One-loop β function of the double sigma model with constant background

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Regular Article - Theoretical Physics


The double sigma model with the strong constraints is equivalent to a classical theory of the normal sigma model with one on-shell self-duality relation. The one-form gauge field comes from the boundary term. It is the same as the normal sigma model. The gauge symmetries under the strong constraints are the diffeomorphism and one-form gauge transformation in the double sigma model. These gauge symmetries are also the same as the Dirac-Born-Infeld (DBI) theory. The main task of this work is to compute one-loop β function to obtain the low energy effective theory of the double sigma model. We implement the self-duality relation in the action to perform the one-loop calculation. At last, we obtain the DBI theory. We also rewrite this theory in terms of the generalized metric and scalar dilaton, and define the generalized scalar curvature and tensor from the equations of motion.


D-branes String Duality 


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  1. [1]
    B. Zwiebach, Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Saadi and B. Zwiebach, Closed String Field Theory from Polyhedra, Annals Phys. 192 (1989) 213 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    P.-M. Ho and Y. Matsuo, M5 from M2, JHEP 06 (2008) 105 [arXiv:0804.3629] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    P.-M. Ho, C.-T. Ma and C.-H. Yeh, BPS States on M5-brane in Large C-field Background, JHEP 08 (2012) 076 [arXiv:1206.1467] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    P.-M. Ho and C.-T. Ma, S-duality for D3-Brane in NS-NS and RR Backgrounds, JHEP 11 (2014) 142 [arXiv:1311.3393] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  6. [6]
    P.-M. Ho and C.-T. Ma, Effective Action for Dp-brane in Large RR (p − 1)-Form Background, JHEP 05 (2013) 056 [arXiv:1302.6919] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    C.-T. Ma and C.-H. Yeh, Supersymmetry and BPS States on D4-brane in Large C-field Background, JHEP 03 (2013) 131 [arXiv:1210.4191] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    B. Jurčo and P. Schupp, Nambu sigma model and effective membrane actions, Phys. Lett. B 713 (2012) 313 [arXiv:1203.2910] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    P. Schupp and B. Jurčo, Nambu Sigma Model and Branes, PoS(CORFU2011)045 [arXiv:1205.2595] [INSPIRE].
  10. [10]
    J.-K. Ho and C.-T. Ma, Dimensional Reduction of the Generalized DBI, arXiv:1410.0972 [INSPIRE].
  11. [11]
    B. Jurčo, P. Schupp and J. Vysoký, On the Generalized Geometry Origin of Noncommutative Gauge Theory, JHEP 07 (2013) 126 [arXiv:1303.6096] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  12. [12]
    B. Jurčo, P. Schupp and J. Vysoký, Extended generalized geometry and a DBI-type effective action for branes ending on branes, JHEP 08 (2014) 170 [arXiv:1404.2795] [INSPIRE].ADSGoogle Scholar
  13. [13]
    K. Lee and J.-H. Park, Partonic description of a supersymmetric p-brane, JHEP 04 (2010) 043 [arXiv:1001.4532] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  14. [14]
    J.-H. Park and C. Sochichiu, Taking off the square root of Nambu-Goto action and obtaining Filippov-Lie algebra gauge theory action, Eur. Phys. J. C 64 (2009) 161 [arXiv:0806.0335] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    O. Hohm, S.K. Kwak and B. Zwiebach, Double Field Theory of Type II Strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    O. Hohm and S.K. Kwak, Frame-like Geometry of Double Field Theory, J. Phys. A 44 (2011) 085404 [arXiv:1011.4101] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  17. [17]
    C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    K. Lee and J.-H. Park, Covariant action for a string indoubled yet gaugedspacetime, Nucl. Phys. B 880 (2014) 134 [arXiv:1307.8377] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    A.A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    A.A. Tseytlin, Duality Symmetric Formulation of String World Sheet Dynamics, Phys. Lett. B 242 (1990) 163 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    N.B. Copland, A Double σ-model for Double Field Theory, JHEP 04 (2012) 044 [arXiv:1111.1828] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    S.D. Avramis, J.-P. Derendinger and N. Prezas, Conformal chiral boson models on twisted doubled tori and non-geometric string vacua, Nucl. Phys. B 827 (2010) 281 [arXiv:0910.0431] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    M.J. Duff, Duality Rotations in String Theory, Nucl. Phys. B 335 (1990) 610 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    J. Berkeley, D.S. Berman and F.J. Rudolph, Strings and Branes are Waves, JHEP 06 (2014) 006 [arXiv:1403.7198] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].ADSMathSciNetGoogle Scholar
  30. [30]
    W. Siegel, Manifest duality in low-energy superstrings, hep-th/9308133 [INSPIRE].
  31. [31]
    C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    G. Aldazabal, W. Baron, D. Marques and C. Núñez, The effective action of Double Field Theory, JHEP 11 (2011) 052 [Erratum ibid. 11 (2011) 109] [arXiv:1109.0290] [INSPIRE].
  34. [34]
    D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, Non-Geometric Fluxes in Supergravity and Double Field Theory, Fortsch. Phys. 60 (2012) 1150 [arXiv:1204.1979] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    T. Kimura, S. Sasaki and M. Yata, World-volume Effective Actions of Exotic Five-branes, JHEP 07 (2014) 127 [arXiv:1404.5442] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring Double Field Theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    C.-T. Ma and C.-M. Shen, Cosmological Implications from O(D, D), Fortsch. Phys. 62 (2014) 921 [arXiv:1405.4073] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    O. Hohm, W. Siegel and B. Zwiebach, Doubled α -geometry, JHEP 02 (2014) 065 [arXiv:1306.2970] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    G. Aldazabal, D. Marques and C. Núñez, Double Field Theory: A Pedagogical Review, Class. Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    D.S. Berman and D.C. Thompson, Duality Symmetric String and M-theory, Phys. Rept. 566 (2015) 1 [arXiv:1306.2643] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    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
  43. [43]
    O. Hohm and H. Samtleben, Exceptional field theory. II. E 7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].ADSGoogle Scholar
  44. [44]
    O. Hohm and H. Samtleben, Exceptional field theory. III. E 8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].ADSGoogle Scholar
  45. [45]
    D.S. Berman and M.J. Perry, Generalized Geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    C.M. Hull, A Geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    C. Albertsson, T. Kimura and R.A. Reid-Edwards, D-branes and doubled geometry, JHEP 04 (2009) 113 [arXiv:0806.1783] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    M. Gualtieri, Generalized complex geometry, math.DG/0401221 [INSPIRE].
  50. [50]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. Oxford Ser. 54 (2003) 281 [math.DG/0209099] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    G.R. Cavalcanti and M. Gualtieri, Generalized complex geometry and T-duality, arXiv:1106.1747 [INSPIRE].
  52. [52]
    M. Hatsuda and T. Kimura, Canonical approach to Courant brackets for D-branes, JHEP 06 (2012) 034 [arXiv:1203.5499] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    T. Asakawa, S. Sasa and S. Watamura, D-branes in Generalized Geometry and Dirac-Born-Infeld Action, JHEP 10 (2012) 064 [arXiv:1206.6964] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    C.-T. Ma, Gauge Transformation of Double Field Theory for Open String, arXiv:1411.0287 [INSPIRE].
  55. [55]
    B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    T. Asakawa, H. Muraki, S. Sasa and S. Watamura, Poisson-generalized geometry and R-flux, arXiv:1408.2649 [INSPIRE].
  57. [57]
    M. Bernstein and J. Sonnenschein, A Comment on the Quantization of Chiral Bosons, Phys. Rev. Lett. 60 (1988) 1772 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    C. Imbimbo and A. Schwimmer, The Lagrangian Formulation of Chiral Scalars, Phys. Lett. B 193 (1987) 455 [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    J.M. Labastida and M. Pernici, On the BRST Quantization of Chiral Bosons, Nucl. Phys. B 297 (1988) 557 [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    R. Floreanini and R. Jackiw, Selfdual Fields as Charge Density Solitons, Phys. Rev. Lett. 59 (1987) 1873 [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    A. Abouelsaood, C.G. Callan Jr., C.R. Nappi and S.A. Yost, Open Strings in Background Gauge Fields, Nucl. Phys. B 280 (1987) 599 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  62. [62]
    C.G. Callan Jr., C. Lovelace, C.R. Nappi and S.A. Yost, String Loop Corrections to β-functions, Nucl. Phys. B 288 (1987) 525 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of Physics, Center for Theoretical Sciences and Center for Advanced Study in Theoretical SciencesNational Taiwan UniversityTaipeiTaiwan

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