The Schwarzian sector of higher spin CFTs

Abstract

Two-dimensional conformal field theories with Virasoro symmetry generically contain a Schwarzian sector. This sector is related to the near-horizon region of the near-extremal BTZ black hole in the holographic dual. In this work we generalize this picture to CFTs with higher spin conserved currents. It is shown that the partition function in the near-extremal limit agrees with that of BF higher spin gravity in AdS2 which is described by a generalized Schwarzian theory. We also provide a spectral decomposition of Schwarzian partition functions via the \( {\mathcal{W}}_N \) fusion kernel and consider supersymmetric generalizations.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    A. Sen, Quantum entropy function from AdS2/CFT1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  2. [2]

    I. Mandal and A. Sen, Black hole microstate counting and its macroscopic counterpart, Class. Quant. Grav. 27 (2010) 214003 [Nucl. Phys. B Proc. Suppl. 216 (2011) 147] [arXiv:1008.3801] [INSPIRE].

  3. [3]

    A. Sen, Microscopic and macroscopic entropy of extremal black holes in string theory, Gen. Rel. Grav. 46 (2014) 1711 [arXiv:1402.0109] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  4. [4]

    J. S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].

  5. [5]

    K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    G. Penington, S. H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].

  8. [8]

    J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. [9]

    G. Sárosi, AdS2 holography and the SYK model, PoS(Modave2017)001 (2018) [arXiv:1711.08482] [INSPIRE].

  10. [10]

    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].

  11. [11]

    P. Saad, S. H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].

  12. [12]

    K. Okuyama and K. Sakai, JT gravity, KdV equations and macroscopic loop operators, JHEP 01 (2020) 156 [arXiv:1911.01659] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    A. Ghosh, H. Maxfield and G. J. Turiaci, A universal Schwarzian sector in two-dimensional conformal field theories, JHEP 05 (2020) 104 [arXiv:1912.07654] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. [15]

    M. R. Gaberdiel and R. Gopakumar, An AdS3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    M. R. Gaberdiel and R. Gopakumar, Minimal model holography, J. Phys. A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    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  Article  Google Scholar 

  18. [18]

    L. Eberhardt, M. R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  19. [19]

    M. R. Gaberdiel, R. Gopakumar, B. Knighton and P. Maity, From symmetric product CFTs to AdS3, arXiv:2011.10038 [INSPIRE].

  20. [20]

    D. J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 05 (2017) 092 [arXiv:1702.08016] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. [21]

    P. Zhang, Y. Gu and A. Kitaev, An obstacle to sub-AdS holography for SYK-like models, JHEP 03 (2020) 094 [arXiv:2012.01620] [INSPIRE].

    ADS  Google Scholar 

  22. [22]

    H. A. González, D. Grumiller and J. Salzer, Towards a bulk description of higher spin SYK, JHEP 05 (2018) 083 [arXiv:1802.01562] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    K. B. Alkalaev, On higher spin extension of the Jackiw-Teitelboim gravity model, J. Phys. A 47 (2014) 365401 [arXiv:1311.5119] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  24. [24]

    N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney and E. Perlmutter, Constraints on higher spin CFT2 , JHEP 05 (2018) 092 [arXiv:1707.07717] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  25. [25]

    L. F. Alday, J.-B. Bae, N. Benjamin and C. Jorge-Diaz, On the spectrum of pure higher spin gravity, JHEP 12 (2020) 001 [arXiv:2009.01830] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. [26]

    N. Drukker, D. Gaiotto and J. Gomis, The virtue of defects in 4D gauge theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. [27]

    M. Gutperle and P. Kraus, Higher spin black holes, JHEP 05 (2011) 022 [arXiv:1103.4304] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  28. [28]

    M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three dimensional higher spin gravity: a review, J. Phys. A 46 (2013) 214001 [arXiv:1208.5182] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    S. Datta and J. R. David, Supersymmetry of classical solutions in Chern-Simons higher spin supergravity, JHEP 01 (2013) 146 [arXiv:1208.3921] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. [30]

    S. Datta and J. R. David, Black holes in higher spin supergravity, JHEP 07 (2013) 110 [arXiv:1303.1946] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. [31]

    M. Bañados, A. Castro, A. Faraggi and J. I. Jottar, Extremal higher spin black holes, JHEP 04 (2016) 077 [arXiv:1512.00073] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  32. [32]

    M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Hypersymmetry bounds and three-dimensional higher-spin black holes, JHEP 08 (2015) 021 [arXiv:1506.01847] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  33. [33]

    S. Datta, J. R. David and S. P. Kumar, Conformal perturbation theory and higher spin entanglement entropy on the torus, JHEP 04 (2015) 041 [arXiv:1412.3946] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    N. J. Iles and G. M. T. Watts, Modular properties of characters of the W3 algebra, JHEP 01 (2016) 089 [arXiv:1411.4039] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    J. R. David, M. R. Gaberdiel and R. Gopakumar, The heat kernel on AdS3 and its applications, JHEP 04 (2010) 125 [arXiv:0911.5085] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. [36]

    M. R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W-symmetry in AdS3, JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  37. [37]

    D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. [38]

    F. M. Haehl and M. Rozali, Effective field theory for chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    W. Li and S. Theisen, Some aspects of holographic W-gravity, JHEP 08 (2015) 035 [arXiv:1504.07799] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  40. [40]

    T. Hartman, C. A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  41. [41]

    S. Banerjee et al., Smoothed transitions in higher spin AdS gravity, Class. Quant. Grav. 30 (2013) 104001 [arXiv:1209.5396] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  42. [42]

    C. Candu and C. Vollenweider, The N = 1 algebra W[μ] and its truncations, JHEP 11 (2013) 032 [arXiv:1305.0013] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. [43]

    T. G. Mertens, G. J. Turiaci and H. L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. [44]

    P. Kraus, Lectures on black holes and the AdS3/CFT2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  45. [45]

    W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev. D 95 (2017) 026009 [Addendum ibid. 95 (2017) 069904] [arXiv:1610.08917] [INSPIRE].

  46. [46]

    A. M. Charles and F. Larsen, A one-loop test of the near-AdS2/near-CFT1 correspondence, JHEP 07 (2020) 186 [arXiv:1908.03575] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  47. [47]

    T. G. Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  48. [48]

    A. Gaikwad, L. K. Joshi, G. Mandal and S. R. Wadia, Holographic dual to charged SYK from 3D gravity and Chern-Simons, JHEP 02 (2020) 033 [arXiv:1802.07746] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  49. [49]

    E. Perlmutter, Bounding the space of holographic CFTs with chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  50. [50]

    J. R. David, S. Khetrapal and S. P. Kumar, Local quenches and quantum chaos from higher spin perturbations, JHEP 10 (2017) 156 [arXiv:1707.07166] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. [51]

    P. Narayan and J. Yoon, Chaos in three-dimensional higher spin gravity, JHEP 07 (2019) 046 [arXiv:1903.08761] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. [52]

    P. Nayak, J. Sonner and M. Vielma, Extended eigenstate thermalization and the role of FZZT branes in the Schwarzian theory, JHEP 03 (2020) 168 [arXiv:1907.10061] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. [53]

    A. Banerjee, A. Kundu and R. Poojary, Maximal chaos from strings, branes and Schwarzian action, JHEP 06 (2019) 076 [arXiv:1811.04977] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. [54]

    K. Alkalaev and X. Bekaert, Towards higher-spin AdS2/CFT1 holography, JHEP 04 (2020) 206 [arXiv:1911.13212] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  55. [55]

    K. Alkalaev and X. Bekaert, On BF-type higher-spin actions in two dimensions, JHEP 05 (2020) 158 [arXiv:2002.02387] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  56. [56]

    C. Peng, N = (0, 2) SYK, chaos and higher-spins, JHEP 12 (2018) 065 [arXiv:1805.09325] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  57. [57]

    J. de Boer and J. I. Jottar, Entanglement entropy and higher spin holography in AdS3, JHEP 04 (2014) 089 [arXiv:1306.4347] [INSPIRE].

    Article  Google Scholar 

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Correspondence to Shouvik Datta.

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ArXiv ePrint: 2101.04980

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Datta, S. The Schwarzian sector of higher spin CFTs. J. High Energ. Phys. 2021, 171 (2021). https://doi.org/10.1007/JHEP04(2021)171

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Keywords

  • Conformal and W Symmetry
  • AdS-CFT Correspondence
  • Field Theories in Lower Dimensions