Abstract
We consider the conformal field theory of N complex massless scalars in 2 + 1 dimensions, coupled to a U(N) Chern-Simons theory at level k. This theory has a ’t Hooft large N limit, keeping fixed λ ≡ N/k. We compute some correlation functions in this theory exactly as a function of λ, in the large N (planar) limit. We show that the results match with the general predictions of Maldacena and Zhiboedov for the correlators of theories that have high-spin symmetries in the large N limit. It has been suggested in the past that this theory is dual (in the large N limit) to the Legendre transform of the theory of fermions coupled to a Chern-Simons gauge field, and our results allow us to find the precise mapping between the two theories. We find that in the large N limit the theory of N scalars coupled to a U(N) k Chern-Simons theory is equivalent to the Legendre transform of the theory of k fermions coupled to a U(k) N Chern-Simons theory, thus providing a bosonization of the latter theory. We conjecture that perhaps this duality is valid also for finite values of N and k, where on the fermionic side we should now have (for N f flavors) a \( \mathrm{U}{(k)_{{{{{N-{N_f}}} \left/ {2} \right.}}}} \) theory. Similar results hold for real scalars (fermions) coupled to the O(N) k Chern-Simons theory.
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M. Moshe and J. Zinn-Justin, Quantum field theory in the large-N limit: a review, Phys. Rept. 385 (2003) 69 [hep-th/0306133] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
I. Klebanov and A. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP 07 (2005) 044 [hep-th/0305040] [INSPIRE].
B. Sundborg, Stringy gravity, interacting tensionless strings and massless higher spins, Nucl. Phys. Proc. Suppl. 102 (2001) 113 [hep-th/0103247] [INSPIRE].
E. Witten, Spacetime Reconstruction, talk at the John Schwarz 60’th Birthday Symposium, Pasadena U.S.A. (2001), http://theory.caltech.edu/jhs60/witten/1.html.
E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [Erratum ibid. B 660 (2003) 403] [hep-th/0205131] [INSPIRE].
E. Fradkin and M.A. Vasiliev, On the Gravitational Interaction of Massless Higher Spin Fields, Phys. Lett. B 189 (1987) 89 [INSPIRE].
M.A. Vasiliev, More on equations of motion for interacting massless fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 285 (1992) 225 [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].
M.A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].
M. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in AdS d , Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
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].
A.C. Petkou, Evaluating the AdS dual of the critical O(N) vector model, JHEP 03 (2003) 049 [hep-th/0302063] [INSPIRE].
S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
S. Giombi and X. Yin, Higher spins in AdS and twistorial holography, JHEP 04 (2011) 086 [arXiv:1004.3736] [INSPIRE].
S.R. Das and A. Jevicki, Large-N collective fields and holography, Phys. Rev. D 68 (2003) 044011 [hep-th/0304093] [INSPIRE].
M.R. Douglas, L. Mazzucato and S.S. Razamat, Holographic dual of free field theory, Phys. Rev. D 83 (2011) 071701 [arXiv:1011.4926] [INSPIRE].
R. de Mello Koch, A. Jevicki, K. Jin and J.P. Rodrigues, AdS 4/CFT 3 Construction from Collective Fields, Phys. Rev. D 83 (2011) 025006 [arXiv:1008.0633] [INSPIRE].
A. Jevicki, K. Jin and Q. Ye, Collective Dipole Model of AdS/CFT and Higher Spin Gravity, J. Phys. A 44 (2011) 465402 [arXiv:1106.3983] [INSPIRE].
S. Giombi et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, D = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].
L. Girardello, M. Porrati and A. Zaffaroni, 3 − D interacting CFTs and generalized Higgs phenomenon in higher spin theories on AdS, Phys. Lett. B 561 (2003) 289 [hep-th/0212181] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, arXiv:1112.1016 [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, arXiv:1204.3882 [INSPIRE].
C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ triality: from higher spin fields to strings, arXiv:1207.4485 [INSPIRE].
S.G. Naculich, H. Riggs and H. Schnitzer, Group level duality in WZW models and Chern-Simons theory, Phys. Lett. B 246 (1990) 417 [INSPIRE].
M. Camperi, F. Levstein and G. Zemba, THE Large-N limit of Chern-Simons gauge theory, Phys. Lett. B 247 (1990) 549 [INSPIRE].
E. Mlawer, S.G. Naculich, H. Riggs and H. Schnitzer, Group level duality of WZW fusion coefficients and Chern-Simons link observables, Nucl. Phys. B 352 (1991) 863 [INSPIRE].
G. Gur-Ari and R. Yacoby, Correlators of large N fermionic Chern-Simons vector models, arXiv:1211.1866 [INSPIRE].
A. Giveon and D. Kutasov, Seiberg duality in Chern-Simons theory, Nucl. Phys. B 812 (2009) 1 [arXiv:0808.0360] [INSPIRE].
A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982) 975 [INSPIRE].
W. Chen, G.W. Semenoff and Y.-S. Wu, Two loop analysis of nonAbelian Chern-Simons theory, Phys. Rev. D 46 (1992) 5521 [hep-th/9209005] [INSPIRE].
S. Giombi, S. Prakash and X. Yin, A note on CFT correlators in three dimensions, arXiv:1104.4317 [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
K. Wilson and J.B. Kogut, The renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].
M.J. Strassler, Field theory without Feynman diagrams: One loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].
A. Niemi and G. Semenoff, Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times, Phys. Rev. Lett. 51 (1983) 2077 [INSPIRE].
A. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions, Phys. Rev. Lett. 52 (1984) 18 [INSPIRE].
A. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].
S. Deser and A. Redlich, CP 1-fermion correspondence in D = 3, Phys. Rev. Lett. 61 (1988) 1541 [INSPIRE].
C. Burgess, C. Lütken and F. Quevedo, Bosonization in higher dimensions, Phys. Lett. B 336 (1994) 18 [hep-th/9407078] [INSPIRE].
E.H. Fradkin and F.A. Schaposnik, The fermion-boson mapping in three-dimensional quantum field theory, Phys. Lett. B 338 (1994) 253 [hep-th/9407182] [INSPIRE].
R. Banerjee, Bosonization in three-dimensional quantum field theory, Phys. Lett. B 358 (1995) 297 [hep-th/9504130] [INSPIRE].
D. Barci, C. Fosco and L. Oxman, On bosonization in three-dimensions, Phys. Lett. B 375 (1996) 267 [hep-th/9508075] [INSPIRE].
N. Banerjee, R. Banerjee and S. Ghosh, NonAbelian bosonization in three-dimensional field theory, Nucl. Phys. B 481 (1996) 421 [hep-th/9607065] [INSPIRE].
S. Ghosh, Bosonization exercise in three-dimensions: gauged massive Thirring model, Phys. Rev. D 59 (1999) 045014 [hep-th/9808058] [INSPIRE].
D. Barci, L. Oxman and S. Sorella, Topological and universal aspects of bosonized interacting fermionic systems in (2 + 1)-dimensions, Phys. Rev. D 59 (1999) 105012 [hep-th/9811068] [INSPIRE].
N. Shaji, R. Shankar and M. Sivakumar, On Bose-Fermi equivalence in a U(1) gauge theory with Chern-Simons action, Mod. Phys. Lett. A 5 (1990) 593 [INSPIRE].
S.K. Paul, R. Shankar and M. Sivakumar, Fermionization of selfinteracting charged scalar fields coupled to Abelian Chern-Simons gauge fields in (2 + 1)-dimensions, Mod. Phys. Lett. A 6 (1991) 553 [INSPIRE].
R. Shankar and M. Sivakumar, Bose-Fermi transmutation in (2 + 1)-dimensions: effect of selfinteractions and the Maxwell term, Mod. Phys. Lett. A 6 (1991) 2379 [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, unpublished.
S. Jain, S.P. Trivedi, S.R. Wadia and S. Yokoyama, Supersymmetric Chern-Simons Theories with Vector Matter, arXiv:1207.4750 [INSPIRE].
S.H. Shenker and X. Yin, Vector Models in the Singlet Sector at Finite Temperature, arXiv:1109.3519 [INSPIRE].
S. Banerjee, S. Hellerman, J. Maltz and S.H. Shenker, Light States in Chern-Simons Theory Coupled to Fundamental Matter, arXiv:1207.4195 [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
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ArXiv ePrint: 1207.4593
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Aharony, O., Gur-Ari, G. & Yacoby, R. Correlation functions of large N Chern-Simons-Matter theories and bosonization in three dimensions. J. High Energ. Phys. 2012, 28 (2012). https://doi.org/10.1007/JHEP12(2012)028
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DOI: https://doi.org/10.1007/JHEP12(2012)028