$\mathbf{AdS_3\times S^3}$ Tree-Level Correlators: Hidden Six-Dimensional Conformal Symmetry

We revisit the calculation of holographic correlators in $AdS_3$. We develop new methods to evaluate exchange Witten diagrams, resolving some technical difficulties that prevent a straightforward application of the methods used in higher dimensions. We perform detailed calculations in the $AdS_3 \times S^3 \times K3$ background. We find strong evidence that four-point tree-level correlators of KK modes of the tensor multiplets enjoy a hidden 6d conformal symmetry. The correlators can all be packaged into a single generating function, related to the 6d flat space superamplitude. This generalizes an analogous structure found in $AdS_5 \times S^5$ supergravity.


Introduction
Correlation functions of local operators are basic observables in holographic CFTs, and as such have been intensely studied since the early days of AdS/CFT. Only recently however have truly efficient computational methods been developed. Broadly speaking, these new techniques are inspired by the modern "on-shell" approach to perturbative gauge theory amplitudes. One focusses on the full holographic correlator, which is a much simpler and more rigid object than individual Witten diagrams. Correlators can be strongly constrained and sometimes completely determined by imposing symmetries and other consistency requirements.
In the paradigmatic AdS/CFT example, namely the dual pair of N = 4 super Yang-Mills theory and IIB string theory on AdS 5 × S 5 , this new approach has led to a compelling methods [32][33][34][35]. For example, the four-point function of the KK modes with lowest conformal dimension was conjectured from a limit of the light-light-heavy-heavy correlators [34].
There is also a clear physical incentive to revisit AdS 3 × S 3 holographic correlators, beyond the mere demonstration that our previous methods can be generalized to this more difficult case. Do these correlators exhibit a hidden conformal symmetry analogous to the one found for AdS 5 × S 5 case [10]? While a conceptually satisfactory explanation is still lacking, in the 10d case such a symmetry appears to hinge crucially on a few facts. First, the AdS 5 × S 5 metric is conformally flat, a feature shared by the AdS 3 × S 3 background but not (for example) by the maximally supersymmetric M-theory cases, namely 5 AdS 7 × S 4 and AdS 4 × S 7 . Second, the flat space 10d superamplitude in IIB supergravity contains a kinematic factor G (10) N δ 16 (Q), which can in some heuristic sense be regarded as a dimensionless coupling. An analogous power-counting applies to superamplitudes in 6d (2, 0) supergravity, where the kinematic factor G (6) N δ 8 (Q) is again dimensionless. Third, the fourpoint superamplitude in 10d flat space IIB supergravity enjoys an accidental 10d conformal symmetry. The 10d amplitude can be viewed as a generating function of all four-point correlators of the full tower of KK modes on AdS 5 × S 5 [10]. The situation in 6d (2, 0) flat space supergravity is more elaborate. As we have mentioned, there are two relevant supermultiplets, the graviton and tensor multiplets. As it turns out, the superamplitude with four external tensors enjoys an accidental 6d conformal symmetry! 6 By analogy with the AdS 5 × S 5 case, it seems natural to anticipate that that all fourpoint correlators of tensor multiplet KK modes in AdS 3 × S 3 can be packaged into a single 6d object. We find strong evidence that this is indeed the case. Our strategy is to develop a systematic position space method similar to the one used in [1,2,18]. New ingredients in AdS 3 include a derivation of the superconformal Ward identity, and the computation of AdS 3 exchange Witten diagrams that require a generalization of the existent techniques. Using this method we compute equal-weight four-point functions of one-half BPS operators that arise as KK modes of the 6d tensor multiplets. We have obtained results for operators with conformal dimensions ∆ = 1, 2, 3, 4. Our result for ∆ = 1 reproduces the recent conjecture of [34]. We also discuss an independent method in Mellin space. The Mellin space method for AdS 3 × S 3 × K3 is not as powerful as in AdS 5 × S 5 and AdS 7 × S 4 , but is still very useful to illustrate the analogy between holographic correlators and scattering amplitudes, which plays a crucial role in formulating a guess for the master formula.
From these concrete examples of correlators, we observe nontrivial evidence of a sixdimensional hidden conformal symmetry. Assuming that such a symmetry persists for arbitrary external weights, it is immediate to write down a simple generating function of all four-point correlators of tensor KK modes by replacing the x 2 ij in the lowest-weight four- 5 Indeed, in the M-theory cases, the radii of the AdS and sphere factors are different. In the case of AdS7 × S 4 one can immediately see that a putative 11d conformal symmetry would be structurally incompatible with the explicit results of [18]. 6 We were not able to find in the literature fully explicit expressions for amplitudes involving external supergravitons in 6d (2, 0) supergravity (see [36] for the state of the art), but we suspect that they do not enjoy such a symmetry.
point function with six dimensional distances. On the other hand, such a hidden conformal symmetry is not present in the four-point functions of scalar one-half BPS operators that arise from the 6d gravity multiplet, as we have checked using the position space method. We hope that the new data obtained here will stimulate a better understanding of the nature of the hidden conformal symmetry, in both the AdS 5 × S 5 and the AdS 3 × S 3 cases.
The rest of the paper is organized as follows. In Section 2 we discuss the superconformal kinematics of scalar one-half BPS four-point functions. In Section 3 we introduce the position space method for AdS 3 and compute several examples of four-point functions. In Section 4 we provide a different perspective in Mellin space. In Section 5, we point out the existence of a six-dimensional hidden conformal symmetry. Using this symmetry we conjecture a formula for all one-half BPS four-point functions. We conclude in Section 6 by mentioning some future directions. The paper also includes three appendices to which we have relegated various technical details.
Note: As we were about to submit this paper to the arXiv, we learnt of an upcoming work [37] that obtains AdS 3 ×S 3 four-point tree-level correlators with pairwise equal weights by generalizing the approach of [34].

Superconformal kinematics
Let us start with the constraints of superconformal invariance. We focus on the one-half The Kaluza-Klein reduction on AdS 3 × S 3 of 6d (2, 0) supergravity coupled to 21 tensor multiplets leads to two different types of one-half BPS scalar operators. The first kind originates from the anti-self-dual tensors with k = 1, 2, . . ., and they transform in the vector representation of the SO(21) flavor symmetry. The second kind comes from 6d supergravity fields with k = 2, 3, . . ., and they are neutral under the flavor symmetry. In this work, we focus on correlation function of half-BPS operators in the tensor multiplet, although the superconformal constraints are the same for both types of operators.
To begin with, it is convenient to keep track of the R-symmetry structure by contracting all the indices with auxiliary spinors v α ,vα We have exploited the fact that the spinors are automatically "null" and the one-half BPS operator is symmetric and traceless (with respect to the tensor) in α i andα j . We note that rescaling preserves the null property of the spinors. This allows us to parameterize the spinors as 3) The goal of this paper is to calculate the four-point function This is then a function of both the spacetime as well as the R-symmetry coordinates. Covariance under the conformal and R-symmetry implies that it is really a function of the cross ratios z = z 12 z 34 z 13 z 24 ,z =z 12z34 z 13z24 , α = y 13 y 24 y 12 y 34 ,ᾱ =ȳ 13ȳ24 y 12ȳ34 , (2.5) and z ij ≡ z i − z j , y ij ≡ y i − y j , etc. Hence we can write it as where the kinematic factor K is given by Following the convention in [1,2], without loss of generality, we order the weights as k 1 ≥ k 2 ≥ k 3 ≥ k 4 . There are two cases: The various γ 0 ij are given by From the above definition, it follows that G I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 (z,z; α,ᾱ) is a polynomial in α andᾱ with the same degree L.
The cross ratios z,z, α,ᾱ are related in a simple way to the cross ratios that appear in four-point correlators of SCFT d≥3 with R-symmetry group SO(d ≥ 5) Here t µ is a d -dimensional null vector satisfying t µ t µ = 0 and t ij ≡ t µ i t jµ . When d = 4, we can construct a 4-dimensional null vector from the spinors, t µ ≡ σ µ αα v αvα . Note that the two sets of cross ratios are inequivalent. Expressing z,z (or α,ᾱ) in terms of U , V (or σ, τ ) is generally ambiguous due to the appearance of square-roots when solving the quadratic equations 7 . In two dimensions, the four-point function G I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 (z,z; α,ᾱ) is only invariant under simultaneously exchanging (z, α) ↔ (z,ᾱ). This is in contrast to higher dimensions where the four-point function is always invariant under exchanging a single pair of cross ratios z ↔z and α ↔ᾱ. That is because in theories with d ≥ 3, d ≥ 5 the four-point functions are functions of x 2 ij and therefore can be written in terms of U and V (and similarly for the R-symmetry). On the other hand, in two dimensions there is one more structure, namely µν x µ ij x ν kl and therefore the four-point function cannot be written as a function of U and V alone. This is the essential new feature of SCFT 2 compared to higher dimensions, which necessary requires to work with (z,z; α,ᾱ).
So far we have only used the bosonic part of the global superconformal group. The fermionic generators impose extra constraints as the superconformal Ward identities 8 These identities can be solved as follows (2.14) where G I 1 I 2 I 3 I 4 0,k 1 k 2 k 3 k 4 (z,z; α,ᾱ) is a special solution which upon twisting becomes purely holomorphic (anti-holomorphic) The function f (z, α) can be further shown to be protected by non-renormalization theorems, by using the argument of [40]. The function H I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 (z,z; α,ᾱ) encodes unprotected dynamical information, and because of the prefactor (1 − zα)(1 −zᾱ), H I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 is a polynomial in both α andᾱ with the reduced degrees L − 1.

Position space
In this section, we develop a concrete position space method to compute holographic fourpoint functions. The method is very similar to the one used in [1,2] for AdS 5 ×S 5 and in [18] for AdS 7 ×S 4 . However, it has also new important ingredients due to the unique features of AdS 3 space. In Section 3.1 we review some elements of the 6d (2, 0) supergravity coupled to tensor multiplets, compactified on AdS 3 × S 3 . Then, in Section 3.2 we outline the position space algorithm and in Section 3.3 we compute the four-point function of the lowest-weight operator. Finally, in Section 3.4 we apply the method to the four-point functions of higher weights. 8 We have derived the superconformal Ward identities using two different methods. The first one uses the analytic superspace formalism, and is parallel to the analysis in [38]. The second method uses a chiral algebra twist [39] on one of the psu(1, 1|2) subalgebra of the small N = 4 superconformal algebra. The second method is conceptually more interesting, and we will elaborate on it further in Appendix A.

KK mode
The spectrum is organized into superconformal multiplets which come in three infinite Kaluza-Klein towers Γ l , Σ l and Θ I l . In the tables, h,h are the SL(2) L , SL(2) R spins, and j,j are the SU (2) L and SU (2) R spins. When the R-symmetry quantum numbers are negative, the corresponding field does not exist. We have only kept the fields that are singlets under the outer automorphism group SO(4) out because in this work we focus only on four-point functions of operators which are singlets. The superconformal primary of the multiplets Γ l which contains the spin-2 fields is the (massive) graviphoton field V µ . The lowest KK multiplet with l = 0 is ultra-short: it contains only the non-dynamical massless graviton and graviphoton. The superconformal primaries of the spin-1 multiplets Σ l and Θ I l are the scalar fields σ and s I respectively. These two multiplets have the same SO(2, 2) and SO(4) R quantum numbers, but Θ I l transform in the vector representation of SO(21) while Σ l are singlets. In terms of 6d fields, Σ l is made of fields from 6d (2, 0) supergravity and Θ I l comes from the anti-self-dual tensors. Moreover, the minimal allowed value for Θ I l is l = −1, and the corresponding super primary has conformal dimension ∆ = 1. The top component (not shown in the table) is an exactly marginal operator and transforms as a vector under SO(4) out . By contrast, Σ l with l = −1 is pure-gauge and does not exist in the spectrum [41]. The cubic couplings of the Kaluza-Klein modes were obtained in [44]. The cubic couplings satisfy the R-symmetry selection rule, and vanish when they are extremal. The quartic and higher-oder vertices have not been worked out in the literature. Moreover, [41,44] showed that the vector fields are described by two Proca-Chern-Simons vector fields supplemented by a first-order constraint. The vector fields couple to the currents made out of scalar fields both electrically and magnetically. We will show in Appendix B that the constraint can be solved in terms of three massive Chern-Simons fields which satisfy first-order equation of motions. After proper field redefinition, all the couplings of the vector fields with currents become electric.

The position space algorithm
We are now ready to formulate the position space method. We start with an ansatz for the four-point function which includes all the possible exchange and contact Witten diagrams The exchange Witten diagrams are subject to the R-symmetry selection rule and the requirement that the cubic coupling is non-extremal. In addition, the contact Witten diagrams should contain no more than two derivatives. This condition comes from the consistency with the flat space limit in which the theory contains only two derivatives. The next step is to evaluate all the diagrams in the ansatz. Compared to the AdS 5 × S 5 case, there are two new kinds of Witten diagrams. The first is the exchange diagram of twist-zero fields which are the massless Chern-Simons and the graviton field. The standard method of [31] is not applicable for these diagrams. 9 We instead evaluate them by solving second order differential equations with appropriate boundary conditions. These differential equations follow from the simple fact that the two-particle quadratic conformal Casimir is the same as the wave equation in the bulk, which collapses the exchange diagram into a contact diagram when acting on the bulk-to-bulk propagator. 10 We leave the details of this method to Appendix C. The second type of new diagram is the exchange diagrams which involve massive Chern-Simons vector fields. This type of diagrams can be evaluated using the method of [31] with slight modifications. All in all, all the Witten diagrams can be evaluated in terms of a finite sum of contact diagrams (D-functions). To impose the superconformal Ward identities (2.13), we exploit the fact thatD-functions can be uniquely decomposed as where Φ(z,z) =D 1111 is the scalar box diagram, and R Φ,U,V,1 are rational functions of z andz. It is clear that the supergravity ansatz admits a similar decomposition. By further using the recursion relation [46] we can similarly decompose the left side of the superconformal Ward identity (2.13) into this basis. The rational coefficient functions R I 1 I 2 I 3 I 3 Ward,X (z,z, α,ᾱ) with X = Φ, U, V, 1 are required to vanish by (2.13), giving rise to a set of linear equations for the unknown coefficients in the ansatz. In contrast to the AdS 5 × S 5 and AdS 7 × S 4 cases, solving superconformal Ward identities in general does not uniquely fix all the relative coefficients. We will see that all the coefficients in A I 1 I 2 I 3 I 4 con parameterizing the quartic vertices are fixed in terms of the coefficients in the exchange part of the ansatz. The remaining unsolved coefficients can be fixed by comparing with the known supergravity cubic couplings.

The lowest-weight four-point function
We now apply the above method to the simplest four-point correlator with k i = k = 1. The cubic coupling selection rules dictate that only the non-dynamical graviton and Chern-Simons gauge field can be exchanged. Therefore we have the following ansatz for the exchange part of the four-point function where Y m andȲm are the SU (2) L and SU (2) R R-symmetry polynomials associated with exchanging the representation (j,j) = (m,m). The function W gr is the exchange Witten diagram of the non-dynamical graviton, and is worked out in Appendix C to be Similarly, W CS,1,+ , W CS,1,− are contributions of the Witten diagrams with the massless Chern-Simons gauge field V + l=0,µ , V − l=0,µ being exchanged. They are given by The ansatz for A t−exch , A u−exch are obtained from A s−exch using crossing symmetry. The contact part of the ansatz takes the from where the sum is restricted by 0 ≤ a, b, a + b ≤ 1. Note that no individual α,ᾱ appears in the ansatz because the four-point function is parity even under separate exchange of z ↔z and α ↔ᾱ. The contribution of the contact diagrams in the other two channels A t-con , A u-con are obtained from A s-con using crossing symmetry. Plugging this ansatz into the superconformal Ward identities (2.13), we find that and all the contact term coefficients are solved in terms of λ gr . Therefore the four-point function is fixed up to an overall coefficient. Rewriting the solution in the form of (2.14) we find that reproducing the result of [34].

Higher-weight four-point functions
Let us move on to the next simplest correlator with k = 2. Our ansatz for the singular part of the four-point function is ) . (3.12) Here W mgr,4 is the exchange diagram of a massive graviton of dimension 4 and W sc,2 is a scalar exchange diagram of dimension 2. Both diagrams can be computed using the method of [31]. The contact part ansatz A s−con contains zero and two-derivative contact Witten diagrams, and is a polynomial in σ and τ of degree 2. Solving the superconformal Ward identities uniquely fixes the coefficients in A s−con in terms of the coefficients appearing in A s−exch . Moreover, the coefficients of the exchange contributions of fields belonging to the same multiplet are fixed up to an overall normalization. The remaining unfixed relative coefficients corresponding to different multiplets can be fixed using the cubic vertices worked out in [44]. The final solution can be rewritten in the form of (2.14) as 14) The case of higher-weight correlators with k > 2 is completely analogous to the above example with k = 2. We have also applied this method to obtain four-point correlators for two more examples with k = 3 and k = 4. We will refrain from writing down the explicit results, since in the Section 5.2 we will present a much more compact way of writing these correlators.

Mellin space
The position space method described in Section 3.2 offers a concrete way to compute fourpoint functions with as little input from supergravity. However, the results in position space do not look particularly illuminating. In this section, we look at the problem from a different perspective using the Mellin representation formalism [27][28][29][30], which offers new intuition to the problem. The Mellin representation formalism was demonstrated to be the most natural language for describing holographic correlators, making manifest their scattering amplitude nature. This formalism unfortunately becomes ill-defined in one dimension due to the nonlinear dependence of the cross ratios 11 . Because the superconformal symmetry forces the chiral cross ratios z,z to appear in the 2d one-half BPS correlator G I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 , one may wonder if the Mellin representation will be particularly useful. Nevertheless, by restricting our attention to certain components of the four-point function we can argue that the Mellin representation is still a good language. In particular, the Mellin representation formalism allows us to easily bootstrap the k i = k = 1 correlator as we demonstrate below.
For k = 1, has no dependence on α andᾱ. The symmetry under z ↔z, α ↔ᾱ implies that H I 1 is a symmetric function of z,z and can be unambiguously expressed in terms of U , V . We therefore have the following inverse Mellin representation is a rational function (this was justified by the previous position space calculation) and therefore does not contribute to the Mellin amplitude 12 . It then follows that the following components of G I 1 I 2 I 3 I 4 k=1 with R-symmetry factors also have a well-defined Mellin representation. By rewriting the superconformal factor we read off the three R-symmetry components of G I 1 I 2 I 3 I 4 k=1 P 1 : , (4.4) , (4.5) . (4.6) They can be expressed in the same form as (4.1) with u = 4 − s − t, by absorbing the multiplicative monomials U m V n via shifting s and t.
The monomials then become difference operators which act as are crossing symmetric. It is convenient to first make the flavor dependence more explicit k=1,I (s, t) . (4.9) Crossing symmetry then implies that (4.10) 12 The rational terms are generated from regularization effects when the integration contours are pinched.
These conditions turn out to be constraining enough, and uniquely fix the Mellin amplitude up to an overall factor Translating the result back into H I 1 I 2 I 3 I 4 , we find which reproduces our previous result (3.11) in position space. In fact the above arguments apply more generally to a class of four-point correlators with have the same extremality 13 , e.g., k 1 = k 2 = n, k 3 = k 4 = 1. For these nearextremal correlators, the auxiliary Mellin amplitudes M I 1 I 2 I 3 I 4 are uniquely determined by the bootstrap conditions up to an overall coefficient, and take the same form as (4.12) with shifted simple poles.
Let us also make two comments about applying the Mellin space method to correlators with higher extremities. First of all, it is necessary to make the assumption that H I 1 I 2 I 3 I 4 is even under α ↔ᾱ, or in other words, can be uniquely expressed in terms of σ and τ . This is needed such that H I 1 I 2 I 3 I 4 can be unambiguously written in terms of U and V , and therefore admits a well-defined Mellin representation. The R-symmetry structure of H I 1 I 2 I 3 I 4 a priori can be more general. However the even parity of H I 1 I 2 I 3 I 4 is observed in all examples computed using the position space method, and we believe is true in general. Second, the bootstrap conditions are not as constraining as the AdS 5 × S 5 case. We expect that M I 1 I 2 I 3 I 4 also takes the form of a sum of simple poles. However, the pole structure of the ansatz makes the condition on analytic structures weaker. In particular, the requirement that M I 1 I 2 I 3 I 4 should have polynomial residues is now trivially satisfied due to the lack 13 Extremality E is defined as E = k2 + k3 + k4 − k1 where k1 is the largest of all ki. of simultaneous poles M I 1 I 2 I 3 I 4 . Therefore, unlike the AdS 5 × S 5 case where the Mellin amplitude is fixed up to an overall factor, we cannot solve all the parameters in the ansatz for the Mellin amplitude. This parallels what we have observed in the position space method, and is an inevitable consequence of the fact that we have fewer supersymmetry for AdS 3 × S 3 × K3.

Hidden conformal symmetry
The general one-half BPS four-point functions of 4d N = 4 super Yang-Mills theory in the supergravity limit were obtained in [1,2] by solving an algebraic bootstrap problem. The formula took a surprisingly simple form and therefore strongly suggested the existence of some elegant underlying structure. Recently, this was made precise by [10] in terms of a conjectured 10d conformal symmetry. In terms of this symmetry, all one-half BPS four-point functions can be organized into one generating function, which is obtained by promoting the 4d distances in the lowest-weight correlator into 10d distances. Though a rigorous understanding is still lacking regarding its origin, some intuitive arguments were given in [10] to motivate the existence of such a symmetry. We will enumerate below some of these arguments and we will see that many features are also shared by AdS 3 × S 3 . First of all, the AdS 5 × S 5 background is conformally equivalent to the flat space R 9,1 . The SO(10, 2) symmetry can be interpreted as the conformal group in R 9,1 . The same statement can be made for the AdS 3 × S 3 background and the conformal group SO(6, 2). 14 Secondly, it was argued that the AdS 5 × S 5 auxiliary Mellin amplitude M of [1,2] , should be identified, in the large Mellin variable limit, with the 1 stu factor in the superamplitude of IIB supergravity in 10d flat space When divided by the dimensionless "coupling" G N δ 16 (Q), the amplitude 1 stu is conformal invariant in ten dimensions, i.e., annihilated by the special conformal transformation generator 15 For AdS 3 × S 3 we found a highly nontrivial analogy. By taking the asymptotic limit of the auxiliary Mellin amplitude M I 1 I 2 I 3 I 4

k=1
, we find that we precisely reproduce the four tensor 14 More precisely, it requires that the AdS space should have the same radius as the sphere. This is true for AdS5 × S 5 and AdS3 × S 3 , but is not true for, e.g., AdS7 × S 4 which is dual to 6d (2, 0) SCFTs. 15 We use momentum conservation to solve p4 in terms of p1, p2, p3, and write 1 stu = scattering amplitude in the theory of 6d (2,0) supergravity coupled to 21 Abelian tensor multiplets [47] A (2,0) ∼ G After dividing by the dimensionless quantity G (6) N δ 8 (Q), the amplitude is also conformally invariant in 6d. Finally, [10] also observed that the form of double-trace anomalous dimensions coincides with the partial wave decomposition coefficients of the 10d scattering amplitudes. Moreover the use of the 10d conformal block diagonalizes the mixing problem of double-trace operators. We have not investigated the counterparts of these problems in six dimensions, but it is likely that the questions will have similar answers. We hope to return to these questions in the future.

Conjecture of four-point functions with general weights
Motivated by the above similarities, we propose that a hidden SO(6, 2) symmetry exists for AdS 3 × S 3 , in the same sense of the AdS 5 × S 5 case. This symmetry will translate into a prescription for writing down a generating function for all one-half BPS four-point functions.
Let us define from H I 1 I 2 I 3 I 4 a crossing-symmetric function can be promoted into a generating function by doing a simple replacement in the arguments All the functions H I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 with higher values of k i are obtained by first expanding H(x i , t i ) in powers of t ij and then collecting all the possible monomials i<j (t ij ) γ ij that appear in H I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 . For example, The last expression is nothing but (3.14). We have also checked that expanding the generating function gives H I 1 I 2 I 3 I 4 kkkk which agree completely with our position space calculations for k = 3, 4.
We also conjecture that G I 1 I 2 I 3 I 4 0,k 1 k 2 k 3 k 4 takes the form of generalized free fields correlator, plus the contributions due to the mixing of the single-trace operator with double-trace operators in the external operators [48,49]. This is based on the explicit examples with k = 1, 2, 3, 4 which we have computed using the position space method. In principle, given H I 1 I 2 I 3 I 4 k 1 k 2 k 3 k 4 the function G I 1 I 2 I 3 I 4 0,k 1 k 2 k 3 k 4 can be fixed by requiring consistency with the conformal block decomposition. For example, if we consider four-point functions with {k 1 , k 2 , k 3 , k 4 } = {n, n, 1, 1}, consistency with the single-trace conformal blocks in the decomposition fixes G I 1 I 2 I 3 I 4 0,nn11 to be (5.10) For n = 1, this reduces to (3.10). When n > 1, we note that GFF only gives the schannel term proportional to δ I 1 I 2 δ I 3 I 4 . The t and u-channel terms are present because the supergravity fields are dual to a mixture of the single-trace operator and double-trace operators. The mixing is needed to ensure that the extremal three-point functions vanish from the supergravity calculation. An important observation is that the above structure based on hidden conformal symmetry only exists for the one-half BPS operators which come from the 6d tensor multiplets. To see this, we can use the same position space method to compute examples of four-point functions where the one-half BPS operators are from the 6d supergravity multiplet. The computation for the one-half BPS operators from the Σ l multiplets is almost identical to that of the Θ I l multiplets. From the selection rules, it is clear that both cases with the same l ≥ 0 have the same exchange Witten diagrams. Moreover, solving the superconformal Ward identities uniquely determines the contact diagrams in terms of the cubic couplings. However the cubic couplings in these two cases are different [44]. The results for Σ l therefore differ from the tensor four-point functions of Θ I l with I i set to be equal, and we do not observe a similar structure.

Future directions
We conclude with an outline of a few research avenues for the future: • The hidden conformal symmetry enjoyed by four-point tree-level correlators of tensor modes in AdS 3 × S 3 is in many respects similar to the one that holds in AdS 5 × S 5 . (There is a unique supermultiplet in 10d IIB supergravity, and the symmetry holds there for four-point tree-level correlators of all KK modes). Both backgrounds are conformally flat, and the relevant flat space superamplitudes enjoy an accidental conformal symmetry, respectively in six and ten dimensions. A third case that should work along very similar lines is AdS 2 × S 2 , where the superamplitude of four external hypermultiplets takes the simple form G N δ 4 (Q) · c. The kinematic prefactor G (4) N δ 4 (Q) is again dimensionless, while c is a just constant, and thus obviously conformally invariant in four dimensions. Assuming the hidden symmetry, it is immediate to write down the generating function of four-point tree-level correlators of all KK modes. It would be interesting to perform some explicit checks of this ansatz.
• Another closely related background is AdS 3 ×S 3 ×T 4 . Upon reducing IIB supergravity on T 4 , one obtains (2, 2) supergravity in 6d. Our methods can be straightforwardly applied to that case.
• It will be important to achieve a first-principles derivation of the hidden conformal symmetry, perhaps along the line of [50], which related Einstein gravity to conformal gravity. Such a conceptual understanding would also elucidate the regime of validity of the symmetry. For example, does it extend to higher-point tree-level correlators in AdS 5 × S 5 ? 16 Is it broken by 1/N corrections and how?
• In this paper, we focussed on the four-point functions of the modes Θ I l , the KK tower that arises from the 6d tensor multiplets. Two additional KK towers, Γ l and Σ l , arise from the 6d (2, 0) supergraviton. We would like to study the most general four-point functions which involve operators from all these multiplets. We have found that fourpoint correlators of Σ l fields are incompatible, at least naively, with 6d conformal symmetry, but perhaps the symmetry is present in a more subtle way.
• The full set of tree-level four-point functions is also needed to solve the mixing problem of double-trace operators and extract the spectrum of anomalous dimensions. These tree-level data can then be used to bootstrap one-loop four-point functions, following the blueprint of [6][7][8][11][12][13]. An interesting open question is if the hidden symmetry for the Θ I l multiplet survives the supergravity loop corrections.
• Related to the previous point, it would also be useful to perform an analysis using the Lorentzian inversion formula, along the lines of [10], a method independent of our supergravity computation.
A Twisting small N = 4 In this section we give a derivation of the superconformal Ward identities from topological twisting. Let us focus on the global part of the small N = 4 superconformal algebra and consider only the holomorphic part. The algebra is psu (1, 1|2), and is captured by the following commutation relations Moreover, one can construct the following twisted sl(2, C) algebra which is Q-exact Let us revisit the one-half BPS operators with SU (2) L indices contracted with spinors When y = z, it amounts to inserting operators in nontrivial Q-cohomology classes at the origin and then twist-translating using Because L −1 is Q-exact, the twist-translated operators remain in the Q-cohomology. Since the twist construction uses only the left-moving part of the 2d algebra, it commutes with the right-moving algebra. By standard arguments, the correlators of such twisted operators have no holomorphic dependence. This directly translates into our superconformal Ward identity (2.13).

B Proca-Chern-Simons versus massive Chern-Simons
In [44], it was shown that the vector fields from the spin-2 multiplet Γ k and spin-1 multiplet Σ k satisfy second-order Proca-Chern-Simons equations. Meanwhile their first-order derivatives satisfy a linear constraint and there are only three independent degrees of freedom. In this appendix we show that we can use field redefinition to solve the constraints, which gives three vector fields described by the first-order massive Chern-Simons equations. Moreover, we point out that the magnetic coupling to currents in [44] disappears after the field redefinition. We will work with the cubic vertices where the vector fields couple to two scalar fields σ. The case with two scalar fields s I is analogous. We start from the equation of motion of the gauge fields with quadratic corrections [44] and P ± m is the differential operator The coefficients W σσA ±

123
, V σσC ± 123 are defined in [44] but their precisely forms are not important to us. One can act with P ∓ k+1 on the first equation and solve for one variable in the second equation to get Proca-Chern-Simons equations for A ± µ and C ± µ . We also notice that the couplings to the current J µ are both electric and magnetic (i.e., V µ J µ and µνρ V µ ν J ρ ).
We now define the following new field In terms of L ± µ we can rewrite (B.2) as where the Q ± µ (σ) on the RHS is the quadratic corrections from σ and does not depend on A ± µ , C ± µ , L ± µ . The explicit expression of Q ± µ (σ) can be derived from above, but we will not write it down here. Therefore, in terms of A ± µ , C ± µ , L ± µ we have a system of first-order massive Chern-Simons equations (B.5), (B.1), (B.6) which can be more compactly written as Here 1 is the unit matrix and M ± is a mass matrix. The vector Q ± µ (σ) contains all the other terms depending on σ. The mass matrix can be diagonalized by using the following linear combinations In terms of V ± i,µ , we have where Λ = diag{k − 1, −k − 1, k + 3} is the diagonalized mass matrix. Moreover, as a result of (B.5) one can check that there is only only electric coupling in Q ± µ (σ), i.e., no µνρ ν J ρ appears. These eigenvectors are the fields appeared in our tables 1, 2. We can identify V ± 1,µ ,

C AdS 3 Witten diagrams
In this appendix we discuss the computation of exchange Witten diagrams which are unique to AdS 3 . Exchange diagrams of other fields, such as scalars, Proca fields, massive symmetric traceless tensors can be computed using the standard method. See, e.g., Appendix A of [2] for a summary of formulae.

C.1 Contact Witten diagrams with three derivatives
Before we start discussing exchange Witten diagrams, it is useful to first look at a special type of contact Witten diagrams which has an odd number of derivatives. The special contact Witten diagram is built from contact vertices of the type and has three derivatives. We will focus on the case where only four scalar fields are involved, though it is straightforward to generalize the result to include more scalar fields. We are looking at the following contact Witten diagram (we have distributed the three derivatives on the external legs 1, 3 and 4) defined by the integral where G ∆ i B∂ (z, x i ) is the bulk-to-boundary propagator The propagator G Proca,k of a Proca field with squared-mass m 2 k = (k − 1) 2 − 2 (and dual dimension ∆ k = k) satisfies (Proca k G Proca,k ) µ;ν = g µν δ(z 1 , z 2 ) + (. . .)δ k,1 , Now consider only the z integral in (C.10) We act on the integral with the differential operator AB , (C.14) where L It follows that Note that the operator on the LHS is nothing but the two-particle quadratic Casimir. The operator on the RHS is the AdS Laplacian with a constant shift [53] Now we can use the equation of motion of the bulk-to-bulk propagator and perform the remaining w-integral. The operator g νρ ρ µ can be ignored because we can integrate by part. Its contribution vanishes since I Proca,k,ν (x 1 , x 2 ; w) is coupled to a conserved current. All in all, we get Casimir (12) where W con Proca is a two-derivative contact diagram Instead of evaluating the diagram using the method of [31], we can alternatively solve the differential equation (C.18). We first need a special solution. This is not difficult for the cases when the method of [31] applies, and the answer is a finite sum of D-functions.
The equation (C.18) has two homogenous solutions, which are the conformal block of the exchanged single-trace operator and its shadow. They can be fixed by imposing boundary conditions. When we decompose W con Proca , it should contain only single-trace and doubletrace blocks, and no shadow conformal block. Moreover in the Euclidean regime, i.e., z = z * , W con Proca is single-valued (as is clear from its integral definition). These conditions uniquely fix the solution.
(C.23) Note that for k = 1, i.e., the massless case, there are only three-derivative contact terms. When k > 1, we can write W con mCS,k = W con mCS,k + 1 k − 1 W Proca,k , (C.24) so that the differential equation for W con mCS,k reduces to the massless form. The special solutions are again easy to guess, and take the general form of (z −z) times a sum of Dfunctions. We will list a few explicit solutions in a moment. The equation (C.22) also admit homogenous solutions which are the conformal block for the single-trace operator and its shadow operator. To fix the solution we require that in the conformal block decomposition, the single-trace conformal block has dimension (h,h) = k+1 2 , k−1 2 for +, and (h,h) = k−1 2 , k+1 2 for −. There is no shadow conformal block. The solution is also single-valued in the Euclidean regime.
Using this method, we can easily compute the massless and massive Chern-Simons exchange diagrams. Let us list the values of the diagrams which appear in this paper. 17

C.3 Exchange Witten diagrams of non-dynamical graviton field
The exchange Witten diagrams of graviton field in AdS 3 also cannot be evaluated using the method of [31]. However, it is straightforward to adapt the method from the previous subsection to the case of non-dynamical gravitons. We will not repeat the analysis but simply write down the solutions for reader's reference. (C. 38)