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Even spin minimal model holography

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Abstract

The even spin \( \mathcal{W}_{\infty}^e \) algebra that is generated by the stress energy tensor together with one Virasoro primary field for every even spin s ≥ 4 is analysed systematically by studying the constraints coming from the Jacobi identities. It is found that the algebra is characterised, in addition to the central charge, by one free parameter that can be identified with the self-coupling constant of the spin 4 field. We show that \( \mathcal{W}_{\infty}^e \) can be thought of as the quantisation of the asymptotic symmetry algebra of the even higher spin theory on AdS3. On the other hand, \( \mathcal{W}_{\infty}^e \) is also quantum equivalent to the \( \mathfrak{s}\mathfrak{o}(N) \) coset algebras, and thus our result establishes an important aspect of the even spin minimal model holography conjecture. The quantum equivalence holds actually at finite central charge, and hence opens the way towards understanding the duality beyond the leading ’t Hooft limit.

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References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  2. M.A. Vasiliev, Higher spin algebras and quantization on the sphere and hyperboloid, Int. J. Mod. Phys. A 6 (1991) 1115 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  3. B. Sundborg, Stringy gravity, interacting tensionless strings and massless higher spins, Nucl. Phys. Proc. Suppl. 102 (2001) 113 [hep-th/0103247] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. E. Witten, Spacetime reconstruction, talk at the John Schwarz 60th birthday symposium, http://theory.caltech.edu/jhs60/witten/1.html.

  5. A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].

  6. E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, arXiv:1112.1016 [INSPIRE].

  8. J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, arXiv:1204.3882 [INSPIRE].

  9. I. Klebanov and A. Polyakov, AdS dual of the critical O(N ) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  10. E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP 07 (2005) 044 [hep-th/0305040] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. S. Giombi and X. Yin, The higher spin/vector model duality, arXiv:1208.4036 [INSPIRE].

  12. M.R. Gaberdiel and R. Gopakumar, An AdS 3 dual for minimal model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].

    ADS  Google Scholar 

  13. M.R. Gaberdiel and R. Gopakumar, Minimal model holography, arXiv:1207.6697 [INSPIRE].

  14. S. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. S. Prokushkin and M.A. Vasiliev, 3D higher spin gauge theories with matter, hep-th/9812242 [INSPIRE].

  16. M.A. Vasiliev, Higher spin gauge theories: star product and AdS space, in The many faces of the superworld, M.A. Shifman ed., World Scientific Singapore (2000), pg. 533 [hep-th/9910096] [INSPIRE].

  17. M. Henneaux and S.-J. Rey, Nonlinear W ∞ as asymptotic symmetry of three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. M.R. Gaberdiel and T. Hartman, Symmetries of holographic minimal models, JHEP 05 (2011) 031 [arXiv:1101.2910] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. B. Khesin and F. Malikov, Universal Drinfeld-Sokolov reduction and matrices of complex size, Commun. Math. Phys. 175 (1996) 113 [hep-th/9405116] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W -symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [arXiv:1107.0290] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. M.R. Gaberdiel and R. Gopakumar, Triality in minimal model holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. A. Kuniba, T. Nakanishi and J. Suzuki, Ferromagnetizations and antiferromagnetizations in RSOS models, Nucl. Phys. B 356 (1991) 750 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. D. Altschuler, M. Bauer and H. Saleur, Level rank duality in nonunitary coset theories, J. Phys. A 23 (1990) L789 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  27. C. Ahn, The large-N ’t Hooft limit of coset minimal models, JHEP 10 (2011) 125 [arXiv:1106.0351] [INSPIRE].

    Article  ADS  Google Scholar 

  28. M.R. Gaberdiel and C. Vollenweider, Minimal model holography for SO(2N), JHEP 08 (2011) 104 [arXiv:1106.2634] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. C. Ahn, The primary spin-4 Casimir operators in the holographic SO(N ) coset minimal models, JHEP 05 (2012) 040 [arXiv:1202.0074] [INSPIRE].

    Article  ADS  Google Scholar 

  30. K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys. C 2 (1991) 787 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  31. K. Thielemans, An algorithmic approach to operator product expansions, W -algebras and W -strings, Ph.D. thesis, KU Leuven, Leuven Belgium (1994) [hep-th/9506159] [INSPIRE].

  32. C. Candu and M.R. Gaberdiel, Duality in N = 2 minimal model holography, arXiv:1207.6646 [INSPIRE].

  33. H. Kausch and G. Watts, A study of W algebras using Jacobi identities, Nucl. Phys. B 354 (1991) 740 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. R. Blumenhagen et al., W algebras with two and three generators, Nucl. Phys. B 361 (1991) 255 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. A. Kliem, The construction of W algebras, Diploma thesis, Universität Bonn, Bonn Germany (1991) [INSPIRE].

  36. P. Bowcock and G. Watts, On the classification of quantum W algebras, Nucl. Phys. B 379 (1992) 63 [hep-th/9111062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  37. K. Hornfeck, Classification of structure constants for W algebras from highest weights, Nucl. Phys. B 411 (1994) 307 [hep-th/9307170] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. A. Honecker, Automorphisms of W algebras and extended rational conformal field theories, Nucl. Phys. B 400 (1993) 574 [hep-th/9211130] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck and R. Hubel, Coset realization of unifying W algebras, Int. J. Mod. Phys. A 10 (1995) 2367 [hep-th/9406203] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  40. P. Bouwknegt, Extended conformal algebras from Kac-Moody algebras, MIT-CTP-1665, M.I.T., Cambridge U.S.A. (1988) [INSPIRE].

  41. A. Honecker, A note on the algebraic evaluation of correlators in local chiral conformal field theory, hep-th/9209029 [INSPIRE].

  42. P. Goddard and D.I. Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys. A 1 (1986) 303 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  43. C. Candu and M.R. Gaberdiel, Supersymmetric holography on AdS 3, arXiv:1203.1939 [INSPIRE].

  44. T. Creutzig, Y. Hikida and P.B. Ronne, N = 1 supersymmetric higher spin holography on AdS 3, arXiv:1209.5404 [INSPIRE].

  45. J. de Boer, L. Feher and A. Honecker, A class of W algebras with infinitely generated classical limit, Nucl. Phys. B 420 (1994) 409 [hep-th/9312049] [INSPIRE].

    Article  ADS  Google Scholar 

  46. W. Eholzer, A. Honecker and R. Hubel, How complete is the classification of W symmetries?, Phys. Lett. B 308 (1993) 42 [hep-th/9302124] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  47. A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical defects in higher spin theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [INSPIRE].

    ADS  Google Scholar 

  48. E. Perlmutter, T. Prochazka and J. Raeymaekers, The semiclassical limit of W N CFTs and Vasiliev theory, arXiv:1210.8452 [INSPIRE].

  49. E. Fradkin and V.Y. Linetsky, Supersymmetric Racah basis, family of infinite dimensional superalgebras, SU(∞ + 1|∞) and related 2D models, Mod. Phys. Lett. A 6 (1991) 617 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  50. P. Bowcock, Quasi-primary fields and associativity of chiral algebras, Nucl. Phys. B 356 (1991) 367 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Institut für Theoretische Physik, ETH Zürich, CH-8093, Zürich, Switzerland

    Constantin Candu, Matthias R. Gaberdiel, Maximilian Kelm & Carl Vollenweider

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  1. Constantin Candu
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  2. Matthias R. Gaberdiel
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  3. Maximilian Kelm
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  4. Carl Vollenweider
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Correspondence to Carl Vollenweider.

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ArXiv ePrint: 1211.3113

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Candu, C., Gaberdiel, M.R., Kelm, M. et al. Even spin minimal model holography. J. High Energ. Phys. 2013, 185 (2013). https://doi.org/10.1007/JHEP01(2013)185

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