2136] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    G. Aldazabal, W. Baron, D. Marques and C. Núñez, The effective action of double field theory, JHEP 11 (2011) 052 [Erratum ibid. 1111 (2011) 109] [arXiv:1109.0290] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    D. Geissbuhler, Double field theory and N = 4 gauged supergravity, JHEP 11 (2011) 116 [arXiv:1109.4280] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    O. Hohm and B. Zwiebach, On the Riemann tensor in double field theory, arXiv:1112.5296 [INSPIRE].
  33. [33]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) × R+ generalised geometry, connections and M-theory, arXiv:1112.3989 [INSPIRE].
  35. [35]
    P. West, Generalised geometry, eleven dimensions and E 11, JHEP 02 (2012) 018 [arXiv:1111.1642] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A. Rocen and P. West, E 11 , generalised space-time and IIA string theory: the RR sector, arXiv:1012.2744 [INSPIRE].
  37. [37]
    P. West, E 11 , generalised space-time and IIA string theory, Phys. Lett. B 696 (2011) 403 [arXiv:1009.2624] [INSPIRE].ADSGoogle Scholar
  38. [38]
    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].ADSCrossRefGoogle Scholar
  39. [39]
    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
  40. [40]
    D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    O. Hohm, T-duality versus gauge symmetry, Prog. Theor. Phys. Suppl. 188 (2011) 116 [arXiv:1101.3484] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  42. [42]
    B. Zwiebach, Double field theory, T-duality and Courant brackets, arXiv:1109.1782 [INSPIRE].
  43. [43]
    J. Scherk and J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979)61 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    G. Aldazabal, P.G. Camara, A. Font and L. Ibáñez, More dual fluxes and moduli fixing, JHEP 05 (2006) 070 [hep-th/0602089] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    G. Aldazabal, E. Andres, P.G. Camara and M. Graña, U-dual fluxes and generalized geometry, JHEP 11 (2010) 083 [arXiv:1007.5509] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    C. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP 05 (2006) 009 [hep-th/0512005] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  49. [49]
    C. Hull and R. Reid-Edwards, Gauge symmetry, T-duality and doubled geometry, JHEP 08 (2008) 043 [arXiv:0711.4818] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    C. Hull and R. Reid-Edwards, Non-geometric backgrounds, doubled geometry and generalised T-duality, JHEP 09 (2009) 014 [arXiv:0902.4032] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  51. [51]
    G. Dall’Agata, N. Prezas, H. Samtleben and M. Trigiante, Gauged supergravities from twisted doubled tori and non-geometric string backgrounds, Nucl. Phys. B 799 (2008) 80 [arXiv:0712.1026] [INSPIRE].
  52. [52]
    H. Samtleben, Lectures on gauged supergravity and flux compactifications, Class. Quant. Grav. 25 (2008) 214002 [arXiv:0808.4076] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    D. Andriot, E. Goi, R. Minasian and M. Petrini, Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory, JHEP 05 (2011) 028 [arXiv:1003.3774] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    D. Andriot, R. Minasian and M. Petrini, Flux backgrounds from twists, JHEP 12 (2009) 028 [arXiv:0903.0633] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  55. [55]
    J. Schon and M. Weidner, Gauged N = 4 supergravities, JHEP 05 (2006) 034 [hep-th/0602024] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    G. Aldazabal, D. Marques, C. Núñez and J.A. Rosabal, On type IIB moduli stabilization and N =4,8 supergravities, Nucl. Phys. B 849(2011) 80 [arXiv:1101.5954][INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G. Dibitetto, A. Guarino and D. Roest, How to halve maximal supergravity, JHEP 06 (2011) 030 [arXiv:1104.3587] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  58. [58]
    M. Graña, R. Minasian, M. Petrini and D. Waldram, T-duality, generalized geometry and non-geometric backgrounds, JHEP 04 (2009) 075 [arXiv:0807.4527] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic string theory. 2. The interacting heterotic string, Nucl. Phys. B 267 (1986) 75 [INSPIRE].
  60. [60]
    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, Heterotic string theory. 1. The free heterotic string, Nucl. Phys. B 256 (1985) 253 [INSPIRE].
  61. [61]
    D. Andriot, Heterotic string from a higher dimensional perspective, Nucl. Phys. B 855 (2012) 222 [arXiv:1102.1434] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    J. Maharana and J.H. Schwarz, Noncompact symmetries in string theory, Nucl. Phys. B 390 (1993) 3 [hep-th/9207016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  63. [63]
    N. Kaloper and R.C. Myers, The odd story of massive supergravity, JHEP 05 (1999) 010 [hep-th/9901045] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA 2012

Authors and Affiliations

  1. 1.Institut de Physique ThéoriqueGif-sur-Yvette CedexFrance

Personalised recommendations