Ad S 3 × S 3 Tree-Level Correlators: Hidden Six-Dimensional Conformal Symmetry
PPrepared for submission to JHEP
PUPT-2587,YITP-SB-2019-17
AdS × S Tree-Level Correlators:Hidden Six-Dimensional Conformal Symmetry
Leonardo Rastelli D ,K , Konstantinos Roumpedakis D , Xinan Zhou D D YITP, Stony Brook University, Stony Brook, NY 11794, USA D Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA K CERN, Theoretical Physics Department, 1211 Geneva 23, Switzerland
Abstract:
We revisit the calculation of holographic correlators in
AdS . We developnew methods to evaluate exchange Witten diagrams, resolving some technical difficultiesthat prevent a straightforward application of the methods used in higher dimensions. Weperform detailed calculations in the AdS × S × K background. We find strong evidencethat four-point tree-level correlators of KK modes of the tensor multiplets enjoy a hidden 6dconformal symmetry. The correlators can all be packaged into a single generating function,related to the 6d flat space superamplitude. This generalizes an analogous structure foundin AdS × S supergravity. a r X i v : . [ h e p - t h ] M a y ontents AdS × S supergravity 73.2 The position space algorithm 83.3 The lowest-weight four-point function 93.4 Higher-weight four-point functions 10 N = 4
18B Proca-Chern-Simons versus massive Chern-Simons 19C
AdS Witten diagrams 20
C.1 Contact Witten diagrams with three derivatives 20C.2 Exchange Witten diagrams of massless and massive Chern-Simons fields 21C.3 Exchange Witten diagrams of non-dynamical graviton field 25
Correlation functions of local operators are basic observables in holographic CFTs, and assuch have been intensely studied since the early days of AdS/CFT. Only recently howeverhave truly efficient computational methods been developed. Broadly speaking, these newtechniques are inspired by the modern “on-shell” approach to perturbative gauge theoryamplitudes. One focusses on the full holographic correlator, which is a much simpler andmore rigid object than individual Witten diagrams. Correlators can be strongly constrainedand sometimes completely determined by imposing symmetries and other consistency re-quirements.In the paradigmatic AdS/CFT example, namely the dual pair of N = 4 super Yang-Mills theory and IIB string theory on AdS × S , this new approach has led to a compelling– 1 –onjecture for all one-half BPS four-point correlators in the tree-level supergravity limit[1, 2]. From these tree-level correlation functions, one can extract the a wealth of O (1 /N ) CFT data about non-protected double-trace operators [6–10]. In turn, these data serveas input in the “AdS unitarity method” [11] to yield one-loop results and thus O (1 /N ) anomalous dimensions [6, 7, 12, 13]. The conjectured tree-level correlators [1, 2], and theCFT data that can be extracted from them, take a remarkably simple form, which has beeninterpreted [10] as arising from a “hidden” (and rather mysterious) 10d conformal symmetry.This success story has been replicated, at least partially, for other supergravity back-grounds. The techniques developed for
AdS × S have been generalized to AdS [18–23], AdS [21] and AdS [19, 24–26] backgrounds with maximal or half-maximal supersymme-try, leading to many interesting new results. By contrast, AdS backgrounds are morechallenging and have so far defied our efforts. The purpose of the present work is to remedythis situation.The greater technical difficulty of the AdS case can be traced to the chiral natureof boundary correlators (and, dually, of bulk Witten diagrams). Four-point correlatorsin a 2d CFT are functions of the usual holomorphic coordinates on the plane, z and ¯ z ,but unlike the situation in higher dimensional CFTs, there is no requirement of symmetryunder the exchange of z and ¯ z . This chiral character limits the applicability of the Mellinformalism [27–30], so the Mellin bootstrap approach of [1, 2, 18, 19, 21] does not generalizeimmediately to AdS . On the other hand, the “position space” method developed in [1, 2]can be applied, but requires some extra work. In the position space approach, one writesan ansatz for the holographic correlator as a sum of Witten diagrams, taking into accountonly general selection rules that follows from the structure of the supergravity theory, butwith arbitrary coefficients. The coefficients are then fixed by imposing superconformalWard identities. In the AdS case, there are new types exchange Witten diagrams forChern-Simons bulk interactions, and the standard techniques of [31] need to be suitablygeneralized. There are further subtle issues in the massless limit of the bulk exchangedfield, even for the standard Maxwell and gravity cases. We resolve all of these technicaldifficulties in the present paper.While our new results for exchange Witten diagrams in AdS are of general applicabil-ity, we will focus on performing detailed calculations in the best studied background, namely AdS × S × K , which arises in the near-horizon limit of the D1-D5 system. By reducingIIB supergravity on K , whose size is taken to be much smaller than the (common) radius ofcurvature of AdS and S , one obtains 6d (2,0) supergravity coupled to 21 tensor multiplets.Upon further reduction on S , the gravity multiplet and the tensor multiplets give rise toinfinite Kaluza-Klein towers of one-half BPS multiplets of the P SU (1 , | × P SU (1 , , | superalgebra. Prior to our work, some partial results had been obtained using indirect Some highly non-trivial checks of this conjecture have been performed in [3–5] by explicit supergravitycalculations. One can further consider stringy corrections to four-point functions, see, e.g. , [14–17] for work in thisdirection. We have in mind the usual kinematic setup of choosing a conformal frame where the positions of threeoperators are fixed, so that the correlator depends only on the coordinates of forth operator. The generalization to
AdS × S × T is straightforward. – 2 –ethods [32–35]. For example, the four-point function of the KK modes with lowest con-formal dimension was conjectured from a limit of the light-light-heavy-heavy correlators[34].There is also a clear physical incentive to revisit AdS × S holographic correlators,beyond the mere demonstration that our previous methods can be generalized to this moredifficult case. Do these correlators exhibit a hidden conformal symmetry analogous to theone found for AdS × S case [10]? While a conceptually satisfactory explanation is stilllacking, in the 10d case such a symmetry appears to hinge crucially on a few facts. First,the AdS × S metric is conformally flat, a feature shared by the AdS × S background butnot (for example) by the maximally supersymmetric M-theory cases, namely AdS × S and AdS × S . Second, the flat space 10d superamplitude in IIB supergravity contains akinematic factor G (10) N δ ( Q ) , which can in some heuristic sense be regarded as a dimen-sionless coupling. An analogous power-counting applies to superamplitudes in 6d (2 , supergravity, where the kinematic factor G (6) N δ ( Q ) is again dimensionless. Third, the four-point superamplitude in 10d flat space IIB supergravity enjoys an accidental 10d conformalsymmetry. The 10d amplitude can be viewed as a generating function of all four-pointcorrelators of the full tower of KK modes on AdS × S [10]. The situation in 6d (2 , flat space supergravity is more elaborate. As we have mentioned, there are two relevantsupermultiplets, the graviton and tensor multiplets. As it turns out, the superamplitudewith four external tensors enjoys an accidental 6d conformal symmetry! By analogy with the
AdS × S case, it seems natural to anticipate that that all four-point correlators of tensor multiplet KK modes in AdS × S can be packaged into a single6d object. We find strong evidence that this is indeed the case. Our strategy is to developa systematic position space method similar to the one used in [1, 2, 18]. New ingredientsin AdS include a derivation of the superconformal Ward identity, and the computation of AdS 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 operatorsthat arise as KK modes of the 6d tensor multiplets. We have obtained results for operatorswith conformal dimensions ∆ = 1 , , , . Our result for ∆ = 1 reproduces the recentconjecture of [34]. We also discuss an independent method in Mellin space. The Mellinspace method for AdS × S × K is not as powerful as in AdS × S and AdS × S , butis still very useful to illustrate the analogy between holographic correlators and scatteringamplitudes, 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 six-dimensional hidden conformal symmetry. Assuming that such a symmetry persists forarbitrary external weights, it is immediate to write down a simple generating function of allfour-point correlators of tensor KK modes by replacing the x ij in the lowest-weight four- Indeed, in the M-theory cases, the radii of the AdS and sphere factors are different. In the caseof
AdS × S one can immediately see that a putative 11d conformal symmetry would be structurallyincompatible with the explicit results of [18]. We were not able to find in the literature fully explicit expressions for amplitudes involving externalsupergravitons in 6d (2 , supergravity (see [36] for the state of the art), but we suspect that they do notenjoy such a symmetry. – 3 –oint function with six dimensional distances. On the other hand, such a hidden conformalsymmetry is not present in the four-point functions of scalar one-half BPS operators thatarise 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 thenature of the hidden conformal symmetry, in both the AdS × S and the AdS × S cases.The rest of the paper is organized as follows. In Section 2 we discuss the superconformalkinematics of scalar one-half BPS four-point functions. In Section 3 we introduce theposition space method for AdS and compute several examples of four-point functions. InSection 4 we provide a different perspective in Mellin space. In Section 5, we point outthe existence of a six-dimensional hidden conformal symmetry. Using this symmetry weconjecture a formula for all one-half BPS four-point functions. We conclude in Section 6by mentioning some future directions. The paper also includes three appendices to whichwe have relegated various technical details. Note:
As we were about to submit this paper to the arXiv, we learnt of an upcomingwork [37] that obtains
AdS × S four-point tree-level correlators with pairwise equal weightsby generalizing the approach of [34]. Let us start with the constraints of superconformal invariance. We focus on the one-halfBPS local operators O α ...α k , ˙ α ... ˙ α k k ( x ) with α i , ˙ α i = 1 , , in the ( j, ¯ j ) = (cid:0) k , k (cid:1) representationof SU (2) L × SU (2) R . These operators have protected conformal dimensions ( h, ¯ h ) = (cid:0) k , k (cid:1) .The Kaluza-Klein reduction on AdS × S of 6d (2 , supergravity coupled to 21 tensormultiplets leads to two different types of one-half BPS scalar operators. The first kindoriginates from the anti-self-dual tensors with k = 1 , , . . . , and they transform in thevector representation of the SO (21) flavor symmetry. The second kind comes from 6dsupergravity fields with k = 2 , , . . . , and they are neutral under the flavor symmetry. Inthis 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 contractingall the indices with auxiliary spinors v α , ¯ v ˙ α O Ik ( x ; y, ¯ y ) = O I,α ...α k , ˙ α ... ˙ α k k v α . . . v α k ¯ v ˙ α . . . ¯ v ˙ α k . (2.1)We have exploited the fact that the spinors are automatically “null” (cid:15) αβ v α v β = (cid:15) ˙ α ˙ β ¯ v ˙ α ¯ v ˙ β = 0 , (2.2)and the one-half BPS operator is symmetric and traceless (with respect to the (cid:15) tensor) in α i and ˙ α j . We note that rescaling preserves the null property of the spinors. This allowsus to parameterize the spinors as v = (cid:32) y (cid:33) , ¯ v = (cid:32) y (cid:33) . (2.3)– 4 –he goal of this paper is to calculate the four-point function G I I I I k k k k = (cid:104)O I k O I k O I k O I k (cid:105) . (2.4)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 thecross ratios z = z z z z , ¯ z = ¯ z ¯ z ¯ z ¯ z , α = y y y y , ¯ α = ¯ y ¯ y ¯ y ¯ y , (2.5)and z ij ≡ z i − z j , y ij ≡ y i − y j , etc . Hence we can write it as G I I I I k k k k = K G I I I I k k k k ( z, ¯ z ; α, ¯ α ) , (2.6)where the kinematic factor K is given by K = (cid:89) i
AdS × S and in [18]for AdS × S . However, it has also new important ingredients due to the unique features of AdS space. In Section 3.1 we review some elements of the 6d (2 , supergravity coupled totensor multiplets, compactified on AdS × S . Then, in Section 3.2 we outline the positionspace algorithm and in Section 3.3 we compute the four-point function of the lowest-weightoperator. Finally, in Section 3.4 we apply the method to the four-point functions of higherweights. We have derived the superconformal Ward identities using two different methods. The first one usesthe analytic superspace formalism, and is parallel to the analysis in [38]. The second method uses a chiralalgebra twist [39] on one of the psu (1 , | subalgebra of the small N = 4 superconformal algebra. Thesecond method is conceptually more interesting, and we will elaborate on it further in Appendix A. – 6 – .1 A brief review of AdS × S supergravity The near horizon limit of Q D1-branes and Q D5-branes wrapping a K surface is de-scribed by IIB supergravity in AdS × S × K when Q Q (cid:29) . As in this limit the sizeof K is much smaller than the size of S , we can reduce IIB supergravity on K3 and get6d (2 , supergravity coupled to n = 21 anti-self-dual tensor multiplets. Then, furthercompactification of the theory on S gives the Kaluza-Klein spectrum [41–44] summarizedin the tables 1, 2, 3 below.KK mode h j ¯ h ¯ j spin (cid:96) R SO (21) ϕ + µν l + 2 l l l V + µ l + 1 l + 1 l l W + µ l + 2 l l + 1 l − ρ + l + 1 l + 1 l + 1 l − Table 1 : Kaluza Klein modes from the spin-2 multiplets Γ l . We have kept only the relevantbosonic field modes which are singlets under the outer automorphism group SO (4) out . Notethat the fields ϕ + µν , V + µ , W + µ , ρ + are also accompanied by ϕ − µν , V − µ , W − µ , ρ − as requiredby parity. The quantum numbers h , ¯ h , j , ¯ j of the − fields are obtained from the quantumnumbers of the + fields by interchanging left and right. The number l labels the Kaluza-Klein levels and l = 0 , , , . . . .KK mode h j ¯ h ¯ j spin (cid:96) R SO (21) Y + µ l + 2 l l + 1 l + 1 σ l + 1 l + 1 l + 1 l + 1 τ l + 2 l l + 2 l Y − µ l + 1 l + 1 l + 2 l Table 2 : Kaluza-Klein modes from the spin-1 multiplet Σ l , l = 0 , , . . . .KK mode h j ¯ h ¯ j spin (cid:96) R SO (21) Z + ,Iµ l + 2 l l + 1 l + 1 s I l + 1 l + 1 l + 1 l + 1 t I l + 2 l l + 2 l Z − ,Iµ l + 1 l + 1 l + 2 l Table 3 : Kaluza-Klein modes from the spin-1 multiplet Θ Il , l = − , , , . . . .The spectrum is organized into superconformal multiplets which come in three infiniteKaluza-Klein towers Γ l , Σ l and Θ Il . 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– 7 –re negative, the corresponding field does not exist. We have only kept the fields that aresinglets under the outer automorphism group SO (4) out because in this work we focus onlyon four-point functions of operators which are singlets. The superconformal primary of themultiplets Γ l which contains the spin-2 fields is the (massive) graviphoton field V µ . Thelowest KK multiplet with l = 0 is ultra-short: it contains only the non-dynamical masslessgraviton and graviphoton. The superconformal primaries of the spin-1 multiplets Σ l and Θ Il are the scalar fields σ and s I respectively. These two multiplets have the same SO (2 , and SO (4) R quantum numbers, but Θ Il 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 , supergravityand Θ Il comes from the anti-self-dual tensors. Moreover, the minimal allowed value for Θ Il is l = − , and the corresponding super primary has conformal dimension ∆ = 1 . The topcomponent (not shown in the table) is an exactly marginal operator and transforms as avector under SO (4) out . By contrast, Σ l with l = − is pure-gauge and does not exist inthe spectrum [41].The cubic couplings of the Kaluza-Klein modes were obtained in [44]. The cubic cou-plings satisfy the R-symmetry selection rule, and vanish when they are extremal. Thequartic 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 vectorfields supplemented by a first-order constraint. The vector fields couple to the currentsmade out of scalar fields both electrically and magnetically. We will show in AppendixB that the constraint can be solved in terms of three massive Chern-Simons fields whichsatisfy first-order equation of motions. After proper field redefinition, all the couplings ofthe vector fields with currents become electric. We are now ready to formulate the position space method. We start with an ansatz for thefour-point function which includes all the possible exchange and contact Witten diagrams A I I I I = δ I I δ I I A s-exch + δ I I δ I I A t-exch + δ I I δ I I A u-exch + δ I I δ I I A s-con + δ I I δ I I A t-con + δ I I δ I I A u-con . (3.1)The exchange Witten diagrams are subject to the R-symmetry selection rule and the require-ment that the cubic coupling is non-extremal. In addition, the contact Witten diagramsshould contain no more than two derivatives. This condition comes from the consistencywith the flat space limit in which the theory contains only two derivatives. The next stepis to evaluate all the diagrams in the ansatz. Compared to the AdS × S case, there aretwo new kinds of Witten diagrams. The first is the exchange diagram of twist-zero fieldswhich are the massless Chern-Simons and the graviton field. The standard method of [31]is not applicable for these diagrams. We instead evaluate them by solving second orderdifferential equations with appropriate boundary conditions. These differential equationsfollow from the simple fact that the two-particle quadratic conformal Casimir is the same as If one naively applies the method of [31], one finds the answer is divergent. The unphysical divergenceis associated with dropping certain boundary terms in the analysis which is not allowed for d = 2 . – 8 –he wave equation in the bulk, which collapses the exchange diagram into a contact diagramwhen acting on the bulk-to-bulk propagator. We leave the details of this method to Ap-pendix C. The second type of new diagram is the exchange diagrams which involve massiveChern-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 termsof a finite sum of contact diagrams ( ¯ D -functions). To impose the superconformal Wardidentities (2.13), we exploit the fact that ¯ D -functions can be uniquely decomposed as ¯ D ∆ ∆ ∆ ∆ = R Φ ( z, ¯ z )Φ( z, ¯ z )+ R V ( z, ¯ z ) log(1 − z )(1 − ¯ z )+ R U ( z, ¯ z ) log( z ¯ z )+ R ( z, ¯ z ) (3.2)where Φ( z, ¯ z ) = ¯ D is the scalar box diagram, and R Φ ,U,V, are rational functions of z and ¯ z . It is clear that the supergravity ansatz admits a similar decomposition. By furtherusing the recursion relation [46] ∂ z Φ = Φ¯ z − z + log(1 − z )(1 − ¯ z ) z (¯ z − z ) + log( z ¯ z )( z − z − ¯ z ) ,∂ ¯ z Φ = Φ z − ¯ z + log(1 − z )(1 − ¯ z )¯ z ( z − ¯ z ) + log( z ¯ z )(¯ z − z − z ) , (3.3)we can similarly decompose the left side of the superconformal Ward identity (2.13) into thisbasis. The rational coefficient functions R I I I I Ward ,X ( z, ¯ z, α, ¯ α ) with X = Φ , U, V, are requiredto vanish by (2.13), giving rise to a set of linear equations for the unknown coefficients inthe ansatz. In contrast to the AdS × S and AdS × S cases, solving superconformalWard identities in general does not uniquely fix all the relative coefficients. We will seethat all the coefficients in A I I I I con parameterizing the quartic vertices are fixed in termsof the coefficients in the exchange part of the ansatz. The remaining unsolved coefficientscan be fixed by comparing with the known supergravity cubic couplings. We now apply the above method to the simplest four-point correlator with k i = k = 1 . Thecubic 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 theexchange part of the four-point function A s − exch = λ gr Y ¯ Y W gr (cid:124) (cid:123)(cid:122) (cid:125) ϕ l =0 ,µν + λ CS ( Y ¯ Y W CS, , + (cid:124) (cid:123)(cid:122) (cid:125) V + l =0 ,µ + ¯ Y Y W CS, , − (cid:124) (cid:123)(cid:122) (cid:125) V − l =0 ,µ ) , (3.4)where Y m and ¯ Y ¯ m are the SU (2) L and SU (2) R R-symmetry polynomials Y m = P m (1 − α ) , ¯ Y ¯ m = P ¯ m (1 − α ) , (3.5)associated with exchanging the representation ( j, ¯ j ) = ( m, ¯ m ) . The function W gr is theexchange Witten diagram of the non-dynamical graviton, and is worked out in AppendixC to be W gr = π U ( U − V −
1) ¯ D ) . (3.6) This fact was also recently exploited in [45] to obtain the conformal block decomposition coefficients ofexchange Witten diagrams and conformal partial waves in the crossed channel. – 9 –imilarly, W CS, , + , W CS, , − are contributions of the Witten diagrams with the masslessChern-Simons gauge field V + l =0 ,µ , V − l =0 ,µ being exchanged. They are given by W CS, , ± = π (cid:0) ∓ ( z − ¯ z ) U ¯ D + log V (cid:1) . (3.7)The ansatz for A t − exch , A u − exch are obtained from A s − exch using crossing symmetry. Thecontact part of the ansatz takes the from A s-con = (cid:88) c (0) ab σ a τ b U ¯ D + (cid:88) c (1 ,s ) ab σ a τ b U ¯ D + (cid:88) c (1 ,t ) ab σ a τ b U ¯ D + (cid:88) c (1 ,u ) ab σ a τ b U ¯ D , (3.8)where the sum is restricted by ≤ a, b, a + b ≤ . Note that no individual α , ¯ α appears inthe ansatz because the four-point function is parity even under separate exchange of z ↔ ¯ z and α ↔ ¯ α . 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 λ CS = 12 λ gr , (3.9)and all the contact term coefficients are solved in terms of λ gr . Therefore the four-pointfunction is fixed up to an overall coefficient. Rewriting the solution in the form of (2.14)we find that G I I I I , = πλ g V ( V δ I I δ I I + U τ δ I I δ I I + U V σδ I I δ I I ) , (3.10) H I I I I = − πλ g V ( δ I I δ I I V ¯ D + δ I I δ I I U ¯ D + δ I I δ I I U V ¯ D ) , (3.11)reproducing the result of [34]. Let us move on to the next simplest correlator with k = 2 . Our ansatz for the singular partof the four-point function is A s − exch = λ (0) gr Y ¯ Y W gr (cid:124) (cid:123)(cid:122) (cid:125) ϕ l =0 ,µν + λ (0) CS ( Y ¯ Y W CS, , + (cid:124) (cid:123)(cid:122) (cid:125) V + l =0 ,µ + ¯ Y Y W CS, , − (cid:124) (cid:123)(cid:122) (cid:125) V − l =0 ,µ )+ λ (2) ϕ Y ¯ Y W mgr, (cid:124) (cid:123)(cid:122) (cid:125) ϕ l =2 ,µν + λ (2) V ( Y ¯ Y W CS, , + (cid:124) (cid:123)(cid:122) (cid:125) V + l =2 ,µ + ¯ Y Y W CS, , − (cid:124) (cid:123)(cid:122) (cid:125) V − l =2 ,µ )+ λ (0) σ Y ¯ Y W sc, (cid:124) (cid:123)(cid:122) (cid:125) σ l =0 + λ (0) Y ( Y ¯ Y W CS, , + (cid:124) (cid:123)(cid:122) (cid:125) Y − l =0 ,µ + ¯ Y Y W CS, , − (cid:124) (cid:123)(cid:122) (cid:125) Y + l =0 ,µ ) . (3.12)Here W mgr, is the exchange diagram of a massive graviton of dimension 4 and W sc, is ascalar exchange diagram of dimension 2. Both diagrams can be computed using the methodof [31]. The contact part ansatz A s − con contains zero and two-derivative contact Witten– 10 –iagrams, and is a polynomial in σ and τ of degree 2. Solving the superconformal Wardidentities uniquely fixes the coefficients in A s − con in terms of the coefficients appearingin A s − exch . Moreover, the coefficients of the exchange contributions of fields belonging tothe same multiplet are fixed up to an overall normalization. The remaining unfixed relativecoefficients corresponding to different multiplets can be fixed using the cubic vertices workedout in [44]. The final solution can be rewritten in the form of (2.14) as G I I I I , = πλ g V ( V δ I I δ I I + U τ δ I I δ I I + U V σ δ I I δ I I ) , (3.13) H I I I I = − πλ g V (cid:0) δ I I δ I I V ( U ¯ D + ¯ D ) + δ I I δ I I U ¯ D + δ I I δ I I U V ¯ D (cid:1) + crossing . (3.14)The case of higher-weight correlators with k > is completely analogous to the aboveexample with k = 2 . We have also applied this method to obtain four-point correlators fortwo more examples with k = 3 and k = 4 . We will refrain from writing down the explicitresults, since in the Section 5.2 we will present a much more compact way of writing thesecorrelators. The position space method described in Section 3.2 offers a concrete way to compute four-point functions with as little input from supergravity. However, the results in position spacedo not look particularly illuminating. In this section, we look at the problem from a differentperspective using the Mellin representation formalism [27–30], which offers new intuitionto the problem. The Mellin representation formalism was demonstrated to be the mostnatural language for describing holographic correlators, making manifest their scatteringamplitude nature. This formalism unfortunately becomes ill-defined in one dimension dueto the nonlinear dependence of the cross ratios . Because the superconformal symmetryforces the chiral cross ratios z , ¯ z to appear in the 2d one-half BPS correlator G I I I I k k k k ,one may wonder if the Mellin representation will be particularly useful. Nevertheless, byrestricting our attention to certain components of the four-point function we can argue thatthe Mellin representation is still a good language. In particular, the Mellin representationformalism allows us to easily bootstrap the k i = k = 1 correlator as we demonstrate below.For k = 1 , H I I I I k =1 has no dependence on α and ¯ α . The symmetry under z ↔ ¯ z , α ↔ ¯ α implies that H I I I I k =1 is a symmetric function of z , ¯ z and can be unambiguouslyexpressed in terms of U , V . We therefore have the following inverse Mellin representation H I I I I k =1 = (cid:90) i ∞− i ∞ ds dt U s V t − (cid:102) M I I I I k =1 ( s, t )Γ ( 2 − s ( 2 − t ( 2 − ˜ u , (4.1) For d ≥ there are two independent conformal cross ratios U , V in four-point functions, while for d = 1 there is only one independent cross ratio z . More generally, there are n ( n − / independent cross ratiosfor a scalar n -point function if n ≤ d + 2 . When n > d + 2 the cross ratios have relations. – 11 –here ˜ u = 2 − s − t . We assume that G I I I I ,k =1 is a rational function (this was justifiedby the previous position space calculation) and therefore does not contribute to the Mellinamplitude . It then follows that the following components of G I I I I k =1 with R-symmetryfactors P ≡ , P ≡ α + ¯ α , P ≡ α ¯ α = τ , (4.2)also have a well-defined Mellin representation. By rewriting the superconformal factor (1 − zα )(1 − ¯ z ¯ α ) as α + ¯ α V − U −
1) + α ¯ αU + α − ¯ α z − ¯ z ) , (4.3)we read off the three R-symmetry components of G I I I I k =1 P : G I I I I k =1 , I =1 ≡ H I I I I k =1 , (4.4) P : G I I I I k =1 , I =2 ≡ ( V − U − H I I I I k =1 , (4.5) P : G I I I I k =1 , I =3 = U H I I I I k =1 . (4.6)They can be expressed in the same form as (4.1) G I I I I k =1 , I = (cid:90) i ∞− i ∞ ds dt U s V t − M I I I I k =1 , I ( s, t )Γ ( 2 − s ( 2 − t ( 2 − u , (4.7)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 (cid:92) U m V n ◦ (cid:102) M I I I I k =1 ( s, t ) = (cid:102) M I I I I k =1 ( s − m, t − n ) (cid:18) − s (cid:19) m (cid:18) − t (cid:19) n (cid:18) s + t − (cid:19) − m − n . (4.8)We are now ready to formulate a bootstrap problem for (cid:102) M I I I I k =1 by enumerating the extraconstraints that should be satisfied by M I I I I k =1 , I .1. Bose symmetry.
The Mellin amplitudes M I I I I k =1 , I are crossing symmetric. It isconvenient to first make the flavor dependence more explicit M I I I I k =1 , I ( s, t ) = δ I I δ I I M ( s ) k =1 , I ( s, t ) + δ I I δ I I M ( t ) k =1 , I ( s, t ) + δ I I δ I I M ( u ) k =1 , I ( s, t ) . (4.9)Crossing symmetry then implies that M ( t ) k =1 , ( s, t ) M ( t ) k =1 , ( s, t ) M ( t ) k =1 , ( s, t ) = − − M ( s ) k =1 , ( t, s ) M ( s ) k =1 , ( t, s ) M ( s ) k =1 , ( t, s ) , M ( u ) k =1 , I ( s, t ) = M ( s ) k =1 , I ( u, t ) . (4.10) The rational terms are generated from regularization effects when the integration contours are pinched.See [2] for details. – 12 –.
Analytic structure.
The four-point function can be computed as a sum of Wittendiagrams. The conformal block decomposition consists of only single-trace operators anddouble-trace operators. The twists of exchanged single-trace operators translate into theposition of simple poles in the Mellin amplitudes, while the double-trace operators arealready accounted by the Gamma function factors. From the supergravity spectrum weexpect that M ( s ) k =1 , I contain only a simple pole at s = 0 due to the s-channel exchangeof the non-dynamical graviton and gauge field. Similarly, M ( t ) k =1 , I and M ( u ) k =1 , I have onlysimple poles at t = 0 and u = 0 respectively. Furthermore, the residue at each pole is apolynomial in the other Mandelstam variable.3. Asymptotics. M ( a ) k =1 , I should grow linearly at large values of the Mandelstamvariables, M ( a ) k =1 , I ( βs, βt ) ∼ O ( β ) , for β → ∞ . (4.11)This comes from the expectation that the Mellin amplitudes M ( a ) k =1 , I in the asymptoticregime should reproduce the flat space scattering amplitudes of tensors in 6d (2 , super-gravity [28].These conditions turn out to be constraining enough, and uniquely fix the Mellin am-plitude up to an overall factor (cid:102) M I I I I k =1 ( s, t ) ∝ δ I I δ I I s + δ I I δ I I t + δ I I δ I I ˜ u . (4.12)Translating the result back into H I I I I , we find H I I I I ∝ δ I I δ I I V − ¯ D + δ I I δ I I U V − ¯ D + δ I I δ I I U V − ¯ D , (4.13)which reproduces our previous result (3.11) in position space.In fact the above arguments apply more generally to a class of four-point correlatorswith have the same extremality , e.g. , k = k = n , k = k = 1 . For these near-extremal correlators, the auxiliary Mellin amplitudes (cid:102) M I I I I are uniquely determined bythe bootstrap conditions up to an overall coefficient, and take the same form as (4.12) withshifted simple poles.Let us also make two comments about applying the Mellin space method to correlatorswith higher extremities. First of all, it is necessary to make the assumption that H I I I I is even under α ↔ ¯ α , or in other words, can be uniquely expressed in terms of σ and τ .This is needed such that H I I I I 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 I I I a priori can be more general. However the even parity of H I I I I is observed inall 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 × S case. We expectthat (cid:102) M I I I I also takes the form of a sum of simple poles. However, the pole structure ofthe ansatz makes the condition on analytic structures weaker. In particular, the requirementthat M I I I I should have polynomial residues is now trivially satisfied due to the lack Extremality E is defined as E = k + k + k − k where k is the largest of all k i . – 13 –f simultaneous poles (cid:102) M I I I I . Therefore, unlike the AdS × S case where the Mellinamplitude is fixed up to an overall factor, we cannot solve all the parameters in the ansatzfor the Mellin amplitude. This parallels what we have observed in the position spacemethod, and is an inevitable consequence of the fact that we have fewer supersymmetry for AdS × S × K . The general one-half BPS four-point functions of 4d N = 4 super Yang-Mills theory inthe 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 existenceof some elegant underlying structure. Recently, this was made precise by [10] in termsof a conjectured 10d conformal symmetry. In terms of this symmetry, all one-half BPSfour-point functions can be organized into one generating function, which is obtained bypromoting the 4d distances in the lowest-weight correlator into 10d distances. Though arigorous understanding is still lacking regarding its origin, some intuitive arguments weregiven in [10] to motivate the existence of such a symmetry. We will enumerate below someof these arguments and we will see that many features are also shared by AdS × S .First of all, the AdS × S background is conformally equivalent to the flat space R , .The SO (10 , symmetry can be interpreted as the conformal group in R , . The samestatement can be made for the AdS × S background and the conformal group SO (6 , . Secondly, it was argued that the
AdS × S auxiliary Mellin amplitude (cid:102) M of [1, 2] (cid:102) M ∼ s − t − u − , (5.1)should be identified, in the large Mellin variable limit, with the stu factor in the superam-plitude of IIB supergravity in 10d flat space A IIB ∼ G (10) N δ ( Q ) × stu . (5.2)When divided by the dimensionless “coupling” G (10) N δ ( Q ) , the amplitude stu is confor-mal invariant in ten dimensions, i.e. , annihilated by the special conformal transformationgenerator K µ = (cid:88) i =1 (cid:18) p iµ ∂∂p νi ∂∂p i,ν − p νi ∂∂p νi ∂∂p µi − d − ∂∂p µi (cid:19) . (5.3)For AdS × S we found a highly nontrivial analogy. By taking the asymptotic limit of theauxiliary Mellin amplitude (cid:102) M I I I I k =1 , we find that we precisely reproduce the four tensor More precisely, it requires that the AdS space should have the same radius as the sphere. This is truefor
AdS × S and AdS × S , but is not true for, e.g. , AdS × S which is dual to 6d (2 , SCFTs. We use momentum conservation to solve p in terms of p , p , p , and write stu = p · p )( p · p )( p · p ) . – 14 –cattering amplitude in the theory of 6d (2,0) supergravity coupled to 21 Abelian tensormultiplets [47] A (2,0) ∼ G (6) N δ ( Q ) (cid:18) δ I I δ I I s + δ I I δ I I t + δ I I δ I I u (cid:19) . (5.4)After dividing by the dimensionless quantity G (6) N δ ( Q ) , the amplitude is also conformallyinvariant in 6d. Finally, [10] also observed that the form of double-trace anomalous di-mensions coincides with the partial wave decomposition coefficients of the 10d scatteringamplitudes. Moreover the use of the 10d conformal block diagonalizes the mixing problemof double-trace operators. We have not investigated the counterparts of these problems insix dimensions, but it is likely that the questions will have similar answers. We hope toreturn to these questions in the future. Motivated by the above similarities, we propose that a hidden SO (6 , symmetry existsfor AdS × S , in the same sense of the AdS × S case. This symmetry will translateinto a prescription for writing down a generating function for all one-half BPS four-pointfunctions.Let us define from H I I I I a crossing-symmetric function H I I I I k k k k = K (cid:18) t t x x (cid:19) − H I I I I k k k k x x x x . (5.5)In particular, for k i = 1 H I I I I = H I I I I x x x x ≡ H I I I I k =1 x x x x , (5.6)is a function of x ij only. Our main contention is that H I I I I can be promoted into a generating function by doing a simple replacement in the arguments H ( x i , t i ) I I I I ≡ H I I I I ( x ij + t ij ) . (5.7)All the functions H I I I I k k k k 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 (cid:81) i
AdS × S is in many respects similar to the one that holds in AdS × S .(There is a unique supermultiplet in 10d IIB supergravity, and the symmetry holdsthere for four-point tree-level correlators of all KK modes). Both backgrounds areconformally flat, and the relevant flat space superamplitudes enjoy an accidental con-formal symmetry, respectively in six and ten dimensions. A third case that shouldwork along very similar lines is
AdS × S , where the superamplitude of four ex-ternal hypermultiplets takes the simple form G (4) N δ ( Q ) · c . The kinematic prefactor– 16 – (4) N δ ( Q ) is again dimensionless, while c is a just constant, and thus obviously confor-mally invariant in four dimensions. Assuming the hidden symmetry, it is immediateto write down the generating function of four-point tree-level correlators of all KKmodes. It would be interesting to perform some explicit checks of this ansatz. • Another closely related background is
AdS × S × T . Upon reducing IIB supergravityon T , one obtains (2 , supergravity in 6d. Our methods can be straightforwardlyapplied to that case. • It will be important to achieve a first-principles derivation of the hidden conformalsymmetry, perhaps along the line of [50], which related Einstein gravity to conformalgravity. Such a conceptual understanding would also elucidate the regime of validityof the symmetry. For example, does it extend to higher-point tree-level correlators in
AdS × S ? Is it broken by /N corrections and how? • In this paper, we focussed on the four-point functions of the modes Θ Il , the KK towerthat arises from the 6d tensor multiplets. Two additional KK towers, Γ l and Σ l , arisefrom the 6d (2 , supergraviton. We would like to study the most general four-pointfunctions which involve operators from all these multiplets. We have found that four-point correlators of Σ l fields are incompatible, at least naively, with 6d conformalsymmetry, 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 problemof double-trace operators and extract the spectrum of anomalous dimensions. Thesetree-level data can then be used to bootstrap one-loop four-point functions, followingthe blueprint of [6–8, 11–13]. An interesting open question is if the hidden symmetryfor the Θ Il multiplet survives the supergravity loop corrections. • Related to the previous point, it would also be useful to perform an analysis usingthe Lorentzian inversion formula, along the lines of [10], a method independent of oursupergravity computation.
Acknowledgments
We thank Nathan Benjamin, Shai Chester, Wolfger Peelaers, Silviu Pufu and Yifan Wangfor discussions. X.Z. also thanks the International Institute of Physics for hospitality duringhis visit and the participants of the workshop “Nonperturbative Methods for ConformalTheories” for useful conversations and comments. The work of L.R. and K.R. is supportedin part by the NSF grant Five-point functions can also be computed using a generalized version of the position space method,see [51] for recent progress. – 17 –
Twisting small N = 4 In this section we give a derivation of the superconformal Ward identities from topologicaltwisting. Let us focus on the global part of the small N = 4 superconformal algebra andconsider only the holomorphic part. The algebra is psu (1 , | , and is captured by thefollowing commutation relations [ L , L ± ] = ∓ L ± , [ L , L − ] = 2 L , [ J i , J j ] = i(cid:15) ijk J k , [ L , G aA ± ] = ∓ G aA ± , [ J i , G aA ± ] = −
12 ( σ i ) a b G bA ± , { G a + , G b −− ) } = 2 (cid:15) ab L − σ abi J i , { G a + − , G b − } = 2 (cid:15) ab L + 2 σ abi J i , { G a + , G b − } = 2 (cid:15) ab L , { G a + − , G b −− } = 2 (cid:15) ab L − , { G a + ± , G b + ± } = { G a −± , G b −± } = 0 , [ L , G ab ] = 0 , [ L , G ab − ] = G ab , [ L − , G ab ] = − G ab − , [ L − , G ab − ] = 0 . (A.1)where ( σ i ) a b are the Pauli matrices and σ abi = ( σ i ) a c (cid:15) bc with (cid:15) + − = (cid:15) + − = 1 . This algebrahas an SU (2) R symmetry as well as an SU (2) A automorphism under which the supercharges G aAn transform in ( , ) . Following [39], we can consider a topological twist by looking atthe cohomology of the nilponent supercharge Q = G ++ − + G − + , { Q , Q } = 0 . (A.2)Operator which are in the Q -cohomology class are one-half BPS under the left-moving psu (1 , | { Q , O (0)] = 0 , O (0) (cid:54) = { Q , O (cid:48) (0)] ⇒ h = j . (A.3)Moreover, one can construct the following twisted sl (2 , C ) algebra which is Q -exact { Q , G −−− } = 2 L − − σ −− i J i ≡ (cid:98) L − , { Q , − G + − } = 2 L − σ ++ i J i ≡ (cid:98) L , { Q , G −− } = 2 L + 2 σ + − i J i ≡ (cid:98) L . (A.4)Let us revisit the one-half BPS operators with SU (2) L indices contracted with spinors O ( z ; y ) = O α ,...,α k ( z ) v α . . . v α k , v α = (1 , y ) . (A.5)When y = z , it amounts to inserting operators in nontrivial Q -cohomology classes at theorigin and then twist-translating using (cid:98) L − O ( z ; z ) = e z (cid:98) L − O (0) e − z (cid:98) L − . (A.6)Because (cid:98) L − is Q -exact, the twist-translated operators remain in the Q -cohomology. Sincethe twist construction uses only the left-moving part of the 2d algebra, it commutes withthe right-moving algebra. By standard arguments, the correlators of such twisted operatorshave no holomorphic dependence. This directly translates into our superconformal Wardidentity (2.13). – 18 – 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 mul-tiplet Σ k satisfy second-order Proca-Chern-Simons equations. Meanwhile their first-orderderivatives satisfy a linear constraint and there are only three independent degrees of free-dom. 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 thefield redefinition. We will work with the cubic vertices where the vector fields couple to twoscalar 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] P ± k − A ± µ + P ± k +3 C ± µ = ± ( W σσA ± + W σσC ± ) J µ , (B.1) P ∓ k +1 P ± k − A ± µ − P ∓ k +1 P ± k +3 C ± µ = ( V σσA ± − V σσC ± ) J µ ± ( W σσA ± P ± k − J µ − W σσC ± P ± k +3 J µ ) , (B.2)where J µ = ∂ µ σ σ − σ ∂ µ σ , (B.3)and P ± m is the differential operator ( P ± m ) µλ = (cid:15) µνλ (cid:79) ν ± mδ λµ . (B.4)The coefficients W σσA ± , W σσC ± , V σσA ± , V σσC ± are defined in [44] but their preciselyforms are not important to us. One can act with P ∓ k +1 on the first equation and solve forone 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 (cid:15) µνρ V µ (cid:79) ν J ρ ).We now define the following new field L ± µ = ±
12 ( P A µ − P C µ ) + 2( k − k − k + 1)( k + 1)( k + 1) J µ . (B.5)In terms of L ± µ we can rewrite (B.2) as ± P L ± µ ∓ A ± µ ∓ C ± µ − ( k − k + 1) A ± µ + ( k + 1)( k + 3) C ± µ = Q ± µ ( σ ) , (B.6)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 willnot write it down here. Therefore, in terms of A ± µ , C ± µ , L ± µ we have a system of first-ordermassive Chern-Simons equations (B.5), (B.1), (B.6) which can be more compactly writtenas ( P + M ± ) L ± µ A ± µ C ± µ = Q ± µ ( σ ) . (B.7)– 19 –ere is the unit matrix and M ± is a mass matrix. The vector Q ± µ ( σ ) contains all theother terms depending on σ . The mass matrix can be diagonalized by using the followinglinear combinations V ± ,µ = k (1 − k ) L ± µ − k + 1) A ± µ + k ( k + 3) C ± µ / k ( k + 2) / , (B.8) V ± ,µ = kL ± µ − ( k + 2) A ± µ / k ( k + 2) / , (B.9) V ± ,µ = A ± µ + C ± µ / k ( k + 2) / . (B.10)In terms of V ± i,µ , we have ( P ± Λ ) V ± ,µ V ± ,µ V ± ,µ = Q (cid:48)± µ ( σ ) . (B.11)where Λ = diag { k − , − k − , k + 3 } is the diagonalized mass matrix. Moreover, as a resultof (B.5) one can check that there is only only electric coupling in Q (cid:48)± µ ( σ ) , i.e. , no (cid:15) µνρ (cid:79) ν J ρ appears. These eigenvectors are the fields appeared in our tables 1, 2. We can identify V ± ,µ , V ± ,µ , V ± ,µ respectively with V ± µ (at level k − ), Y ∓ µ (at level k − ), W ± µ (at level k + 1 ). C AdS Witten diagrams
In this appendix we discuss the computation of exchange Witten diagrams which are uniqueto
AdS . Exchange diagrams of other fields, such as scalars, Proca fields, massive symmetrictraceless 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 specialtype of contact Witten diagrams which has an odd number of derivatives. The specialcontact Witten diagram is built from contact vertices of the type (cid:15) µνρ ( ∂ µ φ ∂ ν φ ∂ ρ φ ) φ φ . . . , (C.1)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 arelooking at the following contact Witten diagram (we have distributed the three derivativeson the external legs 1, 3 and 4) defined by the integral W (134)con ≡ (cid:90) d zz z (cid:15) µνρ ∂ µ G ∆ B∂ ( z, x ) ∂ ν G ∆ B∂ ( z, x ) ∂ ρ G ∆ B∂ ( z, x ) G ∆ B∂ ( z, x ) , (C.2)where G ∆ i B∂ ( z, x i ) is the bulk-to-boundary propagator G ∆ i B∂ ( z, x i ) = (cid:18) z z + ( (cid:126)z − (cid:126)x i ) (cid:19) ∆ i . (C.3)– 20 –his integral can be evaluated to the following W (134)con = x ∆ − ∆ − ∆ − ∆ x − ∆ +∆ +∆ − ∆ x r ∆ +∆ − ∆ − ∆ iπ Γ( ∆ +∆ +∆ +∆ )Γ(∆ )Γ(∆ )Γ(∆ )Γ(∆ ) × ( z − ¯ z )( ¯ D ∆ +2 , ∆ , ∆ +1 , ∆ +1 + ¯ D ∆ +1 , ∆ , ∆ +2 , ∆ +1 − ∆ ¯ D ∆ +1 , ∆ , ∆ +1 , ∆ ) , (C.4)where z and ¯ z are the chiral and anti-chiral cross ratios. The ¯ D -functions are related tothe standard D -functions D ∆ ∆ ∆ ∆ ≡ (cid:90) d zz G ∆ B∂ ( z, x ) G ∆ B∂ ( z, x ) G ∆ B∂ ( z, x ) G ∆ B∂ ( z, x ) , (C.5)via (cid:81) i =1 Γ(∆ i )Γ(Σ − d ) 2 π d D ∆ ∆ ∆ ∆ ( x , x , x , x ) = r Σ − ∆ − ∆ r Σ − ∆ − ∆ r Σ − ∆ r ∆ ¯ D ∆ ∆ ∆ ∆ ( U, V ) , (C.6)where d = 2 and ≡ (cid:80) i =1 ∆ i . C.2 Exchange Witten diagrams of massless and massive Chern-Simons fields
We now move on to the exchange Witten diagram of massless and massive Chern-Simonsfields. The propagator satisfies the equation of motion P ± k − G mCS,k, ± µ ; ν ( z , z ) = ∓ g µν δ ( z , z ) + ( . . . ) δ k, , (C.7)where . . . are suitable terms added to make the differential operator invertible [52]. Thecorresponding vector field has conformal dimension ∆ = k . When k = 1 , the vector fieldis a massless gauge field in the bulk and when k (cid:54) = 1 the vector is massive. The exchangediagram is defined by W CS,k, ± = (cid:90) d zz d ww J µ ( z ; x , x ) G mCS,k, ± µ ; ν ( z, w ) J ν ( w ; x , x ) , (C.8)where J µ ( z ; x , x ) is a conserved current made out of scalar bulk-to-boundary propagators J µ ( z ; x , x ) = ∂ µ G ∆ φ B∂ ( z, x ) G ∆ φ B∂ ( z, x ) − G ∆ φ B∂ ( z, x ) ∂ µ G ∆ φ B∂ ( z, x ) , (cid:79) µ J µ = 0 . (C.9)The massive case can be treated using the method of [31] with slight modifications. Themassless case however requires special attention, and the method of [31] leads to formaldivergences. In this subsection, we will give a different method to compute these exchangediagrams which can be applied to both the massless and the massive case. For simplicity,we will restrict ourselves to the case where the external operators have the same dimension ∆ i = ∆ φ .The idea is to view the exchange Witten diagram as solution to a differential equationwith certain boundary conditions. We first look at Witten exchange diagrams of Procafields as a more familiar example. The exchange diagrams are defined by W Proca ,k = (cid:90) d zz d ww J µ ( z ; x , x ) G Proca ,k,µ ; ν ( z, w ) J ν ( w ; x , x ) , (C.10)– 21 –he propagator G Proca ,k of a Proca field with squared-mass m k = ( k − − (and dualdimension ∆ k = k ) satisfies (Proca k G Proca ,k ) µ ; ν = g µν δ ( z , z ) + ( . . . ) δ k, , (C.11)where (Proca k ) µν ≡ (cid:3) δ νµ − g νρ (cid:79) ρ (cid:79) µ − m k δ νµ . (C.12)Now consider only the z integral in (C.10) I Proca ,k,ν ( x , x ; w ) ≡ (cid:90) d zz J µ ( z ; x , x ) G Proca ,k,µ ; ν ( z, w ) . (C.13)We act on the integral with the differential operator L (1) AB + L (2) AB + L ( w ) AB , (C.14)where L ( i ) AB are the conformal generators of the boundary point x i , and L ( w ) AB is AdS isom-etry generator of the bulk point w . Because the z -integral is conformally covariant, it isannihilated by this operator ( L (1) AB + L (2) AB + L ( w ) AB ) I Proca ,k,ν ( x , x ; w ) = 0 . (C.15)It follows that − (cid:16) L (1) AB + L (2) AB (cid:17) δ νµ I Proca ,k,ν ( x , x ; w ) = − (cid:16) L ( w ) AB (cid:17) δ νµ I Proca ,k,ν ( x , x ; w ) . (C.16)Note that the operator on the LHS is nothing but the two-particle quadratic Casimir. Theoperator on the RHS is the AdS Laplacian with a constant shift [53] − (cid:16) L ( z ) AB (cid:17) δ νµ = ( (cid:3) + 2) δ νµ = (Proca k ) µν + g νρ (cid:79) ρ (cid:79) µ + ( k − δ νµ . (C.17)Now we can use the equation of motion of the bulk-to-bulk propagator and perform theremaining w -integral. The operator g νρ (cid:79) ρ (cid:79) µ can be ignored because we can integrate bypart. Its contribution vanishes since I Proca ,k,ν ( x , x ; w ) is coupled to a conserved current.All in all, we get (cid:16) Casimir (12) − ( k − (cid:17) W Proca ,k = W conProca , (C.18)where W conProca is a two-derivative contact diagram W conProca = (cid:90) d zz J µ ( z ; x , x ) J µ ( z ; x , x ) . (C.19)Instead of evaluating the diagram using the method of [31], we can alternatively solvethe differential equation (C.18). We first need a special solution. This is not difficult forthe 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 theexchanged single-trace operator and its shadow. They can be fixed by imposing boundary– 22 –onditions. When we decompose W conProca , it should contain only single-trace and double-trace blocks, and no shadow conformal block. Moreover in the Euclidean regime, i.e. , ¯ z = z ∗ , W conProca is single-valued (as is clear from its integral definition). These conditionsuniquely fix the solution.To compute the massive Chern-Simons exchange diagrams, we first notice the followingrelations among differential operators ( P − k − P + k − ) µν = (cid:3) δ νµ − (cid:79) µ (cid:79) ν − (( k − − δ νµ (cid:124) (cid:123)(cid:122) (cid:125) (Proca k ) µν . (C.20)It then follows from (C.7) that the massive Chern-Simons propagator can be obtained fromapplying P ∓ k − on (Maxwell) Proca propagators G mCS , ± ,kµ ; ν = ∓ ( P ∓ k − G Proca ,k ) µ ; ν . (C.21)We now act on the massive Chern-Simons exchange diagram with the two-particle quadraticCasimir. Using the same argument and (C.21), we get the following differential equation (cid:16) Casimir (12) − ( k − (cid:17) W CS,k, ± = W conmCS ,k , (C.22)where W conmCS ,k = (cid:90) d zz J µ ( z ; x , x )( P ∓ k − J ) µ ( z ; x , x ) = ± W (134)con − W (234)con ) + ( k − W conProca . (C.23)Note that for k = 1 , i.e. , the massless case, there are only three-derivative contact terms.When k > , we can write W conmCS ,k = (cid:102) W conmCS ,k + 1 k − W Proca ,k , (C.24)so that the differential equation for (cid:102) W conmCS ,k reduces to the massless form. The specialsolutions are again easy to guess, and take the general form of ( z − ¯ z ) times a sum of D -functions. We will list a few explicit solutions in a moment. The equation (C.22) also admithomogenous solutions which are the conformal block for the single-trace operator and itsshadow operator. To fix the solution we require that in the conformal block decomposition,the single-trace conformal block has dimension ( h, ¯ h ) = (cid:0) k +12 , k − (cid:1) for + , and ( h, ¯ h ) = (cid:0) k − , k +12 (cid:1) for − . There is no shadow conformal block. The solution is also single-valuedin the Euclidean regime.Using this method, we can easily compute the massless and massive Chern-Simonsexchange diagrams. Let us list the values of the diagrams which appear in this paper. ∆ φ = 1 W CS, , ± = 1 x x π (cid:0) ∓ ( z − ¯ z ) U ¯ D + log V (cid:1) . (C.25) We have rescaled the exchange diagrams by some overall factors which are unimportant to the positionspace method. – 23 – φ = 2 W CS, , ± = πx x (cid:0) ∓ ( z − ¯ z ) U ( ¯ D + 2 ¯ D ) + log V (cid:1) , (C.26) W CS, , ± = πUx x (cid:0) ± z − ¯ z ) U ¯ D + ¯ D − V ¯ D − ¯ D + ¯ D (cid:1) . (C.27) ∆ φ = 3 W CS, , ± = πx x (cid:0) ∓ ( z − ¯ z ) U ( ¯ D + 3 ¯ D + 3 ¯ D ) + 2 log V (cid:1) , (C.28) W CS, , ± = πUx x (cid:18) ± z − ¯ z ) U (3 ¯ D + 4 ¯ D ) + 3( ¯ D − V ¯ D − ¯ D + ¯ D )+ 4 U ( ¯ D − V ¯ D − ¯ D + ¯ D ) (cid:19) , (C.29) W CS, , ± = πU x x (cid:0) ± ( z − ¯ z ) U ¯ D + ¯ D − V ¯ D − ¯ D + ¯ D (cid:1) . (C.30) ∆ φ = 4 W CS, , ± = πx x (cid:0) ∓ ( z − ¯ z ) U (3 ¯ D + 12 ¯ D + 15 ¯ D + 10 ¯ D ) + 18 log V (cid:1) , (C.31) W CS, , ± = πUx x (cid:18) ± z − ¯ z ) U (12 ¯ D + 20 ¯ D + 15 ¯ D )+12( ¯ D − V ¯ D − ¯ D + ¯ D ) + 20 U ( ¯ D − V ¯ D − ¯ D + ¯ D )+15 U ( ¯ D − V ¯ D − ¯ D + ¯ D ) (cid:19) , (C.32) W CS, , ± = πU x x (cid:18) ± ( z − ¯ z ) U (5 ¯ D + 6 ¯ D ) + 5( ¯ D − V ¯ D − ¯ D + ¯ D )+6 U ( ¯ D − V ¯ D − ¯ D + ¯ D ) (cid:19) , (C.33) W CS, , ± = πU x x (cid:0) ± z − ¯ z ) U ¯ D + 3( ¯ D − V ¯ D − ¯ D + ¯ D ) (cid:1) . (C.34)– 24 – .3 Exchange Witten diagrams of non-dynamical graviton field The exchange Witten diagrams of graviton field in
AdS also cannot be evaluated usingthe method of [31]. However, it is straightforward to adapt the method from the previoussubsection to the case of non-dynamical gravitons. We will not repeat the analysis butsimply write down the solutions for reader’s reference. ∆ φ = 1 W gr = π x x (2 + U ( U − V −
1) ¯ D ) . (C.35) ∆ φ = 2 W gr = πx x (cid:18) U (cid:0) − ¯ D − D + ( U − V − D + 3 ¯ D ) (cid:1)(cid:19) . (C.36) ∆ φ = 3 W gr = πx x (cid:18)
12 + U (cid:0) − D − D −
16 ¯ D +( U − V − D + 6 ¯ D + 5 ¯ D ) (cid:1)(cid:19) . (C.37) ∆ φ = 4 W gr = πx x (cid:18)
288 + U (cid:0) −
36 ¯ D −
60 ¯ D −
45 ¯ D −
165 ¯ D +( U − V − D + 60 ¯ D + 60 ¯ D + 35 ¯ D ) (cid:1)(cid:19) . (C.38) References [1] L. Rastelli and X. Zhou, “Mellin amplitudes for
AdS × S ,” Phys. Rev. Lett. no. 9,(2017) 091602, arXiv:1608.06624 [hep-th] .[2] L. Rastelli and X. Zhou, “How to Succeed at Holographic Correlators Without ReallyTrying,”
JHEP (2018) 014, arXiv:1710.05923 [hep-th] .[3] G. Arutyunov, S. Frolov, R. Klabbers, and S. Savin, “Towards 4-point correlation functionsof any -BPS operators from supergravity,” JHEP (2017) 005, arXiv:1701.00998[hep-th] .[4] G. Arutyunov, R. Klabbers, and S. Savin, “Four-point functions of 1/2-BPS operators of anyweights in the supergravity approximation,” JHEP (2018) 118, arXiv:1808.06788[hep-th] .[5] G. Arutyunov, R. Klabbers, and S. Savin, “Four-point functions of all-different-weight chiralprimary operators in the supergravity approximation,” JHEP (2018) 023, arXiv:1806.09200 [hep-th] .[6] L. F. Alday and A. Bissi, “Loop Corrections to Supergravity on AdS × S ,” Phys. Rev. Lett. no. 17, (2017) 171601, arXiv:1706.02388 [hep-th] . – 25 –
7] F. Aprile, J. M. Drummond, P. Heslop, and H. Paul, “Quantum Gravity from ConformalField Theory,”
JHEP (2018) 035, arXiv:1706.02822 [hep-th] .[8] F. Aprile, J. M. Drummond, P. Heslop, and H. Paul, “Unmixing Supergravity,” JHEP (2018) 133, arXiv:1706.08456 [hep-th] .[9] F. Aprile, J. Drummond, P. Heslop, and H. Paul, “Double-trace spectrum of N = 4 supersymmetric Yang-Mills theory at strong coupling,” Phys. Rev.
D98 no. 12, (2018)126008, arXiv:1802.06889 [hep-th] .[10] S. Caron-Huot and A.-K. Trinh, “All tree-level correlators in AdS ÃŮS supergravity: hiddenten-dimensional conformal symmetry,” JHEP (2019) 196, arXiv:1809.09173 [hep-th] .[11] O. Aharony, L. F. Alday, A. Bissi, and E. Perlmutter, “Loops in AdS from Conformal FieldTheory,” JHEP (2017) 036, arXiv:1612.03891 [hep-th] .[12] F. Aprile, J. M. Drummond, P. Heslop, and H. Paul, “Loop corrections for Kaluza-Klein AdSamplitudes,” JHEP (2018) 056, arXiv:1711.03903 [hep-th] .[13] L. F. Alday and S. Caron-Huot, “Gravitational S-matrix from CFT dispersion relations,” JHEP (2018) 017, arXiv:1711.02031 [hep-th] .[14] V. Gonçalves, “Four point function of N = 4 stress-tensor multiplet at strong coupling,” JHEP (2015) 150, arXiv:1411.1675 [hep-th] .[15] L. F. Alday, A. Bissi, and E. Perlmutter, “Genus-One String Amplitudes from ConformalField Theory,” arXiv:1809.10670 [hep-th] .[16] L. F. Alday, “On Genus-one String Amplitudes on AdS × S ,” arXiv:1812.11783[hep-th] .[17] D. J. Binder, S. M. Chester, S. S. Pufu, and Y. Wang, “ N = 4 Super-Yang-Mills Correlatorsat Strong Coupling from String Theory and Localization,” arXiv:1902.06263 [hep-th] .[18] L. Rastelli and X. Zhou, “Holographic Four-Point Functions in the (2, 0) Theory,”
JHEP (2018) 087, arXiv:1712.02788 [hep-th] .[19] X. Zhou, “On Superconformal Four-Point Mellin Amplitudes in Dimension d > ,” JHEP (2018) 187, arXiv:1712.02800 [hep-th] .[20] P. Heslop and A. E. Lipstein, “M-theory Beyond The Supergravity Approximation,” JHEP (2018) 004, arXiv:1712.08570 [hep-th] .[21] X. Zhou, “On Mellin Amplitudes in SCFTs with Eight Supercharges,” JHEP (2018) 147, arXiv:1804.02397 [hep-th] .[22] S. M. Chester and E. Perlmutter, “M-Theory Reconstruction from (2,0) CFT and the ChiralAlgebra Conjecture,” JHEP (2018) 116, arXiv:1805.00892 [hep-th] .[23] T. Abl, P. Heslop, and A. E. Lipstein, “Recursion relations for anomalous dimensions in the6d (2 , theory,” JHEP (2019) 038, arXiv:1902.00463 [hep-th] .[24] S. M. Chester, “AdS /CFT for unprotected operators,” JHEP (2018) 030, arXiv:1803.01379 [hep-th] .[25] S. M. Chester, S. S. Pufu, and X. Yin, “The M-Theory S-Matrix From ABJM: Beyond 11DSupergravity,” JHEP (2018) 115, arXiv:1804.00949 [hep-th] .[26] D. J. Binder, S. M. Chester, and S. S. Pufu, “Absence of D R in M-Theory From ABJM,” arXiv:1808.10554 [hep-th] . – 26 –
27] G. Mack, “D-independent representation of Conformal Field Theories in D dimensions viatransformation to auxiliary Dual Resonance Models. Scalar amplitudes,” arXiv:0907.2407[hep-th] .[28] J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,”
JHEP (2011) 025, arXiv:1011.1485 [hep-th] .[29] M. F. Paulos, “Towards Feynman rules for Mellin amplitudes,” JHEP (2011) 074, arXiv:1107.1504 [hep-th] .[30] A. L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju, and B. C. van Rees, “A NaturalLanguage for AdS/CFT Correlators,” JHEP (2011) 095, arXiv:1107.1499 [hep-th] .[31] E. D’Hoker, D. Z. Freedman, and L. Rastelli, “AdS / CFT four point functions: How tosucceed at z integrals without really trying,” Nucl. Phys.
B562 (1999) 395–411, arXiv:hep-th/9905049 [hep-th] .[32] A. Galliani, S. Giusto, and R. Russo, “Holographic 4-point correlators with heavy states,”
JHEP (2017) 040, arXiv:1705.09250 [hep-th] .[33] A. Bombini, A. Galliani, S. Giusto, E. Moscato, and R. Russo, “Unitary 4-point correlatorsfrom classical geometries,” Eur. Phys. J.
C78 no. 1, (2018) 8, arXiv:1710.06820 [hep-th] .[34] S. Giusto, R. Russo, and C. Wen, “Holographic correlators in AdS ,” JHEP (2019) 096, arXiv:1812.06479 [hep-th] .[35] A. Bombini and A. Galliani, “AdS four-point functions from -BPS states,” arXiv:1904.02656 [hep-th] .[36] M. Heydeman, J. H. Schwarz, C. Wen, and S.-Q. Zhang, “All Tree Amplitudes of 6D (2 , Supergravity: Interacting Tensor Multiplets and the K Moduli Space,”
Phys. Rev. Lett. no. 11, (2019) 111604, arXiv:1812.06111 [hep-th] .[37] S. Giusto, R. Russo, A. Tyukov, and C. Wen, “Holographic correlators in
AdS withoutWitten diagrams,” to appear .[38] F. A. Dolan, L. Gallot, and E. Sokatchev, “On four-point functions of 1/2-BPS operators ingeneral dimensions,” JHEP (2004) 056, arXiv:hep-th/0405180 [hep-th] .[39] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, “Infinite ChiralSymmetry in Four Dimensions,” Commun. Math. Phys. no. 3, (2015) 1359–1433, arXiv:1312.5344 [hep-th] .[40] M. Baggio, J. de Boer, and K. Papadodimas, “A non-renormalization theorem for chiralprimary 3-point functions,”
JHEP (2012) 137, arXiv:1203.1036 [hep-th] .[41] S. Deger, A. Kaya, E. Sezgin, and P. Sundell, “Spectrum of D = 6, N=4b supergravity onAdS in three-dimensions x S**3,” Nucl. Phys.
B536 (1998) 110–140, arXiv:hep-th/9804166[hep-th] .[42] J. de Boer, “Six-dimensional supergravity on S**3 x AdS(3) and 2-D conformal field theory,”
Nucl. Phys.
B548 (1999) 139–166, arXiv:hep-th/9806104 [hep-th] .[43] J. de Boer, “Large N elliptic genus and AdS / CFT correspondence,”
JHEP (1999) 017, arXiv:hep-th/9812240 [hep-th] .[44] G. Arutyunov, A. Pankiewicz, and S. Theisen, “Cubic couplings in D = 6 N=4b supergravityon AdS(3) x S**3,” Phys. Rev.
D63 (2001) 044024, arXiv:hep-th/0007061 [hep-th] . – 27 –
45] X. Zhou, “Recursion Relations in Witten Diagrams and Conformal Partial Waves,” arXiv:1812.01006 [hep-th] .[46] B. Eden, A. C. Petkou, C. Schubert, and E. Sokatchev, “Partial nonrenormalization of thestress tensor four point function in N=4 SYM and AdS / CFT,”
Nucl. Phys.
B607 (2001)191–212, arXiv:hep-th/0009106 [hep-th] .[47] Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Supersymmetry Constraints and StringTheory on K3,”
JHEP (2015) 142, arXiv:1508.07305 [hep-th] .[48] G. Arutyunov and S. Frolov, “On the correspondence between gravity fields and CFToperators,” JHEP (2000) 017, arXiv:hep-th/0003038 [hep-th] .[49] M. Taylor, “Matching of correlators in AdS(3) / CFT(2),” JHEP (2008) 010, arXiv:0709.1838 [hep-th] .[50] J. Maldacena, “Einstein Gravity from Conformal Gravity,” arXiv:1105.5632 [hep-th] .[51] V. Gonçalves, R. Pereira, and X. Zhou, “ (cid:48) Five-Point Function from
AdS × S Supergravity,” to appear .[52] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Graviton and gaugeboson propagators in AdS(d+1),”
Nucl. Phys.
B562 (1999) 330–352, arXiv:hep-th/9902042 [hep-th] .[53] K. Pilch and A. N. Schellekens, “Formulae for the Eigenvalues of the Laplacian on TensorHarmonics on Symmetric Coset Spaces,”
J. Math. Phys. (1984) 3455.(1984) 3455.