Advertisement

Correlation functions in the D1-D5 orbifold CFT

  • Joan Garcia i Tormo
  • Marika Taylor
Open Access
Regular Article - Theoretical Physics

Abstract

The D1-D5 system has an orbifold point in its moduli space, at which it may be described by an \( \mathcal{N} \) = (4,4) supersymmetric sigma model with target space M N /S(N) where M is \( {\mathbb{T}}^4 \) or K3. In this paper we consider correlation functions involving chiral operators constructed from twist fields: we find explicit expressions for processes involving a twist n operator joining n twist operators of arbitrary twist. These expressions are universal, in that they are independent of the choice of M , and the final results can be expressed in a compact form. We explain how these results are relevant to the black hole microstate programme: one point functions of chiral operators can be used to reconstruct AdS3 near horizon regions of D1-D5 microstates and to match microstates constructed in supergravity with the CFT.

Keywords

AdS-CFT Correspondence Black Holes in String 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. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    O. Lunin and S.D. Mathur, Metric of the multiply wound rotating string, Nucl. Phys. B 610 (2001) 49 [hep-th/0105136] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    O. Lunin and S.D. Mathur, Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, Phys. Rev. Lett. 88 (2002) 211303 [hep-th/0202072] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S.D. Mathur, A proposal to resolve the black hole information paradox, Int. J. Mod. Phys. D 11 (2002) 1537 [hep-th/0205192] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    O. Lunin, S.D. Mathur and A. Saxena, What is the gravity dual of a chiral primary?, Nucl. Phys. B 655 (2003) 185 [hep-th/0211292] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    S.D. Mathur, What Exactly is the Information Paradox?, Lect. Notes Phys. 769 (2009) 3 [arXiv:0803.2030].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    M. Shigemori, Exotic branes and black hole microstates, Int. J. Mod. Phys. Conf. Ser. 21 (2013) 77 [INSPIRE].CrossRefGoogle Scholar
  13. [13]
    O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [INSPIRE].
  14. [14]
    M. Taylor, General 2 charge geometries, JHEP 03 (2006) 009 [hep-th/0507223] [INSPIRE].
  15. [15]
    I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, Phys. Rev. Lett. 98 (2007) 071601 [hep-th/0609154] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, JHEP 04 (2007) 023 [hep-th/0611171] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    S.D. Mathur, A. Saxena and Y.K. Srivastava, Constructing ‘hair’ for the three charge hole, Nucl. Phys. B 680 (2004) 415 [hep-th/0311092] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    I. Bena and P. Kraus, Microscopic description of black rings in AdS/CFT, JHEP 12 (2004) 070 [hep-th/0408186] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    S. Giusto, S.D. Mathur and A. Saxena, 3-charge geometries and their CFT duals, Nucl. Phys. B 710 (2005) 425 [hep-th/0406103] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    O. Lunin, Adding momentum to D1-D5 system, JHEP 04 (2004) 054 [hep-th/0404006] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [INSPIRE].ADSGoogle Scholar
  23. [23]
    V. Balasubramanian, P. Kraus and M. Shigemori, Massless black holes and black rings as effective geometries of the D1-D5 system, Class. Quant. Grav. 22 (2005) 4803 [hep-th/0508110] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    V. Cardoso, O.J.C. Dias, J.L. Hovdebo and R.C. Myers, Instability of non-supersymmetric smooth geometries, Phys. Rev. D 73 (2006) 064031 [hep-th/0512277] [INSPIRE].ADSGoogle Scholar
  26. [26]
    S. Giusto and R. Russo, Adding new hair to the 3-charge black ring, Class. Quant. Grav. 29 (2012) 085006 [arXiv:1201.2585] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    S. Giusto and R. Russo, Perturbative superstrata, Nucl. Phys. B 869 (2013) 164 [arXiv:1211.1957] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    S. Giusto, O. Lunin, S.D. Mathur and D. Turton, D1-D5-P microstates at the cap, JHEP 02 (2013) 050 [arXiv:1211.0306] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    S. Giusto and R. Russo, Superdescendants of the D1D5 CFT and their dual 3-charge geometries, JHEP 03 (2014) 007 [arXiv:1311.5536] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    I. Bena, S. Giusto, R. Russo, M. Shigemori and N.P. Warner, Habemus Superstratum! A constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    B. Chakrabarty, D. Turton and A. Virmani, Holographic description of non-supersymmetric orbifolded D1-D5-P solutions, JHEP 11 (2015) 063 [arXiv:1508.01231] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    S. Giusto, E. Moscato and R. Russo, AdS 3 holography for 1/4 and 1/8 BPS geometries, JHEP 11 (2015) 004 [arXiv:1507.00945] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    I. Bena, E. Martinec, D. Turton and N.P. Warner, Momentum Fractionation on Superstrata, JHEP 05 (2016) 064 [arXiv:1601.05805] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    I. Bena, E. Martinec, D. Turton and N.P. Warner, M-theory Superstrata and the MSW String, JHEP 06 (2017) 137 [arXiv:1703.10171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    E.J. Martinec and S. Massai, String Theory of Supertubes, arXiv:1705.10844 [INSPIRE].
  37. [37]
    A. Bombini and S. Giusto, Non-extremal superdescendants of the D1D5 CFT, JHEP 10 (2017) 023 [arXiv:1706.09761] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    I. Bena, D. Turton, R. Walker and N.P. Warner, Integrability and Black-Hole Microstate Geometries, JHEP 11 (2017) 021 [arXiv:1709.01107] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    I. Bena, P. Heidmann and P.F. Ramirez, A systematic construction of microstate geometries with low angular momentum, JHEP 10 (2017) 217 [arXiv:1709.02812] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    I. Bena et al., Asymptotically-flat supergravity solutions deep inside the black-hole regime, JHEP 02 (2018) 014 [arXiv:1711.10474] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    G. Bossard, S. Katmadas and D. Turton, Two Kissing Bolts, JHEP 02 (2018) 008 [arXiv:1711.04784] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    K. Skenderis and M. Taylor, Kaluza-Klein holography, JHEP 05 (2006) 057 [hep-th/0603016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    A. Galliani, S. Giusto and R. Russo, Holographic 4-point correlators with heavy states, JHEP 10 (2017) 040 [arXiv:1705.09250] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    A. Bombini, A. Galliani, S. Giusto, E. Moscato and R. Russo, Unitary 4-point correlators from classical geometries, Eur. Phys. J. C 78 (2018) 8 [arXiv:1710.06820] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    K. Skenderis and M. Taylor, Holographic Coulomb branch vevs, JHEP 08 (2006) 001 [hep-th/0604169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    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].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    M.R. Gaberdiel and I. Kirsch, Worldsheet correlators in AdS 3 /CF T 2, JHEP 04 (2007) 050 [hep-th/0703001] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    M. Taylor, Matching of correlators in AdS 3 /CF T 2, JHEP 06 (2008) 010 [arXiv:0709.1838] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
  51. [51]
    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].
  52. [52]
    S.G. Avery and B.D. Chowdhury, Emission from the D1D5 CFT: Higher Twists, JHEP 01 (2010) 087 [arXiv:0907.1663] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    S.G. Avery, B.D. Chowdhury and S.D. Mathur, Deforming the D1D5 CFT away from the orbifold point, JHEP 06 (2010) 031 [arXiv:1002.3132] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    S.G. Avery and B.D. Chowdhury, Intertwining Relations for the Deformed D1D5 CFT, JHEP 05 (2011) 025 [arXiv:1007.2202] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    S.G. Avery, B.D. Chowdhury and S.D. Mathur, Excitations in the deformed D1D5 CFT, JHEP 06 (2010) 032 [arXiv:1003.2746] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    B.A. Burrington, A.W. Peet and I.G. Zadeh, Twist-nontwist correlators in M N /S N orbifold CFTs, Phys. Rev. D 87 (2013) 106008 [arXiv:1211.6689] [INSPIRE].
  57. [57]
    B.A. Burrington, A.W. Peet and I.G. Zadeh, Operator mixing for string states in the D1-D5 CFT near the orbifold point, Phys. Rev. D 87 (2013) 106001 [arXiv:1211.6699] [INSPIRE].ADSGoogle Scholar
  58. [58]
    B.A. Burrington, S.D. Mathur, A.W. Peet and I.G. Zadeh, Analyzing the squeezed state generated by a twist deformation, Phys. Rev. D 91 (2015) 124072 [arXiv:1410.5790] [INSPIRE].ADSMathSciNetGoogle Scholar
  59. [59]
    Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the deformation operator in the D1D5 CFT, JHEP 01 (2015) 071 [arXiv:1410.4543] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the twist operator in the D1D5 CFT, JHEP 08 (2014) 064 [arXiv:1405.0259] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  61. [61]
    B.A. Burrington, A.W. Peet and I.G. Zadeh, Bosonization, cocycles and the D1-D5 CFT on the covering surface, Phys. Rev. D 93 (2016) 026004 [arXiv:1509.00022] [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    Z. Carson, S. Hampton and S.D. Mathur, Second order effect of twist deformations in the D1D5 CFT, JHEP 04 (2016) 115 [arXiv:1511.04046] [INSPIRE].ADSMathSciNetGoogle Scholar
  63. [63]
    Z. Carson, S. Hampton and S.D. Mathur, Full action of two deformation operators in the D1D5 CFT, JHEP 11 (2017) 096 [arXiv:1612.03886] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    B.A. Burrington, I.T. Jardine and A.W. Peet, Operator mixing in deformed D1D5 CFT and the OPE on the cover, JHEP 06 (2017) 149 [arXiv:1703.04744] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    Z. Carson, I.T. Jardine and A.W. Peet, Component twist method for higher twists in D1-D5 CFT, Phys. Rev. D 96 (2017) 026006 [arXiv:1704.03401] [INSPIRE].ADSGoogle Scholar
  66. [66]
    B.A. Burrington, I.T. Jardine and A.W. Peet, The OPE of bare twist operators in bosonic S N orbifold CFTs at large N , arXiv:1804.01562 [INSPIRE].
  67. [67]
    A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  68. [68]
    A. Pakman, L. Rastelli and S.S. Razamat, Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds, Phys. Rev. D 80 (2009) 086009 [arXiv:0905.3451] [INSPIRE].ADSGoogle Scholar
  69. [69]
    A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CFT(2), JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    H.J. Boonstra, B. Peeters and K. Skenderis, Duality and asymptotic geometries, Phys. Lett. B 411 (1997) 59 [hep-th/9706192] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  72. [72]
    A. Jevicki, M. Mihailescu and S. Ramgoolam, Gravity from CFT on S N (X): Symmetries and interactions, Nucl. Phys. B 577 (2000) 47 [hep-th/9907144] [INSPIRE].
  73. [73]
    J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    S. Deger, A. Kaya, E. Sezgin and P. Sundell, Spectrum of D = 6, N=4b supergravity on AdS in three-dimensions ×S 3, Nucl. Phys. B 536 (1998) 110 [hep-th/9804166] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  75. [75]
    G. Arutyunov, A. Pankiewicz and S. Theisen, Cubic couplings in D = 6 N = 4b supergravity on AdS 3 × S 3, Phys. Rev. D 63 (2001) 044024 [hep-th/0007061] [INSPIRE].ADSGoogle Scholar
  76. [76]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    M.R. Gaberdiel and M. Kelm, The symmetric orbifold of \( \mathcal{N} \) = 2 minimal models, JHEP 07 (2016) 113 [arXiv:1604.03964] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  79. [79]
    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].ADSCrossRefzbMATHGoogle Scholar
  80. [80]
    L. Eberhardt, M.R. Gaberdiel and W. Li, A holographic dual for string theory on AdS 3 × S 3 × S 3 × S 1, JHEP 08 (2017) 111 [arXiv:1707.02705] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    M.R. Gaberdiel, R. Gopakumar and C. Hull, Stringy AdS 3 from the worldsheet, JHEP 07 (2017) 090 [arXiv:1704.08665] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  82. [82]
    M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefanski and A. Torrielli, Protected string spectrum in AdS 3 /CFT 2 from worldsheet integrability, JHEP 04 (2017) 091 [arXiv:1701.03501] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Mathematical Sciences and STAG Research CentreUniversity of SouthamptonSouthamptonU.K.

Personalised recommendations