The large level limit of Kazama-Suzuki models

Open Access
Regular Article - Theoretical Physics

Abstract

Limits of families of conformal field theories are of interest in the context of AdS/CFT dualities. We explore here the large level limit of the two-dimensional \( \mathcal{N}=\left(2,\ 2\right) \) superconformal \( {\mathcal{W}}_{n+1} \) minimal models that appear in the context of the supersymmetric higher-spin AdS3/CFT2 duality. These models are constructed as Kazama-Suzuki coset models of the form SU(n + 1)/U(n). We determine a family of boundary conditions in the limit theories, and use the modular bootstrap to obtain the full bulk spectrum of \( \mathcal{N}=2 \) super-\( {\mathcal{W}}_{n+1} \) primaries in the theory. We also confirm the identification of this limit theory as the continuous orbifold \( {\mathbb{C}}^n/\mathrm{U}(n) \) that was discussed recently.

Keywords

Field Theories in Lower Dimensions Extended Supersymmetry Conformal and W Symmetry 

Notes

Open Access

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References

  1. [1]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    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].ADSCrossRefMATHGoogle Scholar
  3. [3]
    S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3 − D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].ADSGoogle Scholar
  5. [5]
    T. Creutzig, Y. Hikida and P.B. Rønne, Higher spin AdS 3 supergravity and its dual CFT, JHEP 02 (2012) 109 [arXiv:1111.2139] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  6. [6]
    C. Candu and M.R. Gaberdiel, Supersymmetric holography on AdS 3, JHEP 09 (2013) 071 [arXiv:1203.1939] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  7. [7]
    C. Candu and M.R. Gaberdiel, Duality in N = 2 Minimal Model Holography, JHEP 02 (2013) 070 [arXiv:1207.6646] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    T. Creutzig, Y. Hikida and P.B. Rønne, Three point functions in higher spin AdS 3 supergravity, JHEP 01 (2013) 171 [arXiv:1211.2237] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  9. [9]
    H. Moradi and K. Zoubos, Three-Point Functions in N = 2 Higher-Spin Holography, JHEP 04 (2013) 018 [arXiv:1211.2239] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M.R. Gaberdiel and R. Gopakumar, Large-N = 4 Holography, JHEP 09 (2013) 036 [arXiv:1305.4181] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    M.R. Gaberdiel and R. Gopakumar, Higher Spins & Strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Y. Kazama and H. Suzuki, Characterization of N = 2 Superconformal Models Generated by Coset Space Method, Phys. Lett. B 216 (1989) 112 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    Y. Kazama and H. Suzuki, New N = 2 Superconformal Field Theories and Superstring Compactification, Nucl. Phys. B 321 (1989) 232 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    M.R. Gaberdiel and P. Suchanek, Limits of Minimal Models and Continuous Orbifolds, JHEP 03 (2012) 104 [arXiv:1112.1708] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    I. Bakas and E. Kiritsis, Bosonic Realization of a Universal W Algebra and Z(infinity) Parafermions, Nucl. Phys. B 343 (1990) 185 [Erratum ibid. B 350 (1991) 512] [INSPIRE].
  16. [16]
    I. Bakas and E. Kiritsis, Grassmannian Coset Models and Unitary Representations of W , Mod. Phys. Lett. A 5 (1990) 2039 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    S. Fredenhagen and C. Restuccia, The geometry of the limit of N = 2 minimal models, J. Phys. A 46 (2013) 045402 [arXiv:1208.6136] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  18. [18]
    C. Restuccia, Limit theories and continuous orbifolds, arXiv:1310.6857 [INSPIRE].
  19. [19]
    M.R. Gaberdiel and M. Kelm, The continuous orbifold of \( \mathcal{N}=2 \) minimal model holography, JHEP 08 (2014) 084 [arXiv:1406.2345] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    P. Goddard, A. Kent and D.I. Olive, Virasoro Algebras and Coset Space Models, Phys. Lett. B 152 (1985) 88 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    P. Goddard, A. Kent and D.I. Olive, Unitary Representations of the Virasoro and Supervirasoro Algebras, Commun. Math. Phys. 103 (1986) 105 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  22. [22]
    J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    S. Fredenhagen and V. Schomerus, D-branes in coset models, JHEP 02 (2002) 005 [hep-th/0111189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    S. Fredenhagen and V. Schomerus, On boundary RG flows in coset conformal field theories, Phys. Rev. D 67 (2003) 085001 [hep-th/0205011] [INSPIRE].ADSGoogle Scholar
  25. [25]
    S. Fredenhagen, Organizing boundary RG flows, Nucl. Phys. B 660 (2003) 436 [hep-th/0301229] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    S. Fredenhagen, C. Restuccia and R. Sun, The limit of N = (2, 2) superconformal minimal models, JHEP 10 (2012) 141 [arXiv:1204.0446] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    M.R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, Partition Functions of Holographic Minimal Models, JHEP 08 (2011) 077 [arXiv:1106.1897] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2003).Google Scholar
  29. [29]
    Y. Sugawara, Thermodynamics of Superstring on Near-extremal NS5 and Effective Hagedorn Behavior, JHEP 10 (2012) 159 [arXiv:1208.3534] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    I. Runkel and G.M.T. Watts, A Nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    S. Fredenhagen and V. Schomerus, Boundary Liouville theory at c = 1, JHEP 05 (2005) 025 [hep-th/0409256] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    S. Fredenhagen and D. Wellig, A common limit of super Liouville theory and minimal models, JHEP 09 (2007) 098 [arXiv:0706.1650] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    S. Fredenhagen, Boundary conditions in Toda theories and minimal models, JHEP 02 (2011) 052 [arXiv:1012.0485] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutGolmGermany

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