Holographic Ward identities for symmetry breaking in two dimensions

  • Riccardo Argurio
  • Gaston GiribetEmail author
  • Andrea Marzolla
  • Daniel Naegels
  • J. Anibal Sierra-Garcia
Open Access
Regular Article - Theoretical Physics


We investigate symmetry breaking in two-dimensional field theories which have a holographic gravity dual. Being at large N, the Coleman theorem does not hold and Goldstone bosons are expected. We consider the minimal setup to describe a conserved current and a charged operator, and we perform holographic renormalization in order to find the correct Ward identities describing symmetry breaking. This involves some subtleties related to the different boundary conditions that a vector can have in the three-dimensional bulk. We establish which is the correct prescription that yields, after renormalization, the same Ward identities as in higher dimensions.


AdS-CFT Correspondence Field Theories in Lower Dimensions Spontaneous Symmetry Breaking 


Open Access

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  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  4. [4]
    I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Goldstone, A. Salam and S. Weinberg, Broken Symmetries, Phys. Rev. 127 (1962) 965 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S.-k. Ma and R. Rajaraman, Comments on the Absence of Spontaneous Symmetry Breaking in Low Dimensions, Phys. Rev. D 11 (1975) 1701 [INSPIRE].ADSGoogle Scholar
  10. [10]
    N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    P.C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158 (1967) 383 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    S.R. Coleman, R. Jackiw and H.D. Politzer, Spontaneous Symmetry Breaking in the O(N) Model for Large N , Phys. Rev. D 10 (1974) 2491 [INSPIRE].ADSGoogle Scholar
  13. [13]
    D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].ADSGoogle Scholar
  14. [14]
    E. Witten, Chiral Symmetry, the 1/n Expansion, and the SU(N) Thirring Model, Nucl. Phys. B 145 (1978) 110.ADSCrossRefGoogle Scholar
  15. [15]
    D. Anninos, S.A. Hartnoll and N. Iqbal, Holography and the Coleman-Mermin-Wagner theorem, Phys. Rev. D 82 (2010) 066008 [arXiv:1005.1973] [INSPIRE].ADSGoogle Scholar
  16. [16]
    D. Marolf and S.F. Ross, Boundary Conditions and New Dualities: Vector Fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249,ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    T. Faulkner and N. Iqbal, Friedel oscillations and horizon charge in 1D holographic liquids, JHEP 07 (2013) 060 [arXiv:1207.4208] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    T. Andrade, J.I. Jottar and R.G. Leigh, Boundary Conditions and Unitarity: the Maxwell-Chern-Simons System in AdS 3 /CF T 2, JHEP 05 (2012) 071 [arXiv:1111.5054] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    D.J. Gross, I.R. Klebanov, A.V. Matytsin and A.V. Smilga, Screening versus confinement in (1+1)-dimensions, Nucl. Phys. B 461 (1996) 109 [hep-th/9511104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P. Minces and V.O. Rivelles, Scalar field theory in the AdS/CFT correspondence revisited, Nucl. Phys. B 572 (2000) 651 [hep-th/9907079] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS 3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Bertolini, D. Musso, I. Papadimitriou and H. Raj, A goldstino at the bottom of the cascade, JHEP 11 (2015) 184 [arXiv:1509.03594] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    R. Argurio, A. Marzolla, A. Mezzalira and D. Musso, Analytic pseudo-Goldstone bosons, JHEP 03 (2016) 012 [arXiv:1512.03750] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  25. [25]
    M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
  26. [26]
    J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J.M.S. Wu, Two-point Functions in a Holographic Kondo Model, JHEP 03 (2017) 039 [arXiv:1612.02005] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    M. Gell-Mann, R.J. Oakes and B. Renner, Behavior of current divergences under SU(3) × SU(3), Phys. Rev. 175 (1968) 2195 [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Riccardo Argurio
    • 1
  • Gaston Giribet
    • 2
    • 3
    Email author
  • Andrea Marzolla
    • 1
  • Daniel Naegels
    • 1
  • J. Anibal Sierra-Garcia
    • 4
  1. 1.Physique Théorique et Mathématique and International Solvay InstitutesUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Martin Fisher School of PhysicsBrandeis UniversityWalthamU.S.A.
  3. 3.Physics DepartmentUniversity of Buenos Aires FCEN-UBA and IFIBA-CONICETBuenos AiresArgentina
  4. 4.Department of Particle Physics and IGFAEUniversity of Santiago de CompostelaSantiago de CompostelaSpain

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