Abstract
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.
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
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].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
J. Goldstone, A. Salam and S. Weinberg, Broken Symmetries, Phys. Rev. 127 (1962) 965 [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259.
S.-k. Ma and R. Rajaraman, Comments on the Absence of Spontaneous Symmetry Breaking in Low Dimensions, Phys. Rev. D 11 (1975) 1701 [INSPIRE].
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].
P.C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158 (1967) 383 [INSPIRE].
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].
D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
E. Witten, Chiral Symmetry, the 1/n Expansion, and the SU(N) Thirring Model, Nucl. Phys. B 145 (1978) 110.
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].
D. Marolf and S.F. Ross, Boundary Conditions and New Dualities: Vector Fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249,
T. Faulkner and N. Iqbal, Friedel oscillations and horizon charge in 1D holographic liquids, JHEP 07 (2013) 060 [arXiv:1207.4208] [INSPIRE].
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].
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].
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].
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].
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].
R. Argurio, A. Marzolla, A. Mezzalira and D. Musso, Analytic pseudo-Goldstone bosons, JHEP 03 (2016) 012 [arXiv:1512.03750] [INSPIRE].
M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
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].
M. Gell-Mann, R.J. Oakes and B. Renner, Behavior of current divergences under SU(3) × SU(3), Phys. Rev. 175 (1968) 2195 [INSPIRE].
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Argurio, R., Giribet, G., Marzolla, A. et al. Holographic Ward identities for symmetry breaking in two dimensions. J. High Energ. Phys. 2017, 7 (2017). https://doi.org/10.1007/JHEP04(2017)007
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DOI: https://doi.org/10.1007/JHEP04(2017)007