Integrability and the Conformal Field Theory of the Higgs branch

In the context of the AdS$_3$/CFT$_2$ correspondence, we investigate the Higgs branch CFT$_2$. Witten showed that states localised near the small instanton singularity can be described in terms of vector multiplet variables. This theory has a planar, weak-coupling limit, in which anomalous dimensions of single-trace composite operators can be calculated. At one loop, the calculation reduces to finding the spectrum of a spin-chain with nearest-neighbour interactions. This CFT$_2$ spin-chain matches precisely the one that was previously found as the weak-coupling limit of the integrable system describing the AdS$_3$ side of the duality. We compute the one-loop dilatation operator in a non-trivial compact subsector and show that it corresponds to an integrable spin-chain Hamiltonian. This provides the first direct evidence of integrability on the CFT$_2$ side of the correspondence.


Introduction
The AdS/CFT correspondence [1,2] can be understood quantitatively through integrability methods in certain classes of AdS 5 /CFT 4 and AdS 4 /CFT 3 dual pairs. Reviews of this approach and references to the literature can be found in [3]. Superstrings on AdS 3 backgrounds with 16 supersymmetries are known to be classically integrable [4,5], and so one may wonder whether integrability underlies the AdS 3 /CFT 2 correspondence. 1 Recently, we have found the complete non-perturbative world-sheet integrable S matrix of string theory on such AdS 3 backgrounds, including the hitherto unaccounted for massless modes [7][8][9][10]. 2 It appears now very likely that the AdS 3 /CFT 2 correspondence with 16 supersymmetries is described, in the planar limit [12], through a holographic quantum integrable system.
As a consequence, there exists an all-loop Bethe Ansatz (BA) whose solutions determine the closed-string spectrum in these backgrounds [4,[13][14][15]. 3 It is known that this BA, in the weak-coupling limit, encodes the energy spectrum of an integrable spin chain with local interactions [13,15,16]. Based on experience with higher-dimensional of an effective action obtained by dropping the kinetic terms of the vector multiplet and integrating out the fundamental-valued hypermultiplets [26,27]. This produces new, nonlocal kinetic terms and interactions for the vector multiplet fields. As a result, such fields no longer have canonical dimensions, rather they have geometric scaling dimensions.
In this paper we propose to identify the origin of the IR Higgs branch with the point at which the quantum integrable system reduces to the local spin-chain constructed in [13,15,16]. At this point on the Higgs branch the dynamics can be described in terms of adjoint-valued vector-and hyper-multiplets, with 1/N f being the effective coupling constant of the CFT 2 . In this setting we identify operators with the correct charges and match them to the field content of the spin-chain constructed in [13,15,16]. In a closed so(4) sub-sector of the CFT 2 we calculate the one-loop anomalous dimension of singletrace operators in the planar limit (N c → ∞) at small 't Hooft coupling λ ≡ N c /N f ≪ 1. We show that the resulting computation reduces in the conventional way [17] to calculating the energy spectrum of an integrable spin-chain. As far as we are aware, this constitutes the first direct evidence of integrability on the CFT side of the duality. This paper is organised as follows. In section 2 we briefly review the UV description of the gauge theory in terms of the D1-D5 system and obtain the UV Lagrangian of the two-dimensional gauge theory by dimensional reduction from six dimensions. In section 3 we describe the conformal field theory at the origin of the Higgs branch arising in the IR, following [26,27]. In section 4 we show how the field content of that theory fits into representations of psu(1, 1|2) 2 , which is the rigid part of the small N = (4, 4) superconformal algebra, and how such fields are represented in a spin-chain that matches the one introduced in [13,16]. In section 5 we compute the one-loop dilatation operator in the bosonic so(4) subsector of the theory, showing that it corresponds to an integrable Hamiltonian. We conclude in section 6 and relegate some technical material to the appendices.

The D1-D5 system
The UV description of the dual pair is encoded in the dynamics of the open strings ending on a stack of D1 and D5-branes, N f D5-branes: 012345, N c D1-branes: 01, with the directions 2345 compactified on a four-torus [1]. The D1-and D5-branes separately preserve supersymmetry corresponding to N = (8,8) in two dimensions. However, the intersection of the two stacks of branes only preserve N = (4,4). Open strings stretching between the branes give rise to several multiplets of this symmetry. The D1-D1 strings correspond to an N = (8,8) U(N c ) vector multiplet, which splits into an N = (4, 4) vector multiplet and an adjoint hypermultiplet. The D1-D5 strings give bi-fundamental U(N c ) × U(N f ) hypermultiplets. Finally, the D5-D5 strings give an N = (8,8) U(N f ) vector multiplet. In the near-horizon limit the D5-D5 strings decouple and the U(N f ) symmetry becomes global. Furthermore, the center-of-mass U(1) vector multiplet also decouples and the gauge group is given by SU(N c ).
The two-dimensional N = (4, 4) multiplets can be obtained by dimensional reduction of the corresponding six-dimensional N = (1, 0) multiplets. The six-dimensional supersymmetry algebra has an R-symmetry that we will denote by su(2) • . 4 Upon reduction to two dimension we get an additional su(2) L ⊕ su(2) R symmetry corresponding to rotations in the four compact dimensions. We will also introduce an additional su(2) • symmetry under which only the fields in the adjoint hypermultiplet transform. In the brane-system, the su(2) L ⊕ su(2) R symmetry corresponds to rotations in the directions 6-9, while su(2) • ⊕ su(2) • gives rotations in the directions 2-5.

Field content in the UV
The vector multiplet consists of the gauge field A µ , two left-moving fermions ψ αȧ L , two right-moving fermions ψα˙a R , four scalars φ αα and three auxiliary fields D˙a˙b. Under the global symmetries the fields transform as The fundamental hypermultiplet contains two complex scalars H˙a, a doublet of leftmoving fermions λα L , a doublet of right-moving fermions λ α R , as well as two auxiliary complex scalars F˙a. The charges of these fields are given by 4 This notation will be useful to make contact with the isometries of AdS 3 × S 3 × T 4 as described in refs. [7,8]. 5 Our index conventions are as follows. We indicate so(1, 1) chiralities by L and R. For the IR Rsymmetry so(4) algebra we hence write so(4) = su(2) L ⊕ su(2) R , with su(2) L,R ⊂ psu(1, 1|2) L,R and we represent the su(2) subalgebras using Greek indicesα,β, . . . and α, β, . . . , respectively. The other so(4) is decomposed as su(2) • ⊕ su(2) • following [7], with each subalgebra by indicesȧ,ḃ, . . . and a, b, . . . , respectively.
The field content of the adjoint hypermultiplet is essentially the same as above , but can be written in terms of real fields by having the full multiplet transforming as a doublet under the global symmetry su(2) • . We denote the scalars by T aȧ , the fermions by χα a L and χ αa R and the auxiliary field by G aȧ . The corresponding charges are given by

Two-dimensional action by dimensional reduction
To obtain the Lagrangian of the two-dimensional UV field theory one can start with sixdimensional supersymmetric Yang-Mills theory and dimensionally reduce to two dimensions [18]. In six dimensions, the fermions in the vector multiplet satisfy the symplectic Majorana condition (ψ˙a) * = Bǫ˙a˙bψ˙b, (2.1) and are chiral Γ 012345 ψ˙a = +ψ˙a. (2. 2) The action for the vector multiplet is The hypermultiplet fermion λ is complex and anti-chiral The Lagrangian for the hypermultiplet and its couplings to the vector multiplet is Above we have written the Lagrangian for a hypermultiplet transforming in the fundamental representation of the gauge group. The Lagrangian for adjoint-valued hypermultiplets is well-known in the literature (see for example [28]) and we will not write it explicitly here. When we dimensionally reduce from six to two dimensions the so(1, 5) Lorentz symmetry is broken to so(1, 1) ⊕ so(4) which we write as so(1, 1) ⊕ su(2) L ⊕ su(2) R . The gauge field then splits into a two-dimensional gauge field A µ and four real scalars φ αα that form a vector of so (4).
Dimensionally reducing the kinetic term for the anti-chiral fermion λ to two dimensions we find iλ / Our gamma-matrix conventions can be found in appendix A. From this expression we see that worldsheet chirality is correlated with so(4) chirality. Hence, we split the fermion λ as so that the kinetic term becomes The fermion ψ˙a in the vector multiplet is chiral. The same sort of calculation as above therefore gives The action for the fundamental hypermultiplet and its couplings to the vector multiplet then takes the form (2.10)

Effective action at the origin of the Higgs branch
In the previous section we wrote down the two-dimensional N = (4, 4) SYM action for the D1-D5 system in the UV. In the context of the AdS/CFT correspondence, we are instead interested in the IR dynamics on the Higgs branch. At generic points of the Higgs branch, the CFT 2 is described by a sigma model whose target-space metric is that of the ADHM instanton moduli space. However, near the origin, where the instanton size goes to zero, this metric is singular and so the sigma model description breaks down. Witten [26] made a proposal for how to capture the physics at the origin of the Higgs branch, which we briefly review in this section. The two-dimensional Yang-Mills coupling is dimensionful making the kinetic term for the vector multiplet (2.3) irrelevant. Hence, such a term can be dropped while flowing to the IR. The resulting Lagrangian is conformal provided we assign the fields in the vector multiplet the following geometric scaling dimensions The fields in the hypermultiplets retain their canonical dimensions. The degrees of freedom at the origin of the Higgs branch are the adjoint-valued vector multiplet and hypermultiplet fields [26,27]. We can obtain the effective action for these adjoint-valued fields by integrating out the fundamental hypermultiplets. Since only the vector multiplet couples directly to the fundamental hypermultiplets, the Lagrangian becomes where above we wrote φ and T as short-hand for all fields in the two adjoint multiplets. L T is the standard, gauge-invariant kinetic term for the adjoint hypermultiplet. The effective Lagrangian for the vector multiplet has an overall factor of N f from the sum over flavours in the fundamental hypermultiplets. 6 As a consequence of integrating out the fundamentals, the vector multiplet fields become dynamical: L eff contains non-standard kinetic terms that account for the scaling dimensions given in equation (3.1). Conformal invariance then dictates the form of the two-point functions of the fields in the vector multiplet. For example, the dimension one scalar φ has a two-point function of the form where C is a normalisation constant. The non-standard kinetic terms in L eff modify the dynamical degrees of freedom in the vector multiplet. In a theory with a conventional kinetic term, the two-dimensional gauge field A µ carries no physical degrees of freedom. In L eff , on the other hand, A µ has a non-standard kinetic term, and so the gauge field will now carry one degree of freedom. Similarly, the field D ab is auxiliary in the standard Lagrangian, but is a physical field of dimension 2 in the effective theory. N = (4, 4) supersymmetry [30], should be powerful enough to determine the kinetic terms of the full vector multiplet from the kinetic term of the scalars φ. 7 L eff also contains higher-order terms in the vector multiplet fields. These arise from interactions between vector multiplet and fundamental hyper multiplets where the only external fields are from the vector multiplet. Such vector-hyper interactions follow from the Lagrangian (2.10). In the following, we will find it useful to express higher-order terms in L eff in terms of the underlying vector-hyper interactions. Two examples of this have been drawn in equations (5.7) and (5.8) below.
We conclude this section by noting that there are two parameters in the theory: the number of flavours N f and the number of colours N c . In the effective Lagrangian, N f plays the role of the Yang-Mills coupling 1/g 2 Y M . Hence, it is natural to consider the weakly coupled 't Hooft limit

Superconformal representations of the fields
In this section we show how the field content of the theory described above fits in representations of psu(1, 1|2) 2 , which is the rigid part of the small N = (4, 4) superconformal algebra, and how these match with the representations used in the spin-chain constructed in [13,16]. We start by briefly reviewing the u(1, 1|2) superalgebra.

The u(1, 1|2) superalgebra
The u(1, 1|2) superalgebra consists of the su(2) R-symmetry generators R a b , the translation P, the supercharges Q a andQ a , the dilatation D, the special conformal transformation K, the conformal supercharges S a andṠ a , the central charge C and the hypercharge B. We write the su(2) commutation relations as where R is an arbitrary generator. The su(1, 1) algebra takes the form The action of P and K on the supercharges is given by and the non-trivial anti-commutators by The (conformal) supercharges carry dimension (D) and hypercharge (B) (4.5) Since the hypercharge B never appears on the right hand side of the commutation relations we can drop it to obtain the algebra su(1, 1|2). If the central charge C vanishes we instead get pu(1, 1|2). Combining these two conditions we obtain the psu(1, 1|2) algebra.

psu(1, 1|2) L ⊕ psu(1, 1|2) R decomposition of the fields
The maximal finite-dimensional subalgebra of the small N = (4, 4) superconformal algebra is psu(1, 1|2) L ⊕ psu(1, 1|2) R . To distinguish the generators of the two copies of psu(1, 1|2) we use subscripts L and R. We also introduce the u(1) generators The states appearing in the representation 8 2 ) R of this algebra were used to construct the all-loop massive S matrix of the AdS 3 × S 3 × T 4 superstring [13]. We collect them in table 1. To compare these states with the ones appearing in the field theory, we identify the su(2) L ⊕ su(2) R ⊂ psu(1, 1|2) L ⊕ psu(1, 1|2) R with the so(4) we got by reducing from six to two dimensions. We further let D measure the dimension and S the Lorentz spin. Finally we identify J • with the u(1) part of the su(2) • R-symmetry. We will now see how the states in the (− 1 2 ; 1 2 ) L ⊗ (− 1 2 ; 1 2 ) R match the field theory states. 8 We denote by (− 1 2 ; 1 2 ) the short representation of psu(1, 1|2) having highest weights − 1 2 and 1 2 for its bosonic sub-algebras su(1, 1) and su(2), respectively. The representation (− 1 2 ; 1 2 ) * differs from (− 1 2 ; 1 2 ) by having a fermionic, rather than bosonic, highest weight state. Table 1: Charges of the states in the representation (− 1 2 ; 1 2 ) L ⊗ (− 1 2 ; 1 2 ) R and their relation to the degrees of freedom of the spin-chain and CFT 2 . We use φ ± L/R to denote the su(2) L/R doublet of scalars in the (− 1 2 ; 1 2 ) L/R representation, while ψ ± L/R denotes the two fermions. The su(1, 1) L/R descendants are obtained by acting on these states with D L/R . Finally, there are four bosons of dimension 2, which are only charged under su(2) • , which decompose into a singlet and a triplet. The triplet can be identified with the auxiliary field D˙a˙b, while the singlet comes from the field strength ǫ µν F µν = 2ǫ µν ∇ µ A ν . Hence, the field strength multiplet perfectly fits into (− 1 2 ; 1 2 ) L ⊗ (− 1 2 ; 1 2 ) R . The multiplet discussed here is very similar to the field strength multiplet of N = 4 SYM (see e.g. ref. [31]). However, the two-dimensional field strength has only a single component F tx = −F xt . Further, as seen above the field strength multiplet contains the normally auxiliary field D˙a˙b which now is dynamical. In table 1 we also summarise the identifications of the spin-chain and CFT 2 variables.
The states in the adjoint hypermultiplet can also be organised into psu (

Spin-chain states
We can now construct local operators using the states in the field strength multiplet and adjoint hypermultiplet. The simplest such state is the 1/2-BPS ground state tr (φ ++ ) L . (4.7) Starting with this state we can obtain excited states by replacing the field at a site by any other adjoint field. Let us first restrict ourselves to the fields in the field strength multiplet. Such an operator will consist of L sites each containing a state from the (− 1 2 ; 1 2 ) L ⊗ (− 1 2 ; 1 2 ) R representation of psu(1, 1|2) L ⊕ psu(1, 1|2) R . This is exactly the operators that appear in the integrable spin-chain constructed in [13]. The Hamiltonian acting on the spin-chain is given by the charge H = D − J. In the next section we will calculate the one-loop correction to this Hamiltonian in a closed sub-sector.
The ground state (4.7) is the highest weight state in the short results in a state with D − J = 1 and M = +1. We can also consider the excitations which carry charges D −J = 1 and M = −1. These two sets of excitations form two multiplets of the centrally extended psu(1|1) 4 c.e. used to construct the S matrix in [15]. The remaining fields in the field strength multiplet can be though of as composite excitations constructed out of the eight fundamental excitations. Let us now consider excitations created from fields in the adjoint hypermultiplet. The four bosonic and four fermionic fields all carry charges D − J = 0 and M = 0. They can be inserted into the operator with no cost of energy, and hence correspond to gapless excitations. Since the scalars T aȧ form singlets under psu(1, 1|2) L ⊕ psu(1, 1|2) R the resulting spin-chain states are exactly of the type that appears in the reducible spin-chain discussed in [16]. The excitations fit perfectly in the massless psu(1|1) 4 c.e. ⊕ su(2) • representation discussed in [7,8]. While the operators containing these fields form reducible short representations at tree-level, we expect them to join into long irreducible representations once we include the interactions. We hope return to the investigation of the exact details of this mechanism in the future.

One-loop dilatation operator in the so(4) sector
In this section we will compute, in the planar limit, the one-loop anomalous dimensions of operators in the effective CFT 2 proposed by Witten. We restrict ourselves to operators made up of scalar fields from the vector multiplet, which form a one-loop closed subsector charged under so(4). 9 We find that the computation of these anomalous dimensions can be re-phrased in terms of finding the spectrum of a nearest-neighbour Hamiltonian for a homogenous so(4) spin chain, in a manner similar to the seminal work of Minahan and Zarembo [17]. As we will show explicitly, such an Hamiltonian has precisely the right form to be integrable.
Witten's prescription for the dynamics at the origin of the Higgs branch is closely related to the "large-N" approach 10 widely used in the study of sigma models [33]. In the present setting it is N f that plays the role of "large-N", while taking N c large will ensure that we can restrict to single-trace operators. Let us then consider single-trace operators made out of scalar fields from the vector multiplet, Since these operators are charged only under so(4) = su(2) L ⊕ su(2) R we refer to it as the so(4) sector. Our goal is to find the one-loop dilatation operator acting on these operators. To extract the dilatation operator we consider the one-loop corrections to the local operators. These corrections are generally UV divergent, which means we need to introduce counter-terms and renormalised operators The matrix Z cancels the UV divergences in the one-loop corrections, and introduces a mixing between different operators. In a perturbative calculation it can be extracted as 9 We will return to the analogous computation for the complete theory in an upcoming work. 10 See [32] for reviews and a more extensive list of references. minus the divergent piece of the sum of all Feynman diagrams. In dimensional regularisation we consider the theory in D = d + 2ǫ dimensions and divergences appear as poles in ǫ. For non-zero ǫ the coupling constant is dimensionful. Hence we introduce as mass parameter µ so that the full coupling constant can be written as µ 2ǫ λ. The corrections to the dilatation operator can then be extracted as In the planar limit, the one-loop dilatation operator acts locally on two neighbouring fields of the operators. From so(4) symmetry we expect it to take the form where M is an so(4) invariant tensor, which can be built out of the identity 1, the permutation P and the trace K. The dilatation operator should preserve supersymmetry. In particular it should annihilate chiral primary operators such as (4.7), since the dimension of these operators is protected. Since this operator is symmetric and traceless, we can impose supersymmetry by requiring the tensor M to take the form for some constants c 1 and c 2 . Hence, we only need to consider Feynman diagrams that act non-trivially in flavour space.

"One-loop" Feynman diagrams
We now proceed to compute the Feynman diagrams in order to determine the form of the spin-chain Hamiltonian. We while the gluon-scalar interaction is given by Above, the single dashed and solid lines correspond to fundamental fermions and scalars, respectively. We refer to the diagrams in (5.6) as one-loop diagrams, because they all give a contribution proportional to the coupling λ = N c /N f . In practice, to compute them we need to plug in the effective vertices (5.7), (5.8), yielding several loops for each diagram. Let us see in more detail how this counting of λ goes for the diagrams in (5.6). We first note that the above vertices all contain a single loop of fields from the fundamental hypermultiplet, which gives an overall factor of N f . Furthermore, the effective action for the fields in the vector multiplet is proportional to N f . Hence, each double line propagator in (5.6) gives a factor 1/N f . Finally, in each diagram there is a single sum over colour indices in the loop, which gives a factor of N c . Since the number of effective vertices in each diagram is one less than the number of internal propagators the overall factor for each diagram is N c /N f . In principle we should also consider the diagrams above, but with the fields from the fundamental hypermultiplet replaced by the corresponding fields in the adjoint hypermultiplet. However, in that case the sums over flavours in the loops are replaced by a sum over colours. Hence, the resulting diagrams are suppressed at small λ. The counting of factors of N c and N f is illustrated in figure 1.
Let us now look at the flavour structure of the diagrams in (5.6). The second and fourth diagram give no non-trivial so(4) flavour interactions. As discussed above, the coefficient of the so(4) identity tensor 1 in the dilatation operator can be determined using supersymmetry. Hence, we do not need to calculate those diagrams. The third diagram in (5.6) gives an so(4) trace in flavour space. By expanding the effective vertices in this diagram in terms of the one-loop diagrams with a fundamental hyper, we find four different diagrams. However, all four vanish due to symmetry as detailed in appendix C. Hence, the first diagram in (5.6) is the only one we have to compute.

Computation of one-loop dilatation operator
In this sub-section we consider the first diagram in (5.6). As was shown above, it is only this diagram that has a potentially non-trivial so(4) flavour structure and hence is needed to determine the dilatation operator. We may expand the diagram by plugging in the effective vertices (5.7) and (5.8) In fact, the first of these diagram again has a trivial flavour structure and so we do not need to calculate it. The second diagram gives an so(4) trace in flavour space, and we compute it in appendix C. There we show that the corresponding integral is UV-finite, and therefore does not contribute to the mixing matrix. We are only left with the last diagram, the fermion box The computation of the divergence for each of these diagrams is somewhat involved due to the propagator (3.3). It is straightforward, however, to show that the difference of the two resulting integrals is finite, see appendix C. To perform the calculation for the sum of the diagrams we work in two steps. We first compute the integral in dimension d using the propagator of (3.3), that in momentum space is The resulting integral has a UV log-divergence in any d > 2. We then check that the diagram remains UV divergent also when taking d → 2. As the diagram (5.10) is logarithmically divergent in the UV it contributes to the dilatation operator. We again relegate the details of such calculation to appendix C. We have concluded that the divergent part of the fermion-box diagram yields a contribution to the mixing matrix M of the form To simplify this we introduce the identity operator 1 and permutation P acting on the tensor product of two spin-1/2 su (2) representations. If Ψ a is an su(2) doublet this action can be written as The combination 1 2 (1 − P ) then represents a projector onto the su(2) singlet Since the divergence of both fermion box diagrams shown in figure 2 is the same, the flavour structure of the mixing matrix M can then be written as Since the above operator annihilates the supersymmetric vacuum (4.7) by itself, and since the diagrams that we have refrained from computing so far do not, we find that these must cancel by supersymmetry.
We conclude that the one-loop dilatation operator in this sector is given by , (5.16) in the so(4) sector, up to an overall numerical constant. This is precisely the Hamiltonian of the integrable so(4) spin-chain [34].

Conclusions
In this paper we have investigated the CFT 2 that arises in the IR limit of N = (4, 4) SYM with matter. At the origin of the Higgs branch, this theory has a well-defined planar, weak-coupling limit. The Higgs branch CFT 2 is expected to be dual to Type IIB string theory on AdS 3 × S 3 × T 4 , for which recently an all-loop integrable S-matrix has been constructed [7][8][9][10]. At weak coupling this all-loop integrable system reduces to a local integrable spin-chain [13,15,16]. In this paper we have proposed to identify this point of the quantum integrable system with the weakly-coupled planar limit of the CFT 2 at the origin of the Higgs branch. Our proposal is motivated by the observation that there the Higgs-branch theory is described by a vector multiplet and adjoint-valued hypermultiplet. While the theory is governed by a non-local effective action [26], the composite operators in it are simply the conventional gauge-invariant sequences of adjoint-valued fields. Remarkably, the field content in this effective theory matches precisely the representations that appear in the local integrable spin-chain [13,15,16]. For this matching to occur, it is necessary for the auxiliary fields present in the vector multiplet to become dynamical. This property is characteristic of supermultiplets with non-canonical kinetic terms, such as the ones that appear in the action introduced in [26].
In the large-N c limit of the CFT 2 one may restrict to single-trace operators. We calculated the anomalous dimensions of such operators in the scalar so(4) sub-sector of the theory. We have shown that this computation is described by an integrable Hamiltonian for the vector representation of so(4) in a manner that is reminiscent of [17]. This is the first appearance of integrability on the CFT 2 side of the duality. Together with the evidence for integrability that has been found on the string theory side [7][8][9][10], it provides a powerful argument for a holographic integrability description of AdS 3 /CFT 2 .
In a forthcoming paper, we will return to the computation of the complete CFT 2 spinchain Hamiltonian. There, we will also address the issue of length-changing effects that are expected to occur in the complete spin-chain, much as they do in N = 4 SYM [35]. This should allow us to understand better, from the weakly-coupled CFT 2 point of view, how the massless modes enter into the reducible spin-chain description introduced in [16]. It will also allow for a CFT 2 -based understanding of the S matrix constructed in [7,8]. In this context it will be particularly illuminating to extend the analysis of the chiral-ring of the CFT 2 that was initiated in [16].
It is widely believed that the Sym N (T 4 ) orbifold can be found at some point in the moduli space of the N = (4, 4) CFT 2 that we have been investigating in this paper. However, it appears to be much harder to understand the spin-chain picture starting from the Sym N (T 4 ) orbifold point [36], though many suggestive connections seem to exist. It would be interesting to see whether it is possible to deform the spin-chain [13,16] to approach the Sym N (T 4 ) orbifold point. It is not clear to us whether the connection to the orbifold can be fully understood in the small λ limit. 11 Throughout this paper we have focused on the CFT 2 description of the D1-D5 system, or in other words on string geometries supported by R-R three-form flux only. More generally, the AdS 3 /CFT 2 correspondence is believed to hold for configurations involving NS5-branes and fundamental strings in addition to the D1-and D5-branes. In this case the string geometry is supported by a mixture of NS-NS and R-R three-form flux, giving a family of backgrounds related by Type IIB S-duality. The all-loop worldsheet S matrix for these backgrounds is known [9,37], and we expect also these dualities to be governed by a quantum integrable system. It would be interesting to understand whether an integrable spin-chain picture emerges in this setting too. It would also be interesting to understand how recent work on higher-spin limits of AdS 3 /CFT 2 , such as [38], can be related to our findings. Recent progress on understanding the large N = (4, 4) AdS 3 /CFT 2 duality [39] opens up the possibility of establishing the expected connections to integrability more firmly.
The "large-N" method (which here means large N f ) is particularly useful in the study of the effective theory that exists at the origin of the Higgs branch, as was originally advocated by Witten [26]. In this paper, we have used this approach to perform some simple computations of anomalous dimensions at leading order. But large-N techniques have been very widely exploited for a long time and an extensive set of tools exists to perform calculations in this approach. As a result, we expect that these methods, combined with integrability, will lead to significant advances in understanding generic unprotected quantities in the AdS 3 /CFT 2 correspondence in the near future. -string theory duality". A.S.'s work is funded by the People Programme (Marie Curie Actions) of the European Union, Grant Agreement No. 317089 (GATIS). A.S. also acknowledges the hospitality at APCTP where part of this work was done, as well as hospitality byÉcole Normale Supérieure in Paris and interesting discussions with Benjamin Basso, Volodya Kazakov and Jan Troost. B.S. acknowledges funding support from an STFC Consolidated Grant "Theoretical Physics at City University" ST/J00037X/1.

A Gamma matrices
We use the gamma matrices In this basis a generic six-dimensional spinor can be written as a tri-spinor Ψ α 0 α 1 α 2 .

C Regularisation of Feynman diagrams
In this appendix we present the details of the Feynman diagram calculation discussed in section 5. Since we are in two dimensions the IR properties of the theory are rather delicate [26]. What is more, the effective propagator for the vector multiplet fields that follows from the two-point function (3.3) is logarithmic in momentum space. All this makes it difficult to evaluate the loop-momenta integrals. To overcome such technical complications we propose the following method. We will compute the one-loop anomalous dimensions in a large N f expansion for a general dimension d > 2, following which we will take the limit d → 2. 12 This method is useful because the effective propagator for the scalars φ in dimension d > 2 has a simple form in momentum space (5.11). It also allows us to retain much better control over the IR region of momentum integration. We believe this prescription is physically well motivated, because we are interested in the anomalous dimensions of the CFT 2 which are related to the UV properties of the theory. Furthermore, within the large-N approach anomalous dimensions are often computed in a similar manner for generic d [40] and, when finite, the d → 2 limit can be taken. The leading order diagrams that we are interested in are given in equation (5.6). The diagrams that give non-trivial flavour interactions are , (C.1) and .
(C. 2) in the loop. This leads to an integral of the form tr / k(/ k − / p 1 )(/ k + / p 2 )(/ k − / q) In the above calculation the dimension d has been introduced in the propagator to avoid IR divergences, while D has been introduced in the integrations to avoid UV divergences.
Hence we now need to take the limit where d and D approach 2 with 2 < D < d. (C.18) A natural way to do this is to first send D → d, and then d → 2, which results in so that the fermionic box diagram (C.2) is UV divergent. This expression also appears IR divergent when we send µ to zero. However, in the full fermion box integral with the momentum assignment we have made there is no such divergence, so this IR divergence is cancelled by a contribution from the UV convergent integral in (C.15) that we have dropped.