Universality for shape dependence of Casimir effects from Weyl anomaly

  • Rong-Xin MiaoEmail author
  • Chong-Sun Chu
Open Access
Regular Article - Theoretical Physics


We reveal elegant relations between the shape dependence of the Casimir effects and Weyl anomaly in boundary conformal field theories (BCFT). We show that for any BCFT which has a description in terms of an effective action, the near boundary divergent behavior of the renormalized stress tensor is completely determined by the central charges of the theory. These relations are verified by free BCFTs. We also test them with holographic models of BCFT and find exact agreement. We propose that these relations between Casimir coefficients and central charges hold for any BCFT. With the holographic models, we reproduce not only the precise form of the near boundary divergent behavior of the stress tensor, but also the surface counter term that is needed to make the total energy finite. As they are proportional to the central charges, the near boundary divergence of the stress tensor must be physical and cannot be dropped by further artificial renormalization. Our results thus provide affirmative support on the physical nature of the divergent energy density near the boundary, whose reality has been a long-standing controversy in the literature.


AdS-CFT Correspondence Classical Theories of Gravity Conformal Field Theory 


Open Access

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  1. [1]
    H.B.G. Casimir, On the attraction between two perfectly conducting plates, Indag. Math. 10 (1948) 261 [Kon. Ned. Akad. Wetensch. Proc. 51 (1948) 793] [Front. Phys. 65 (1987) 342] [Kon. Ned. Akad. Wetensch. Proc. 100N3-4 (1997) 61] [INSPIRE].
  2. [2]
    G. Plunien, B. Müller and W. Greiner, The Casimir effect, Phys. Rept. 134 (1986) 87 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    M. Bordag, U. Mohideen and V.M. Mostepanenko, New developments in the Casimir effect, Phys. Rept. 353 (2001) 1 [quant-ph/0106045] [INSPIRE].
  4. [4]
    T. Appelquist and A. Chodos, The quantum dynamics of Kaluza-Klein theories, Phys. Rev. D 28 (1983) 772 [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    T. Appelquist and A. Chodos, Quantum effects in Kaluza-Klein theories, Phys. Rev. Lett. 50 (1983) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    K.A. Milton, The Casimir effect: recent controversies and progress, J. Phys. A 37 (2004) R209 [hep-th/0406024] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    V.V. Dodonov, Current status of the dynamical Casimir effect, Phys. Scripta 82 (2010) 038105 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    D. Deutsch and P. Candelas, Boundary effects in quantum field theory, Phys. Rev. D 20 (1979) 3063 [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    A. Lewkowycz and E. Perlmutter, Universality in the geometric dependence of Rényi entropy, JHEP 01 (2015) 080 [arXiv:1407.8171] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    X. Dong, Shape dependence of holographic Rényi entropy in conformal field theories, Phys. Rev. Lett. 116 (2016) 251602 [arXiv:1602.08493] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    K. Jensen and A. O’Bannon, Constraint on defect and boundary renormalization group flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    D. Fursaev, Conformal anomalies of CFT’s with boundaries, JHEP 12 (2015) 112 [arXiv:1510.01427] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    C.P. Herzog, K.-W. Huang and K. Jensen, Universal entanglement and boundary geometry in conformal field theory, JHEP 01 (2016) 162 [arXiv:1510.00021] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    G. Kennedy, R. Critchley and J.S. Dowker, Finite temperature field theory with boundaries: stress tensor and surface action renormalization, Annals Phys. 125 (1980) 346 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    G. Kennedy, Finite temperature field theory with boundaries: the photon field, Annals Phys. 138 (1982) 353 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    T. Takayanagi, Holographic dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    R.-X. Miao, C.-S. Chu and W.-Z. Guo, New proposal for a holographic boundary conformal field theory, Phys. Rev. D 96 (2017) 046005 [arXiv:1701.04275] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    C.-S. Chu, R.-X. Miao and W.-Z. Guo, On new proposal for holographic BCFT, JHEP 04 (2017) 089 [arXiv:1701.07202] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    W. Song, Q. Wen and J. Xu, Generalized gravitational entropy for warped anti-de Sitter space, Phys. Rev. Lett. 117 (2016) 011602 [arXiv:1601.02634] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Nozaki, T. Takayanagi and T. Ugajin, Central charges for BCFTs and holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    J.S. Dowker and G. Kennedy, Finite temperature and boundary effects in static space-times, J. Phys. A 11 (1978) 895 [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    T. Emig, A. Hanke, R. Golestanian and M. Kardar, Probing the strong boundary shape dependence of the Casimir force, Phys. Rev. Lett. 87 (2001) 260402 [cond-mat/0106028] [INSPIRE].
  24. [24]
    M. Schaden, Dependence of the direction of the Casimir force on the shape of the boundary, Phys. Rev. Lett. 102 (2009) 060402 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    M.A. Rajabpour, Classification of the sign of the critical Casimir force in two dimensional systems at asymptotically large separations, Phys. Rev. D 94 (2016) 105029 [arXiv:1609.06279] [INSPIRE].ADSGoogle Scholar
  26. [26]
    J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
  27. [27]
    D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J. Cardy, private communication.Google Scholar
  29. [29]
    M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  30. [30]
    C.-S. Chu and R.-X. Miao, Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity, JHEP 12 (2016) 036 [arXiv:1608.00328] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    L. Bianchi, S. Chapman, X. Dong, D.A. Galante, M. Meineri and R.C. Myers, Shape dependence of holographic Rényi entropy in general dimensions, JHEP 11 (2016) 180 [arXiv:1607.07418] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  32. [32]
    L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Balakrishnan, S. Dutta and T. Faulkner, Gravitational dual of the Rényi twist displacement operator, Phys. Rev. D 96 (2017) 046019 [arXiv:1607.06155] [INSPIRE].ADSGoogle Scholar
  34. [34]
    D. Seminara, J. Sisti and E. Tonni, Corner contributions to holographic entanglement entropy in AdS 4 /BCFT 3, JHEP 11 (2017) 076 [arXiv:1708.05080] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Physics DivisionNational Center for Theoretical Sciences, National Tsing-Hua UniversityHsinchuTaiwan
  2. 2.Department of PhysicsNational Tsing-Hua UniversityHsinchuTaiwan

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