Large-N limits of 2d CFTs, Quivers and AdS_3 duals

We explore the large-N limits of 2d CFTs, focusing mostly on WZW models and their cosets. The $SU(N)_k$ theory is parametrized in this limit by a 't Hooft-like coupling. We show a duality between strong coupling, where the theory is described by almost free fermions, and weak coupling where the theory is described by bosonic fields by an analysis of spectra and correlators. The AdS$_3$ dual is described, and several quantitative checks are performed. Besides the more standard states that should correspond to bulk black holes we find ground states with large degeneracy that can dominate the standard Cardy entropy at weak coupling and are expected to correspond to regular horizonless semiclassical bulk solutions.


Introduction and Outlook
Two dimensional conformal field theories (CFTs) have been studied extensively over the years. There are many solvable examples that admit a large-N limit. The exact solvability of these theories in generic regimes of parameter space makes them very attractive as toy models for questions that are typically very hard for analogous field theories in higher dimensions. One can compute the exact spectrum and in some cases also exact correlation functions. In this paper we revisit a special set of two dimensional CFTs: WZW models (and cosets thereof) based on the SU(N) group. Our interest in these models stems from the following observations: (i) On general grounds it is expected that conformal field theories with a large-N expansion have a dual description in terms of a string theory on AdS. For the AdS 3 /CF T 2 correspondence the canonical example is provided by the symmetric orbifold of N copies of the 2d sigma model with target space M 4 , which has been argued to be dual to type IIB string theory on AdS 3 × S 3 × M 4 (the large-N diagrammatic expansion of the symmetric orbifold theory has been discussed recently in [1]). In order to gain a deeper insight into the general principles underlying the AdS/CFT correspondence it is important to extend the list of known examples. Situations involving solvable CFTs are of obvious interest and hence a natural question is whether it is possible to identify the string theory duals of well-known exact 2d CFTs. In the process one hopes to uncover qualitatively new features of the AdS/CFT correspondence.
Moreover, we will show that this theory has two dual descriptions. At weak coupling, λ → 0, the weakly coupled description is the conventional WZW model, written in terms of a bosonic field g that transforms as a bifundamental under the current algebra symmetry SU(N) L × SU(N) R . The theory resembles somewhat the chiral Lagrangian in four dimensions with important differences: there is no chiral symmetry breaking and the IR theory is conformal.
At strong coupling, λ → ∞, the weakly coupled description is in terms of the IR limit of N copies of massless Dirac fermions transforming in the fundamental representation of a U(k) gauge group in which the overall U(1) should be thought of as gauged baryon number.
This looks like a conventional gauge theory 1 and in this language k is color, N is flavor and λ = N k is the Veneziano ratio. The bosonic field g corresponds to the fermion mass operator in the strongly coupled regime. In this sense this theory can be thought of as a gauge theory with quarks where although there is confinement, the theory is conformal in the IR and there is no chiral symmetry breaking. The notion of weak or strong coupling is strictly tied to the definition of λ. The weakly coupled WZW theory is strongly coupled from the point of view of the fermionic formulation.
The analogous and more conventional SU(k) gauge theory on the other hand will give rise to the U(N) k WZW CFT, with U(1) L ×U(1) R the conventional axial and vector baryon number symmetries, both of which are non-anomalous in two dimensions. Such a theory is expected to have an AdS 3 dual with SU(N) L × SU(N) R symmetry.
The closed string sector is expected to be trivial and the dependence on the metric is induced on the boundary by the proper boundary conditions on the open string sector as recognized already in [16]. The open string sector is expected to be realized in a way similar to the one advocated for flavor in higher dimensions. D 2 andD 2 branes generate 1 Although the Yang-Mills action is irrelevant in the IR in two dimensions. the gauge symmetry which at low energy is realized by two CS actions with couplings of opposite sign. A direct computation of the effective action for currents is in agreement with the CFT calculation using the WZW model.
The study of scaling dimensions indicates that in the weak coupling limit, the spectrum of ground states can be made to have a large gap from the stringy states. This suggests that in that limit, a "gravity" description of the physics is possible. In the strong coupling limit all dimensions are of the same order and therefore a stringy bulk description is necessary.
It is in this limit that the physics is described in terms of almost free fermions. The Pauli principle and the existence of a Fermi surface are not visible in the gravity description and a stringy description is necessary in order to accommodate this stringy exclusion principle, very much like in [17].
The ground states are generated by a bifundamental field T that should correspond to an open string stretching between D 2 andD 2 and is dual to the generating group field in the WZW model. This is a picture analogous to the one in [18]. Other ground states in the CFT correspond to multi-particle states. Sources for T correspond to a mass matrix in the fermionic language. They generate a flow that drives the theory to an IR fixed point equivalent to SU(N − r) k where r the rank of the source of T .
The states with scaling dimensions of O(N 2 ) have masses that can be comparable to M planck . We find that they are of the order of M planck at strong coupling and much larger at weak coupling. Therefore, at weak coupling these states could correspond to macroscopic smooth solutions of the bulk theory with associated flavor hair. This hair is responsible for their multiplicity, and in this case and at this energy scale this multiplicity dominates the Cardy entropy.
The cosets are also very interesting CFTs with several possible large-N limits that we analyze. They also have dual versions, between gauged-WZW models and quiver gauge theories coupled to massless quarks.
The SU(M + N) k /SU(N) k theory can be thought of as the following quiver. The Finally the U(k 1 + k 2 ) N /U(k 1 ) N × U(k 2 ) N CFT can be thought of as a quiver gauge theory with gauge group U(k 1 ) × U(k 2 ) × SU(N) and massless quarks transforming as ( , 1, ) and (1, , ) under the gauge group. This description automatically explains the level-rank duality symmetry that states that The subclass of coset models , giving rise to the W N minimal models, has been analyzed recently in [19]. Its closed string sector has been argued to correspond to the quantum Hamiltonian reduction SL(N, C) → W N . Such theories provide more complex examples of large-N limits but we will only touch upon them in this paper.
There are several issues that remain open in this direction. The first concerns a more organized control of the spectrum via the partition function. The AdS/CFT correspondence is fundamentally a relation between partition functions that reads In the canonical formulation of the partition function on the torus If fermions are present in the theory we also need to specify their periodicities around the two cycles of the torus. The imaginary part of τ plays the role of inverse temperature in the bulk, and the real part is a chemical potential for angular momentum. There is a reasonably good understanding of this partition function for the WZW model and the interesting question is whether it can be recast in a way that makes it interpretable as the partition function on AdS, Z AdS . At low energies the Hilbert space of the gravitational theory comprises of a gas of particles moving on AdS. At high energies we encounter black holes. There have been several attempts to make sense of Z AdS as a sum over all saddle points of the full bulk effective action I (including in principle all string and loop corrections), i.e. to recast Z AdS as The most concrete realization of this programme is the Farey tail expansion of [20]. In that case, instead of considering the full partition function, one focuses on a BPS subsector and computes the elliptic genus. Then, one observes that the elliptic genus admits an expansion that is suggestive of a supergravity interpretation in terms of a sum over geometries. It would be interesting to explore if there is a similar expansion of the full partition function of the SU(N) k WZW model. Practically, a more promising context for this idea is to analyze the elliptic genus of N = 2 gauged WZW models, e.g. N = 2 Kazama-Suzuki models (the supersymmetric Grassmannian coset may be an interesting example).
As we have seen there are many configurations in such CFTs that have macroscopic entropy and are therefore expected to correspond to smooth bulk solutions carrying flavor hair. It would be interesting to investigate the existence of such solutions. Moreover, our analysis indicates that at least for the SU(N) k theory at strong coupling, such solutions should be thought of as describing a fermionic ground state (fermi surface) of almost free fermions.
CFTs like the WZW models have a class of features that are interesting to explore in an AdS/CFT setup. They contain current algebra null vectors that are responsible for the truncation of the spectrum (affine cutoff). It is interesting that although such non-trivial relations are counter-intuitive in the weak coupling limit, they are simply explained in the strong coupling limit where the theory can be described in terms of kN Dirac fermions.
From the basic formula g a,b = 1 it is then obvious that Fermi statistics forbids symmetrized powers (g ab ) p with p > k. The current algebra null vectors are responsible for the existence of "instanton" corrections to the partition function (namely terms that behave as e −N ). The study of these effects is interesting as they should match with D-instanton effects in the dual string theory.
A related set of null vectors are the Knizhnik-Zamolodchikov ones, whose content is based on the fact that the stress tensor is quadratic in the currents. They are the key tools in computing the correlation functions. We have seen that the affine-Sugawara construction is an avatar of the proper boundary conditions for the CS theory in 3d. It should be possible to derive the analogue of KZ equations from the bulk.
The bulk description, once developed, will provide a concrete tool for the study of RG flows between different fixed points. A lot is known in 2d CFTs about such flows, but it is expected that the bulk description will provide more efficient tools in this analysis. An intermediate step in this programme is the understanding of the bulk effective actions for the scalar fields, something that needs further study. As a byproduct, this approach would allow a more thorough study of the thermodynamics of 2d QFTs.
A related issue is the holographic dual of a c-function, a fact that is firmly established for 2d CFTs. It would be interesting to find classes of examples where one could follow the flows between different fixed points, exploring in such a way the landscape of 2d CFTs.
Such large landscapes of flows exist in 2d CFTs, [21], with the flows and c-function known exactly, although the intermediate non-conformal theories in that case are non-relativistic Hamiltonian theories that flow to standard relativistic CFTs at the fixed points. Recent ideas on "order" and distance in such a landscape [22] may prove useful.

Solvable 2d CFTs
In two dimensions we have extensive knowledge of a large class of solvable CFTs. These theories are essentially composed of WZW models based on compact affine Lie groups, [10] and cosets, [12,13]. All of these theories can be solved exactly. Solvable extensions include some non-compact theories, like Liouville theory, [23,24] and some semisimple groups [25,26]. Larger classes of irrational CFTs were found by generalizations of the GKO construction, [14,15], but no non-trivial CFT in this class has been solved so far.
In what follows we will consider the simplest classes of WZW and coset models which possess large-N limits. But before doing this it is appropriate to make some general comments on "color" vs "flavor" degrees of freedom.
We define color as degrees of freedom which are gauged. Gauge fields in two dimensions have two types of IR dynamics. The standard, namely YM kinetic terms, are irrelevant in the IR, and never play a role in CFTs. The quadratic gauge terms that play a role are non-propagating -the gauged WZW models provide an example where these terms play a defining role. As a result, pure gauge dynamics in 2d is trivial and the main role of the gauge group is kinematic confinement and removal of colored degrees of freedom from the spectrum. In this sense, gauge interactions in two dimensions reduce the number of degrees of freedom the most because of confinement, compared with 3d or 4d gauge dynamics.
On the other hand, we will define flavor degrees of freedom as those degrees of freedom that are not affected by gauge interactions, in the sense that they are not gauged. For example, a set of N f free Majorana fermions has a maximal flavor symmetry O(N f ) L × O(N f ) R that is promoted to an affine current algebra in two dimensions.

The SU(N) k WZW model
This is an interesting case which seems to contain only flavor. The flavor symmetry is SU(N) L × SU(N) R and as it usually happens with non-trivial compact global symmetries in CFTs, it is extended to a full affine algebra. Note that this not the flavor symmetry we would obtain from N massless Dirac fermions in two dimensions. That symmetry is larger This theory has two parameters, N which is the number of flavors and k that plays the role of the σ-model coupling constant. We will see in the subsequent section that there are several possible large-N limits that can be defined here. This is a CFT whose spectrum is conveniently represented using current algebra representations. There are "ground states" that coincide with the primary affine representations As will be explained in section 4, the SU(N) k theory can be thought of as the IR limit of a theory of N copies of massless Dirac fermions transforming in the fundamental of the gauge group U(k).

The SU(N) k /SU(N) k gauged WZW model
This is the simplest theory that contains SU(N) color but no flavor. The gauge degrees of freedom remove essentially all states in this theory. In particular the theory is topological and has central charge c = 0. It has a finite number of ground states that are in one-to-one correspondence with all integrable representations of the SU(N) k affine algebra. In this sense, this Hilbert space can be thought of as the space of states of a point particle moving on a fuzzy group manifold.
The dual AdS theory is the topological SU(N) Chern-Simons (CS) theory at level k.
This theory is topological and has a finite number of states that are also in one-to-one correspondence with all integrable representations of the SU(N) k affine algebra, [2]. This is the simplest topological open string theory in 3d. The relevant closed string sector is a topological string and has no propagating states. From the interpretation of the previous section we gather that the present theory can be thought of as a simple quiver: two gauge groups U(k) and U(N) and Dirac fermions in the bifundamental. There are no Regge trajectories in this case.
In the 't Hooft limit, N, k ≫ 1 with the ratio λ = N k fixed, and for M fixed the central charge becomes to leading order in M/N c ∼ 2 + λ (1 + λ) 2 NM .

2.4
The U (k 1 +k 2 ) N U (k 1 ) N ×U (k 2 ) N gauged WZW model This theory can be thought of as a theory with U(k 1 + k 2 ) flavor symmetry whose U(k 1 ) × U(k 2 ) part is gauged. The coupling constant is N. This CFT is dual to the coset by level-rank duality, [6]. In this dual version of the CFT one starts from an SU(N)× SU(N) flavor symmetry and then gauges the diagonal subgroup.
From the available parameters we can build two independent 't Hooft couplings, λ i = N/k i . The central charge becomes to leading order in the 't Hooft couplings c = N 2 + subleading indicating that this model is similar to a gauge theory coupled to an adjoint scalar. It is therefore interesting to compare it to the N = 2 one-adjoint A n+1 CSM theories of [7]. Some parallels between these theories are: (a) Both are controlled by three discrete parameters. In the CSM case, these three parameters are: k the level of the CS interaction, N the rank of the U(N) gauge group, and n + 1 the power of the single-trace operator TrX n+1 that appears in the action as a superpotential deformation.
(b) A crucial effect of the superpotential deformation in the CSM theory is that it truncates the chiral ring. The levels k i play an analogous role in the WZW model truncating the spectrum.
(c) Both theories exhibit a non-trivial duality. The U(N) k A n+1 CSM theory is Seiberg- We can also think of the U (k 1 +k 2 ) N U (k 1 ) N ×U (k 2 ) N theory as a quiver gauge theory with gauge group U(k 1 ) × U(k 2 ) × U(N) and massless quarks transforming as ( , 1, ) and (1, , ) under the gauge group. This description automatically explains the level-rank duality symmetry.

On large-N limits
There are several large-N limits that are possible in the CFTs we have mentioned above.
They have been discussed in different contexts in the literature and we will go through them for comparison. As the CFTs in question are solvable, we will be able to characterize explicitly the nature of each of these limits.

The 't Hooft large-N limit
The characteristic feature of the 't Hooft limit is that the coupling constant is rescaled so that it compensates for the increase of degrees of freedom. Another characteristic is that for adjoint theories the normalized n-point functions behave as N 1− n 2 . This implies in particular that the central charge c ∼ O(N 2 ). For the SU(N) k theory the 't Hooft limit kept fixed, [27,19]. When λ ≪ 1 we are in a perturbative regime. In this regime the α ′ -perturbation theory is applicable. Since in this limit, k ≫ N, the affine cutoff [28] is not visible when we consider representations with ∼ O(N) columns in the Young tableau or less. Therefore, the fusion algebra of such low-lying representations is "perturbative": it coincides with the classical Glebsch-Gordan decomposition.
In the opposite limit, λ ≫ 1, we are in the strong coupling regime. The σ-model semiclassical expansion breaks down and since k ≪ N the affine cutoff is felt at relatively low representations. This implies algebraic relations between primary fields (the vanishing of fields with spin higher than k) well before the distinction between single-trace and double-trace operators sets in.
The central charge can be written in this limit as and it is indeed O(N 2 ) as advertised. It does remain so at weak 't Hooft coupling but at strong coupling is parametrically smaller than N 2 and behaves as O(N) for finite k. As the central charge is a quantum measure of the number of degrees of freedom, this indicates that there is a drastic reduction of degrees of freedom at strong 't Hooft coupling mimicking a similar situation predicted by the AdS 4 /CF T 3 correspondence for three dimensional conformal field theories. An important difference is that here this reduction is explicitly calculable. 2 Another difference is that the reduction observed here is by a factor of N while in three dimensions it is by a factor of √ N. 3 Focusing at the conformal dimensions with an O(1) number of boxes of the Young tableau, we obtain (see appendix A) To leading order at 1/N and at weak coupling they all asymptote to zero in agreement with the fact that in the classical theory all primary operators have vanishing scaling dimension.
In general, in the 't Hooft limit the dimensions of primaries are where m i are the Dynkin indices provided the sums are O(1). Otherwise, the full formula (4.7) should be used. All m i take values 0 ≤ m i ≤ N/2. In all cases, k i=1 m i is the total number of boxes in the Young tableau of the associated representation.
At strong 't Hooft coupling on the other hand they asymptote to half-integers Note that this is the same spectrum as in the naive large-N limit discussed in section 3.2.
The maximal dimension is obtained when m i = N/2 ∀ i. In that case, As we will later see, this state is expected to correspond to a regular horizonless semiclassical bulk solution.
On the other hand, for the maximal symmetric tensor, m i = 1 ∀ i ≤ k, For the maximal antisymmetric representation, m 1 = N/2 and all other m i 's zero, (3.10)

The simple large-N limit
Another possibility is to take N → ∞ while keeping k fixed. This limit has been studied previously in specific examples, in [31,32], in order to produce representations of the W ∞ algebra. In this limit and in this sense it looks like the theory is reducing in this case to a vectorial large-N theory. The large-N limit of the dimensions of primary fields gives Therefore the primary field dimensions become half-integers which hints at a free-fermionic formulation in terms of 2kN free fermions (as also suggested by the central charge). This is indeed true as analyzed in appendix B.
On the other hand, the maximal symmetric representation can only have k boxes because of the affine cutoff. We therefore have (3.14) Hence, we have dimensions scaling as O (N) and dimensions scaling as O (1).
In coset theories primary field dimensions can also be of the order of O 1 N as shown in [31,19]. Consider the coset In the naive large-N limit we may rewrite the central charge as which is finite in the large-N limit.
There is an interesting symmetry in this theory, stemming from level-rank duality, that indicates that this CFT is equivalent to a dual one 4 The dimensions of the coset are associated with three representations: we obtain in the large-N limit The '+integer' on the right hand side of (3.18) denotes a possible contribution that appears in some cases because the coset primary is not the top state of the representation but one of the descendants. For representations with an O(1) number of boxes in the Young tableau this dimension is of order O 1 N . For example, consider the case We will still have also dimensions that scale like O (N). For example, Therefore in such CFTs, there are primary operators in class A with dimensions O (N), It was shown in [31] that one can construct a class of operators of dimension O (1) out of operators of the class C. The construction involves operators as in (3.20) and the limit Such operators were shown to have abelian OPEs and generate the analogue of the discrete series operators in pp-wave CFTs as shown in [26].
Note that this large-N limit has a dual version in the CFT (3.17) as a weak coupling limit where all current algebra levels go to infinity. Therefore the σ-model is a flat space to leading order with 2k 1 k 2 dimensions, and the states in this theory, correspond to the class B operators as well as the class C operators that can be made to have O(1) dimensions as explained earlier.
Note also that the U(N) 1 theory is equivalent to a collection of 2N free fermions is obtained from the free fermion theory by coseting the overall U(1).

The BMN large-N limit
This is a large-N limit with a tuning that keeps some dimensions finite, [33,26]. We will consider the following example: The dimension of a generic primary is Consider a representation of type A with x i N +ξ i boxes in the i-th column, with 0 < x i < 1, i = 1, 2, · · · , k and ξ i of order O(1). This representation has a dimension of order O(N).
We also pick the U(1) charges q i so that they cancel the O(N) piece of the previous In the large-N limit  It can appear as an IR fixed point in many CFTs, including the SU(N) chiral model modified by the addition of a WZ term, [10]. It can also appear as the IR fixed point of 2-dimensional massless QCD with gauge group U(k) and N Dirac flavors of quarks [38,39], as explained in appendix B. In such a description the YM action is becoming irrelevant in the IR and the theory flows to the U(kN) 1 /U(k) N coset that is equivalent to the SU(N) k CFT. Therefore k can be identified as the number of colors and N as the number of massless quark flavors.
We will denote the ratio of flavor to color numbers as and we will call it the 't Hooft coupling, although from the fermionic point of view, it is the Nc . In the σ-model picture this ratio does look more like the conventional 't Hooft coupling.
In the gauge theory picture the global flavor symmetry is SU(N) L × SU(N) R and is manifest in the theory. This is an example of a confining gauge theory without chiral symmetry breaking. The reason that such a theory can still be a CFT is that in two dimensions non-abelian gauge fields carry no propagating degrees of freedom and confinement is essentially kinematical. It can be implemented by the (generalized) Gauss law of the appropriate current algebra on the states of the un-gauged theory.
Denoting the currents of the SU(N) k WZW model as J a (z),J a (z) we have the OPEs with a corresponding expression for the right-moving currentsJ a (which we will systematically omit).
The stress-energy tensor satisfies the affine Sugawara construction The central charge in this theory is given by In the large 't Hooft limit, λ ≫ 1, c ≃ N 2 λ = kN. One can observe the same phenomenon that has been observed in 3d CFTs dual to M2 brane geometries. At strong 't Hooft coupling there is a reduction of the number of degrees of freedom; in this case, from O(N 2 ) to O(N). As shown in appendix B, in this regime of N ≫ k, the theory is approximately described by kN free massless Dirac fermions and the YM interaction can be treated perturbatively. In the opposite regime, λ ≪ 1 or N ≪ k, the theory is well described by the (weakly coupled) WZW theory, describing string propagation over a large volume group manifold.
We now proceed to analyze the spectrum and conformal dimensions. The spectrum is composed of affine (spinless) primary states transforming in the (R,R) integrable representations of the global SU(N) L × SU(N) R group, with one copy per representation.
On top of these primary states the whole affine representation is built by acting with the lowering operators of the current algebra, the negative J a −n current modes. Were it not for the existence of non-trivial current algebra null vectors, the multiplicity of the states inside representations would be the same as that of the related U(1) N 2 −1 free theory. The current algebra null vectors provide non-perturbative effects in k, and disappear as k → ∞.
The (left-moving) conformal dimensions for the affine primary fields of the SU(N) k theory, transforming in the R irreducible unitary representation of the SU(N) algebra are given by where C 2 (R) is the quadratic Casimir, defined and analyzed in appendix A. To find the large-N limits we must analyze the scaling of the quadratic Casimir for SU(N) representations, with the result in the N → ∞ limit for representations with an O(1) number of boxes in their Young tableau.
We can give a general formula for the quadratic Casimir C 2 (R) = C 2 (R), corresponding to a Young tableau with m 1 boxes in the first column, m 2 boxes in the second column etc, We use the prescription that the proper Young tableau is the one with the minimum number of boxes. In this notation the Casimir is .
For m i ∼ O(1) and n ∼ O(1) we obtain which are half-integers. In particular, this number is the total number of boxes in the Young tableau, divided by two.
We finally obtain In the strong coupling limit λ → ∞, the dimensions are half-integers, reflecting the fact that the theory asymptotes to a theory of kN free massless fermions (see appendix B and the next section). In that case one can think of a rectangular box of dimensions N × k divided into kN compartments (boxes). A choice of "occupied" boxes defines a Young tableau and therefore an integrable representation R of SU(N) k . Not surprisingly, the set of "ground-states" of the theory, namely the primary fields, is in one-to-one correspondence with filling some of the boxes using the kN fermions while abiding to the Pauli principle.
We will now consider representations with large scaling dimensions. The maximum scaling dimension is given for representations around the "half-box", with scaling dimension For example, the maximal representation corresponding to the "half-box" in (4.10) with n i = 0 has dimension The logarithm of the multiplicity of this ground state is derived in appendix A as log D 2 = 4(λ + 1) log 2 − λ 2 log λ + 2(λ + 1) 2 log(λ + 1) − (λ + 2) 2 log(λ + 2) N 2 2λ 2 + (4.14) At strong coupling, λ ≫ 1, while at weak coupling, λ ≪ 1, We should also consider states whose scaling dimensions scale as O(N) in the large-N limit. In section 3.1 we have mentioned two such representations; the maximal symmetrized representation with all Dynkin indices m 0<i<k+1 = 1, and the "maximal" antisymmetrized representation, m 1 = N 2 and all other m i 's zero. The scaling dimension of the maximal symmetric tensor is while for the antisymmetric one Note that ∆ sym vanishes in the strong coupling limit. This is a consequence of Fermistatistics, as this representation is built out of fermions as k i=1 ψ a i i × cc, and therefore one cannot obtain a symmetric object in the flavor indices, a i . On the other hand, ∆ a → N 8 in the strongly coupled limit as in this case a similar state, can be achieved in terms of free fermions. Such states resemble baryons, and indeed their "masses" are of order N. Of course there are many more representations of this type, beyond the ones discussed above.
Note that the N dependence of scaling dimensions that arises in (4.5), (4.7) contains contributions from tree-level, disk-level and one-loop. On the other hand, the central charge has only a single one-loop contribution. The λ dependence although simple indicates the presence of a full perturbative series of corrections both at small and large λ.
We have seen that the Hilbert space decomposes into a finite number of unitary irreducible representations of the current algebra. For each representation there is a ground state, transforming in the (R,R) of SU(N) L × SU(N) R corresponding to the primary field, and the rest of the states are generated from the primary field by the action of the current oscillators. The different primary operators can be thought of as products of the basic WZW field g a,b (z,z). Using appropriate normal ordering and symmetrizations or antisymmetrizations of the indices we may construct any integrable primary of the algebra. This fact will be important when we analyze the AdS 3 dual.

The four-point function
An important observable in conformal field theory are correlation functions.
We will analyze the large-N limits of the four-point function of the basic primary fields of the SU(N) k WZW model, namely the fundamental g a,b (z,z) and its conjugate g −1 b,a (z,z) following [11] and then study its large-N limit.
The four-point function has been calculated solving the KZ equation and reads B (x) . A detailed analysis is presented in appendix C. In what follows we summarize the main results.
The associated conformal blocks are where F is the hypergeometric function and In the 't Hooft limit we obtain Note that in the symmetric channels G 11 and G 22 are powerlike at leading order in 1/N suggesting the existence of a single block. In the asymmetric channel G 12 there is non-trivial structure at subleading order in 1/N. Finally in the strong coupling limit λ → ∞ the result simplifies to This is indeed compatible with the claim that the theory is described by free fermions in that limit. In particular, the properly normalized fundamental field of the WZW can be written in the strong coupling limit as where the tilde indicates right-movers. From (4.31) the 4-point function in (4.30) follows.

Non-commutativity of large-N and large-λ limits.
An interesting question in any large-N theory is the commutativity of the large-N and the large 't Hooft coupling limit. In the conventional definition we first take the large-N limit, and then let λ become large. In our example we can study these limits explicitly and we will show that they do not commute. More details can be found in appendix C.
The particular observable to study is the OPE coefficient in (4.27). This OPE coefficient has a double expansion If we first take the large-N limit while keeping λ fixed we re-organize the double expansion as Next we take the large-λ limit. The functions Z n have the following behavior in this limit from which we read in the range 1 ≤ n ≤ 3, −2 ≤ m ≤ 0 all other being zero in that range. ψ(x) = d ln Γ(x) dx is the standard digamma function.
We will now take the opposite sequence of limits; the large-λ limit coming first. We rewrite the double expansion as and subsequently take the large-N limit of the functions H n . We obtain lim N →∞ from which we deduce while the rest are zero.
Comparing the expressions (4.36) and (4.40) we observe that the two limits do not commute.

The effective action for sources
We will derive here the effective action of the WZW theory once we couple the currents to sources. The WZW action is given by where the second integral is over a 3d manifold with our two dimensional space as its boundary. The action satisfies the Polyakov-Wiegmann relation, [29] I(gh −1 ) = I(g) + I(h) + 1 16π The associated path integral for SU(N) k is defined as Z = Dg e −k I(g) (4.43) The left and right-moving currents for this theory are given by We can couple external sources to them and define their effective action as (4.46) We may now parameterize without loss of generality the two sources A z,z in terms of two scalar functions, h,h, This does not imply, in particular, that the sources have a flat field strength. On the other hand we may write the group elements as non-local functions of the gauge fields where C z is a path that links the point (z,z) to a reference fixed point.
Using this parametrization we may rewrite the source action as T r d 2 ξA z Az (4.49) We may now perform the path integral in (4.45), by changing variables from g →h −1 gh and noting that the path integral measure is invariant under left and right group transformations to obtain where in the second step we used (4.42). This is the final factorized action for the sources.
Variation with respect to A z , Az will give the current correlators.

Holography on AdS 3
There has been a lot of evidence for the holographic correspondence between 2d CFTs and string theories on AdS 3 (for a review see [16]). where ℓ is the AdS 3 radius. The general formula for the central charge (5.1) was derived in [36] using the low-energy gravity description. Mass and angular momentum are related to the CFT data by The calculation of the central charge in the gravity theory is expected to be reliable in the semi-classical regime ℓ ≫ ℓ p . Therefore, the result (5.1) should be viewed as the leading term in an expansion in ℓ p /ℓ. The formulae (5.2), however, are expected to be a universal feature of the correspondence.
Assuming that the full string theory dual spacetime is of the form where M p is a p-dimensional compact manifold with volume V = ℓ M ℓs p , the threedimensional Planck length can be written in terms of the string coupling g s and the string length ℓ s as 1 where g 3 is the three-dimensional string coupling.
Before going further, it is worth recalling the more familiar situation of the F 1-NS5 system (with p F 1's and k NS5's) that involves string theory on AdS 3 × S 3 . In this case The volume of the internal manifold S 3 and g s behave as Defining N ≡ kp (so that c ∼ N 2 ) and λ ≡ g s N (5.7) we obtain λ = k and V (λ) ∼ λ If we assume the validity of (5.1) for the dual of the SU(N) k WZW model we obtain The next ingredient is the relation of the string coupling to λ. To obtain some intuition we will try to establish first in which region we expect a gravitational description. Recall that the scaling dimensions of representations with a finite number of boxes were given in (4.10) as where the integers n,n are the contributions of the current oscillators. We would like to study the gap in dimensions between the primaries and their descendants. In the limit λ → ∞, the primary dimensions are half integers and therefore there is no adjustable gap separating them from the excited states. On the other hand, as λ → 0 the primary field dimensions vanish and this creates an adjustable gap. This suggests that the gravity limit will be reliable when λ → 0. We will adjust ℓ s so that the scaling dimensions in the field theory limit are constant while the excited states' dimensions vary with λ. This gives We may then estimate the three-dimensional string coupling as The equation above suggests that the analog of g s N is never small, both for large and small λ. We have made several assumptions to derive the previous results, including the fact that the string theory dual is a three-dimensional non-critical string theory. This is motivated from the realization of the CFT and previous experience that suggests that extra adjoint matter will generate extra dimensions, whereas fundamental matter is induced by spacefilling flavor branes. In that sense, g s N is a schematic notation for whatever parameter in the underlying string theory controls the spectrum.
We expect that the bulk string theory has a trivial closed string sector. The reason is two-fold. First, the closed string sector should contain the pure gauge theory states and these are trivial in two dimensions. Second, as we will see, the open string sector associated with flavor will generate the necessary correlators of the stress tensor and other closed string fields. This does not imply that there is no non-trivial gravitational action for the metric but that there will be no non-trivial fluctuations here and no stringy states.
The non-trivial string sectors are associated with the flavor symmetry SU(N) L × SU(N) R . This should be a symmetry that is realized as a bulk gauge symmetry. It will be realized by two sectors of open strings associated to N D 2 branes and ND 2 as in higher-dimensional realizations of flavor. The gauge fields L µ , R µ associated with flavor symmetry will have an action that starts with the CS action as where the ellipsis indicates higher derivative terms starting from with the YM action and The sign of the coupling constants is implied by the parity invariance of the CFT.
In Poincare coordinates the AdS 3 metric reads 14) The asymptotic expansion near the boundary, r → 0, for solutions of the gauge fields is 5 , where we chose the gauge A r = 0. The equations of motion, including the higher order terms imply that A (0) is flat They also imply that it is only the CS terms that contribute to boundary terms. In particular the boundary current is whereĝ is the induced metric on the boundary.
The correct variational principle states that in complex boundary coordinates one of the two A z,z should vanish at the boundary. The boundary action that imposes this condition is We therefore have the boundary conditions The boundary value of the stress tensor can also be computed from (5.18) to be from where we obtain the affine Sugawara stress tensors

The effective action
We will now establish the effective action for the sources, and match eventually to the one obtained in section 4.3 equation (4.50). To do this we follow the analysis in [3]. In this work, it is shown how one can write the gauge fields on-shell, in terms of a two-dimensional flat connection. The end result will be that the on-shell value of the effective action agrees with (4.50).
The equations of motion of the CS theory imply that in the bulk whose solution is with h,h functions of r and the two dimensional coordinates z,z.
Evaluating the CS action on the solution of (5.22) we obtain Adding this to the boundary action (5.18) where h,h are evaluated at the boundary, we obtain perfect agreement with the effective action obtained from the CFT (4.50).
We may also compute the boundary stress tensor. The bulk action does not contribute as the CS term is metric independent, and the YM term gives vanishing boundary contributions as explained in appendix D. The only contribution comes from the counterterm, which is traceless and we used (D.20) assuming k > 0. In the absence of the source A z = 0, and F zz = 0, implies ∂ z Jz = 0, and The chiralities are reversed when k < 0.
In general for k > 0, we should think of Az as the current induced by the source A z .
As they are flat we can parameterize them as We may write where the path starts from a reference point C ∞ and ends up at (z,z). Then the relation between source and current is from where we obtain the response function which is the proper two point function of the anti-holomorphic currents.

The bulk scalar
From the non-trivial ground states of the WZW theory associated to primaries, only one can be considered as the generating operator dual to a complex bulk scalar T ij , that transforms in the bi-fundamental under the bulk gauge group SU(N) L × SU(N) R . The reason is that all other primary ground states can be considered as composites (multi-particle states) of the fundamental scalar g under OPE, since they arise as appropriately regularized algebraic functions (products) of the fundamental operator.
Its mass is given by the standard formula that connects it to the scaling dimension of the dual operator The situation is similar to that of tachyon condensation in D −D systems realizing flavor in holography, [18]. The simplest two-derivative quadratic action compatible with the symmetries is where the last term is the standard quadratic boundary counterterm and The vev associated with the CFT vacuum is T = 0, which is the only choice that keeps the chiral symmetry unbroken. This action will have higher order corrections in T in order to generate the multipoint correlation functions of the WZW theory. It is plausible that the full non-linear theory is described by a DBI-like action along the lines of [34], [35].
Turning on a source for T should correspond to perturbations of the WZW theory by g.
In the strong coupling limit this corresponds to turning on a mass matrix for the fermions.
Therefore, the theory is expected to flow to SU(N − r) k theory where r is the rank of the mass-matrix. This amounts to a reduction of the values of the 't Hooft coupling λ, therefore flowing towards weak coupling. The flow is also visible in the bulk theory, (5.13), (5.33).

Multiparticle states and entropy
We have seen in section 4 that ground states that are multiparticle states can have large multiplicities. Their scaling dimensions are of order O(N 2 ) and therefore may be expected to correspond to macroscopic solutions in the bulk theory. Since however the group degrees of freedom are unconfined (and therefore observable), such states would not correspond to highly entropic states as they are distinguished by their SU(N) 2 quantum numbers. It is therefore expected, but not completely clear, that they should correspond to regular horizonless semiclassical bulk solutions.
The associated mass for the "half-box" states in (4.13) is given by where we have used eqs. (5.1), (5.2). All other primary masses are suppressed by 1/N or more. Observe that in the weak coupling limit, λ → 0, the mass becomes much larger than the Planck scale, M Mp ∼ 1 λ . Hence, these are truly macroscopic configurations. We recall that this is also the limit where the zero mode theory (gravity+CS+scalar) is expected to give a reliable description of the physics.
On the other hand, for λ → ∞, the mass of these states is bounded, M Mp ∼ 6π by the Planck scale. Therefore, in this limit the ground states remain Planckian and do not generate macroscopic states.
The multiplicity of the "half-box" state was calculated in (4.14). We can compare it with the Cardy entropy for a state with the same scaling dimension as the conformal dimensions ∆ =∆ = N 2 8λ in (4.12) We observe that at strong coupling the dimension multiplicity in (4.15) is subleading to (5.36) as π √ 3 > 2 log 2. Such states have subdominant entropy, and never dominate the Cardy entropy. On the other hand, at weak coupling the Cardy entropy is subdominant.
This suggests that in this intermediate energy regime it is this class of states that dominate the partition function. Of course for asymptotically high energies it will be the Cardy entropy that will dominate.
We conclude with the following picture. The theory possesses ground states of different scaling dimensions and degeneracy. Some of them, like the half-box states, are highly massive and can dominate the partition function, but do not give rise to highly entropic states in the bulk. There are also massive states with high entropy at the Regge trajectories of each ground state. These are expected to correspond to bulk black holes. The large degeneracy of some ground states, like the half-box states, implies the existence of a correspondingly large class of ultra-massive black holes in the bulk.

A. Casimirs and dimensions for the SU (N ) algebra
In this appendix we will tabulate some useful results on SU(N) representations namely dimensions, the Dynkin index and the quadratic Casimir that enters in conformal dimensions of primary fields.
We will denote the dimension of the representation R by D(R). The irreducible representation R is completely specified with its Young tableau. Conjugation by an ǫ-tensor acts on any column of length s ≤ N of the Young tableau replacing it with a column of length N − s. We use the convention to have the minimal number of boxes for an SU(n) representation. For example, if we discuss the that can be described also by N − 1 antisymmetrized boxes we substitute instead the fundamental with one box. Therefore, a representation represented by a generic Young tableau has m 1 boxes in the first column, m 2 boxes in the second column etc, with m 1 ≥ m 2 ≥ m 3 · · · ≥ m n . We will denote this representation as (m 1 , m 2 , · · · , m n ).
The affine cutoff implies that only integrable representations can be present as primary states. For SU(N) k , all representations with more than k columns are not integrable.
Therefore n ≤ k above.
In a tensor product of irreducible representations R 1 ⊗ R 2 = k R k we have the following relation for their dimensions from which the well known formulae for dimensions of an arbitrary Young tableau can be calculated.
The Dynkin index S 2 (R) of a representation R is defined as, [37] T where T a R are the Lie algebra generators of the representation R. We will normalize them here so that S 2 ( ) = 1 2 . Again for a tensor product we have the following relations that allow the calculation of all Dynking indices Finally the quadratic Casimir is defined as and is related to the Dynking index by by that follows from (A.2) and (A.4).
The general formula for the Casimir for an arbitrary representation (m 1 , m 2 , · · · , m n ) is The quadratic Casimir is related to the conformal dimensions of primary fields as The large-N limit of dimensions is obtained from the large-N limit of the Casimir with the 't Hooft coupling defined in (3.1). We obtain The dimension, Dynkin index, Casimir and ∆ R (∞) for some common representations are tabulated in table A. We will now consider representations with large dimensions and large multiplicities.
The maximum is when for which we compute .11) and the scaling dimension The dimension of the associated SU(N) representation when n i = 0 for all i is The logarithm of this dimension is (A.14) Using Stirling's formula log(n!) = n log n − n + 1 2 log(2πn) + O(n −1 ) (A. 15) we obtain We may now use the summation formula to obtain log D = 4(λ + 1) log 2 − λ 2 log λ + 2(λ + 1) 2 log(λ + 1) − (λ + 2) 2 log(λ + 2) The left-right symmetric character-valued partition function is where Q is the appropriately normalized U(1) charge (defined when the associated current has central term equal to one), and ∆ R is the conformal weight of the SU(N) 1 reps.
For the j-index antisymmetric representation of SU(N), Its (properly normalized) U(1) charge is ±j/ √ N . Thus, summing up the two contributions in eq. (B.6) we obtain conformal weight j/2 which is in agreement with the interpretation above.
Consider now the two spinor representations. Their U(1) charges can be easily figured out by bosonizing pairwise the complex fermions In this basis, the U(1) current is The spinor and conjugate spinor are generated by the following vertex operators with ǫ k = ±1. The spinor corresponds to N K=1 ǫ k = 1 and the conjugate spinor to It is not difficult to see that the spectrum of U(1) charges coming from C and S is given by When the U(1) charge is Q k , the accompanying SU(N) representation is the k-index antisymmetric representation, which is the correct conformal weight of the spinors.
The quantum of U(1) charge is 1 √ N in the unit-vector sector and 1 2 √ N in the spinor sector. The number of states in the Dirac spinor is 2 N = 2 · 2 N −1 . This is the number of all antisymmetric representation states.

B.1 The fermionic current algebra
We define the U(N) currents J ij in terms of the fermions that satisfy The properly normalized overall U(1) current is The currents are uncharged under the zero mode J 0 .
They realize the tensor product CFT U(N) k 1 . We may construct the U(N) k currents as as well as U(k) N currents The two groups are not independent as they share the same overall U(1) Moreover, the two current algebras are not commuting. A general commutator gives other currents of the maximal O(2kN) 1 current algebra.
The global subalgebra of the U(k) N algebra acts non-trivially on the (U(N) 1 ) k /U(N) k coset. To leading order it will be a symmetry of U(N) k .
In this respect, the present theory is given by . (B.25) In the limit N → ∞, with k fixed, the coset SU(k) N has central charge of order O(1).
Hence, to leading order The same is true in the 't Hooft limit with λ ≫ 1. Therefore, in these limits the WZW model reduces to a theory of kN free complex fermions.
Moreover, (B.25) can be translated to the statement that the conformal U(k) gauge theory of N flavors of massless fermions is equivalent to the SU(N) k WZW model. In this respect, the 't Hooft limit k → ∞, N → ∞ with N/k = λ fixed can be interpreted as a Veneziano limit with λ being the ratio of flavors to colors.
(B.25) can also be written in the alternative form: which indicates the full flavor symmetry of an SU(N) k gauge theory including the U(1) L × U(1) R flavor symmetries. Note that unlike four dimensions the U(1) A is not anomalous in two dimensions as the anomaly is quadratic in the group generators and it vanishes because of the tracelessness of the generators of the color group.
Several of the above issues were discussed in early papers on the realization of chiral symmetry in two dimensions, [38,39].

C. Analysis of a four-point function in SU (N ) k
In this appendix we present the details of the analysis of the SU(N) k four-point function of the fundamental g a,b (z,z) and its conjugate g −1 b,a (z,z) following [11]. The result is and bars stand for complex conjugation.
The function G can be decomposed into group channels as and conformal block channels as The conformal blocks have been calculated by solving the Knizhnik-Zamolodchikov equations, [11] and are given by where F is the hypergeometric function, are the conformal dimensions of the fundamental and adjoint affine primaries and the ( , , Adjoint) OPE coefficient h is given by Now we take the 't Hooft limit of the correlation function, defined in section 3.1, to obtain Using the above the results we find In the limit λ → ∞ of the leading order in 1/N result we obtain (C.21) where we have used F (1, 1, 1, x) = 1 1−x . This is the free-fermion four-point function. In the limit λ → 0 we obtain instead G 11 = 1 − 2λ log |x| + · · · , G 22 = 1 − 2λ log |1 − x| + · · · , G 12 = − 1 N + · · · . (C.23) C.1 On the large-N and large-λ limits.
Finally, we consider the commutativity of the two limits N → ∞ and λ → ∞, in one of the dynamical functions of the WZW model, namely the structure constant h, and show that the two limits do not commute.
We start from (C.11) and expand the OPE coefficient in a double series Taking the large-N limit first while keeping λ fixed we obtain (C.27) The λ → ∞ limit of these expressions gives from which we read in the range 1 ≤ n ≤ 3, −2 ≤ m ≤ 0 with all other coefficients being zero in that range. ψ(x) = d ln Γ(x) dx is the standard digamma function.
On the other hand, taking the large-λ limit first we obtain The large-N limit of these expressions gives lim N →∞ from which we deduce with the rest of the coefficients zero.
Comparing the results (C. 30) and (C. 36) we observe that the two limits do not commute.
the action transforms as follows is the boundary metric.
Putting everything together we obtain (D.12)

D.2 Equations of motion
The relevant variations leading to the equations of motion are √ gg rr g ij F ri δA j (D.14) δS ct = − |k| 8π ∂M d 2 x √ g g ij A i δA j (D. 15) In total we have δS = δS 3d + δS 2d (D.16) (D. 19) It will be shown later that the YM boundary terms vanish at the boundary. From (D. 19) we also obtain the boundary currents ∂ r (rF rz ) − r∂ z F zz + mF rz = 0 (D. 26) We now fix the gauge A r = 0, which implies F rz = ∂ r A z , F rz = ∂ r Az. Equations We obtain the on-shell action S on−shell = S bulk + S ct as as the first contributions vanishes at the boundary. This is the full answer for the abelian theory. In the non-abelian theory the boundary term generated by the YM action will also vanish. In the CS action what will remain is the cubic term as described in the main part of this paper.
Note that if we generalize the YM action to contain any non-linear function of the field strength and its derivatives, all its contributions on-shell will give boundary terms that will vanish.
For the constant mode k 1 = k 2 = 0 we obtain instead as the only non-trivial equations ∂ r (r∂ r A 1 ) + im∂ r A 2 = 0 , ∂ r (r∂ r A 2 ) − im∂ r A 1 = 0 (D.62) with general solution Both of these solutions are singular at the boundary.

D.5 Chern-Simons action only
We now consider the non-abelian action The bulk equations of motion are obtained from the variation and are the flatness condition with general solution where U(r, z,z) is a group element. Therefore where Γ(U) is the properly normalized WZ term. The counterterm is Therefore, In total S on−shell = S W ZW (U) (D.73)