Advertisement

On marginal operators in boundary conformal field theory

  • Christopher P. Herzog
  • Itamar ShamirEmail author
Open Access
Regular Article - Theoretical Physics
  • 33 Downloads

Abstract

The presence of a boundary (or defect) in a conformal field theory allows one to generalize the notion of an exactly marginal deformation. Without a boundary, one must find an operator of protected scaling dimension ∆ equal to the space-time dimension d of the conformal field theory, while with a boundary, as long as the operator dimension is protected, one can make up for the difference d − ∆ by including a factor z−d in the deformation where z is the distance from the boundary. This coordinate dependence does not lead to a reduction in the underlying SO(d, 1) global conformal symmetry group of the boundary conformal field theory. We show that such terms can arise from boundary flows in interacting field theories. Ultimately, we would like to be able to characterize what types of boundary conformal field theories live on the orbits of such deformations. As a first step, we consider a free scalar with a conformally invariant mass term z2φ2, and a fermion with a similar mass. We find a connection to double trace deformations in the AdS/CFT literature.

Keywords

Boundary Quantum Field Theory Conformal Field Theory Conformal and W Symmetry 

Notes

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

References

  1. [1]
    T.C. Lubensky and M.H. Rubin, Critical phenomena in semi-infinite systems. 2. Mean-field theory, Phys. Rev.B 12 (1975) 3885 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    D. Carmi, L. Di Pietro and S. Komatsu, A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP01 (2019) 200 [arXiv:1810.04185] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    I. Buhl-Mortensen, M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, A Quantum Check of AdS/dCFT, JHEP01 (2017) 098 [arXiv:1611.04603] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys.B 363 (1991) 486 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    C.P. Herzog and I. Shamir, On Marginal Operators in Boundary Conformal Field Theory, arXiv:1906.11281 [INSPIRE].
  6. [6]
    S.S. Gubser and I. Mitra, Double trace operators and one loop vacuum energy in AdS/CFT, Phys. Rev.D 67 (2003) 064018 [hep-th/0210093] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    S.S. Gubser and I.R. Klebanov, A Universal result on central charges in the presence of double trace deformations, Nucl. Phys.B 656 (2003) 23 [hep-th/0212138] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Allais, Double-trace deformations, holography and the c-conjecture, JHEP11 (2010) 040 [arXiv:1007.2047] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [INSPIRE].
  10. [10]
    R. Aros and D.E. Diaz, Determinant and Weyl anomaly of Dirac operator: a holographic derivation, J. Phys.A 45 (2012) 125401 [arXiv:1111.1463] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    D. Rodriguez-Gomez and J.G. Russo, Boundary Conformal Anomalies on Hyperbolic Spaces and Euclidean Balls, JHEP12 (2017) 066 [arXiv:1710.09327] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    D. Rodriguez-Gomez and J.G. Russo, Free energy and boundary anomalies on Sa× Hbspaces, JHEP10 (2017) 084 [arXiv:1708.00305] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    N. Bobev, K. Pilch and N.P. Warner, Supersymmetric Janus Solutions in Four Dimensions, JHEP06 (2014) 058 [arXiv:1311.4883] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Gutperle and J. Samani, Holographic RG-flows and Boundary CFTs, Phys. Rev.D 86 (2012) 106007 [arXiv:1207.7325] [INSPIRE].ADSGoogle Scholar
  15. [15]
    G.T. Horowitz, N. Iqbal, J.E. Santos and B. Way, Hovering Black Holes from Charged Defects, Class. Quant. Grav.32 (2015) 105001 [arXiv:1412.1830] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    C.P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP10 (2017) 189 [arXiv:1707.06224] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    E.V. Gorbar, V.P. Gusynin and V.A. Miransky, Dynamical chiral symmetry breaking on a brane in reduced QED, Phys. Rev.D 64 (2001) 105028 [hep-ph/0105059] [INSPIRE].ADSGoogle Scholar
  18. [18]
    S.-J. Rey, Quantum Phase Transitions from String Theory, talk at Strings 2007, Madrid, Spain, 25–29 June 2007.Google Scholar
  19. [19]
    D.B. Kaplan, J.-W. Lee, D.T. Son and M.A. Stephanov, Conformality Lost, Phys. Rev.D 80 (2009) 125005 [arXiv:0905.4752] [INSPIRE].ADSzbMATHGoogle Scholar
  20. [20]
    W.-H. Hsiao and D.T. Son, Duality and universal transport in mixed-dimension electrodynamics, Phys. Rev.B 96 (2017) 075127 [arXiv:1705.01102] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    W.-H. Hsiao and D.T. Son, Self-Dual ν = 1 Bosonic Quantum Hall State in Mixed Dimensional QED, arXiv:1809.06886 [INSPIRE].
  22. [22]
    S. Teber, Field theoretic study of electron-electron interaction effects in Dirac liquids, habilitation, Paris, LPTHE (2017) [arXiv:1810.08428] [INSPIRE].Google Scholar
  23. [23]
    L. Di Pietro, D. Gaiotto, E. Lauria and J. Wu, 3d Abelian Gauge Theories at the Boundary, JHEP05 (2019) 091 [arXiv:1902.09567] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  24. [24]
    C.P. Herzog, K.-W. Huang, I. Shamir and J. Virrueta, Superconformal Models for Graphene and Boundary Central Charges, JHEP09 (2018) 161 [arXiv:1807.01700] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    H.-W. Diehl, Field-theoretical approach to critical behaviour at surfaces, in Phase Transitions and Critical Phenomena (Vol. 10), C. Domb and J.L. Lebowitz eds., pp. 75–267, Academic Press, London (1986).Google Scholar
  26. [26]
    K. Binder, Critical behaviour at surfaces, in Phase Transitions and Critical Phenomena (Vol. 8), C. Domb and J.L. Lebowitz eds., pp. 1–144, Academic Press, London (1983).Google Scholar
  27. [27]
    H.W. Diehl, The Theory of boundary critical phenomena, Int. J. Mod. Phys.B 11 (1997) 3503 [cond-mat/9610143] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    L. Di Pietro, N. Klinghoffer and I. Shamir, On Supersymmetry, Boundary Actions and Brane Charges, JHEP02 (2016) 163 [arXiv:1502.05976] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys.B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    E. D’Hoker, D.Z. Freedman and L. Rastelli, AdS/CFT four point functions: How to succeed at z integrals without really trying, Nucl. Phys.B 562 (1999) 395 [hep-th/9905049] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    P. Liendo, L. Rastelli and B.C. van Rees, The Bootstrap Program for Boundary CFTd, JHEP07 (2013) 113 [arXiv:1210.4258] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M. Kwaśnicki, Ten equivalent definitions of the fractional laplace operator, Fract. Calc. Appl. Anal.20 (2017) 7 [arXiv:1507.07356].MathSciNetCrossRefGoogle Scholar
  34. [34]
    M. Kamela and C.P. Burgess, Massive scalar effective actions on Anti-de Sitter space-time, Can. J. Phys.77 (1999) 85 [hep-th/9808107] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    G. Basar and G.V. Dunne, A Gauge-Gravity Relation in the One-loop Effective Action, J. Phys. A 43 (2010) 072002 [arXiv:0912.1260] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  36. [36]
    R. Aros, D.E. Diaz and A. Montecinos, A Note on a gauge-gravity relation and functional determinants, J. Phys.A 43 (2010) 295401 [arXiv:1004.1394] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  37. [37]
    E.W. Barnes, On the theory of multiple gamma functions, Trans. Cambridge Phil. Soc.19 (1904) 374.Google Scholar
  38. [38]
    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
  39. [39]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP10 (2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    T. Kawano and K. Okuyama, Spinor exchange in AdSd+1 , Nucl. Phys.B 565 (2000) 427 [hep-th/9905130] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    Y. Fujii and K. Yamagishi, Killing spinors on spheres and hyperbolic manifolds, J. Math. Phys.27 (1986) 979 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys.20 (1996) 1 [gr-qc/9505009] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    H. Lü, C.N. Pope and J. Rahmfeld, A Construction of Killing spinors on S n , J. Math. Phys.40 (1999) 4518 [hep-th/9805151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    S. Teber and A.V. Kotikov, Field theoretic renormalization study of reduced quantum electrodynamics and applications to the ultrarelativistic limit of Dirac liquids, Phys. Rev.D 97 (2018) 074004 [arXiv:1801.10385] [INSPIRE].ADSGoogle Scholar
  45. [45]
    F. Coradeschi, P. Lodone, D. Pappadopulo, R. Rattazzi and L. Vitale, A naturally light dilaton, JHEP11 (2013) 057 [arXiv:1306.4601] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentKing’s College LondonLondonU.K.
  2. 2.C.N. Yang Institute for Theoretical Physics, Department of Physics and AstronomyStony Brook UniversityStony BrookU.S.A.
  3. 3.SISSA and INFNTriesteItaly

Personalised recommendations