Lifting at higher levels in the D1D5 CFT

Abstract

The D1D5P system has a large set of BPS states at its orbifold point. Perturbing away from this ‘free’ point leads to some states joining up into long supermultiplets and lifting, while other states remain BPS. We consider the simplest orbifold which exhibits this lift: that with N = 2 copies of the free c = 6 CFT. We write down the number of lifted and unlifted states implied by the index at all levels upto 6. We work to second order in the perturbation strength λ. For levels upto 4, we find the wavefunctions of the lifted states, their supermultiplet structure and the value of the lift. All states that are allowed to lift by the index are in fact lifted at order O(λ2). We observe that the unlifted states in the untwisted sector have an antisymmetry between the copies in the right moving Ramond ground state sector, and extend this observation to find classes of states for arbitrary N that will remain unlifted to O(λ2).

A preprint version of the article is available at ArXiv.

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].

    ADS  MathSciNet  MATH  Google Scholar 

  2. [2]

    C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591 [hep-th/9602043] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  3. [3]

    S.R. Das and S.D. Mathur, Comparing decay rates for black holes and D-branes, Nucl. Phys. B 478 (1996) 561 [hep-th/9606185] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  4. [4]

    J.M. Maldacena and A. Strominger, Black hole grey body factors and D-brane spectroscopy, Phys. Rev. D 55 (1997) 861 [hep-th/9609026] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  5. [5]

    C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435 [hep-th/9512078] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  6. [6]

    R. Dijkgraaf, Instanton strings and hyperKähler geometry, Nucl. Phys. B 543 (1999) 545 [hep-th/9810210] [INSPIRE].

    ADS  MATH  Google Scholar 

  7. [7]

    N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  8. [8]

    F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1–D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  9. [9]

    G.E. Arutyunov and S.A. Frolov, Virasoro amplitude from the SN R24 orbifold sigma model, Theor. Math. Phys. 114 (1998) 43 [hep-th/9708129] [INSPIRE].

    MATH  Google Scholar 

  10. [10]

    G.E. Arutyunov and S.A. Frolov, Four graviton scattering amplitude from SN R8 supersymmetric orbifold sigma model, Nucl. Phys. B 524 (1998) 159 [hep-th/9712061] [INSPIRE].

    ADS  MATH  Google Scholar 

  11. [11]

    A. Jevicki, M. Mihailescu and S. Ramgoolam, Gravity from CFT on SN (X ): Symmetries and interactions, Nucl. Phys. B 577 (2000) 47 [hep-th/9907144] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  12. [12]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  13. [13]

    J.M. Maldacena, G.W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [INSPIRE].

  14. [14]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  15. [15]

    S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  16. [16]

    I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  17. [17]

    I. Bena and N.P. Warner, Black holes, black rings and their microstates, in Lecture Notes in Physics 755, Springer (2008), pp. 1–92 [hep-th/0701216] [INSPIRE].

  18. [18]

    B.D. Chowdhury and A. Virmani, Modave Lectures on Fuzzballs and Emission from the D1–D5 System, in proceedings of the 5th Modave Summer School in Mathematical Physics, Modave, Belgium, 17–21 August 2009, arXiv:1001.1444 [INSPIRE].

  19. [19]

    E. Gava and K.S. Narain, Proving the PP-wave/CFT2 duality, JHEP 12 (2002) 023 [hep-th/0208081] [INSPIRE].

    ADS  Google Scholar 

  20. [20]

    B. Guo and S.D. Mathur, Lifting of level-1 states in the D1D5 CFT, JHEP 03 (2020) 028 [arXiv:1912.05567] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  21. [21]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  22. [22]

    S.G. Avery, B.D. Chowdhury and S.D. Mathur, Excitations in the deformed D1D5 CFT, JHEP 06 (2010) 032 [arXiv:1003.2746] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  23. [23]

    A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  24. [24]

    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].

    ADS  Google Scholar 

  25. [25]

    A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CFT2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  26. [26]

    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].

    ADS  Google Scholar 

  27. [27]

    B.A. Burrington, A.W. Peet and I.G. Zadeh, Twist-nontwist correlators in MN /SN orbifold CFTs, Phys. Rev. D 87 (2013) 106008 [arXiv:1211.6689] [INSPIRE].

    ADS  Google Scholar 

  28. [28]

    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].

    ADS  MathSciNet  Google Scholar 

  29. [29]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  30. [30]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  31. [31]

    Z. Carson, S. Hampton and S.D. Mathur, One-Loop Transition Amplitudes in the D1D5 CFT, JHEP 01 (2017) 006 [arXiv:1606.06212] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  32. [32]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  33. [33]

    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].

    ADS  Google Scholar 

  34. [34]

    Z. Carson, S.D. Mathur and D. Turton, Bogoliubov coefficients for the twist operator in the D1D5 CFT, Nucl. Phys. B 889 (2014) 443 [arXiv:1406.6977] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. [35]

    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].

    ADS  MATH  Google Scholar 

  36. [36]

    L.P. Kadanoff, Multicritical behavior at the Kosterlitz-Thouless critical point, Ann. Phys. 120 (1979) 39.

    ADS  Google Scholar 

  37. [37]

    R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, On Moduli Spaces Of Conformal Field Theories With c ≥ 1, in Perspectives in String Theory, World Scientific, Copenhagen Denmark (1987), pp. 117–137 [INSPIRE].

    Google Scholar 

  38. [38]

    J.L. Cardy, Continuously Varying Exponents and the Value of the Central Charge, J. Phys. A 20 (1987) L891 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  39. [39]

    D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  40. [40]

    H. Eberle, Twistfield perturbations of vertex operators in the2-orbifold model, Ph.D. Thesis, University of Bonn, Bonn Germany (2006) [JHEP 06 (2002) 022] [hep-th/0103059] [INSPIRE].

  41. [41]

    M.R. Gaberdiel, A. Konechny and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42 (2009) 105402 [arXiv:0811.3149] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  42. [42]

    D. Berenstein and A. Miller, Conformal perturbation theory, dimensional regularization, and AdS/CFT correspondence, Phys. Rev. D 90 (2014) 086011 [arXiv:1406.4142] [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    D. Berenstein and A. Miller, Logarithmic enhancements in conformal perturbation theory and their real time interpretation, Int. J. Mod. Phys. A 35 (2020) 2050184 [arXiv:1607.01922] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  44. [44]

    M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  45. [45]

    S. Hampton, S.D. Mathur and I.G. Zadeh, Lifting of D1–D5–P states, JHEP 01 (2019) 075 [arXiv:1804.10097] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  46. [46]

    C.A. Keller and I.G. Zadeh, Lifting \( \frac{1}{4} \)-BPS States on K 3 and Mathieu Moonshine, Commun. Math. Phys. 377 (2020) 225 [arXiv:1905.00035] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  47. [47]

    C.A. Keller and I.G. Zadeh, Conformal Perturbation Theory for Twisted Fields, J. Phys. A 53 (2020) 095401 [arXiv:1907.08207] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  48. [48]

    A. Sevrin, W. Troost and A. Van Proeyen, Superconformal Algebras in Two-Dimensions with N = 4, Phys. Lett. B 208 (1988) 447 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  49. [49]

    B. Guo and S.D. Mathur, Lifting of states in 2-dimensional N = 4 supersymmetric CFTs, JHEP 10 (2019) 155 [arXiv:1905.11923] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  50. [50]

    N. Benjamin, A Refined Count of BPS States in the D1/D5 System, JHEP 06 (2017) 028 [arXiv:1610.07607] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  51. [51]

    N. Benjamin and S.M. Harrison, Symmetries of the refined D1/D5 BPS spectrum, JHEP 11 (2017) 091 [arXiv:1708.02244] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  52. [52]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

    I. Bena, E. Martinec, D. Turton and N.P. Warner, Momentum Fractionation on Superstrata, JHEP 05 (2016) 064 [arXiv:1601.05805] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  54. [54]

    I. Bena et al., Smooth horizonless geometries deep inside the black-hole regime, Phys. Rev. Lett. 117 (2016) 201601 [arXiv:1607.03908] [INSPIRE].

    ADS  Google Scholar 

  55. [55]

    I. Bena et al., Asymptotically-flat supergravity solutions deep inside the black-hole regime, JHEP 02 (2018) 014 [arXiv:1711.10474] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  56. [56]

    N. Čeplak, R. Russo and M. Shigemori, Supercharging Superstrata, JHEP 03 (2019) 095 [arXiv:1812.08761] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  57. [57]

    P. Heidmann and N.P. Warner, Superstratum Symbiosis, JHEP 09 (2019) 059 [arXiv:1903.07631] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  58. [58]

    T. Eguchi and A. Taormina, Unitary Representations of N = 4 Superconformal Algebra, Phys. Lett. B 196 (1987) 75 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  59. [59]

    T. Eguchi and A. Taormina, Character Formulas for the N = 4 Superconformal Algebra, Phys. Lett. B 200 (1988) 315 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  60. [60]

    J.L. Petersen and A. Taormina, Characters of the N = 4 Superconformal Algebra With Two Central Extensions, Nucl. Phys. B 331 (1990) 556 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  61. [61]

    J.L. Petersen and A. Taormina, Characters of the N = 4 Superconformal Algebra With Two Central Extensions: 2. Massless Representations, Nucl. Phys. B 333 (1990) 833 [INSPIRE].

  62. [62]

    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].

    ADS  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bin Guo.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2008.01274

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, B., Mathur, S.D. Lifting at higher levels in the D1D5 CFT. J. High Energ. Phys. 2020, 145 (2020). https://doi.org/10.1007/JHEP11(2020)145

Download citation

Keywords

  • Conformal Field Theory
  • Extended Supersymmetry
  • AdS-CFT Correspondence
  • Black Holes in String Theory