Correlators of large N fermionic Chern-Simons vector models
TL;DR: In this article, it was shown that the Chern-Simons theory is dual to the Legendre-transformed theory of scalar fields coupled to ChernSimons gauge interactions at the level of planar 3-point functions.
Abstract: We consider the large N limit of three-dimensional U(N)
k
Chern-Simons theory coupled to a Dirac fermion in the fundamental representation. In this limit, we compute several correlators to all orders in the ‘t Hooft coupling λ ≡ N/k. It was suggested recently that this theory is dual to the Legendre-transformed theory of scalar fields coupled to Chern-Simons gauge interactions. Our results show that this duality holds for any value of the ‘t Hooft coupling, at least at the level of the planar 3-point functions. In addition, we determine the sign in the duality transformation of the Chern-Simons level, as well as the relation between the “triple-trace” deformation which exists in the bosonic Chern-Simons theory and in the Legendre-transformed fermionic theory.
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TL;DR: In this article, the authors considered the conformal field theory of N complex massless scalars coupled to a U(N) Chern-Simons theory at level k, and they showed that the theory is equivalent to the Legendre transform of the theory of k fermions coupled to the U(k)
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.
335 citations
TL;DR: In this paper, it was shown that in its simplest form the duality maps SU(N ) theories to U(k) theories, though there is also another version relating SU(n ) to U (k), and this precise form strongly affects the mapping under the Duality of baryon and monopole operators.
Abstract: There is significant evidence for a duality between (non-supersymmetric) U(N ) Chern-Simons theories at level k coupled to fermions, and U(k) Chern-Simons theories at level N coupled to scalars. Most of the evidence comes from the large N ’t Hooft limit, where many details of the duality (such as whether the gauge group is U(N ) or SU(N ), the precise level of the U(1) factor, and order one shifts in the level) are not important. The main evidence for the validity of the duality at finite N comes from adding masses and flowing to pure Chern-Simons theories related by level-rank duality, and from flowing to the non-supersymmetric duality from supersymmetric dualities, whose finite N validity is well-established. In this note we clarify the implications of these flows for the precise form of the duality; in particular we argue that in its simplest form the duality maps SU(N ) theories to U(k) theories, though there is also another version relating U(N ) to U(k). This precise form strongly affects the mapping under the duality of baryon and monopole operators, and we show, following arguments by Radicevic, that their mapping is consistent with our claims. We also discuss the implications of our results for the additional duality between these Chern-Simons matter theories and (the UV completion of) high-spin gravity theories on AdS
4. The latter theories should contain heavy particles carrying electric and/or magnetic charges under their U(1) gauge symmetry.
235 citations
Additional excerpts
...Various quantities were computed exactly as a function of this ’t Hooft coupling in the large N limit of these non-supersymmetric field theories [24, 3, 25, 26, 27, 28, 29, 10, 30, 31, 32, 33, 34, 35, 20, 22, 36], and were all found to be consistent with the duality....
[...]
TL;DR: In this article, the authors compare the regularized sum of the 3-sphere free energy of higher spin theories with the corresponding calculations in higher spin theory in Euclidean AdS4 and show that for the minimal Vasiliev theory including fields of only even spin, the regularised sum vanishes.
Abstract: Vasiliev’s type A higher spin theories in AdS4 have been conjectured to be dual to the U(N) or O(N) singlet sectors in 3-d conformal field theories with N-component scalar fields. We compare the $ \mathcal{O} $
(N
0) correction to the 3-sphere free energy F in the CFTs with corresponding calculations in the higher spin theories. This requires evaluating a regularized sum over one loop vacuum energies of an infinite set of massless higher spin gauge fields in Euclidean AdS4. For the Vasiliev theory including fields of all integer spin and a scalar with Δ = 1 boundary condition, we show that the regularized sum vanishes. This is in perfect agreement with the vanishing of subleading corrections to F in the U(N) singlet sector of the theory of N free complex scalar fields. For the minimal Vasiliev theory including fields of only even spin, the regularized sum remarkably equals the value of F for one free real scalar field. This result may agree with the O(N) singlet sector of the theory of N real scalar fields, provided the coupling constant in the Vasiliev theory is identified as G
N
~ 1/(N − 1). Similarly, consideration of the USp(N) singlet sector for N complex scalar fields, which we conjecture to be dual to the husp(2; 0|4) Vasiliev theory, requires G
N
~ 1/(N + 1). We also test the higher spin AdS3
/CFT2 conjectures by calculating the regularized sum over one loop vacuum energies of higher spin fields in AdS3. We match the esult with the $ \mathcal{O} $
(N
0) term in the central charge of the W
N
minimal models; this requires a certain truncation of the CFT operator spectrum so that the bulk theory contains two real scalar fields with the same boundary conditions.
222 citations
TL;DR: In this paper, the thermal free energy for all renormalizable Chern Simon the-Ories coupled to a single fundamental bosonic and fermionic field in the T Hooft large N limit was derived.
Abstract: We compute the thermal free energy for all renormalizable Chern Simon the- ories coupled to a single fundamental bosonic and fermionic field in the 't Hooft large N limit. We use our results to conjecture a strong weak coupling duality invariance for this class of theories. Our conjectured duality reduces to Giveon Kutasov duality when restricted to N = 2 supersymmetric theories and to an earlier conjectured bosonization duality in an appropriate decoupling limit. Consequently the bosonization duality may be regarded as a deformation of Giveon Kutasov duality, suggesting that it is true even at large but finite N.
183 citations
TL;DR: In this paper, the thermal free energy in large N U(N) Chern-Simons-matter theories with matter fields (scalars and/or fermions) in the fundamental representation, in the large temperature limit, was computed.
Abstract: We compute the thermal free energy in large N U(N) Chern-Simons-matter theories with matter fields (scalars and/or fermions) in the fundamental representation, in the large temperature limit. We note that in these theories the eigenvalue distribution of the holonomy of the gauge field along the thermal circle does not localize even at very high temperatures, and this affects the computation significantly. We verify that our results are consistent with the conjectured dualities between Chern-Simons-matter theories with scalar fields and with fermion fields, as well as with the strong-weak coupling duality of the $\mathcal{N}=2$
supersymmetric Chern-Simons-matter theory.
155 citations
References
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TL;DR: In this paper, the authors study the vacua of field theories where some of the gauge symmetry is broken by expectation values of scalar fields, and show how to calculate them from the behavior of perturbations to the AdS background near the boundary.
1,674 citations
TL;DR: In this article, the authors analyzed two-dimensional massless fermion field theories with quartic interactions and showed that symmetry breaking occurs in these models for any value of the coupling constant.
Abstract: Two-dimensional massless fermion field theories with quartic interactions are analyzed. These models are asymptotically free. The models are expanded in powers of $\frac{1}{N,}$ where $N$ is the number of components of the fermion field. In such an expansion one can explicitly sum to all orders in the coupling constants. It is found that dynamical symmetry breaking occurs in these models for any value of the coupling constant. The resulting theories produce a fermion mass dynamically, in addition to a scalar bound state and, if the broken symmetry is continuous, a Goldstone boson. The resulting theories contain no adjustable parameters. The search for symmetry breaking is performed using functional techniques, the new feature here being that a composite field, say $\overline{\ensuremath{\psi}}\ensuremath{\psi}$, develops a nonvanishing vacuum expectation value. The "potential" of composite fields is discussed and constructed. General results are derived for arbitrary theories in which all masses are generated dynamically. It is proved that in asymptotically free theories the dynamical masses must depend on the coupling constants in a nonanalytic fashion, vanishing exponentially when these vanish. It is argued that infrared-stable theories, such as massless-fermion quantum electrodynamics, cannot produce masses dynamically. Four-dimensional scalar field theories with quartic interactions are analyzed in the large-$N$ limit and are shown to yield unphysical results. The models are extended to include gauge fields. It is then found that the gauge vector mesons acquire a mass through a dynamical Higgs mechanism. The higher-order corrections, of order $\frac{1}{N}$, to the models are analyzed. Essential singularities, of the Borel-summable type, are discovered at zero coupling constant. The origin of the singularities is the ultraviolet behavior of the theory.
1,409 citations
TL;DR: In this article, the requirements of conformal invariance for two-and three-point functions for general dimension d on flat space are investigated and a compact group theoretic construction of the threepoint function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor.
926 citations
"Correlators of large N fermionic Ch..." refers methods in this paper
...Using the known conformal structures [18] for these correlators, we find that the general 2-point functions are given by 〈J (x)J (0)〉 = τ0 |x|4 , (65)...
[...]
TL;DR: In this article, the Euler-Heisenberg effective action due to fermions coupled to a homotopically nontrivial gauge transformation with winding number n is calculated in the SU(2) theory.
Abstract: The effective gauge field action due to fermions coupled to $\mathrm{SU}(N)$ gauge fields in three dimensions is found to change by $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}|n|$ under a homotopically nontrivial gauge transformation with winding number $n$. This gauge noninvariance can be eliminated by adding a parity-violating topological term to the action, or by regulating the theory in a way which produces this term automatically in the effective action. The Euler-Heisenberg effective action is calculated in the SU(2) theory and in QED.
647 citations
"Correlators of large N fermionic Ch..." refers background in this paper
...The normalization of the Chern-Simons action (2) is such that the theory is gauge invariant if k ∈ Z, up to a ±(1) 2 shift due to the parity anomaly [8, 9, 10]; this shift will not matter to us since we work in the large k limit....
[...]
TL;DR: In this paper, a compact group theoretic construction of the three point function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor.
Abstract: The requirements of conformal invariance for two and three point functions for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor. The restrictions arising from the associated conservation equations are investigated. It is shown that there are, for general $d$, three linearly independent conformal invariant forms for the three point function of the energy momentum tensor, although for $d=3$ there are two and for $d=2$ only one. The form of the three point function is also demonstrated to simplify considerably when all three points lie on a straight line. Using this the coefficients of the conformal invariant point functions are calculated for free scalar and fermion theories in general dimensions and for abelian vector fields when $d=4$. Ward identities relating three and two point functions are also discussed. This requires careful analysis of the singularities in the short distance expansion and the method of differential regularisation is found convenient. For $d=4$ the coefficients appearing in the energy momentum tensor three point function are related to the coefficients of the two possible terms in the trace anomaly for a conformal theory on a curved space background.
634 citations