Holographic Four-Point Functions in the (2, 0) Theory

We revisit the calculation of holographic correlators for eleven-dimensional supergravity on $AdS_7\times S^4$. Our methods rely entirely on symmetry and eschew detailed knowledge of the supergravity effective action. By an extension of the position space approach developed in [1, 2] for the $AdS_5\times S^5$ background, we compute four-point correlators of one-half BPS operators for identical weights $k=2, 3, 4$. The $k=2$ case corresponds to the four-point function of the stress-tensor multiplet, which was already known, while the other two cases are new. We also translate the problem in Mellin space, where the solution of the superconformal Ward identity takes a surprisingly simple form. We formulate an algebraic problem, whose (conjecturally unique) solution corresponds to the general one-half BPS four-point function.


Introduction
Correlation functions of local operators in holographic CFTs were among the first observables to be computed using the AdS/CFT dictionary, but only recently [1,2] efficient calculational methods have begun to be developed. The traditional recipe is based on a perturbative expansion in Witten diagrams, which becomes very cumbersome (already at tree level) for n-point correlators with n 4. Prior to our work, only a few four-point correlators of Kaluza-Klein (KK) modes were known, to wit (focussing for definiteness on the maximally supersymmetric backgrounds): a handful of cases in the AdS 5 × S 5 background [3][4][5][6][7][8][9]; just the four-point function of the lowest KK mode (the stress-tensor multiplet) in the AdS 7 × S 4 background [10]; and no results whatsoever in the AdS 4 × S 7 background.
The traditional method has two sources of computational complexity: the need for explicit expressions of the vertices in the supergravity effective action; and the proliferation of exchange Witten diagrams as the KK level is increased. In [1,2] we introduced new calculational tools to circumvent these difficulties, for the case of AdS 5 × S 5 supergravity. A first approach, which we refer to as the "position space method", leverages the special feature of the AdS 5 × S 5 background that all exchange Witten diagrams can be written as finite sums of contact diagrams. One can then write an ansatz for the four-point correlator as a sum of contact diagrams, and determine their relative coefficients by imposing superconformal symmetry, with no need for a detailed knowledge of the effective supergravity action. While simpler than the standard perturbative recipe, the position space method also quickly runs out of steam as the KK level is increased. What's worse, the answer takes a completely unintuitive form, with no simpe general pattern. The second, more powerful approach of [1,2] uses the Mellin representation of conformal correlators [11,12]. Tree-level holographic correlators in AdS 5 × S 5 are rational functions of Mandelstam-like invariants, with poles and residues controlled by OPE factorization, in close analogy with tree-level flat space scattering amplitudes. Superconformal symmetry is made manifest by solving the superconformal Ward identity in terms of an "auxiliary" Mellin amplitude. The consistency conditions that this amplitude must satisfy define a very constrained algebraic problem, which very plausibly admits a unique solution. While the position space method is implemented on a case-by-case basis for different correlators, the Mellin algebraic problem takes a universal form. We were able to solve the problem in one fell swoop for all half-BPS four-point function with arbitrary weights -a feat extremely difficult to replicate in position space.
The goal of this note is to extend these techniques to eleven-dimensional supergravity on AdS 7 × S 4 . This is a background of extraordinary physical interest, as it provides a dual description of the mysterious six-dimensional (2, 0) theory at large n, and a prime target for our methods, since we expect maximal supersymmetry to constrain the tree-level holographic correlators uniquely. At a more technical level, this background enjoys the same "truncation conditions" as AdS 5 × S 5 , such that exchange Witten diagrams can be written as finite sums of contact diagrams -equivalently, tree-level Mellin amplitudes are rational functions of the Mandelstam invariants. By contrast, the truncation conditions do not hold for the AdS 4 × S 7 background, and new tools are needed [13].
In this paper we focus on four-point functions of identical one-half BPS local operators, (1.1) The operator O (k) is the superconformal primary of the short multiplet 1 D 2k; 0, 0, 0; 0, k . It has conformal dimension ∆ = 2k and transforms in the symmetric traceless representation of the R-symmetry group SO(5) ∼ = U Sp (4). The integer k 2 corresponds to the Kaluza-Klein level on the supergravity side. In particular O (2) is the superprimary of the stress tensor multiplet, whose four-point function has been computed by Arutyunov and Sokatchev [10], while operators with k 3 are dual to massive Kaluza-Klein modes. No four-point functions for massive KK modes have ever been computed, partly because of the extraordinary challenge of evaluating the quartic vertices of the AdS 7 effective action from the non-linear KK reduction.
The position space method of [1,2] admits a straightforward extension to the AdS 7 × S 4 background. Using this method we confirm the result of [10] for the stress-tensor multiplet four-point function and obtain two new explicit cases with KK levels k = 3 and k = 4. An important test of our results is that they are compatible with the expected chiral algebra structure identified in [16,17]. By performing a certain "twist" that identifies spacetime and R-symmetry cross ratios, correlators of a 6d (2, 0) superconformal theory map to correlators of a 2d chiral algebra. In [16], the chiral algebra associated to the (2, 0) theory of type A n was identified with the W n algebra with central charge c 2d = 4n 3 − 3n − 1. In particular, threepoint functions of one-half BPS operators, previously computed at large n from supergravity, were shown to agree with the structure constants of W n→∞ . Here we check that our newly obtained holographic four-point functions reduce to four-point functions of the W n→∞ algebra upon performing the chiral algebra twist. 2 We also reformulate the problem in Mellin space. The same strategy of [1,2] applies, but the AdS 7 case is significantly more involved. The solution of the superconformal Ward identity takes a much more intricate form in position space, and its translation to Mellin space requires some non-trivial manipulations. Nevertheless, when the dust settles, we find a surprisingly compact way to express the constraints of superconformal invariance in Mellin space, structurally similar to the AdS 5 case. The Mellin amplitude can be written in terms of an auxiliary amplitude, acted upon by a certain difference operator. We formulate a purely algebraic problem based on symmetries and consistency conditions that we believe should determine uniquely the Mellin amplitude, but unlike the AdS 5 case, we have not yet been able to conjecture a general solution. We have translated our position space results into Mellin space and found that the auxiliary Mellin amplitude takes a succinct form for k = 2 and k = 3, and already a rather complicated form for k = 4 (even if significantly more compact than the position space expression). We were not able to guess a pattern for the amplitude for general k. This remains the main outstanding challenge. If that can be overcome, we can look forward to an extension to the AdS 7 × S 4 background of the ideas and techniques that have been recently applied to the AdS 5 × S 5 case, see, e.g., [18][19][20][21][22][23].
The remainder of the paper is organized as follows. In Section 2 we set up notations and review the superconformal kinematics of one-half BPS four-point functions. We formulate the position space method in Section 3.1 and present the results of the first three low-lying correlators in the subsequent subsections. In Section 4.1 we translate the constraints of superconformal symmetry into Mellin space. Using this formalism we formulate in Section 4.2 an algebraic problem that is expected to completely determine the one-half BPS four-point functions. In Section 4.3 we present the Mellin space translation of our three position space 2 To perform this test, we need to independently compute the requisite meromorphic four-point functions of Wn→∞ by purely two-dimensional method. This is easily done by fixing their poles and residues from OPE factorization and requiring crossing symmetry. This calculation uses as an input the structure constants of Wn→∞, so compatibility of our holographic four-point function with the Wn→∞ structure is not really an independent check of the identification [16] of Wn as the correct chiral algebra. It is however a very non-trivial check of our computations.
solutions. Technical details (including a computation of W n→∞ four-point functions) are relegated to the three appendices.

Superconformal kinematics of four-point functions
We begin by reviewing the kinematic constraints of superconformal invariance on one-half BPS four-point functions. The relevant superalgebra osp(8 * |4), whose bosonic subalgebra is the direct sum of the conformal algebra so(6, 2) and of the R-symmetry algebra so(5).
One-half BPS operators O (k) ..I k transform in the symmetric traceless representation of the so(5) R-symmetry. They can be conveniently made index-free by contracting the so(5) indices with five-dimensional null vectors t I , The four-point function is thus a function of the spacetime coordinates x i and of the "internal" coordinates t i . Covariance under so(5) and so (6,2) can be exploited to pull out a kinematic factor, such that the reduced correlator G k depends only on the conformal cross ratios U and V and on the R-symmetry cross ratios σ and τ . The cross ratios have the standard definitions The correlator G k (x i , t i ) is a polynomial in the R-symmetry invariants t ij . It is easy to see that the reduced correlator G k (U, V ; σ, τ ) is a degree-k polynomial of σ and τ . So far we have only imposed the constraints that arise from the bosonic subalgebra of the full superalgebra. The fermionic generators imply further constraints, which link the dependence on the R-symmetry and conformal cross ratios. After making the convenient change of variables the superconformal Ward identity reads [24] ( We can first obtain a partial solution to the superconformal Ward identity by restricting the four-point function to a special slice of R-symmetry cross ratios such that α = α = 1/χ . Then the superconformal Ward identity (2.7) reduces to whose solution is simply any "holomorphic" function 3 of χ, Up to kinematic factors, the function f (χ) coincides with the four-point correlator of the two-dimensional chiral algebra associated to the (2, 0) theory by the cohomological procedure introduced in [16,17]. There is a compelling conjecture [16] that the chiral algebra associated to the (2, 0) theory of type A n is the familiar W n algebra, with central charge c 2d = 4n 3 −3n−1.
In our holographic setting, we are instructed to take a suitable large n limit of the W n algebra, as explained in detail in [16]. In that limit, the structure constants of the W n algebra were matched with the three-point functions of the one-half BPS operators computed holographically by standard supergravity methods. In this work, we will use our new method to compute holographic four-point functions of one-half BPS operators. As an important consistency check, we will match their "holomorphic" piece f (χ) with the corresponding fourpoint functions in the W n→∞ algebra. The full solution of the superconformal Ward identity (2.7) was found in [24]. We reproduce it here with a few crucial typos fixed. A general solution G k of (2.7) can be written as where F k and K k are respectively an "inhomogeneous" solution" and a "homogenous" solution. By this we mean that upon performing the "twist" α = α = 1/χ , F k becomes a purely "holomorphic" function of χ, while K k must vanish identically. The homogenous part K k can further be expressed in terms of a differential operator Υ acting on an unconstrained function H(U, V ; σ, τ ), which is a polynomial in σ and τ of degree k − 2. Explicitly, 4 3 In Euclidean signature, the variables χ and χ are complex conjugate of each other, so this terminology is appropriate. In Lorentzian signature χ and χ are instead real independent variables. Hence "holomorphic" in quotes. 4 Here and below, the parameter takes the fixed value 2. We keep it as to facilitate comparing with the expressions in [24], but we stress that the solution to the superconformal Ward identity takes this particular form only for d = 6.
where the differential operator D is defined as While the expression of the differential operator Υ is not very transparent, its transformation properties under crossing however are surprisingly simple. Let g 1 , g 2 be the two generators of the crossing-symmetry group S 3 under which the cross ratios transform as (2.13) We have found that Υ satisfies 5 where γ and ρ are arbitrary parameters. We can always find a decomposition of G k such that the two functions F k and H k do not mix into each other under crossing. Then the full correlator G k and and the inhomogenous part F k have the same crossing properties (2.15) Using the crossing identities obeyed by the operator Υ, it is then easy to find the crossing relations obeyed by the unconstrained function H k , In closing, we should emphasize that the decomposition (2.10) is not unique, since obviously one can add any "homogeneous" term to F k and subtract the same term from K k . In the case of N = 4 super Yang-Mills, where the solution of the superconformal Ward identity takes a similar form, there is a natural choice for F k , namely the value of the correlator in the free field limit: F k is then a simple rational function of U and V . A priori there is no reason that an analogous natural choice for F k should exist in the (2, 0) theory, but we will find experimentally that there is one, even in the absence (obvious) connection with free field theory.

The position space method
In this subsection we describe a "position-space method" to compute one-half BPS correlators for the AdS 7 × S 4 background, analogous to the one used in [1,2] for the AdS 5 × S 5 case. This method mimics the standard recipe of evaluating holographic correlators as sums of Witten diagrams (see [2] for a quick review of the traditional method), but has the great advantage that no detailed knowledge of the supergravity effective action is required.
By the standard AdS/CFT dictionary, the boundary correlator G k is computed as a sum of bulk Witten diagrams. At tree level, the requisite Witten diagrams (Figure 1) are either contact diagrams or exchange diagrams. They are evaluated by assigning a bulk-to-boundary propagator G ∆ i B∂ to each external leg and a bulk-to-bulk propagator G ∆ BB to the internal line. These propagators are Green's functions in AdS with appropriate boundary conditions. The simplest diagram is a contact Witten diagram without derivatives in the quartic vertex ( Figure  1a). The integral expression of such a diagram defines the so-called "D-function", is the scalar bulk-to-boundary propagator [25]. Exchange Witten diagrams ( Figure 1b) are more complicated objects. However, when the twist (τ := ∆− ) of the exchanged field satisfies a special condition with respect to the external dimensions 6 , a trick discovered in [26] applies, and the exchange Witten diagrams can then be traded for a finite sum of contact Witten diagrams. More details on this technique and an explanation of its mechanism in Mellin space are reviewed in [2]. While the traditional method requires the input of the precise effective Lagrangian, our position space method relies only on some qualitative features of the supergravity theory, to wit, the spectrum of supergravity fields, the selection rules encoded in the cubic vertices and a bound on the number of derivatives that can appear in the quartic vertices. The spectrum of eleven dimensional supergravity compactified on AdS 7 × S 4 has long been worked out in [27] by diagonalizing the quadratic terms in the AdS 7 effective action. The spectrum is organized in infinite Kaluza-Klein towers, with increasing U Sp(4) quantum numbers within each tower.  Clearly the allowed exchanges between two pairs of half-BPS scalar KK modes are only a finite subset of the whole spectrum. There are two selection rules. The first selection rule is the R-symmetry selection rule which follows from elementary U Sp(4) representation theory: the tensor product of two irreducible representations [0, k] of U Sp(4) yields a finite direct sum of irreps, The fields with the admissible representations have been collected in Table 1. However not all fields a priori allowed by the R-symmetry selection rule will appear in the exchange channel. Additionally, there is another selection rule, which restricts the twist τ of a field that can couple to two scalars of dimensions ∆ 1 and ∆ 2 , In the cases of interest here, ∆ 1 = ∆ 2 = 2k. The origin of this twist selection rule was discussed at length in [2] and will not be repeated here. The spectrum and selection rules in the AdS 7 × S 4 supergravity background are such that truncation conditions are satisfied and all exchange diagrams are given by finite sums of D-functions.
We also make an assumption on the quartic vertices. We assume that quartic vertices with at most two spacetime derivatives effectively contribute to the four-point functions 7 . This is necessary in order to recover the correct flat-space limit as the radius of AdS 7 × S 4 is sent to infinity. A further technical simplification (c.f. Appendix D of [2]) is that when ∆ i = d = 6 the contribution from the zero-derivative vertices can be absorbed, by a redefinition 7 The same assumption was made in [1,2] for AdS5 × S 5 based on the same reasoning, and was confirmed by explicit computation in [28]. of parameters, into the contribution of the two-derivative vertices. This will be useful in concrete computations to avoid redundancies in the parameterization of the quartic vertices.
The position space method proceeds as follows. We use the above information on spectrum and vertices to write a general ansatz for the supergravity amplitude which consists of an exchange part and a contact part, Here we are working with the reduced correlator which depends only on the cross ratios and is obtained by stripping off the kinematic factor t 12 t 34 In this ansatz the exchange amplitude is summed over the three channels, related to one another by crossing, The s-channel exchange amplitude A k,s−ex is given by the sum of all s-channel exchange Witten diagrams ompatible with the selection rules for the cubic vertices. Schematically, Here the label X runs over the exchanged fields, E X denotes the corresponding exchange Witten diagram, Y R X (σ, τ ) is the polynomial associated with the irreducible representation R X of the field X and finally λ X are the unknown coefficients to be determined. The formulae for evaluating the requisite exchange diagrams in terms contact diagrams have been given in Appendix C of [2], while a list of the relevant R-symmetry polynomials is given in Appendix A. For the reader's convenience, we have also include these explicit expressions in the Mathematica notebook available from the ArXiv version of this paper. The discussion of contribution from contact diagrams should distinguish two difference cases, as was explained in Appendix D of [2]. When ∆ i = d, we should include in the ansatz both the contribution from contact vertices with no derivatives and with two-derivatives. When ∆ i = d, the zero-derivative contribution can be absorbed into the two-derivative contribution by redefining the parameters in the ansatz. Because of crossing symmetry, it is also convenient to split the two-derivative contribution into channels. The full contact vertex is the crossing symmetrization of the following s-channel contribution, where α i collectively denote the indices of the symmetric traceless representations and S α 1 α 2 α 3 α 4 is an unspecified tensor symmetric under (1 ↔ 2, 3 ↔ 4). When contracted with the Rsymmetry null vectors the above vertices lead to the following contribution to the s-channel contact amplitude, (3.10) Here we have used the so-calledD-functions 8 defined by stripping off some kinematic factors from the D-functions, The coefficients c ab in (3.10) are symmetric thanks to the exchange symmetry (1 ↔ 2, 3 ↔ 4). When ∆ i = d, we need to also include the zero-derivative contribution The crossed channel contributions A k,t−cont and A k,u−cont can then be obtained from A k,s−cont using the crossing relation (3.7). Putting all these pieces together, we now have an anstaz A k (U, V ; σ, τ ) of the four-point function as a finite sum ofD-functions. It has polynomial dependence on σ and τ and contains linearly all the unspecified coefficients λ X , c ab , c ab . These coefficients must to be fine-tuned in order to satisfy the superconformal Ward identity (2.7), (3.14) 8 We emphasize thatD-functions are independent of the spacetime dimension d. This is clearest from their Mellin-Barnes representation, where d completely drops out.
The ansatz A k is not yet in a form such the superconformal Ward identity can be conveniently exploited. Fortunately, allD-functions that appear in the ansatz can be reached from the basic D-functionD 1111 with the repetitive use of six differential operators, The special functionD 1111 is in fact the familiar scalar one-loop box integral in four dimensions and will be denoted as Φ from now on. It has a well-known representation in terms of dilogarithms, and enjoys the following beautiful differential recursion relations [29] (3.17) Using the above properties ofD-functions, we can unambiguously decompose the supergravity ansatz into a basis spanned by Φ, log U , log V and 1, where the four coefficients functions R Φ , R log U , R log V and R 1 are rational functions of χ, χ and polynomials of α, α . This decomposition makes it straightforward to enforce the superconformal Ward identity (2.7) on A k . Upon acting on A k (χ, χ ; α, α ) with the differential operator (χ ∂ χ − 2α ∂ α ) from (2.7) and setting α = 1/χ , a new set of coefficient functions with the help of the differential recursion relation of Φ. The superconformal Ward identity then dictates the following conditions  (2) has the correct value. The details of this calculation are discussed in Appendix B.

The k = 2 four-point function
In the rest of this section we will put the above position method to work. We start with the k = 2 one-half BPS operator O (2) which sits in the same short supermultiplet as the stress tensor. Its four-point function G 2 was first calculated in [10] and we will reproduce their result. By the two selection rules of cubic vertices the allowed exchanges are identified to be all the fields that belong to the k = 2 family in Table 1. Explicitly, the exchange Witten diagrams in the s-channel contribute (3.20) As was discussed above, the contribution of contact Witten diagrams can be split into channels and then cross-symmetrized. Moreover, because ∆ i = d we can absorb the contribution of the zero-derivative terms into the two-derivative terms. Hence we have the following s-channel ansatz for the contact contributions, where c ab = c ba follows from symmetry under exchanging operators 1 and 2. The total amplitude A 2 is obtained from cross-symmetrizing the above s-channel amplitude,  Decomposing this ansatz into the basis of functions Φ, log U , log V and 1 and enforcing the superconformal Ward identity (3.19), we find enough constraints to fix all the coefficients up to an overall factor ξ, By setting the R-symmetry cross-ratios to the special value α = α = 1/χ we find the following "holomorphic" correlator, As we show in Appendix C, it reproduces the corresponding W ∞ four-point function.
The full four-point function can also be massaged into a form consistent with the solution to the superconformal Ward identity 9 . We found that the result can be written as where the differential operator Υ was defined in (2.11). This agrees with the result obtained by Arutynov and Sokatchev [10]. Remarkably, the inhomogeneous term F 2 is a rational function of the cross ratios, just as in N = 4 super-Yang Mills, where the analogous inhomogeneous term can be identified with the free-field limit of the correlator. Of course, one could always choose the inhomogeneous term F to be rational, at the expense of making the dynamical function H more complicated. 10 The non-trivial statement is that we can cast the correlator into a form where F is rational and H is a linear combination ofD-functions. This property will persist in the k = 3 and k = 4 examples that we study below. We conjecture that it holds in general.

The k = 3 four-point function
We now proceed to the next Kaluza-Klein level. The allowed exchanges for G 3 include the three component fields of the k = 2 family in Table 1 and all other fields of the k = 4 family except for the field t 4 . This field is ruled out because it has twist 12, which violates the twist upper bound. The k = 3 family, on the other hand, is absent because of the Rsymmetry selection rule. For the contact diagrams we notice that in this case the conformal dimension of the external operators coincides with the boundary spacetime dimension ∆ = d. As was discussed in the Appendix D of [2], the zero-derivative contribution can no longer be reabsorbed into the two-derivative contribution. We need to include in our ansatz both set of parameters for the quartic vertices, even if this will lead to some (harmless) ambiguities in fixing the coefficients of contact vertices when we use the superconformal Ward identity.
The s-channel ansatz is again given by an exchange part A 3, s−exchange and a contact part In principle this can be done solely usingD-functions identities as in [10,30], but in practice it more efficient to guess and then check. The guesswork starts with the easier step of obtaining H k from M k (see below in Section 4.1). We then check this guess by decomposing into the basis spanned by Φ, log U , log V and 1: the difference G − Υ • H should give a rational function. 10 The only requirement on F is that upon setting α = α = 1/χ it should reduce to the chiral algebra correlator. It is always possible to find a rational function with this property.
where the coefficients c ab = c ba , c ab = c ba are symmetric. The total amplitude ansatz is obtained by further including the t-channel and u-channel contributions which are obtained from the above s-channel contribution by crossing. Due to the ambiguity in the parameterizing of contact vertices, the superconformal Ward identity fixes only the coefficients of the exchange diagrams, leaving a subset of c ab , c ab unfixed,  These unfixed coefficients are actually redundant: the corresponding expressions are proportional to a sum ofD-functions which is zero in disguise, thanks toD-function identities. We can then set them to zero (or to any convenient value). In Appendix B we fix the last coefficient by enforcing the correct value of the OPE coefficient of O (2) , which gives ξ = 1080000 n 3 π 3 . (3.30) By setting α = α = 1/χ , we extract another "holomorphic" correlator which agrees with the expected result from W ∞ (see Appendix C). As was the case for k = 2, the k = 3 supergravity four-point function can also be put into a compact form that manifestly solves the superconformal Ward identity, Here F 3,conn is a simple rational function of the cross ratios, while H 3 is given in terms ofD-functions by the following expression,

Mellin space
The position space method discussed in the previous section is a rigorous algorithm to compute four-point functions without a priori knowing the vertices. Though much easier than the traditional approach, this method also runs out of steam as one moves up in the Kaluza-Klein towers due to unmanageable computational complexity and does not appear promising to provide a general solution. What's more, the position space method presents the final answer as an unintuitive sum of contact Witten diagrams. The most suitable language to address both problems is the Mellin representation initiated by Mack [11] and developed by Penedones and others [12,[31][32][33][34][35][36]. 11 We refer the reader to [2] for a detailed review of this formalism. We will be very brief here. Using the inverse Mellin transformation, we express the connected part of the four-point function as a double integral (4.1) 11 For other applications and recent developments, see [1,[37][38][39][40][41][42][43][44][45][46][47][48]).

For notational convenience we have introduced
A := x 2 12 x 2 34 , B := x 2 13 x 2 24 , C := x 2 14 x 2 23 , a := t 12 t 34 , b := t 13 t 24 , c := t 14 t 23 . (4. 2) The variables s, t and u are interpreted as "Mandelstam variables" 12 and satisfy the constraint s + t + u = 8k. The integration of the two independent variables s, t is performed along the imaginary axis. We further define the Mellin amplitude M(s, t; σ, τ ) a sum of the partial amplitudes from M k, LM N , N (s, t) .
The Mellin amplitude bears several formal similarities with the flat space S-matrix. Being analogous to amputated on-shell tree-level Feynman diagrams in flat space, tree-level Witten diagrams admit extremely simple representations in Mellin space. If we temporarily suppress the dependence on R-symmetry variables, then (4.1) reduces to (4.6) The contact Witten diagram with a nonderivative vertex is just a constant in Mellin space. More generally, the contact Witten diagram with a 2n-derivatives vertex yields a polynomial of degree n in the Mandelstam variables [12]. Exchange diagrams also have a simple structure in Mellin space, as a sum over simple poles with polynomial residues plus another regular polynomial piece [33], Here we have considered an s-channel exchange Witten diagram with an exchanged field of conformal dimension ∆ and spin J (and hence twist τ = ∆ − J). The functions Q J,m (t) and P J−1 (s, t) are polynomials of degrees J and J − 1, respectively. When the twist of the exchanged field satisfies the condition a remarkable simplification occurs: the infinite sum in (4.7) truncates to finitely many terms (see Section 3.2 of [2] for a detailed discussion). This is the Mellin-space counterpart of the fact that certain exchange Witten diagrams can be evaluated as finitely many contact Witten diagrams. The spectrum condition (4.8) is precisely met by the eleven dimensional supergravity on AdS 7 × S 4 . For this reason the analytic structure of the Mellin amplitude of the AdS 7 × S 4 four-point function is extremely simple: it has just finitely many simple poles and some extra polynomial terms.

Superconformal symmetry in Mellin space
In position space, superconformal symmetry of the four-point function is encoded in the solution of the superconformal Ward identity, 13 Our position space results have led us to conjecture that in the supergravity limit we can take H k to be a linear combination ofD-functions, and F k,conn to be rational function of the cross ratios. The details of this section are very technical, but the main result is easy to summarize. We are going to find that Mellin version of (4.9) takes the form where M k is an "auxiliary" Mellin amplitude defined from the dynamical function H k,conn , see (4.12), and Θ a somewhat complicated difference operator defined in (4.22). We start by writing the Mellin transformation of the left-hand side of (4.9) using (4.1), Consider now the right-hand side of (4.9). The Mellin transform of the rational function F k,conn is ill-defined. As explained in [2], it can be defined as "zero". 14 On the other hand, we can write a Mellin representation for the dynamical function H k in terms of an "auxiliary" Mellin amplitude M k (s, t; σ, τ ) 13 As the Mellin transformation of the disconnected part of the correlator is ill-defined, we focus on the connected part. 14 F k,conn can be recovered as a subtle regularization effect in properly defining the contour integrals of the inverse Mellin transformation [2]. whereũ = u − 6. As we demonstrate the shift has the virtue of giving simple transformation properties to M k (s, t; σ, τ ) under crossing, In the auxiliary amplitude M k (s, t; σ, τ ), the triplet variables (s, t,ũ) replaces (s, t, u) to become the set of variables that permute under crossing. This becomes especially evident after we restore all the factors of x 2 ij and t ij . Restoring this factor will also facilitate the extraction of the difference operator. Let us see this in detail.
We first note that the combination is crossing invariant, since it has the same crossing properties as G k . Upon inserting the inverse Mellin transformation (4.12) into (4.13) and decomposing the auxiliary amplitude with respect to the R-symmetry monomials, we find Here the factorΓ 3 (s, t) is short-hand for . We now let the differential operator Υ act on the monomial U s/2+1 V t/2−2k+1 σ m τ n , leading to the following integral where U = A/B, V = C/B, σ = b/a, τ = c/a have been substituted back into the expression. The factor Θ is a polynomial of A, B, C, a, b, c and s, t obtained from the action of the differential operator Υ on the monomial U s/2+1 V t/2−2k+1 σ m τ n . To write an explicit expression for Θ, it is useful further to introduce the following combinations of Mandelstam variables, (4.17) In terms of these variables, Θ reads +ABC a 2 (Y + 2)(Z + 2) + b 2 (X + 2)(Y + 2) + c 2 (X + 2)(Z + 2) + bc(2 + X)(4 + X) + ab(2 + Y )(4 + Y ) + ac(2 +Z)(4 +Z) . (4.18) The reader should not focus on this complicated expression because further manipulations will soon lead to a major simplification. At this stage we only want to point out that the above expression of Θ can be checked to be crossing invariant under any permutation of the triplets (a, A, s) , Crossing invariance of (4.16) implies the following crossing identities for M k, lmn (s, t),  from which the crossing identities (4.13) of M k (s, t; σ, τ ) immediately follow. As in [1,2], we should reinterpret the monomials of A, B, C in (4.18) as difference operators acting on functions of s, t in the integrand, thus promoting the factor Θ to an operator Θ. This operator Θ can be written in a compact form if the respective shift on X, Y andZ has first been performed, as we now show. All monomials that appear in Θ have the form A α B 3−α−β C β . Multiplying an inverse Mellin integral by such a monomial, we have (The shift of the integration contour is important in producing rational terms by the mechanism discussed in [2]. Here we are focusing on the Mellin amplitude and ignore contour issues.) Note that in the first term we use the shifted Mandelstam variableũ = u − 6 = 8k − s − t − 6, while the unshifted u appears in the second term. We conclude that multiplication by the monomial A α B 3−α−β C β corresponds in Mellin space to a difference operator that shifts s → s − 2α and t → t − 2β.
Interpreting every monomials in Θ in this fashion we find a difference operator Θ. We can make the expression of Θ very compact by performing the shift in two stages: first we shift on the factor of X, Y ,Z multiplying each monomial A α B 3−α−β C β and bring it the left; then A α B 3−α−β C β remains an operator to act on whatever is in the integrand on the right. We arrive at the following simple expression, where we defined an "unshifted" Z variable Z :=Z + 6 = u + 4m − 4k + 2 (4.23) and the crossing-invariant factor R is given by 15 The expressions AR, BR, CR are written shorthands that should be understood as follows: one first expands AR, BR, CR into monomials A α B 3−α−β C β and then regards each of them as the operator A α B 3−α−β C β . These operators will only act on objects multiplied from the right and will no longer shift the X, Y , Z factors multiplied from the left.
We can now give the explicit action of A α B 3−α−β C β as an operator that transforms a term of M k into a term of M k , .

(4.26)
This action is obtained by applying the aforementioned shift of s and t on the integrand and taking into consideration the difference of Gamma function factors between the definitions (4.1) and (4.16).
To summarize, the superconformal Ward identity implies that the full Mellin amplitude M k can be written in terms of an auxiliary amplitude M k acted upon by the difference operator Θ, The operator Θ is given by (4.22) where each monomial operator acts as in (4.26). 15 Curiously, this factor is closely related to the analogous factor that R that appears in the solution of the 4d N = 4 superconformal Ward identity, namely by R = a 2 B 2 . We don't have a deep understanding of this observation.

Consistency conditions and the algebraic problem
We now take stock and summarize the conditions on the Mellin amplitude that follow from our discussion in the previous sections: 1. Crossing symmetry: As the external operators are identical bosonic operators, the Mellin amplitude M k satisfies the crossing relations where the action of Θ has been defined in the previous subsection.
The name of the game is to find a function M k (s, t; σ, τ ) such that all the conditions are simultaneously satisfied. We leave a detailed analysis of this very constrainted "bootstrap" problem for the future. As in the four-dimensional case analyzed in [1,2], we find it very plausible that this problem has a unique solution (up to overall rescaling).

Solutions from the position space method
In this subsection we give solutions to the algebraic problem defined above for k = 2, 3, 4.
These solutions are obtained from the position space results, but look much simpler in Mellin space when the prescription (4.31) is implemented.
We start from the simplest example of k = 2. In this case, the homogenous part H 2 is a degree-0 polynomial of σ and τ . Therefore there is only one R-symmetry structure in the auxiliary amplitude M 2 . The answer is given by which is manifestly symmetric under the permutation of s, t andũ.
Moving on to the next simplest case of k = 3, we know that H 3 is a degree-one polynomial of σ and τ and therefore consists of three terms. The three R-symmetry monomials σ, τ , Finally, M 3,100 is given by  The two independent auxiliary amplitudes turn out to be The above expression is much more complicated than those for k = 2 and k = 3, but the auxiliary amplitude is still significantly simpler than the full amplitude. We have not yet been able to discern a pattern that would allow to guess the auxiliary amplitude for general k.

B Fixing the overall constant
In Section 3 we have shown in examples how the superconformal Ward identity determines all the coefficients of the position space ansatz, up to an overall scaling factor ξ. To fix the overall normalization, we are going to demand that the singular terms of the four-point function that correspond to exchanged one-half BPS operators in (say) the s-channel OPE have the correct coefficients. Those coefficients are read off from the one-half BPS three-point functions, which have been computed both in supergravity [50,51] and using the W ∞ algebra [16], where unit normalized two-point functions are assumed. Here C form a basis of symmetric traceless tensors of the R-symmetry group and we labeled such representations with α i . The factor C α 1 C α 2 C α 3 is the unique scalar contraction of C I 1 ...I k i α i . More details on the C-algebra can be found in, e.g., [52].
To keep track of various normalization factors, we start recalling that a scalar field φ with canonically normalized action has the following bulk-to-boundary propagator where ∆(∆ − d) = m 2 . In this normalization, the two-point function of the dual operatorÕ ∆ is [53]) Assuming a cubic bulk vertex of the form the three-point function is Finally the four-point function for the s-channel exchange of O ∆ Here E ∆ denotes the exchange Witten diagram for four-external scalar operators of dimension ∆ and one internal scalar of dimension ∆ , normalized as in (A.1) of [2]. We now change the operator normalization such that the new operator O ∆ is unitnormalized, We specialize these formulae to our supergravity calculation. We restore the α i indices and recall that the operator O (k) α has dimension ∆ = 2k. Then and where the collective index α 5 is summed over. Comparing (B.1) with (B.9) solves the cubic coupling µ 2k,2k,2k . Inserting µ 2k,2k,2k gives us the precise contribution of the exchange Witten diagram in the four-point function.
We now recall the unfixed parameter λ s k in Section 3 (for k = 2, λ s 2 ≡ ξ) appears in the t i -contracted four-point function as, Here the factor To make comparison with (B.10), we only need to contract (B.10) with the null vectors. Note the tensor C I 1 ...I k α is normalized to be with the completeness relation 16 ..I k ,(J 1 ...J k ) + (mix within I, J) . (B.13) 16 Here we did not write down the extra term with mixed i and j because we will only need to match the σ n + τ n term.
The null vector t 1 of operator 1 is contracted to C α 1 via another C α 1 (t where the above completeness relation has been used. Because t 1 is a null vector, the implicit mixed part in the completeness relation does not contribute and the right side of (B.14) is exact. We contract with the null vectors t 2,3,4 in the same way. After some tedious algebra, we find where n R is the ratio of t i -contracted C α 1 C α 2 C α 5 C α 5 C α 3 C α 4 and Y (t 1 , t 2 , t 3 , t 4 ). Note our convention for Y (t 1 , t 2 , t 3 , t 4 ) is always of the form Y (t 1 , t 2 , t 3 , t 4 ) = (t 1 · t 2 t 3 · t 4 ) n (σ n + τ n + . . .) (B.16) where ". . ." stands for the terms of the form σ a τ b with a < n and b < n. Using the above completeness relation (B.14), it is not difficult to solve the combinatoric problem of n R where C α 1,2,3,4 have k I-indices and C α 5 has k I-indices. All in all, the final answer for n R depends only on k (note due to R-symmetry selection rules, k is always an even integer) As a consistency check, let us use the above formula and compute the ratios of unfixed parameters λ s i in the position space ansatz. For k = 3, there are two such parameters λ s 2 and λ s 4 . We find the ratio is Happily, they all agree with ratios computed independently using our position space method.

C Four-point functions of the W n→∞ algebra
In this Appendix we use the "holomorphic bootstrap" method of [54] to compute four-point functions of the W n→∞ algebra, for the cases relevant to our supergravity computation. We start by recalling that in a chiral algebra the four-point function of identical quasiprimary operators, The function F(χ) can be written as a sum of the SL(2, R) blocks Here C hhi is the three-point coupling in and all two-point functions are normalized to unity. Combining the crossing equation with the conformal block decomposition, we find that the singularities of F at χ = 1 and χ → ∞ are Here α n and β n are computable numbers given the three-point functions coefficients C hhi with h i < 2h. The meromorphic F(χ) is completely determined by these singularities and by its value at χ = 0, F(χ) = 1 + Alternatively, we can cast F(χ) into the following more convenient parameterization [54], which is manifestly crossing-symmetric and has the same singularity structure, where the finitely many constants c n are determined from the conformal block decomposition at χ = 0.

k = 2
We first look at the k = 2 case which corresponds to 2d stress tensor. The general form of F(χ) for h = k = 2 reads To fix the constants c n we only need to match with the OPE coefficients of T T 1 and T T T C T T 1 = 1 , C T T T = 2 3/2 c −1/2 . (C.9) We find c 0 = 1 , (C.10) In the 1/c expansion, the four-point function with the above solution of c n simply reads Notice the leading term is nothing but the disconnected piece of the full four-point function under the chiral algebra twist and is anticipated from the large-c factorization. The subleading term in 1/c on the other hand reproduces precisely the holomorphic four-point function we obtained from the supergravity computation, upon recalling that c ∼ 4n 3 in the large n limit.
The general F(χ) for k = 4 can be written as (C.15) By matching with the OPE coefficients of W 4 W 4 1, W 4 W 4 T , W 4 W 4 W 4 , we found that again in perfect agreement with our supergravity computation.