Advertisement

Protected string spectrum in AdS3/CFT2 from worldsheet integrability

  • Marco Baggio
  • Olof Ohlsson Sax
  • Alessandro Sfondrini
  • Bogdan StefańskiJr.
  • Alessandro Torrielli
Open Access
Regular Article - Theoretical Physics

Abstract

We derive the protected closed-string spectra of AdS3/CFT2 dual pairs with 16 supercharges at arbitrary values of the string tension and of the three-form fluxes. These follow immediately from the all-loop Bethe equations for the spectra of the integrable worldsheet theories. Further, representing the underlying integrable systems as spin chains, we find that their dynamics involves length-changing interactions and that protected states correspond to gapless excitations above the Berenstein-Maldacena-Nastase vacuum. In the case of AdS3 × S3 × T4 the degeneracies of such operators precisely match those of the dual CFT2 and the supergravity spectrum. On the other hand, we find that for AdS3 × S3 × S3 × S1 there are fewer protected states than previous supergravity calculations had suggested. In particular, protected states have the same su(2) charge with respect to the two three-spheres.

Keywords

AdS-CFT Correspondence Bethe Ansatz Conformal Field Theory 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS 3/CF T 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  2. [2]
    P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 10 (2012) 109 [arXiv:1207.5531] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    A. Cagnazzo and K. Zarembo, B-field in AdS 3/CF T 2 Correspondence and Integrability, JHEP 11 (2012) 133 [Erratum ibid. 1304 (2013) 003] [arXiv:1209.4049] [INSPIRE].
  4. [4]
    J.R. David and B. Sahoo, Giant magnons in the D1 − D5 system, JHEP 07 (2008) 033 [arXiv:0804.3267] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic SU(1|1)2 S-matrix for AdS 3/CF T 2, JHEP 04 (2013) 113 [arXiv:1211.5119] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  7. [7]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefaski, Jr. and A. Torrielli, The all-loop integrable spin-chain for strings on AdS 3 × S 3 × T 4: the massive sector, JHEP 08 (2013) 043 [arXiv:1303.5995] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  8. [8]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Jr., Towards the All-Loop Worldsheet S Matrix for AdS 3 × S 3 × T 4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  9. [9]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Jr., The complete AdS 3 × S 3 × T 4 worldsheet S matrix, JHEP 10 (2014) 66 [arXiv:1406.0453] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  10. [10]
    T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Jr., The complete worldsheet S matrix of superstrings on AdS 3 × S 3 × T 4 with mixed three-form flux, Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  11. [11]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Jr., The AdS3 × S3 × S3 × S1 worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [INSPIRE].zbMATHGoogle Scholar
  12. [12]
    A. Sfondrini, Towards integrability for AdS3/CFT2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski, Jr. and A. Torrielli, Dressing phases of AdS 3/CF T 2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].Google Scholar
  14. [14]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski, Jr., A. Torrielli and O. Ohlsson Sax, On the dressing factors, Bethe equations and Yangian symmetry of strings on AdS 3 × S 3 × T 4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].ADSzbMATHGoogle Scholar
  15. [15]
    J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M.C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS 3 asymptotic Bethe ansatz, Phys. Rev. D 93 (2016) 106006 [arXiv:1512.08761] [INSPIRE].ADSGoogle Scholar
  17. [17]
    R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Jr., On the spectrum of AdS 3 × S 3 × T 4 strings with Ramond-Ramond flux, J. Phys. A 49 (2016) 41LT03 [arXiv:1605.00518] [INSPIRE].
  18. [18]
    O. Ohlsson Sax and B. Stefanski, Jr., Integrability, spin-chains and the AdS 3/CF T 2 correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [INSPIRE].zbMATHGoogle Scholar
  19. [19]
    R. Borsato, O. Ohlsson Sax and A. Sfondrini, All-loop Bethe ansatz equations for AdS 3/CF T 2, JHEP 04 (2013) 116 [arXiv:1212.0505] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  20. [20]
    M. Beccaria, F. Levkovich-Maslyuk, G. Macorini and A.A. Tseytlin, Quantum corrections to spinning superstrings in AdS 3 × S 3 × M 4: determining the dressing phase, JHEP 04 (2013) 006 [arXiv:1211.6090] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  21. [21]
    P. Sundin, Worldsheet two- and four-point functions at one loop in AdS 3/CF T 2, Phys. Lett. B 733 (2014) 134 [arXiv:1403.1449] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  22. [22]
    N. Gromov and G. Sizov, Exact Slope and Interpolating Functions in N = 6 Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M.C. Abbott, Comment on Strings in AdS 3 × S 3 × S 3 × S 1 at One Loop, JHEP 02 (2013) 102 [arXiv:1211.5587] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M.C. Abbott, The AdS3 × S3 × S3 × S1 Hernández-López phases: a semiclassical derivation, J. Phys. A 46 (2013) 445401 [arXiv:1306.5106] [INSPIRE].ADSzbMATHGoogle Scholar
  26. [26]
    P. Sundin and L. Wulff, Worldsheet scattering in AdS 3/CF T 2, JHEP 07 (2013) 007 [arXiv:1302.5349] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  27. [27]
    R. Roiban, P. Sundin, A. Tseytlin and L. Wulff, The one-loop worldsheet S-matrix for the AdS n × S n × T 10−2n superstring, JHEP 08 (2014) 160 [arXiv:1407.7883] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    P. Sundin and L. Wulff, One- and two-loop checks for the AdS 3 × S 3 × T 4 superstring with mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].ADSzbMATHGoogle Scholar
  29. [29]
    P. Sundin and L. Wulff, The AdS n × S n × T 10−2n BMN string at two loops, JHEP 11 (2015) 154 [arXiv:1508.04313] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    P. Sundin and L. Wulff, The complete one-loop BMN S-matrix in AdS3 × S3 × T4, JHEP 06 (2016) 062 [arXiv:1605.01632] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1 − D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CF T 2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  34. [34]
    O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Jr., Integrability and the Conformal Field Theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  35. [35]
    S. Gukov, E. Martinec, G.W. Moore and A. Strominger, The search for a holographic dual to AdS 3 × S 3 × S 3 × S 1, Adv. Theor. Math. Phys. 9 (2005) 435 [hep-th/0403090] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    D. Tong, The holographic dual of AdS 3 × S 3 × S 3 × S 1, JHEP 04 (2014) 193 [arXiv:1402.5135] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    J. de Boer, Six-dimensional supergravity on S 3 × AdS 3 and 2 − D conformal field theory, Nucl. Phys. B 548 (1999) 139 [hep-th/9806104] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    J. de Boer, Large-N elliptic genus and AdS/CFT correspondence, JHEP 05 (1999) 017 [hep-th/9812240] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [INSPIRE].
  40. [40]
    J. de Boer, A. Pasquinucci and K. Skenderis, AdS/CFT dualities involving large 2 − D N = 4 superconformal symmetry, Adv. Theor. Math. Phys. 3 (1999) 577 [hep-th/9904073] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    S. Gukov, E. Martinec, G.W. Moore and A. Strominger, An Index for 2 − D field theories with large-N = 4 superconformal symmetry, hep-th/0404023 [INSPIRE].
  42. [42]
    T. Lloyd and B. Stefanski, Jr., AdS 3/CF T 2, finite-gap equations and massless modes, JHEP 04 (2014) 179 [arXiv:1312.3268] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    M.C. Abbott and I. Aniceto, Macroscopic (and Microscopic) Massless Modes, Nucl. Phys. B 894 (2015) 75 [arXiv:1412.6380] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    A. Prinsloo, D1 and D5-brane giant gravitons on AdS 3 × S 3 × S 3 × S 1, JHEP 12 (2014) 094 [arXiv:1406.6134] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].ADSzbMATHGoogle Scholar
  47. [47]
    N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    L. Eberhardt, M.R. Gaberdiel, R. Gopakumar and W. Li, BPS spectrum on AdS 3 × S 3 × S 3 × S 1, JHEP 03 (2017) 124 [arXiv:1701.03552] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M. Günaydin, J.L. Petersen, A. Taormina and A. Van Proeyen, On the Unitary Representations of a Class of N = 4 Superconformal Algebras, Nucl. Phys. B 322 (1989) 402 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    E. Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  53. [53]
    L. Göttsche and W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993) 235.MathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  55. [55]
    H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  57. [57]
    O. Ohlsson Sax, B. Stefanski, Jr. and A. Torrielli, On the massless modes of the AdS 3/CF T 2 integrable systems, JHEP 03 (2013) 109 [arXiv:1211.1952] [INSPIRE].ADSzbMATHGoogle Scholar
  58. [58]
    B. Hoare, A. Stepanchuk and A.A. Tseytlin, Giant magnon solution and dispersion relation in string theory in AdS 3 × S 3 × T 4 with mixed flux, Nucl. Phys. B 879 (2014) 318 [arXiv:1311.1794] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  59. [59]
    M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. [60]
    M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  61. [61]
    Z. Bajnok, R.A. Janik and T. Lukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys. B 816 (2009) 376 [arXiv:0811.4448] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. [62]
    D. Bombardelli, A next-to-leading Luescher formula, JHEP 01 (2014) 037 [arXiv:1309.4083] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  63. [63]
    G. Arutyunov and S. Frolov, On String S-matrix, Bound States and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. [64]
    G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: A Proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  67. [67]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 Mirror Model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N} \) = super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS 5/CFT 4, JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, The full Quantum Spectral Curve for AdS 4/CF T 3, arXiv:1701.00473 [INSPIRE].
  71. [71]
    A. Sfondrini and S.J. van Tongeren, Lifting asymptotic degeneracies with the Mirror TBA, JHEP 09 (2011) 050 [arXiv:1106.3909] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  72. [72]
    R.A. Janik, Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections, Lett. Math. Phys. 99 (2012) 277 [arXiv:1012.3994] [INSPIRE].
  73. [73]
    D. Bombardelli et al., An integrability primer for the gauge-gravity correspondence: An introduction, J. Phys. A 49 (2016) 320301 [arXiv:1606.02945] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  74. [74]
    M.P. Heller, R.A. Janik and T. Lukowski, A New derivation of Lüscher F-term and fluctuations around the giant magnon, JHEP 06 (2008) 036 [arXiv:0801.4463] [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    G. Arutyunov, S. Frolov and A. Sfondrini, Exceptional Operators in N = 4 super Yang-Mills, JHEP 09 (2012) 006 [arXiv:1205.6660] [INSPIRE].ADSCrossRefGoogle Scholar
  76. [76]
    M. Taylor, Matching of correlators in AdS 3/CF T 2, JHEP 06 (2008) 010 [arXiv:0709.1838] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  78. [78]
    O. Lunin and S.D. Mathur, Three point functions for M N /S N orbifolds with N = 4 supersymmetry, Commun. Math. Phys. 227 (2002) 385 [hep-th/0103169] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  79. [79]
    M.R. Gaberdiel and I. Kirsch, Worldsheet correlators in AdS 3/CF T 2, JHEP 04 (2007) 050 [hep-th/0703001] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    A. Dabholkar and A. Pakman, Exact chiral ring of AdS 3/CF T 2, Adv. Theor. Math. Phys. 13 (2009) 409 [hep-th/0703022] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  81. [81]
    A. Pakman and A. Sever, Exact N = 4 correlators of AdS 3/CF T 2, Phys. Lett. B 652 (2007) 60 [arXiv:0704.3040] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  82. [82]
    J. de Boer, J. Manschot, K. Papadodimas and E. Verlinde, The Chiral ring of AdS 3/CF T 2 and the attractor mechanism, JHEP 03 (2009) 030 [arXiv:0809.0507] [INSPIRE].CrossRefGoogle Scholar
  83. [83]
    M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  84. [84]
    N. Beisert, M. Bianchi, J.F. Morales and H. Samtleben, Higher spin symmetry and N = 4 SYM, JHEP 07 (2004) 058 [hep-th/0405057] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  85. [85]
    N. Beisert, The Dilatation operator of N = 4 super Yang-Mills theory and integrability, Phys. Rept. 405 (2004) 1 [hep-th/0407277] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  86. [86]
    S. Elitzur, O. Feinerman, A. Giveon and D. Tsabar, String theory on AdS 3 × S 3 × S 3 × S 1, Phys. Lett. B 449 (1999) 180 [hep-th/9811245] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  87. [87]
    J.M. Maldacena and H. Ooguri, Strings in AdS 3 and SL(2, ) WZW model 1.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Instituut voor Theoretische Fysica, KU LeuvenLeuvenBelgium
  2. 2.Nordita, Stockholm University and KTH Royal Institute of TechnologyStockholmSweden
  3. 3.Institut für Theoretische Physik, ETH ZürichZürichSwitzerland
  4. 4.Centre for Mathematical Science, City, University of LondonLondonU.K.
  5. 5.Department of MathematicsUniversity of SurreyGuildfordU.K.

Personalised recommendations