More on holographic correlators: Twisted and dimensionally reduced structures
MMore on holographic correlators:
Twisted and dimensionally reduced structures
Connor Behan, a Pietro Ferrero, a Xinan Zhou b a Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quar-ter, Woodstock Road, Oxford, OX2 6GG, U.K. b Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA
E-mail: [email protected],[email protected],[email protected]
Abstract:
Recently four-point holographic correlators with arbitrary external BPS op-erators were constructively derived in [1, 2] at tree-level for maximally superconformal the-ories. In this paper, we capitalize on these theoretical data, and perform a detailed study oftheir analytic properties. We point out that these maximally supersymmetric holographiccorrelators exhibit a hidden dimensional reduction structure `a la
Parisi and Sourlas. Thisemergent structure allows the correlators to be compactly expressed in terms of only scalarexchange diagrams in a dimensionally reduced spacetime, where formally both the AdSand the sphere factors have four dimensions less. We also demonstrate the superconformalproperties of holographic correlators under the chiral algebra and topological twistings. For
AdS × S and AdS × S , we obtain closed form expressions for the meromorphic twistedcorrelators from the maximally R-symmetry violating limit of the holographic correlators.The results are compared with independent field theory computations in 4d N = 4 SYMand the 6d (2 ,
0) theory, finding perfect agreement. For
AdS × S , we focus on an infinitefamily of near-extremal four-point correlators, and extract various protected OPE coeffi-cients from supergravity. These OPE coefficients provide new holographic predictions to bematched by future supersymmetric localization calculations. In deriving these results, wealso develop many technical tools which should have broader applicability beyond studyingholographic correlators. a r X i v : . [ h e p - t h ] J a n ontents d > W -algebra correlators at large N 184.2 Matching from Mellin space 21 W -algebra correlators at large N 245.2 Matching from Mellin space 28 A.1 Four dimensions 55A.2 Six dimensions 57
B The infinite sum over descendants 58
B.1 Integers vs half-integers 58B.2 Translation to a finite sum 59– 1 –
Introduction
Historically, the holographic computation of superconformal correlators using AdS super-gravity has been an extremely difficult task, even just at tree-level and for four half-BPSoperators (see, e.g. , [3–7] for early progress). For example, the complete quartic verticesfor
AdS × S IIB supergravity were worked out in [8] and occupied 15 pages. The sheercomplexity of the vertices presents a daunting challenge for the standard diagrammaticexpansion method, rendering it practically useless for generic correlators. However, animportant breakthrough was made in [9, 10], where a different line of attack was intro-duced by incorporating bootstrap ideas. It was argued that holographic correlators havevery rigid structures, and can be therefore completely determined by using only supersym-metry and consistency conditions. This led to several efficient new methods to computeholographic correlators, which avoid inputting the details of the complicated supergravityeffective actions altogether. Most spectacularly, the bootstrap methods gave a completesolution to all tree-level four-point functions of half-BPS operators for
AdS × S IIB su-pergravity [9, 10], without computing a single Witten diagram. On the other hand, thesuccess of these methods in other maximally supersymmetric backgrounds (
AdS × S and AdS × S ) was much more modest. While many new results, unattainable by brute force,were generated by using these methods [11–13], the algebraic bootstrap problems were toodifficult to solve in general. It was only recently that a different constructive method wasdeveloped [1, 2] which completed the program for tree-level half-BPS four-point functionsin maximally supersymmetric theories. The constructive method starts with a special R-symmetry polarization configuration, dubbed “maximally R-symmetry violating” in [1, 2],where correlators drastically simplify and can be easily computed. The full correlators arethen reconstructed from this limit by using symmetries. This method applies to any space-time dimension, and leads to a closed form formula for all tree-level four-point functionsin all maximally supersymmetric theories [1, 2].The solution of general tree-level four-point functions generates a wealth of new theo-retical data. The purpose of the current paper is to capitalize on these data and extractuseful physical information. Our analysis will include two complementary aspects. Thefirst is to find new hidden structures in the correlators. Identifying these hidden structuresnot only leads to more compact expressions, but also suggests new symmetry properties ofthe bulk theory. In particular, we will show that the correlators exhibit an emergent di-mensional reduction structure, which is closely related to the Parisi-Sourlas supersymmetry[14, 15]. The second aspect concerns certain protected subsectors of the superconformalfield theories. Their information can be cleanly isolated by focusing on special twistedconfigurations of the supergravity correlators. We also perform independent boundary cal-culations, which perfectly match the bulk predictions. This provides non-trivial checks forseveral conjectures in the protected sector. Below we provide a more detailed summary ofour main results, embedded in brief reviews of related backgrounds. For an overview of these bootstrap methods, see section 2 of [2]. – 2 – mergent dimensional reduction
It is well known that scattering amplitudes in flat space often contain surprising structureswhich reveal unexpected symmetries hidden from the Lagrangian description. Viewed asscattering amplitudes in anti de Sitter space, it is not surprising that holographic correlatorscan similarly exhibit hidden structures which cannot be seen from the diagrammatic ex-pansion. A beautiful example is the hidden ten dimensional conformal symmetry observedin
AdS × S [16], which organizes all tree-level four-point functions into a generating func-tion. The generating function is nothing but the stress tensor four-point function afterreplacing four dimensional distances with ten dimensional distances. A similar six dimen-sional version was also discovered for AdS × S × K AdS × S and AdS × S ).In this paper, we will point out a different hidden structure in tree-level four-pointfunctions that involves dimensional reduction, and is present in all maximally superconfor-mal theories. We recall from [1, 2] that four-point functions in AdS d +1 × S d − are sums ofexchange amplitudes over finitely many supergravity multiplets, with no additional contactinteractions. We will show that the exchange amplitude of a multiplet p can be simplifiedas a differential operator acting on only three scalar exchange amplitudes. Schematically,we have Diff ◦ ( M ( d ) (cid:15)p + M ( d ) (cid:15)p +2 + M ( d ) (cid:15)p +4 ) (1.1)where (cid:15) = ( d − /
2, and M ( d )∆ is the scalar exchange amplitude in AdS d +1 with internaldimension ∆. Note that in addition to superconformal primary of the multiplet with∆ = (cid:15)p , there are also new scalars with shifted conformal dimensions. The spinningfield contributions in the multiplet are generated by the differential operator action. Quitemagically, the combination of diagrams in (1.1) turns out to be just a single scalar exchangediagram in a lower dimensional AdS d − space M ( d ) (cid:15)p + M ( d ) (cid:15)p +2 + M ( d ) (cid:15)p +4 = M ( d − (cid:15)p . (1.2)A similar pattern can be found for the R-symmetry part, which leads to a dimensionallyreduced internal space S d − . These observations allow us to repackage the full multipletamplitude into the following simple formDiff (cid:48) ◦ ( Y d − p, M ( d − (cid:15)p ) (1.3)where Y d − p, is the R-symmetry polynomial of the rank- p symmetric traceless representationof the reduced R-symmetry group SO ( d −
4) associated with S d − . Consequently, thecorrelators in the original theory are now fully captured by the correlators of a simplescalar theory in the lower dimensional spacetime AdS d − × S d − !The appearance of dimensionally reduced spacetimes is quite curious, and we do nothave a good understanding of its physical origin. Nevertheless, we find strong evidenceindicating that this reduction phenomenon is intimately related to the Parisi-Sourlas di-mensional reduction, which relates a d dimensional theory with Parisi-Sourlas supersym-metry and a d − AdS d +1 and AdS d − ,which was shown in [19] as the consequence of the holographically realized Parisi-Sourlassupersymmetry. It turns out that using the dimensional reduction formula twice givesprecisely (1.2). Twisted correlators and protected sectors
Common to the maximally superconformal theories considered in this paper is the existenceof a protected subsector of operators. These operators are formed by restricting a certainclass of operators (half-BPS operators are such examples) to a two dimensional plane ora one dimensional line, and “twisting” the operators by giving them special R-symmetrypolarizations specified by the locations of the insertions [20–23]. Crucially, the subsectorinvolves only members of short multiplets, closes on itself under OPE, and does not dependon marginal deformations (if there are any) [20]. Therefore this construction isolates a fullyprotected subsector. The protected subsector has the form of a unitary chiral algebra insix dimensions [21], a non-unitary chiral algebra in four dimensions [20] and topologicalquantum mechanics in three dimensions [22, 23]. The presence of the protected subsectorimmediately imposes strong constraints on the correlation functions, as they must becomemeromorphic or topological in the twisted configurations. It was also quickly recognizedthat non-trivial unitarity bounds are encoded in these rigid structures analytically [20, 24–26]. Along the same lines, one can use chiral algebra or topologically protected datato greatly increase the power of the numerical conformal bootstrap [22, 27–36]. Whencombined with supersymmetric localization, these techniques can also be used to predictnew holographic dualities [37–39].While the existence of a 6d/2d or 4d/2d correspondence, in the sense described above,is a proven fact, a precise description of the lower dimensional theory is not always readilyavailable. In well studied cases, e.g. , theories of Argyres-Douglas type or class S , muchof the recent progress in understanding their chiral algebras has been guided by a setof compelling conjectures [20, 21, 40–42]. The 3d/1d correspondence is on a somewhatdifferent footing due to the explicit Lagrangians derived in [43–45]. Nevertheless, applyingthese results to ABJM theory requires the use of conjectural dualities [46, 47] and eventhen the calculations can be very involved.In this work, we further strengthen the evidence for these conjectures in the case ofholographic CFTs with maximal supersymmetry. We perform independent calculationsfrom both the field theory side and the supergravity side, finding perfect agreement. For6d and 4d, where the twisting leads to meromorphic functions, we compute four-point chi-ral correlators on the field theory side by improving the holomorphic bootstrap method of[48]. The basic idea is that meromorphic functions are determined by singularities whichare dictated by the OPE of the chiral algebra. However, the improved algorithm allowsus to write down closed form expressions for all chiral four-point functions, in a form thatforeshadows the structures anticipated from the supergravity side. We then reproducethese results in Mellin space by taking residues, from the MRV limit of the supergravityamplitudes [1, 2]. For the 3d case, the topological nature of the twisted correlators makes– 4 –he situation inevitably more involved. Therefore, we will content ourselves with mostlyfocusing on the next-to-next-to-extremal correlators of the form (cid:104)O O O k O k (cid:105) . Moreover,instead of attempting to find closed form expressions for the topological correlators, we willfocus on identifying the finitely many operators which contribute to a topological correla-tor, and computing their OPE coefficients in a 1 /c T expansion. Similarly to the 6d and 4dcases, we will approach the 3d problem both from the boundary TQFT description, andfrom the bulk side using the Mellin space results. To summarize, the calculations outlinedabove provide non-trivial checks for the conjectures in the protected sector. Given thatmany aspects of the protected subsectors appear to be within reach of a formal proof, itmight also be helpful to turn the logic around and view our results as a tree-level check ofAdS/CFT. In performing these checks, we have developed many technical tools for study-ing holographic four-point functions. We expect these tools will have broader applicability.The rest of the paper is organized as follows. We set the stage in section 2 by reviewingsome basic superconformal kinematics. In section 3, we show that the holographic corre-lators have an emergent dimensional reduction structure, and explain the connection withthe Parisi-Sourlas supersymmetry. Discussions of the protected subsectors begin in section4, where we show the agreement between the field theory and supergravity calculations for6d. In section 5, we present a parallel check for the 4d case. The 3d case is discussed insection 6 to 8. We explain in section 6 the reason why the topological property makes thecalculations different. The field theory computations are carried out in section 7, and thesupergravity analysis is done in section 8. We conclude in section 9 with a brief discussionof future directions. Various technical details are relegated to the two appendices. To start, we will use this section to fix our notation and review some basic superconformalkinematics of four-point correlation functions. In particular, we will review a powerfulkinematic constraint, namely the superconformal Ward identities in diverse dimensions[49, 50], which served as a precursor to the SCFT/chiral algebra correspondence. d >
AdS × S , AdS × S and AdS × S , which havethe maximal amount of superconformal symmetry. In these backgrounds all supergravitysingle-particle states are dual to components of the half-BPS supermultiplets. The superprimaries of these half-BPS multiplets are scalar operators of the form O I ...I k k ( x ), whichtransform in rank- k symmetric traceless representations of the R-symmetry group. Thescaling dimensions of the half-BPS operators are fixed by the R-symmetry charges∆ = (cid:15)k, (cid:15) ≡ d −
22 (2.1)where k ∈ { , , . . . } . We will saturate the R-symmetry indices of these operators withpolarization vectors t I that are null in order to respect tracelessness O k ( x, t ) = O I ...I k k ( x ) t I . . . t I k , t · t = 0 . (2.2)– 5 –he central objects of this paper are the four-point correlation functions of these operators (cid:104)O ( x , t ) O ( x , t ) O ( x , t ) O ( x , t ) (cid:105) . (2.3)The extremality for a correlator is defined as E = (cid:40) k ¯1 + k ¯2 + k ¯3 − k ¯4 , k ¯1 + k ¯4 ≥ k ¯2 + k ¯3 k ¯1 , k ¯1 + k ¯4 < k ¯2 + k ¯3 (2.4)where k ¯ ı is the i -th smallest element of { k , k , k , k } . The standard conformal and R-symmetry cross-ratios are U = x x x x , V = x x x x , σ = t t t t , τ = t t t t , (2.5)with x ij ≡ x i − x j and t ij ≡ t i · t j . We will exploit conformal symmetry and R-symmetry toextract a kinematic factor such that the correlator becomes a function of the cross-ratios,and depends on σ and τ as a polynomial of total degree E . If the weights are ordered as k ≤ k ≤ k ≤ k , a convention that accomplishes this is (cid:104)O ( x , t ) O ( x , t ) O ( x , t ) O ( x , t ) (cid:105) = (cid:18) t x (cid:15) (cid:19) ( k + k − k − k ) (cid:18) t x (cid:15) (cid:19) ( k + k − k − k ) (cid:18) t x (cid:15) (cid:19) ( k + k + k − k ) −E (cid:18) t x (cid:15) (cid:19) k −E (cid:18) t t x (cid:15) x (cid:15) (cid:19) E G ( U, V ; σ, τ ) . (2.6)However, we should stress that when working with various correlators, we will specifythe ordering and often it will be different from k ≤ k ≤ k ≤ k . The formulas weintroduce in this section and section 3, which form the basis for later results, will be validfor arbitrary ordering of k i as long as the σ and τ dependence remains polynomial withdegree E . In practice, we implement this by interchanging the four labels in (2.6) accordingto the permutation which takes us from k ≤ k ≤ k ≤ k to whatever the new orderingis. The dynamical function G ( U, V ; σ, τ ) will have two important pieces in this work –the disconnected part and the tree-level part. We are allowed to use these terms becauseholographic CFTs admit a “large N limit”, and disconnected and tree-level correspondto the first two orders in the 1 /N expansion. In AdS/CFT, N is typically the numberof branes that give rise to the near-horizon geometry that has an AdS d +1 factor. Sincetree-level corrections scale with a different power of N in each dimension, namely N − − (cid:15) ,it is useful to quote results in terms of the central charge. c (6D) T = 70(4 N − N −
1) (2.7) c (4D) T = 30( N −
1) (2.8) CFTs in even dimension have an additional notion of central charge because they exhibit a Weylanomaly when placed on a curved manifold [51]. A universal prefactor, computed in [20, 21], relates thecoefficient of the Euler density to the corresponding Virasoro central charge which we will call c d in latersections. – 6 – (3D) T ≈ − π − (8 N + 9) (cid:18) π (cid:19) − (cid:48) (cid:20)(cid:0) N − (cid:1) (cid:16) π (cid:17) (cid:21) (cid:20)(cid:0) N − (cid:1) (cid:16) π (cid:17) (cid:21) (2.9) ≈ √ π N The 3d expression, which includes all perturbative terms, can be computed using thetechniques of [52, 53]. Note that we are using conventions such that (cid:104) T µν ( x ) T ρσ (0) (cid:105) = dd − (cid:0) d (cid:1) π d (cid:20) I µσ I νρ + 12 I µρ I νσ − d δ µν δ σρ (cid:21) c T x d (2.10) I µν ≡ δ µν − x µ x ν x which means that a single free boson has c T = 1 [54]. The four-point functions (2.6), by construction, obey all of the Ward identities from or-dinary conformal symmetry and R-symmetry. In addition to this, we must consider su-perconformal Ward identities associated with the fermionic generators. The action of thefermionic generators generically relate correlation functions of superconformal primaries tothose of their super descendants – a complicated bootstrap problem in general (see, e.g. ,[55]). The situation becomes nicer when the external operators are the super primariesof the half-BPS multiplets. In this case, Ward identities from superconformal symmetrylead to constraints which only involve the original four-point function. More precisely, thisconstraint takes the universal form [50] (cid:20) χ (cid:48) ∂∂χ (cid:48) − (cid:15)α (cid:48) ∂∂α (cid:48) (cid:21) G ( χ, χ (cid:48) ; α, α (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) α (cid:48) = χ (cid:48) = 0 (2.11)which is often called the superconformal Ward identity . Here we have introduced the changeof variables U = χχ (cid:48) , V = (1 − χ )(1 − χ (cid:48) ) , σ = αα (cid:48) , τ = (1 − α )(1 − α (cid:48) ) . (2.12)These new variables will be identified with cross-ratios in one dimension ( i.e. , for sl (2) and su (2)), as we will be more specific about in the next subsection.A direct consequence of (2.11) is that there are certain loci in cross-ratio space, onwhich the four-point functions are topological along certain directions. We will call thesespecial choices of cross-ratio configurations twisted configurations . These configurationsexist in d = 3 , ,
6, with the prescription being slightly different in each case. We will beinterested in seeing how this structure is captured in the results of
AdS × S , AdS × S and AdS × S supergravity respectively.Starting with d = 6, (2.11) implies (cid:15) = 2 : ∂∂χ (cid:48) G (cid:18) χ, χ (cid:48) ; 1 χ (cid:48) , χ (cid:48) (cid:19) = 0 . (2.13)– 7 –his is the statement that the twisted correlator is a holomorphic function. It is now un-derstood that, far from being an arbitrary holomorphic function, the holomorphic functionis really a four-point function of an auxiliary chiral algebra in two dimensions [21]. Sincethis conclusion required us to have two independent R-symmetry cross-ratios, it is easy tosee that it only applies to the 6d superconformal algebra osp (8 ∗ | d = 4 according to (cid:15) = 1 : ∂∂χ (cid:48) G (cid:18) χ, χ (cid:48) ; α, χ (cid:48) (cid:19) = 0 . (2.14)This time, the structure is somewhat richer since we have a holomorphic dependence on α as well. This means there is still a notion of R-symmetry for the twisted correlator. Asexplained in [20], this is a consequence of the chiral algebra including super-Virasoro gen-erators whenever the 4d theory has psu (2 , |
4) or su (2 , |
3) symmetry. The chiral algebra isnon-supersymmetric for su (2 , | α is not an independentcross-ratio in these theories.Finally, for d = 3 we have (cid:15) = 12 : ∂∂χ (cid:48) G (cid:18) χ (cid:48) , χ (cid:48) ; α, χ (cid:48) (cid:19) = 0 . (2.15)This indicates that, if we take a diagonal limit first, the remaining spatial cross-ratio dropsout after the twist. More precisely, the correlator only depends on space-time positionsthrough the ordering since the superconformal Ward identity is a local statement. This iswhat gives the one dimensional protected subsector the structure of topological quantummechanics [22, 23]. To still interpret this object as a CFT in one dimension, it would haveto consist solely of dimension zero operators which can be thought of as conserved currentsfor a higher-spin algebra [38]. This 3d/1d correspondence applies to osp ( N |
4) with
N ≥ α is always present.Although (2.11) is a statement about half-BPS four-point functions, the results of[20–23] apply to a larger family of operators in short multiplets, and also to higher-pointfunctions. From a holographic point of view, the easiest way to study correlators of theseoperators would be to perform the OPE on half-BPS correlators. However, this will notbe pursued here. It is also worth noting that we will encounter the d = 2 superconformalWard identity in section 5 when we construct psu (1 , |
2) blocks. The implications of thisfor protected quantities were discussed in [17].
Correlators of operators in lower dimensions emerge from correlators in higher dimensionaltheories after performing superconformal twisting, as we have seen from the discussionon superconformal Ward identities in the last subsection. In this subsection, we give asummary of the kinematics of correlators of local operators on the plane (or restricted toa line).Let us consider an operator with conformal weight h and su (2) spin j . The su (2) refersto the possible residual R-symmetry after twisting. We can denote the operator as O h,j ( z, y ) = O h,j | a ...a j ( z ) y a . . . y a j (2.16)– 8 –here we have similarly used su (2) spinors y a to keep track of the R-symmetry indices. For su (2) and sl (2), only one independent cross-ratio can be formed from four points. Theselower dimensional cross-ratios are defined in terms of the coordinates as χ = z z z z , α = y y y y , (2.17)and can be identified with the symbols in (2.12). Note that z ij ≡ z i − z j and y ij ≡ y ai y bj (cid:15) ab .Covariance under sl (2) and su (2) constrains the possible terms in the OPE of theoperators (2.16). To discuss these constraints individually, let us pretend for a momentthat the y and z dependence can be separated. O h ( z ) O h ( z ) = (cid:88) O λ O ∞ (cid:88) m =0 ( h + h ) m m !(2 h ) m ∂ m O h ( z ) z h + h − h − m (2.18a) O j ( y ) O j ( y ) = (cid:88) O λ O j ! (cid:88) σ ∈ S j O j | a σ (1) ...a σ (2 j ) y a . . . y a j + j y a j + j . . . y a j (2.18b)In both cases, the outer sum runs over (quasi)primaries while the inner sum covers thestates other than the highest weight. The chiral OPE (2.18a), first applied to the searchfor W -algebras in [58], leads to the familiar s -channel expansion of the four-point function. (cid:104)O h ( z ) O h ( z ) O h ( z ) O h ( z ) (cid:105) = (cid:18) z z (cid:19) h (cid:18) z z (cid:19) h F ( χ ) z h + h z h + h (2.19) F ( χ ) = (cid:88) O λ O λ O g h ,h h ( χ )where the sl (2) blocks are given by g h ,h h ( χ ) = χ h F ( h − h , h + h ; 2 h ; χ ) . (2.20)In fact, the coefficients in (2.18a) can be fixed by demanding that (2.19) is reproduced. Aconvenient result, which can be shown with embedding space methods [59, 60], is that thesecond OPE (2.18b) leads to an entirely analogous expression with h (cid:55)→ − j everywhere. (cid:104)O j ( y ) O j ( y ) O j ( y ) O j ( y ) (cid:105) = (cid:18) y y (cid:19) j (cid:18) y y (cid:19) j y j + j y j + j F ( α ) (2.21) F ( α ) = (cid:88) O λ O λ O g j ,j − j (cid:18) α (cid:19) = (cid:88) O λ O λ O ( − j ( j + j )! P j − j ,j − j j + j (1 − α )( − α ) j ( j − j + 1) j + j Clearly, a pure su (2) representation – one with j as the only non-zero quantum number – cannot satisfythe unitarity bounds. We will also not see operators like this in the non-unitary chiral algebra discussedin section 5. Instead, to apply (2.18a) and (2.18b) to a reasonable theory, they should be combined into asingle OPE. Note however that non-unitary multiplets, even when absent from the theory, play a role indetermining the superconformal blocks [56, 57]. This is far from the only reason to consider the OPE directly in embedding space [61]. – 9 –e have used a standard hypergeometric function identity in the last line, relating sl (2)blocks and Jacobi polynomials, in order to make contact with another convention in theliterature. Note also that the kinematic y ij factors we extracted here are different from theones in (2.6). In this section, we point out interesting hidden structures in tree-level maximally super-symmetric four-point Mellin amplitudes. We start by reviewing the results of [1, 2] insection 3.1. We then show in section 3.2 that correlators can be expressed as finite lin-ear combinations of scalar exchange Witten diagrams, acted on by differential operatorsof cross-ratios. The combination coefficients are highly special and exhibit an emergentParisi-Sourlas symmetry. This allows the sum of Witten diagrams in each multiplet to bewritten in terms of a single scalar exchange Witten diagram in an AdS space with fourdimensions fewer. An analogous structure exists for the R-symmetry dependence, wherethe dimension of the internal manifold is similarly reduced by four. In section 3.3 we usethese observations to provide a compact way to rewrite the holographic correlators, whichmanifests supergraph-like structures. The form of the result also suggests a lower dimen-sional scalar seed theory which encodes all the essential data. Correlators of the originaltheory can be obtained by dressing the seed theory correlators with differential operators.
The natural language for holographic correlators is the Mellin representation [62–64] whichmakes the analytic structure manifest. In the four-point case, the Mellin amplitude involvesMandelstam-like variables satisfying s + t + u = (cid:80) i =1 ∆ i . Tree-level Mellin amplitudes fora CFT with a weakly coupled gravity dual take the form M ( s, t ) = M ( s ) ( s, t ) + M ( t ) ( s, t ) + M ( u ) ( s, t ) + M contact ( s, t ) . (3.1)Here M ( s ) , M ( t ) , M ( u ) correspond to exchange contributions, and have simple poles inthe Mandelstam variable of the respective channel. M contact is regular and accounts foradditional contact interactions. Upon transforming back to position space, the poles inthe Mellin amplitude – from exchange Witten diagrams – produce conformal blocks foran internal single-particle operator. Poles corresponding to double-particle operators areinstead contained in the measure for the inverse Mellin transformation which is ideallysuited to theories at large N . One often hears these types of operators referred to as “single-trace” and “double-trace” respectively.Strictly speaking, this is only correct in the strict N → ∞ limit. Even if one is not concerned with anymicroscopic Lagrangian, there is a well defined basis of double-particle operators which have correctionterms in addition to normal ordered squares of single-particle operators [65, 66]. – 10 –estoring the dependence on σ and τ , the inverse Mellin transformation is given bythe following contour integral G conn1234 ( U, V ; σ, τ ) = (cid:90) i ∞− i ∞ d s d t (4 πi ) U s − a s V t − a t M ( s, t ; σ, τ ) (3.2) × Γ (cid:20) ∆ + ∆ − s (cid:21) Γ (cid:20) ∆ + ∆ − t (cid:21) Γ (cid:20) ∆ + ∆ − u (cid:21) × Γ (cid:20) ∆ + ∆ − s (cid:21) Γ (cid:20) ∆ + ∆ − t (cid:21) Γ (cid:20) ∆ + ∆ − u (cid:21) where a s = (cid:15) k + k ) − (cid:15) E , a t = (cid:15) E − (cid:15) k − k ) , (3.3)and we recall that ∆ i = (cid:15)k i . To make sense of this, we need to realize that the shape of thecontour is different from the schematic one above. The well known prescription is to choosea contour which makes the real part positive in the arguments of all six gamma functions.This is always possible for a G conn which does not require regularization. This restrictionon the contour, based on double-particle poles, is not sufficient because the critical stripdescribed above usually contains some of the single-particle poles of the Mellin amplitudeitself. To resolve this ambiguity, we must put single-particle and double-particle exchangeson the same footing, i.e. , by choosing a contour such that the poles of M ( s ) and M ( t ) lieto the right while those of M ( u ) lie to the left. In other words, whether we are talkingabout the gamma functions or Mellin amplitudes, we keep semi-infinite sequences of polesthat increase the exponents but not semi-infinite sequences of poles that decrease them. Clearly, these towers have integer spacing because an exchanged primary appears togetherwith its descendants. The Mellin amplitudes of interest to us were constructively derived for all externalhalf-BPS operators in [1, 2] for all three maximally supersymmetric backgrounds. Thederivation exploited a special R-symmetry configuration, dubbed maximally R-symmetryviolating (MRV) in [1, 2], where major simplifications occur. The full amplitudes werethen obtained from the MRV amplitudes by using symmetries. A remarkable feature inthese results is that all the contact terms in the amplitudes vanish, after using a naturalprescription to symmetrize the exchange amplitudes recovered from the MRV limit. There-fore the full amplitudes can be written as the sum over only exchange amplitudes in threechannels. Specifically, [2] gives a formula for the Mellin amplitudes as a sum over simple The 4 πi , which might look strange, becomes 2 πi again after mapping s , t and u to the variables whichgeneralize more easily to arbitrary n -point correlators. In the case of
AdS × S and AdS × S , the spectrum is such that the single-particle sequences are infact finite [67]. The truncation occurs as a consistency condition of the 1 /N expansion [10]. The residuesof M ( s, t ; σ, τ ) must conspire to stop the single-particle and double-particle poles from overlapping whichwould lead to higher order singularities that are unphysical at tree-level. This phenomenon was historicallyvery important in the holographic correlator program before methods were developed in [12] to crack the AdS × S case as well. We should note that this logic only defines a correlator when there is some notion of which poles belongtogether. In the non-perturbative context [68], a Mellin amplitude is only useful if the appropriate contouris specified along with it. – 11 –oles M ( s )1234 ( s, t ; σ, τ ) = max {| k | , | k |} +2 E− (cid:88) p =max {| k | , | k |} +2step 2 ∞ (cid:88) m =0 (cid:88) ≤ i + j ≤E σ i τ j R p,m ; i,j ( t, u ) s − m − (cid:15)p (3.4)with M ( t )1234 ( s, t ; σ, τ ) = τ E M ( s )3214 (cid:18) t, s ; στ , τ (cid:19) M ( u )1234 ( s, t ; σ, τ ) = σ E M ( s )4231 (cid:18) u, t ; 1 σ , τσ (cid:19) (3.5)determining the other channels. The integer p labels the exchanged supergravity mul-tiplets whose super primaries are dual to the half-BPS operators O p . The R-symmetryselection rule, and the requirement of the effective action being finite, restrict the range of p to the finite set as indicated in the above sum.For the reader’s convenience, we now give explicit expressions for the residues. In [1, 2],they took the form R i,jp,m ( t, u ) = K i,jp ( t, u ) L i,jp,m N i,jp , (3.7)where we have dropped the position labels as we will often do when they are clear fromthe context. The factor of (3.7) which encodes all the t and u dependence is K i,jp = 2 i (2 i + κ u ) t − t + + 2 j (2 j + κ t ) u − u + − j ( (cid:15) − κ u ) t + u − − i ( (cid:15) − κ t ) u + t − + 14 (2 p − κ t − κ u )(2 p + (cid:15) − κ t + κ u )( u − t − + 4 (cid:15) ij )+ (cid:15) ( κ u + κ t − p )( κ u + κ t + 2 p + (cid:15) − it − + ju − )+ 4 (cid:15)ij ( t + ( κ u + (cid:15) −
2) + u + ( κ t + (cid:15) − − ijt + u + , (3.8)where we have defined κ s ≡ | k + k − k − k | , κ t ≡ | k + k − k − k | , κ u ≡ | k + k − k − k | , (3.9)and introduced the shorthand notation u ± = u ± (cid:15) κ u − (cid:15) , t ± = t ± (cid:15) κ t − (cid:15) k + k + k + k .For the other two factors in (3.7), we will find it convenient to rewrite them in a waywhich is manifestly proportional to the OPE coefficients of the exchanged super primaries O p . In the resulting expression R i,jp,m ( t, u ) = C k k p C k k p K i,jp ( t, u ) H i,jp,m , (3.11) Identities for the s -channel amplitude itself, which might be useful to keep in mind, are M ( s )1234 ( s, t ; σ, τ ) = M ( s )3412 ( s, t ; σ, τ ) = M ( s )2134 ( s, u ; τ, σ ) . (3.6) – 12 –e have H i,jp,m = 2 − (cid:16) (cid:15) − (cid:15) +9 (cid:17) Γ[ (cid:15)p ]Γ[ (cid:15) ( p + (cid:15) − p + k ]Γ[ p − k ]Γ[ p + k ]Γ[ p − k ]Γ[ p ]Γ[ p + (cid:15) − (cid:15) ( p + k )]Γ[ (cid:15) ( p − k )]Γ[ (cid:15) ( p + k )]Γ[ (cid:15) ( p − k )] (3.12) × ( − i + j + p − κt − κu Γ[ κ u +2+2 i ] − Γ[ ( (cid:15) − p + Σ − κ s − l )]Γ[ κ t +2+2 j ] − i ! j ! m !Γ[2 − (cid:15) + m + (cid:15)p ]Γ[ (cid:15) ( k + k − p )2 − m ]Γ[ p +2) − Σ+ κ s +4 l ]Γ[ (cid:15) ( k + k − p )2 − m ] . and l = E − i − j , which is the analogue of u for the R-symmetry. The OPE coefficients,which come from the cubic couplings in the supergravity effective action, are given by[69–71] C k k k = πN − β − Γ[ β +1] (cid:81) i =1 √ Γ[ k i +2]Γ[ αi +12 ] , d = 3 √ k k k N , d = 4 β − ( πN ) Γ[ β ] (cid:81) i =1 Γ[ α i + ] √ Γ[2 k i − , d = 6 (3.13)with α = 12 ( k + k − k ) , α = 12 ( k + k − k ) , α = 12 ( k + k − k ) , (3.14)and β = α + α + α . The above answer gives explicit expressions for holographic correlators of four arbitraryhalf-BPS operators in all three maximally supersymmetry theories. The fact that they canbe cast into the same form as functions of (cid:15) already shows an unexpected universality ofamplitudes in diverse dimensions. In this subsection, we further analyze these amplitudesto uncover more interesting properties. We will show that the maximally supersymmetricMellin amplitudes exhibit a surprising hidden structure, where a dimensionally reducedspacetime naturally arises. The emergence of a lower dimensional spacetime in these am-plitudes is highly reminiscent of the Parisi-Sourlas dimensional reduction.Let us expose this structure by separating the amplitudes into different parts. Westart from the AdS part of the amplitudes, which concerns only the dependence on theMandelstam variables. First we observe that the Mandelstam variables appear in K i,jp ( t, u )as polynomials. We can bring this factor outside of the Mellin representation (3.2) asdifferential operators via the following identification U ∂ U ↔ ( s − a s ) × , V ∂ V ↔ ( t − a t ) × . (3.15)Therefore, we only need to focus on the sum over poles in s which are labelled by m . Letus isolate the m -dependence from H i,jp,m , which reads A p,m = 1 m !Γ[2 − (cid:15) + m + (cid:15)p ]Γ[ (cid:15) ( k + k − p )2 − m ]Γ[ (cid:15) ( k + k − p )2 − m ] , (3.16)and denote the sum as F p ( s ) = ∞ (cid:88) m =0 A p,m s − (cid:15)p − m . (3.17)– 13 – priori , we do not expect the function F p ( s ) to have any special property. Remarkably,however, we find that the sum can always be written as a linear combination of exactly three scalar exchange Witten diagrams! More precisely, we have F p ( s ) = N (cid:0) M ( d ) (cid:15)p ( s ) + c M ( d ) (cid:15)p +2 ( s ) + c M ( d ) (cid:15)p +4 ( s ) (cid:1) (3.18)where c = − (cid:0) (cid:15) ( p + k )2 (cid:1)(cid:0) (cid:15) ( p − k )2 (cid:1)(cid:0) (cid:15) ( p + k )2 (cid:1)(cid:0) (cid:15) ( p − k )2 (cid:1) (cid:15)p ) (cid:0) (cid:15) ( p − (cid:1) ,c = (cid:0) (cid:15) ( p + k )2 (cid:1) (cid:0) (cid:15) ( p − k )2 (cid:1) (cid:0) (cid:15) ( p + k )2 (cid:1) (cid:0) (cid:15) ( p − k )2 (cid:1) (1 + (cid:15) ( p − ( (cid:15)p ) (2 + (cid:15) ( p − , (3.19)with k ij = k i − k j . The overall coefficient N is not important here. M ( d )∆ ( s ) is the Mellinamplitude of a scalar exchange Witten diagram in AdS d +1 of dimension ∆, and withexternal dimensions ∆ i . These scalar exchange Witten diagram amplitudes are given by M ( d )∆ ( s ) = ∞ (cid:88) m =0 ( − − (cid:15) ]Γ[ m + ∆ − (cid:15) ] − m !Γ[ ∆+∆ ]Γ[ ∆ − ∆ ]Γ[ ∆+∆ ]Γ[ ∆ − ∆ ] × ∆ +∆ − ∆ − m ]Γ[ ∆ +∆ − ∆ − m ]( s − ∆ − m ) (3.20)where ∆ ij = ∆ i − ∆ j .Meanwhile, we can make another striking observation: F p ( s ) is also secretly the scalarexchange Witten diagram M ( d − (cid:15)p ( s ) in four dimensions lower , with internal dimension (cid:15)p and the same external dimensions! The appearance of AdS d − in an AdS d +1 supergrav-ity amplitude is quite unexpected. However, this equivalence across dimensions can benaturally explained in terms of an emergent Parisi-Sourlas supersymmetry, as we will seebelow.We first recall that Parisi-Sourlas supersymmetry is a geometrization of a hidden sym-metry of stochastic equations. This supersymmetry is key to the celebrated conjecture ofParisi and Sourlas [14], and links together the IR fixed points of three seemingly unrelatedmodels: a random field model in d dimensions, a supersymmetric field theory without dis-order in d dimensions, and a model without disorder in d − d − g ( d − ,(cid:96) = g ( d )∆ ,(cid:96) + C , g ( d )∆+2 ,(cid:96) + C , − g ( d )∆+1 ,(cid:96) − + C , − g ( d )∆ ,(cid:96) − + C , − g ( d )∆+2 ,(cid:96) − (3.21)– 14 –here the C i,j are pure numbers found in [15]. The combination on the RHS can beinterpreted as a single superconformal block under Parisi-Sourlas supersymmetry. Laterit was pointed out in [19] that the Parisi-Sourlas supersymmetry can also be realizedholographically. This gives rise to similar dimensional reduction identities for exchangeWitten diagrams, obtained by replacing conformal blocks with the corresponding exchangeWitten diagrams in AdS. Here we will only focus on the relevant case with (cid:96) = 0. Therelation (3.21) simplifies into g ( d − , = g ( d )∆ , + C , g ( d )∆+2 , , (3.22)with C , = − (∆ − ∆ )(∆ + ∆ )(∆ − ∆ )(∆ + ∆ )4( d − − d − − . (3.23)The corresponding Witten diagram relation is [19] M ( d − = M ( d )∆ + C , M ( d )∆+2 . (3.24)By using the relation twice on the AdS d − scalar exchange amplitude, we reach AdS d +1 and reproduce precisely the combination (3.18).A similar dimensional reduction structure is also present in the R-symmetry part.Since i , j appear in K i,jp ( t, u ) polynomially, we can treat them as differential operatorsacting on the monomial σ i τ j in (3.4) σ∂ σ ↔ i × , τ ∂ τ ↔ j × , (3.25)on the same footing as the Mandelstam variables. The remaining i -, j -dependence iscontained in H i,jp,m , which we denote as B i,jp = ( − i + j + p − κt − κu Γ[ ( (cid:15) − p + Σ − κ s − l )] i ! j !Γ[ κ u +2+2 i ]Γ[ p +2) − Σ+ κ s +4 l ]Γ[ κ t +2+2 j ] . (3.26)We now resum i and j to get (cid:80) i,j B i,jp σ i τ j . It turns out that the sum is nothing but theR-symmetry polynomial associated with the exchange of the rank- p symmetric tracelessrepresentation Y p = ( − p − κt + κu Γ[ p − κ t − κ u ]Γ[ p − κ t + κ u ]Γ[ κ t ] Y p, (cid:12)(cid:12)(cid:12)(cid:12) d = (cid:15) , (3.27)but for a different R-symmetry group SO ( (cid:15) ). Here the polynomial Y p, was given in [1, 2]for any SO ( d ) R-symmetry group Y p, = (cid:88) i,j ( − p − κt + κu σ i τ j (cid:20) Γ[ p + κ u − κ t ] i ! j ! ( κ u )!Γ[ d + p −
1] (3.28) × ( κ t + κ u − p ) i + j ( κ t + κ u +2 p +2 d − ) i + j Γ[ − d +2 p + κ t + κ u ](1 + κ u ) i (1 + κ t ) j (cid:21) , Similar generalizations are also found for two-point functions in boundary CFTs [19], and CFTs on realprojective space [72]. – 15 –nd we have changed its normalization to make Y p more symmetric. For the physicalspacetime dimensions (cid:15) = , ,
2, the new R-symmetry dimension d = (cid:15) is the same asshifting the original one by 4 d → d − . (3.29)This implies the dimension of the internal sphere is reduced by four, which perfectly par-allels the dimensional reduction structure for the AdS part.The above dimensional reduction may lead to negative dimensions. However, thisdoes not give rise to problems. We formally view d and d as continuous parameters, andanalytically continue them to negative values. Clearly, Mellin amplitudes and R-symmetrypolynomials can be defined for any values of d and d . Moreover, we regard U , V and σ , τ as independent variables, as such is the case for d > d > It is time to take stock and put all the ingredients together. Using observations from thelast subsection, we will now give a compact new look to the results of [1, 2]. We first restorethe kinematic factor in (2.6) to manifest the Bose symmetry, and define Y ( s ) p ( t i ) = (cid:89) i In this section we focus on the case of 11d supergravity on AdS × S , which is conjecturallydual to a 6d N = (2 , 0) SCFT of type g = A N − at large N . As discussed in [21], the osp (8 ∗ | 4) superconformal algebra contains an su (1 , | 2) subalgebra, and therefore accordingto the construction of [20] such a theory admits a protected subsector of operators whose(twisted) correlation functions restricted to a plane form a chiral algebra. In [21] it wasfurthermore conjectured that the chiral algebra associated to the type g = A N − theoryis W g : precisely the algebra appearing on the 2d side of the AGT correspondence [74]. Inthis case, the full higher-spin algebra is known: it is the quantization of the classical W N algebra, which in turn is the outcome of the Drinfel’d-Sokolov reduction of sl ( N ). Thisalso appears as the asymptotic symmetry algebra of AdS higher-spin gravity, as discussedin [75, 76].In [21] two tests of this conjecture were proposed: the first based on the superconfor-mal index, the second on the computation of the three-point functions between half-BPSoperators at large N , comparing the latter with certain three-point functions in the chi-ral algebra W g . Here we focus on the second kind of test, providing further evidence forthe validity of the conjecture. In the following, we shall focus on the sector of half-BPSoperators and compute the four-point functions between the dual operators in the chiralalgebra. Then, we shall use the four-point functions computed in [1, 2] at large N , performthe twist described in section 2, and derive the same four-point functions directly fromMellin space, finding perfect agreement. W -algebra correlators at large N As discussed in [21], the ring of half-BPS operators in the 6d N = (2 , 0) theory is isomorphicto the ring that is freely generated by the Casimir invariants of g . This led to the conjecturethat the chiral algebra associated with the 6d theory is W g , generated precisely by themeromorphic currents associated to the half-BPS operators. It was also shown that thecentral charge of W g is given by c d = 4 N − N − . (4.2)The chiral algebra generators, which we shall refer to as W ( k ) , with k ≥ 2, are bosonicprimaries of dimension k . The three-point functions of such operators read (cid:104) W ( k ) ( z ) W ( k ) ( z ) W ( k ) ( z ) (cid:105) = C k k k z k + k − k z k + k − k z k + k − k , (4.3)where the coefficients C k k k are given in the supergravity approximation by (3.13) (in thecase d = 6). Note that these three-point functions are non-vanishing only when k = | k | + 2 , | k | + 4 , ... , k + k − . (4.4) The R-symmetry decomposes as su (2) × u (1) ⊂ usp (4) and all operators that can contribute to thechiral algebra are neutral under u (1). The “twisting-translating” procedure O ( z ) = O b ...b j ( z, ¯ z, u b (¯ z ) . . . u b j (¯ z ) , u (¯ z ) ≡ (1 , ¯ z ) (4.1)defines their representatives away from the origin. – 18 –his means that, to this order, the OPE between generators is W ( k ) ( z ) W ( k ) ( z ) = δ k ,k z k + k + k − (cid:88) p = | k | +2step 2 C k k p k + k − p − (cid:88) m =0 ( k + p ) m m !(2 p ) m ∂ m W ( p ) ( z ) z k + k − p − m + . . . (4.5)where we have specialized (2.18a) to this case and explicitly singled out the contributionof the identity. The terms hidden by . . . in (4.5) come in two types – those suppressed byhigher powers of N and/or those that are regular in z . A careful accounting of regular andsingular terms will be important for the calculation of four-point functions which we nowdiscuss. In particular, for operators that lead to a singular contribution to the four-pointfunction (found by taking the OPE twice), we must keep track of whether they belong tosingular terms of both OPEs or just one of them.We are concerned with solving for the dynamical part of a four-point function F ( χ )which should be expanded in the sl (2) blocks given by (2.20). Once this is known, thekinematic prefactor can be restored by (2.19). The chiral algebra four-point functions forequal weights ( k i = k ) have been computed in [11] for k = 2 , , 4, using the holomorphicbootstrap method of [48]. Being a bootstrap approach, this exploits the fact that thedynamical function must satisfy F ( χ ) = ( − Σ χ k + k (1 − χ ) − k − k F (1 − χ ) (4.6a) F ( χ ) = ( − Σ χ k + k F (cid:18) χ (cid:19) (4.6b)which generate the S group of crossing transformations. Here we compute such correla-tors using an alternative (but equivalent) method that we shall summarize below. This alsoallows us to give closed form expressions for the four-point functions with general weights.To illustrate the idea, let us first recall the basic fact that chiral algebra correlators aremeromorphic functions of the coordinates. Here, after stripping off the kinematic prefactor,we obtain meromorphic functions of the cross-ratio χ , whose only possible singularities inthe complex χ -plane are poles at χ = 0 , , ∞ , corresponding to singular terms in the OPE.This is, at least in principle, enough to fix the correlation functions completely. To exploitthis information in a convenient way, we make an ansatz for the four-point function as F ( χ ) = F ( s )1234 ( χ ) + F ( t )1234 ( χ ) , (4.7)where F ( s ) contains the singular terms coming from the s -channel OPE, while F ( t ) is itscrossing-symmetric completion. By this, we mean that F ( t ) is the image of F ( s ) under(4.6a). This ensures that (4.7) is crossing symmetric. To fix the correlators, all that isleft to do is then to constrain the form of F ( s ) . As anticipated, this is done by requiringthat, in the small- χ expansion, it contains the correct powers of χ with coefficients thatcorrespond to singular terms in the s -channel OPE. The choice to focus on the crossed configuration (4.6a) which switches (1 ↔ ↔ s - and in the t -channel OPE. – 19 – comment is now in order. We would like to find these correlators purely from theknowledge of the three-point functions (4.3), but this is possible only if, at large N , theseare the only contributions to singular terms in the OPE. The OPE between two generators W ( k ) and W ( k ) contains, in general, two kinds of sl (2) multiplets: those built on top ofanother chiral generator, W ( p ) , but also those coming from normal ordered products ofgenerators. At large N , the former have OPE coefficients that scale with N − , while thelatter generically scale with more negative powers of N starting at N − . There is, however,an exception: the operator : W ( k ) W ( k ) :, which appears with an order 1 coefficient. Thiswill contribute to F ( s ) if it is a singular term in the four-point function as a whole.Let us then look at the four-point function (cid:104) W ( k ) W ( k ) W ( k ) W ( k ) (cid:105) . There are twoways one can have a contribution of order N − from potentially singular terms in the s -channel OPE: either the exchange of a generator W ( p ) , or that of : W ( k ) W ( k ) :. Thelatter appears with an OPE coefficient of order N − in W ( k ) × W ( k ) , and order 1 in W ( k ) × W ( k ) , hence we are allowed to ignore it only if it is a regular term, which happensif k + k ≤ k + k . A similar story applies to the t -channel OPE, which leads to therequirement that k + k ≤ k + k . Hence, we should carry out the computation withthese two assumptions, so that we will be able to neglect all the normal ordered productsof operators in the OPE. All other orderings can be obtained from this using crossingsymmetry.Under the assumptions above, the chiral algebra generators W ( p ) and their sl (2) de-scendants are the only singular terms relevant to tree-level in the s - and the t -channelOPEs. Hence, we make an ansatz for F ( s ) as F ( s )1234 ( χ ) = min { k + k ,k + k }− (cid:88) n =max {| k | , | k |} +2 a n χ n , (4.9)for some coefficients a n to be determined, and we demand that this satisfies the right OPE,namely that F ( s )1234 ( χ ) = k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p g k ,k p ( χ ) trunc , (4.10)where the subscript “trunc” refers to the fact that we should expand the expression enclosedin brackets for small χ , and truncate it at an order corresponding to the highest powerof χ that appears in (4.9), namely k + k − 1. This allows us to extract the coefficients a n in a direct way: we simply project the right hand side of (4.10) onto each power of χ The prefactor chosen for the four-point function is such that (cid:104) W ( k ) (0) W ( k ) ( χ ) W ( k ) (1) W ( k ) ( ∞ ) (cid:105) = χ − k − k F ( χ ) , (4.8)so that : W ( k ) W ( k ) : is always a regular term, while : W ( k ) W ( k ) : is regular only when k + k ≤ k + k .The crossing relation chosen in (4.6a), instead, is such that : W ( k ) W ( k ) : is always a regular term, while: W ( k ) W ( k ) : is regular only when k + k ≤ k + k . – 20 –ppearing in the ansatz (4.9), for instance using the orthogonality relation (cid:73) C dχ πi χ χ n − m = δ n,m , (4.11)where C is a contour the encircles χ = 0 in the complex plane. A direct computation thengives the result a n = k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p ( p − k ) n − p ( p + k ) n − p (2 h ) n − p ( n − p )! . (4.12)As discussed above, this expression is valid only under the assumption that k + k ≤ k + k and k + k ≤ k + k . Using crossing symmetry one can extend these results to anyconfiguration. The four-point function that follows from (4.9), with these coefficients, cannow be recast into the form (cid:68) W ( k ) ( z ) W ( k ) ( z ) W ( k ) ( z ) W ( k ) ( z ) (cid:69) = (cid:18) z z (cid:19) k (cid:18) z z (cid:19) k F ( χ ) z k + k z k + k k + k ≤ k + k , k + k ≤ k + k (4.13)with F ( χ ) = χ k + k k + k − (cid:88) p = | k | +2step 2 C k k p C k k p k + k − p − (cid:88) m =0 ( p − k ) m ( p + k ) m m !(2 p ) m χ k + k − p − m + k + k − (cid:88) p = | k | +2step 2 C k k p C k k p k + k − p − (cid:88) m =0 ( p + k ) m ( p + k ) m m !(2 p ) m (1 − χ ) k + k − p − m . (4.14)This will be convenient for the next subsection.We give a final reminder that, due to the δ k ,k in (4.5), there is one more term thatshould be added when the weights are pairwise equal. This is a = 1, in contrast to theother ones that are proportional to C k k p C k k p = O ( N − ). It is easy to see that this givesthe same disconnected correlator that we get after applying the superconformal twist togeneralized free theory in six dimensions. We now turn our attention to the Mellin amplitudes presented in (3.11) for AdS × S . Wewish to derive the same chiral algebra correlators (4.14) by evaluating (3.2) in a twistedconfiguration. Due to the inequalities in (4.13), we will focus on the ordering k ≤ k ≤ To explain the abbreviated notation, notice that the sums in (4.14) actually do not start until p =max {| k | , | k |} + 2 and p = max {| k | , | k |} + 2. This is because the C k k k are only given by (3.13)when the triangle inequality is satisfied. As stated in (4.4), they are zero otherwise. – 21 – ≤ k . Based on this, the Mellin amplitudes quoted in (3.4) are associated with thecorrelator that comes from switching (1 ↔ 2) in (2.6). (cid:104)O ( x , t ) O ( x , t ) O ( x , t ) O ( x , t ) (cid:105) = (cid:18) t x (cid:19) ( k + k − k − k ) (cid:18) t x (cid:19) ( k + k − k − k ) (cid:18) t x (cid:19) ( k + k + k − k ) −E (cid:18) t x (cid:19) k −E (cid:18) t t x x (cid:19) E G (cid:18) UV , V ; τ, σ (cid:19) (4.15)On the other hand, we should change to a different prefactor to facilitate direct comparisonsto (4.14). (cid:104)O ( x , t ) O ( x , t ) O ( x , t ) O ( x , t ) (cid:105) = (cid:18) t x (cid:19) − k (cid:18) t x (cid:19) k − k (cid:18) t x (cid:19) k (cid:18) t x (cid:19) ( k + k ) (cid:18) t x (cid:19) ( k + k ) F ( U, V ; σ, τ ) (4.16)Relating one function to the other and going to the Mellin representation gives F ( U, V ; σ, τ ) = ( σU ) ( k + k ) −E G (cid:18) UV , V ; τ, σ (cid:19) (4.17)= (cid:90) i ∞− i ∞ d s d t (4 πi ) U s V − s − t + k + k σ ( k + k ) −E M ( s, t ; τ, σ )Γ { k i } . Again, the contour keeps poles such that the exponents on U and V form increasingsequences. In this case, that means the s -channel and the u -channel.After we implement the superconformal twist, the integrand of (4.17) will depend on χ (cid:48) . The key observation we make is that, by meromorphy of the final answer, χ (cid:48) may beset to any value which makes the evaluation of (4.17) convenient. This does not constituteany additional assumption on the four-point function apart from the statement that thesuperconformal Ward identity (2.13) holds. We will now take χ (cid:48) → s -channel MRV limit. We should therefore expect the resulting functionto decompose into the two pieces of (4.7).After we do this, the largest pole in s that we need is s = 2 k + 2 k . Due to theordering being considered, this is also the smallest pole in s . It just barely contributes ifwe set τ E− j σ j (cid:55)→ χ (cid:48)− E in the polynomial and drop all lower degree terms. We now get F ( χ ) = 12 Γ( κ s ) (cid:90) i ∞− i ∞ d t πi χ k + k (1 − χ ) k − t M (2 k + 2 k , t ; χ (cid:48)− , χ (cid:48)− ) × Γ (cid:20) k + k − t (cid:21) Γ (cid:20) k + k − t (cid:21) Γ (cid:20) k + t (cid:21) Γ (cid:20) k + t (cid:21) . (4.18)Since the function is guaranteed to have the form F ( χ ) = F ( s ) ( χ ) + F ( t ) ( χ ), it is enoughto compute F ( t ) ( χ ) by extracting the terms that have a negative power of 1 − χ . Thesecome from the single-particle poles in u – namely t = 2 k + 2 k − m − p . Clearly, We could just as well consider k ≤ k ≤ k ≤ k since (4.14) is manifestly symmetric under (1 ↔ – 22 – ( s ) ( χ ) can then be computed using crossing symmetry. Looking at the expression for R p,m ;0 ,j (2 k + 2 k − m − p, k + 2 k ), F ( t ) ( χ ) = χ k + k Γ( κ s )16 k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p k + k − p − (cid:88) m =0 π − p − m !(2 p ) m (1 − χ ) k + k − p − m Γ (cid:104) p + k (cid:105) Γ (cid:104) p − k (cid:105) Γ (cid:104) p + k (cid:105) Γ (cid:104) p − k (cid:105) Γ( p − k )Γ( p − k )( p + k ) − m ( p + k ) − m E (cid:88) i =0 B ,jp K ,jp (cid:18) k + 2 k − m − p, k + 2 k (cid:19) (4.19)where K ,jp (2 k + 2 k − m − p, k + 2 k ) = K (cid:48) + jK (cid:48)(cid:48) (4.20) K (cid:48) ≡ − κ s ( κ s − κ u − m − p )(2 p − κ s − κ u )(2 p − κ s + κ u ) K (cid:48)(cid:48) ≡ κ s ( κ s − κ s + κ u − m − p ) + 2 κ s (2 p − κ s − κ u )(2 p − κ s + κ u ) . Note that, compared to (3.8), the permutation of the four points causes κ s to appear inplace of κ u and κ u to appear in place of κ t . There are now O (1) and O ( j ) contributionsto the inner sum of (4.19). For O (1), E (cid:88) j =0 B ,jp K (cid:48) = Γ (cid:0) − κ s (cid:1) Γ (cid:16) p − − κ u − E + k + k (cid:17) Γ (cid:16) p − − E + k + k (cid:17) Γ (cid:0) κ s +22 (cid:1) Γ (cid:16) p +2+ κ u +2 E− k − k (cid:17) Γ (cid:16) p +2+2 E− k − k (cid:17) K (cid:48) π = 2 κ s − κ s − κ s + 1) Γ (cid:16) p − + κ s ± κ u (cid:17) Γ (cid:16) p +22 − κ s ± κ u (cid:17) K (cid:48) √ π = 2 κ s − κ s − κ s + 1) Γ (cid:16) p − − k (cid:17) Γ (cid:16) p − − k (cid:17) Γ (cid:16) p +2+ k (cid:17) Γ (cid:16) p +2+ k (cid:17) K (cid:48) √ π . (4.21)In the second line, we have used a gamma function identity. We have also used the fact that,even though E has a piecewise dependence on the weights, it can be traded for κ u whichalso does. In the third line, we have used the fact that { κ s + κ u , κ s − κ u } = {− k , − k } even if we do not know which is which. The analysis for O ( j ) proceeds similarly with E (cid:88) j =0 j B ,jp K (cid:48)(cid:48) = 2 κ s − Γ( κ s + 1) (2 p − κ s + κ u )(2 p − κ s − κ u ) × Γ (cid:16) p − − k (cid:17) Γ (cid:16) p − − k (cid:17) Γ (cid:16) p +2+ k (cid:17) Γ (cid:16) p +2+ k (cid:17) K (cid:48)(cid:48) √ π . (4.22)– 23 –utting (4.21) and (4.22) back into the original formula, F ( t ) ( χ ) = χ k + k k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p k + k − p − (cid:88) m =0 ( p + k ) m ( p + k ) m m !(2 p ) m (1 − χ ) k + k − p − m × ( κ s − K (cid:48) + (2 p − κ s + κ u )(2 p − κ s − κ u ) K (cid:48)(cid:48) κ s ( p + k )( p + k )( p − k − p − k − . (4.23)Since this is almost the desired expression, we just have to go back to (4.20) to verify thatthe fraction at the end is indeed 1.As a check of our result, the inverse Mellin transform can also be arranged so that wecompute the dynamical function all at once instead of F ( s ) and F ( t ) individually. This isexplained in Appendix A. Let us now move to the case of Type IIB supergravity on AdS × S , which is conjecturallydual to 4d N = 4 super Yang-Mills (SYM) with gauge algebra g = A N − , at large N . Therelevant superconformal algebra in this case is psu (2 , | 4) which, as discussed in [20], admitsa chiral algebra subsector whose currents come from Schur multiplets of the 4d theory. Furthermore, when N > d = 4, the chiral algebra contains supercurrents of its own.This leads to a global subalgebra of psu (1 , | 2) when N = 4. The minimal extension ofthis to a chiral algebra is the “small” N = 4 superconformal algebra, which turns out tobe the full answer for N = 2. In more general cases, the proposed chiral algebra for 4d N = 4 SYM is an N = 4 super W -algebra with rank( g ) generators that are in one-to-onecorrespondence with the Casimir invariants of g . As an initial check of this conjecture, theauthors of [20] showed that it predicts the correct superconformal index. They also showedthat the chiral algebra for a superconformal gauge theory can in principle be constructedfrom BRST quantization.More recent results include free-field realizations for N = 2 , , N = 4 SYM at large N and an AdS higher-spin Chern-Simons theory in [37]. The latter echoes the discussion in section 4 wherein the 6d N =(2 , 0) theory is associated with the higher-spin algebra hs [ µ ]. Here we perform a test basedon computing the four-point functions between chiral algebra generators at large N , andcomparing them with the holographic correlators between half-BPS operators [9, 10] usingMellin space techniques. Again, the symmetric form of the Mellin amplitudes found in[1, 2] will be most convenient for our purposes. W -algebra correlators at large N As in the case of the 6d N = (2 , 0) theory, the ring of half-BPS operators is freely generatedby the Casimir invariants of the gauge algebra g . In the case g = A N − , the central charge Again, the twisted translation (4.1) must be applied to the Schur operators. This time, the maximalsubalgebra is su (2) × su (2) × u (1) ⊂ su (4) which means that an su (2) R-symmetry survives in the chiralalgebra. In general there can be Schur operators with non-zero u (1) charge but not in the OPE betweentwo scalars [20]. – 24 –s c d = − c d = − N − . (5.1)The global psu (1 , | 2) of the full higher-spin algebra has an su (2) R-symmetry, and thechiral algebra generators form non-trivial irreducible representations of this algebra. Fur-thermore, they are superconformal primaries of half-BPS multiplets with respect to theglobal psu (1 , | J ( k ) a ...a j and contract them with polar-ization vectors as in (2.16). Their sl (2) and su (2) spins (which must be equal) are givenby h = j = k . Note that for odd k they are fermionic, having half-integer dimension. Inthis notation, their three-point functions can be written as (cid:104) J ( k ) ( z , y ) J ( k ) ( z , y ) J ( k ) ( z , y ) (cid:105) = C k k k (cid:16) z y (cid:17) k k − k (cid:16) z y (cid:17) k k − k (cid:16) z y (cid:17) k k − k , (5.2)where C k k k to the order we need should be read off from (3.13) (in the d = 4 case). Onceagain, the third operator weight satisfies the selection rule (4.4) in terms of the other two,which we restate as k = | k | + 2 , | k | + 4 , ... , k + k − . (5.3)As anticipated at the end of section 2, four-point functions in this chiral algebra mustsatisfy an an additional constraint, namely (cid:20) α ∂∂α − ∂∂χ (cid:21) F ( χ ; α ) (cid:12)(cid:12)(cid:12)(cid:12) α = χ = 0 (5.4)in terms of the cross-ratios (2.17). This is the superconformal Ward identity (SCWI)for the global superalgebra psu (1 , | J ( k ) are annihilated byhalf of the supercharges of a “small” N = 4 superconformal algebra, there are 2 directionsby which we can descend and therefore 2 conformal primaries in their superconformalmultiplet. However, when J ( k ) is Grassman even (odd), it will only be the state obtained byacting with both supercharges which is again Grassman even (odd) and therefore admissiblein the same OPE. This super descendant, which we will denote by T ( k ) a ...a j , has the quantumnumbers h = k +22 and j = k − . Notice that it is precisely the stress tensor for the case of k = 2. In the primary and descendant terms of the superconformal block, the cross-ratio χ must appear through the sl (2) block (2.20). Due to the “isomorphism” with su (2), thecross-ratio α will appear in the same function, continued to negative values of the conformal– 25 –imensions. We arrive at G k ,k h ( χ ; α ) = (cid:104) g ( k / ,k / h ( χ ) g ( − k / , − k / − h ( α − )+ (cid:0) h − k (cid:1) (cid:0) h − k (cid:1) h (1 − h ) g ( k / ,k / h +1 ( χ ) g ( − k / , − k / − h ( α − ) (cid:105) (5.5)for a J (2 h ) multiplet exchanged in (cid:104) J ( k ) J ( k ) J ( k ) J ( k ) (cid:105) . Knowing the relative coefficientin (5.5) enables us to write the singular OPE of two generators. Combining (2.18a) forspace-time and (2.18b) for R-symmetry, we find J ( k ) ( z , y ) J ( k ) ( z , y ) = δ k ,k ( z /y ) k + k + k − (cid:88) p = | k | +2step 2 C k k p k k − p − (cid:88) m =0 (cid:16) k + p (cid:17) m m !( p ) m y k k − p z k k − p − m ∂ m k ! (cid:88) σ ∈ S p J ( p ) a σ (1) ...a σ ( p ) ( z ) y a . . . y a ( p + k / y a ( p + k / . . . y a p − k + k − (cid:88) p = | k | +2step 2 C k k p ( p + k )( p − k )4 p (cid:112) p − k k − p − (cid:88) m =0 (cid:16) k + p +22 (cid:17) m m !( p + 2) m y k k − p +22 z k k − p − − m ∂ m ( p − (cid:88) σ ∈ S p − T ( p ) a σ (1) ...a σ ( p − ( z ) y a . . . y a ( p + k − / y a ( p + k / . . . y a p − (5.6)plus terms involving normal ordered products that are suppressed by at least 1 /N .We are now ready to compute the four-point functions in the chiral algebra. Again,it is possible to use (an adaptation of) the holomorphic bootstrap of [48], but instead itturns out to be better to use a similar strategy to the one of section 4. That is, we startby making an ansatz F ( χ ; α ) = F ( s )1234 ( χ ; α ) + F ( t )1234 ( χ ; α ) , (5.7)where F ( t ) is identified as the completion of F ( s ) under crossing symmetry, which is gen-erated by F ( χ ; α ) = ( α χ ) k k (cid:18) α (1 − χ ) α − (cid:19) − k k F (cid:18) − χ ; αα − (cid:19) (5.8a) F ( χ ; α ) = ( α χ ) k k F (cid:18) χ ; 1 α (cid:19) . (5.8b)As in section 4, we shall work under the assumption that k + k ≤ k + k and k + k ≤ k + k , so that we can neglect normal ordered products of operators in the OPE. By using(5.8a) and (5.8b), all other choices for the weights are within reach. If we fix the terms in F ( s ) such that each power of χ appears multiple times, once for each power of z in (5.6),– 26 –his leads us to a four-point function with the same structure as (4.14): (cid:68) J ( k ) ( z , y ) J ( k ) ( z , y ) J ( k ) ( z , y ) J ( k ) ( z , y ) (cid:69) = (cid:18) z y z y (cid:19) k (cid:18) z y z y (cid:19) k (cid:18) y z (cid:19) k k (cid:18) y z (cid:19) k k F ( χ ; α ) k + k ≤ k + k , k + k ≤ k + k (5.9)where F ( χ ; α ) χ k k = k + k − (cid:88) p = | k | +2step 2 C k k p C k k p g k , k − p ( α − ) k k − p − (cid:88) m =0 (cid:16) k + p (cid:17) m (cid:16) k + p (cid:17) m m !( p ) m χ k k − p − m (5.10)+ ( p − k )( p − k )16 p (1 − p ) g k , k − p ( α − ) k k − p − (cid:88) m =0 (cid:16) k + p +22 (cid:17) m (cid:16) k + p +22 (cid:17) m m !( p + 2) m χ k k − p − − m + ( α − k k α k k + k − (cid:88) p = | k | +2step 2 C k k p C k k p g k , k − p (1 − α − ) k k − p − (cid:88) m =0 (cid:16) k + p (cid:17) m (cid:16) k + p (cid:17) m m !( p ) m (1 − χ ) k k − p − m + ( p − k )( p − k )16 p (1 − p ) g k , k − p (1 − α − ) k k − p − (cid:88) m =0 (cid:16) k + p +22 (cid:17) m (cid:16) k + p +22 (cid:17) m m !( p + 2) m (1 − χ ) k k − p − − m . However, computing the coefficient for a given power of χ all at once will lead us to asignificant simplification. An ansatz for F ( s ) which facilitates this latter approach andtakes into account the singular OPE is then F ( s )1234 ( χ ; α ) = 12 (min { k + k ,k + k }− (cid:88) n =1+ 12 max {| k | , | k |} P n ( α ) χ n , (5.11)where P n ( α ) are functions of α that we can fix by requiring consistency with the OPE,including only the chiral algebra generators. Namely, we demand that F ( s )1234 ( χ ; α ) = k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p G k ,k p/ ( χ ; α ) trunc , (5.12)where the subscript “trunc” refers to the fact that we should expand the expression enclosedin brackets for small χ , and truncate it at an order corresponding to the highest power of χ that appears in (5.11), namely ( k + k − χ ,we can extract the coefficients P n ( α ), which turn out to be given by P n ( α ) = k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p I p/ ,n ( α ) , (5.13)– 27 –here we have introduced I h,n ( α ) = (cid:73) C dχ πi χ χ − n G k ,k h ( χ ; α )= ( h − k / n − h ( h + k / n − h ( n − h )!(2 h ) n − h (cid:104) g ( − k / , − k / − h ( α − )+ ( h − n )(2 h + k )(2 h − k )8 h (2 h − h + n ) g ( − k / , − k / − h − ( α − ) (cid:105) . (5.14)While the sum over I h,n ( α ) looks complicated a priori , with P n ( α ) being in principlearbitrary polynomials of degree n , for the specific values of n that enter the sum (5.13)they turn out to have a particularly simple form, in which the dependence on α is almostcompletely factorized: P n ( α ) = √ k k k k N α n − (cid:18) k − k − k + k − n (1 − α ) (cid:19) . (5.15)As in section 4, using crossing, we can continue the results above to all configurations ofthe { k i } , and therefore this fixes all the four-point functions between chiral generators attree-level. The next step is to show that the same answer in encoded in the Mellin amplitudes (3.11)for AdS × S . Our discussion here will run parallel to that of subsection 4.2. As explainedthere, we will consider weights that satisfy k ≤ k ≤ k ≤ k . The correlators we derivewill initially take the form (5.10) but the simplification apparent in (5.15) will turn out tohave an interpretation in Mellin space as well.Since the convention that will allow the most direct comparison is (cid:104)O ( x , t ) O ( x , t ) O ( x , t ) O ( x , t ) (cid:105) = (cid:18) t x (cid:19) − k (cid:18) t x (cid:19) k − k (cid:18) t x (cid:19) k (cid:18) t x (cid:19) ( k + k ) (cid:18) t x (cid:19) ( k + k ) F ( U, V ; σ, τ ) , (5.16)we can solve for the dynamical part as F ( U, V ; σ, τ ) = ( σU ) ( k + k ) −E G (cid:18) UV , V ; τ, σ (cid:19) (5.17)= (cid:90) i ∞− i ∞ d s d t (4 πi ) U s V − s − t + ( k + k ) σ ( k + k ) −E M ( s, t ; τ, σ )Γ { k i } . Evaluating (5.17) in the twisted configuration will again be based on taking χ (cid:48) → 0. Clearly,the freedom to set χ (cid:48) to a convenient value is a consequence of the superconformal Ward This can be argued from the relation between the sl (2) blocks with negative arguments appearing in(5.14) and the Jacobi polynomials (see (2.21)). – 28 –dentity. We have been able to prove analytically that it holds for all weights by showingthat the Mellin amplitudes in (3.11) can be written in the form M ( s, t ; σ, τ ) = ˆ R ◦ (cid:102) M ( s, t ; σ, τ ) (5.18)where the operator ˆ R is the Mellin space version of (1 − χα )(1 − χα (cid:48) )(1 − χ (cid:48) α )(1 − χ (cid:48) α (cid:48) ).It should come as no surprise that the auxiliary Mellin amplitude on the right hand sidewhich accomplishes this is the one conjectured in [9, 10] with the normalization from [77].Based on our chosen ordering, the superconformal twist with χ (cid:48) → s = k + k is the only pole that contributes (effectively giving τ E− j σ j (cid:55)→ χ (cid:48)−E α j ( α − E− j ).After this, the u -channel poles of t = k + k − m − p that lead to singularities can beextracted to give the second term in F ( χ ; α ) = F ( s ) ( χ ; α ) + F ( t ) ( χ ; α ). Continuing, theexpression for R p,m ;0 ,j ( k + k − m − p, k + k ) allows us to write F ( t ) ( χ ; α )( αχ ) k k = Γ( κ s )16 k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p k k − p − (cid:88) m =0 Γ[ p + m + 1] − m !(1 − χ ) k k − p − m Γ (cid:20) k + p + 2 m (cid:21) Γ (cid:20) k + p + 2 m (cid:21) E (cid:88) j =0 (1 − α − ) E− j B ,jp K ,jp (cid:18) k + k − m − p,k + k (cid:19) (5.19)in terms of the factors from (3.11). By scrutinizing the gamma functions in (3.26), we cansee that the inner sum only goes up to E − k + k − p instead of E . This allows us to performthe reflection j (cid:55)→ E − k + k − p − j which yields E (cid:88) j =0 (1 − α − ) E− j B ,jp K ,jp (cid:18) k + k − m − p,k + k (cid:19) = Γ( p )Γ (cid:0) κ s +22 (cid:1) − Γ (cid:16) p + k +22 (cid:17) Γ (cid:16) p + k +22 (cid:17) (5.20) (cid:18) αα − (cid:19) p − k − k E− k k − p (cid:88) j =0 (1 − α − ) j (cid:16) k − p (cid:17) j (cid:16) k − p (cid:17) j j !(1 − p ) j K , E− k k − p − jp (cid:18) k + k − m − p,k + k (cid:19) . To evaluate (5.20), we need to plug in K , E− k k − p − jp ( k + k − m − p, k + k ) (5.21)= 12 κ s ( m + p )[( κ s − p ) − κ u ] + 12 κ s j [ κ u − κ s + 4 κ s (2 m + p ) − p ]= κ s (cid:20) p − j )( p + m ) p ( p + k )( p + k ) + 2 jmp ( p − k )( p − k ) (cid:21) . The fact that it is an even function of κ u is what allows us to write it in terms of theweights analytically. It is now straightforward to see that our expression for the t -channel– 29 –ingularity becomes F ( t ) ( χ ; α )( αχ ) k k = 18 k + k − (cid:88) p =max {| k | , | k |} +2step 2 C k k p C k k p k k − p − (cid:88) m =0 (1 − χ ) p +2 m − k − k × Γ( p )Γ (cid:16) k +2 m + p (cid:17) Γ (cid:16) k +2 m + p (cid:17) (cid:16) αα − (cid:17) p − k − k m !Γ( p + m + 1)Γ (cid:16) k +2+ p (cid:17) Γ (cid:16) k +2+ p (cid:17) ∞ (cid:88) j =0 (cid:16) k − p (cid:17) j (cid:16) k − p (cid:17) j j !(1 − p ) j (cid:18) α − α (cid:19) j × (cid:20) p − j )( p + m ) p ( p + k )( p + k ) + 2 jmp ( p − k )( p − k ) (cid:21) . (5.22)In the splitting that we have chosen for (5.21), the first term has the property that allfactors may be readily absorbed into the gamma functions. This shifts their arguments toexactly the values that we expect for a superconformal primary. From this point of view, itis reassuring that the second term is proportional to m . This allows us to reindex the sumover m so that the (1 − z ) p − k − k singularity, which cannot come from a super descendant,is manifestly absent. When this is done, F ( t ) ( χ ; α )( αχ ) k k = k + k − (cid:88) p = | k | +2step 2 C k k p C k k p g k , k − p (1 − α − )(1 − α − ) − k − k k k − p − (cid:88) m =0 (cid:16) k + p (cid:17) m (cid:16) k + p (cid:17) m m !( p ) m (1 − χ ) k k − p − m +( p − k )( p − k )16 p (1 − p ) g k , k − p (1 − α − )(1 − α − ) − k − k k k − p − (cid:88) m =0 (cid:16) k + p +22 (cid:17) m (cid:16) k + p +22 (cid:17) m m !( p + 2) m (1 − χ ) k k − p − − m (5.23)which is nothing but the second part of (5.10).We have already mentioned in (5.18) that these Mellin amplitudes can be obtained fromthe action of a difference operator ˆ R on an auxiliary Mellin amplitude (cid:102) M ( s, t ; σ, τ ). Re-turning to this point can help us understand the absence of α n − , α n − , . . . terms in (5.15).Consider the poles of the s -channel amplitude delineated by (3.4). Although every valuein (max {| k | , | k |} + 2 , . . . , min { k + k , k + k } − 2) appears as a pole of M ( s ) ( s, t ; σ, τ ),closer inspection reveals that each one has at most two R-symmetry monomials σ i τ j inits residue. This non-trivial fact follows from a key property of the auxiliary amplitudebootstrapped in [9, 10]. Namely that, for a fixed monomial σ i τ j , it has exactly one polein each Mandelstam variable. The action of the difference operator is then only able toproduce one additional pole in the full result.We can see how this structure appears by changing the order of the sums in (3.4).Specifically, we should focus on a specific pole s = s (taken to be even for simplicity) andthen set m = s − p . Performing the sum over p and keeping u as an independent variable– 30 –eads us toRes s → s M i,j ( s, t ) = ( − i + j − κt + κu √ k k k k Γ (cid:0) s +22 (cid:1) − i ! j !Γ (cid:16) κ u +2+2 i (cid:17) Γ (cid:16) κ t +2+2 j (cid:17) Γ (cid:16) k + k − s (cid:17) Γ (cid:16) k + k − s (cid:17) × (cid:34) K i,j ( t, u )Γ (cid:0) i + j − E + Σ − κ s (cid:1) Γ (cid:0) s (cid:1) − (4 i + 4 j − E − κ s + Σ − s )Γ (cid:0) − i − j + E − Σ − κ s (cid:1) − s ( s + 2) − i + 4 j − E + Σ − κ s )( s + 4)(4 i + 4 j − E − κ s + Σ − s ) (4 + 4 i + 4 j − E − κ s + Σ − s ) ( t − − j )( u − − i )Γ (cid:0) s +22 + i + j − E + Σ − κ s (cid:1) Γ (cid:0) − s (cid:1) − Γ( s )Γ (cid:0) s − − i − j + E − Σ − κ s (cid:1) (cid:35) . (5.24)We now observe that the first term forces us to have 2 s = 4 i + 4 j − E − κ s + Σ because1 − i − j + E − Σ − κ s , which appears in the gamma function, is a non-positive number. By a similar argument, the second term allows us to have either this value or 2 s =4 + 4 i + 4 j − E − κ s + Σ.Now that we know M i,j ( s, t ) can be written as a sum of only two residues, it is thisform of the Mellin amplitude, when subjected to the procedure of this subsection, thatproduces the polynomial (5.15). Indeed, we can look at all of the single-particle poles in s which can contribute singular terms to F ( s ) . These have the form s = p + 2 m which, asshown above, can only appear with τ j σ ( p +2 m +2 E− k − k ) / − j and τ j σ ( p +2 m +2 E− k − k − / − j .If we then perform the superconformal twist with χ (cid:48) → 1, we pick out only those monomialswith j = 0. This causes the integrand of (5.17) to become a linear combination of α p + m and α p + m − as required. For the SCFTs studied in sections 4 and 5, we managed to find order 1 /c T four-pointfunctions of arbitrary strong generators in the associated chiral algebra. Much like inhigher dimensions, four-point functions in a 2d CFT (whether chiral or not) receive acontribution from infinitely many quasiprimary operators. This is in agreement with thefact that there are infinitely many 4d and 6d superconformal multiplets in the cohomologyof the supercharge used in the chiral algebra construction.We can easily list those 6d N = (2 , 0) multiplets determined in [21]. In the notationof [78], they are B [ j , j , [ r ,r ]∆ for r = 0, C [ j , , [ r ,r ]∆ for r ≤ D [0 , , [ r ,r ]∆ for r ≤ 2. Similarly, the Schur multiplets of 4d N = 4 super Yang-Mills are B ¯ B [0 , R ∆ , A ¯ A [ j, ¯ ] R ∆ , A ¯ B [ j, R ∆ and B ¯ A [0 , ¯ ] R ∆ where there are no restrictions on the R-symmetry rep-resentation beyond those required for unitarity. In particular, certain Lorentz quantumnumbers are allowed to be arbitrarily large. Of course, the derivations of (4.14) and (5.10) The only exception occurs when i = j = 0 and all the weights are identical. But then, K , ( t, u ) = 0anyway. In CFTs with less supersymmetry, this way of writing the Schur multiplets is still correct as long as weadditionally specify that the primaries have zero u (1) r-charge. The detailed calculation was done in [20]for N = 2, [30] for N = 3 and [42] for N = 4. – 31 –id not require us to deal with infinite sums over the spin explicitly. We instead exploitedthe fact that chiral algebra correlators, being meromorphic functions, are uniquely fixedby a finite number of singularities.This situation is quite different for the topological correlators predicted by the 3d su-perconformal Ward identity (2.15). On the one hand, one sees from the multiplet structureof osp ( N | 4) that there is no longer any room for spinning operators to contribute to anabsolutely protected subsector. As a result, adding up every term in the OPE becomes aviable way to compute twisted four-point functions. On the other hand, due to the lackof any nice meromorphic property, it might be the only viable way. We can see a simpleexample of spinning operators dropping out in the four-point functions at c T = ∞ : (cid:104)O E ( x , t ) O E ( x , t ) O k ( x , t ) O k ( x , t ) (cid:105) = (cid:18) t | x | (cid:19) k −E (cid:18) t t | x || x | (cid:19) E (cid:104) δ k E U E (cid:16) σ E + τ E V − E (cid:17)(cid:105) . (6.1)Here, without loss of generality, we have labelled two operators by E since the extremalityis the smaller of the two weights. The superconformal twist instructs us to take t t | x || x | U σ → sgn( z )sgn( z ) y y t t | x || x | U V τ → sgn( z )sgn( z ) y y (6.2)which no longer require infinitely many conformal blocks in the s -channel to reproduce.Instead, we just need one exchanged multiplet for each of the su (2) spins from 0 to E . The way forward To compute twisted four-point functions in a way which does not rely on crossingsymmetry, we have already mentioned the method in Appendix A. While this has beenpresented mainly as a check of the results in sections 4 and 5, one could imagine using itas a starting point for 3d calculations as well. Unfortunately, we can already see that thisapproach will not allow us to compute a four-point function all at once anymore. Considersetting χ = χ (cid:48) and taking them both to zero. In Appendix A, we needed this step to makeone of the integrals localize onto a single term. For the 3d case, however, such a limitwould prevent us from seeing the sgn( z ) and sgn( z ) terms which provide a non-zerocontribution above.Since four-point functions in the twisted configuration will have to be assembled frommultiple pieces either way, we have found it more intuitive to do this using the OPE. Forthis reason, we will devote the rest of this paper to understanding the OPE coefficientsthat couple two half-BPS operators to a third (exchanged) operator in three dimensions.The 3d N = 8 multiplets, which will feature heavily in this analysis, are listed in Table 1.They obey the R-symmetry selection rule[0 , , k , ⊗ [0 , , k , 0] = k k (cid:77) j I = | k | j I (cid:77) j II = | k | [0 , j I − j II , j II , 0] (6.3)– 32 –ame Primary Unitarity Bound Null State L [ j ] [ r ,r ,r ,r ]∆ ∆ > j + r + r + ( r + r ) + 1 A [ j ] [ r ,r ,r ,r ]∆ , j ≥ j + r + r + ( r + r ) + 1 [ j − [ r +1 ,r ,r ,r ]∆+ A [0] [ r ,r ,r ,r ]∆ ∆ = r + r + ( r + r ) + 1 [0] [ r +2 ,r ,r ,r ]∆+1 B [0] [ r ,r ,r ,r ]∆ ∆ = r + r + ( r + r ) [1] [ r +1 ,r ,r ,r ]∆+ Table 1 : Unitary multiplets of the 3d N = 8 superconformal algebra which is a real form of osp (8 | B [0] [0 , ,k, k . To summarize, we wish to solve for topological four-point functions by identifying thefinitely many operators that contribute in a given channel and then computing their OPEcoefficients in a 1 /c T expansion. Section 7 will approach this question from the TQFTpoint of view while in section 8 we will return to Mellin space. In both cases, the E = 2correlators (cid:104)O O O k O k (cid:105) , and permutations thereof, will be our primary interest. Beforemoving onto the next section, we will get the process started with generalized free theory.This exercise consists of setting E = 2 in (6.1) and projecting onto the SO (8) harmonicpolynomials of [49] in the t -channel. A straightforward calculation leads to λ B [0] [0000]0 = 1 , λ B [0] [0200]2 = 323 , λ B [0] [0040]2 = 163 (6.4) λ kB [0] [0 , ,k − , k +22 = 2 k +2 k − k + 1 , λ kB [0] [0 , ,k, k +22 = 2 k +3 k + 2 , λ kB [0] [0 , ,k +2 , k +22 = 2 k +3 ( k + 1)( k + 2)as the non-zero protected OPE coefficients. We should now investigate how the 3d/1dcorrespondence allows us to recover (6.4) along with some OPE data at the next order. As shown in [22, 23], any 3d N = 4 SCFT admits correlators that are topological onthe line. In order to construct them, one passes to the cohomology of a certain linearcombination of Poincar´e and conformal supercharges. Consider the following subalgebraof the N = 8 R-symmetry. su (2) I × su (2) II × su (2) III × su (2) IV ⊂ so (8) (7.1) Apart from the Roman numerals, there can still be external labels that run from 1 to 4 in a four-pointfunction. The half-BPS operator at position i has j I i = j II i = k i = ∆ i . Note that we are taking the “numerics-inspired” convention g ∆ ,(cid:96) ( z, ¯ z ) ∼ (cid:0) z ¯ z (cid:1) ∆2 ∼ ( ρ ¯ ρ ) ∆2 instead of g ∆ ,(cid:96) ( z, ¯ z ) ∼ ( z ¯ z ) ∆2 ∼ (16 ρ ¯ ρ ) ∆2 which is also common. – 33 –e will regard these factors as R, flavour, flavour, R from the N = 4 point of view.Standard branching rules show that the su (2) spins are j I = r + 2 r + r + r , j II = r , j III = r , j IV = r r , r , r , r ] Dynkin labels of so (8). After focusing on irreps that can beexchanged by two half-BPS operators, we find j III = j IV = 0 while j I and j II are preciselythe quantities indicated in (6.3). A result of [22, 23] is that superconformal primariesof the B [0] [0 ,j I − j II , j II , j I multiplets survive the cohomological prescription. Since they areinvariant under su (2) IV , one may refer to them as Higgs branch operators. Their positiondependence is given by (2.16) after a “twisting-translating” procedure with u ( x ) = (cid:0) , x r (cid:1) . O ( x, y ) = O a ...a j II ( x ) y a . . . y a j II = O b ...b j I a ...a j II ( x, , u b ( x ) . . . u b j I ( x ) y a . . . y a j II (7.3)In what follows, we will compactify the x -direction on a circle of radius r . If the scaling dimension of the original operator is a half-integer, it turns out that the cor-responding twisted-translated operator on the circle is effectively fermionic. In particular,their two-point and three-point functions may be expressed as (cid:104)O A ( ϕ , y ) O B ( ϕ , y ) (cid:105) = δ AB B A y j A (sgn ϕ ) A (7.4) (cid:104)O A ( ϕ , y ) O B ( ϕ , y ) O C ( ϕ , y ) (cid:105) = C ABC y j A + j B − j C y j B + j C − j A y j C + j A − j B (sgn ϕ ) ∆ B +∆ A − ∆ C (sgn ϕ ) ∆ A +∆ C − ∆ B (sgn ϕ ) ∆ C +∆ B − ∆ A . This structure leads to the OPE O A ( ϕ , y ) O B ( ϕ , y ) = (cid:88) O C AB O B O ( − j + j AB +∆+∆ AB (cid:104) y , y (cid:105) j A + j B − j (sgn ϕ ) ∆ A +∆ B − ∆ j )! (cid:88) σ ∈ S j O a σ (1) ...a σ (2 j ) ( ϕ ) y a . . . y a j + jAB y a j + jAB +1 . . . y j (7.5)which is a straightforward modification of (2.18b). We can solve for four-point functionsby using it twice. In terms of the cross-ratio α , the s -channel result is (cid:104)O A ( ϕ , y ) O B ( ϕ , y ) O C ( ϕ , y ) O D ( ϕ , y ) (cid:105) = (cid:88) O C AB O C CD O B O ( α − j DC ( − j g j AB ,j DC − j (cid:18) − α (cid:19)(cid:18) y y (cid:19) j AB (cid:18) y y (cid:19) j CD α j DC y j A + j B y j C + j D (sgn ϕ ) ∆ A +∆ B − ∆ (sgn ϕ ) ∆ C +∆ D − ∆ . (7.6)Recalling (2.21), the first line of (7.6) can be written as C AB O C CD O B O ( j + j CD )!( j − j CD + 1) j + j CD P j AB − j CD ,j BA − j CD j + j CD (2 α − . (7.7)– 34 –t is shown in [22] that the coefficient of the Jacobi polynomial in (7.7) is nothing but thesuperconformal block coefficient 4 − ∆ λ AB O λ CD O in three dimensions. This allows us towrite down crossing equations that only involve the operators in (7.6).Let us pick an E = 2 (next-to-next-to-extremal) correlator which has 6 multiplets inthe topological four-point function. More precisely, we will pick a family of them given by∆ A = ∆ C = j A = j C = 1, ∆ B = ∆ D = j B = j D = k . To derive a crossing equation, let uschoose the ϕ < ϕ < ϕ < ϕ ordering so that all of the sign functions above come outpositive. Equating (7.6) to the one other channel that preserves this cyclic ordering tellsus that α − (cid:34)(cid:32) λ kB [0] [0 , ,k − , k − + 4 λ kB [0] [0 , ,k − , k + λ kB [0] [0 , ,k − , k +22 (cid:33) P ,k − (2 α − 1) (7.8)+ (cid:32) λ kB [0] [0 , ,k, k + λ kB [0] [0 , ,k, k +22 (cid:33) P ,k − (2 α − 1) + λ kB [0] [0 , ,k +2 , k +22 P ,k − (2 α − (cid:35) must be invariant under α ↔ αα − . This constraint is solved by32 λ kB [0] [0 , ,k − , k − + 8 λ kB [0] [0 , ,k − , k + 8 λ kB [0] [0 , ,k, k + 2 λ kB [0] [0 , ,k − , k +22 + 2 λ kB [0] [0 , ,k, k +22 − k ( k + 3) λ kB [0] [0 , ,k +2 , k +22 = 0 . (7.9)This does not need to be modified for k = 2 which has the additional consideration of Bosesymmetry. In this case, we will just get λ B [0] [0100]1 = λ B [0] [0120]2 = 0 automatically. Onecan permute the correlator with ∆ A = ∆ B = j A = j B = 1 and ∆ C = ∆ D = j C = j D = k .Studying this next, α − k − (cid:34)(cid:32) λ kB [0] [0 , ,k − , k − − λ kB [0] [0 , ,k − , k + λ kB [0] [0 , ,k − , k +22 (cid:33) P ,k − (2 α − 1) (7.10)+ (cid:32) λ kB [0] [0 , ,k, k − λ kB [0] [0 , ,k, k +22 (cid:33) P ,k − (2 α − 1) + λ kB [0] [0 , ,k +2 , k +22 P ,k − (2 α − (cid:35) = (cid:18) α − α (cid:19) (cid:20)(cid:16) λ B [0] [0000]0 λ kkB [0] [0000]0 + λ B [0] [0200]2 λ kkB [0] [0200]2 (cid:17) P , (cid:18) α + 1 α − (cid:19) +4 λ B [0] [0020]1 λ kkB [0] [0020]1 P , (cid:18) α + 1 α − (cid:19) + λ B [0] [0040]2 λ kkB [0] [0040]2 P , (cid:18) α + 1 α − (cid:19)(cid:21) . The left hand side of (7.10) has some sign differences compared to (7.8). This is because C O C kk O becomes C k O C k O under crossing which needs to be put back in order. Solving– 35 –7.10) and taking the identity to be unit-normalized, we have k λ B [0] [0040]2 λ kkB [0] [0040]2 = 323 λ kB [0] [0 , ,k − , k − − λ kB [0] [0 , ,k − , k + 23 λ kB [0] [0 , ,k − , k +22 (7.11a) + 8( k − λ kB [0] [0 , ,k, k − k − λ kB [0] [0 , ,k, k +22 + k ( k − λ kB [0] [0 , ,k +2 , k +22 k +2 λ B [0] [0020]1 λ kkB [0] [0020]1 = − λ kB [0] [0 , ,k − , k − + 8 λ kB [0] [0 , ,k − , k − λ kB [0] [0 , ,k − , k +22 (7.11b) +8 λ kB [0] [0 , ,k, k − λ kB [0] [0 , ,k, k +22 + k ( k + 3) λ kB [0] [0 , ,k +2 , k +22 k − λ B [0] [0200]2 λ kkB [0] [0200]2 = 323 λ kB [0] [0 , ,k − , k − − λ kB [0] [0 , ,k − , k + 23 λ kB [0] [0 , ,k − , k +22 − k +3 (7.11c) − k + 2)3 λ kB [0] [0 , ,k, k + k + 23 λ kB [0] [0 , ,k, k +22 + ( k + 2)( k + 3)3 λ kB [0] [0 , ,k +2 , k +22 . The specialization of (7.11) to k = 3 agrees with the set of equations given in [33] eventhough we have chosen a different basis. It is convenient that we get enough crossingequations to completely fix the new OPE coefficients in terms of the ones that alreadyappear in (7.9).The number of topological crossing equations we obtain, whether of linear or quadratictype, becomes arbitrarily large as we increase the external weights. Consider ∆ A = ∆ C = j A = j C = E , ∆ B = ∆ D = j B = j D = k which is not next-to-next-to-extremal anymore.This is a more general setup which still has only squared OPE coefficients (linear type). Ifwe define the weighted sums¯ λ m ≡ k + E (cid:88) p = m + k −E − p λ E kB [0] [ ,p − m − k −E , m + k −E , ] p , (7.12)crossing symmetry for this correlator may be compactly stated in terms of a generalizedhypergeometric function. ¯ λ n = E (cid:88) m =0 K nm ¯ λ m (7.13) K nm ≡ ( − E E !( −E ) n (2 n + k − E + 1) n !( n + k − E + 1) E +1 4 F (cid:34) − m , n − E , − − n − k , 1 + m + k − E −E , −E (cid:35) This comes from explicitly projecting one Jacobi polynomial onto another term-by-term. An empirical observation is that this crossing matrix gives half of its geometric multiplicityto the +1 eigenvalue rounded up which leads to (cid:6) E (cid:7) crossing equations. Basic consistency conditions like (7.9) and (7.11) need to hold (order-by-order when c T can be tuned) in any 3d N = 8 SCFT. A more precise check of AdS/CFT can be done if Hypergeometric functions with the argument suppressed are understood to be evaluated at 1. Since the conformal blocks for this problem are the same as those for a spatial cross-ratio, analyticallycontinued to negative integer weights, it should be possible to obtain (7.13) as a limit of the continuouscase in [79]. – 36 –e compute topologically protected data that are unique to ABJM theory at level 1. Animportant tool for this is an explicit Lagrangian, derived in [43] via localization, whichdescribes the topological subsector for the fixed-point of any theory consisting of N = 4hyper and vector multiplets. Even though it is a Chern-Simons matter theory, ABJMcan be realized in this way due to its duality with a U ( N ) gauge theory consisting of twohypermultiplets [46, 47].Since each representation of the gauge group must be accompanied by its conjugate,the superconformal primary of a hypermultiplet contributes two fields to a gauge theoryLagrangian. Instead of writing down these fields in three dimensions, we will jump rightto their twisted-translated counterparts in the topological theory. To wit, a fundamentalhypermultiplet gives rise to 1d fields Q and ˜ Q – one for each N dimensional representationof U ( N ) – and an adjoint hypermultiplet analogously gives rise to X and ˜ X . These arethe ingredients needed for ABJM theory. Their actions are given by S Q = − πr (cid:90) π − π d ϕ ˜ Q α (cid:104) ˙ Q + σQ (cid:105) α S X = − πr (cid:90) π − π d ϕ ˜ X αβ (cid:104) ˙ X + [ σ, X ] (cid:105) βα (7.14)where σ = diag( σ , . . . , σ N ) (7.15)describes the Cartan of the U ( N ) gauge group. Computing functional determinants withthe proper reality condition, the one for S Q depends on individual matrix elements of (7.15)while the one for S X depends on their differences [43]. Z σ = N (cid:89) α =1 (cid:89) n ∈ Z + ( in + σ α ) − N (cid:89) α,β =1 (cid:89) n ∈ Z + ( in + σ αβ ) − = N (cid:89) α =1 πσ α ) N (cid:89) α,β =1 πσ αβ ) − (7.16)Integrating out Q and ˜ Q , local operators on the circle will be built from gauge invariantproducts of X and ˜ X . Their correlators at fixed σ , which should be fed into (cid:104)O ( ϕ ) . . . O n ( ϕ n ) (cid:105) = 1 Z N N ! (cid:90) d N σ (cid:89) α<β ( πσ αβ ) Z σ (cid:104)O ( ϕ ) . . . O n ( ϕ n ) (cid:105) σ , (7.17)follow from Wick’s theorem. Since σ plays the role of a mass, the relevant propagatorshould be read off from the inverse Fourier transform of ( in + σ ) − . The result of this is (cid:68) X αβ ( ϕ ) ˜ X δγ ( ϕ ) (cid:69) σ = − δ αγ δ δβ e − σ αβ ϕ πr [sgn ϕ + tanh( πσ αβ )] (7.18) (cid:68) X αβ ( ϕ , y ) X δγ ( ϕ , y ) (cid:69) σ = δ αγ δ δβ e − σ αβ ϕ πr [sgn ϕ + tanh( πσ αβ )] y . – 37 –n the second line, we have recognized that ( X, ˜ X ) forms a doublet under su (2) II whichmeans we can define X by saturating this doublet with a polarization vector. Althoughmost of our calculations will be done at large N , it is important to note that operatorsfrom the interacting theory should only involve the traceless partˆ X ≡ X − N Tr X . (7.19)The trace is necessarily a decoupled free multiplet as can be seen from the action or therepresentation theory of osp (8 | j I in ˆ X and degree 2 j II in y . For us, this degeneracy is resolved by the fact that each tracesuppresses an operator further at large N . We have already mentioned that the mostsensible external operators in a Witten diagram expansion are primaries of a half-BPSmultiplet. To reflect their single-particle character, we will write O k ( ϕ, y ) = Tr (cid:104) ˆ X ( ϕ, y ) k (cid:105) (7.20)in this section. However, it is important to remember that multi-trace operators also con-tribute to single-particle states. Their contribution just receives additional powers of 1 /N .The internal operators come in various types which we will now discuss for (cid:104)O O k O O k (cid:105) .In fact, it is not difficult to predict the powers of N at which they first appear. λ kB [0] [0 , ,k − , k − , λ kB [0] [0 , ,k − , k = O (cid:0) N − (cid:1) , λ kB [0] [0 , ,k, k = O (cid:16) N − (cid:17) (7.21) λ kB [0] [0 , ,k − , k +22 , λ kB [0] [0 , ,k, k +22 , λ kB [0] [0 , ,k +2 , k +22 = O (1)First, the O (1) operators are the ones that survive in generalized free theory so this meansthey are double-trace. However, O k +2 has the same quantum numbers as [ O O k ] [0 , ,k +2 , so, already at tree-level, the squared coefficient for B [0] [0 , ,k +2 , k +22 should really be considereda sum of two squared OPE coefficients from the field theory point of view. It once againbecomes a single squared OPE coefficient (in the right basis), if we regard the externaloperator not as O k but as the single-particle state having maximal overlap with O k . Thisis because extremal correlators of operators dual to KK-modes vanish at all orders [65, 66].Next, B [0] [0 , ,k, k first contributes at tree-level because its dimension is too low to in-clude a double-trace between O and O k as one of its components. The two remainingcoefficients vanish even at tree-level. For B [0] [0 , ,k − , k , this is because double-trace com-ponents are ruled out by scaling dimension while single-trace components are ruled outby the fact that j I (cid:54) = j II . For B [0] [0 , ,k − , k − , this is because extremal correlators vanishin the supergravity basis. As we will see, we indeed get a contribution at tree-level if weconsider the external operator to be purely O k rather than a linear combination of O k and[ O O k − ] [0 , ,k, . It is instructive to check that this linear combination does not affect the O (cid:16) N − (cid:17) coefficient. Notice that there would be trace relations if we worked at finite N . Even if one is careful not toovercount operators, such relations make a grading of the spectrum by the number of traces ambiguous. Indeed, once the indices of (7.20) are restored, there is no way to contract them because su (2) does nothave any invariant tensors that are totally symmetric. – 38 – .3 Single-trace OPE coefficients The next step is to explore what the integral (7.17) can tell us in detail. After solving forsingle-trace OPE coefficients at the first non-vanishing order following [38], we will turnto the analogous calculation for double-trace operators. This will reproduce (6.4) but alsoextend this result to the most general O (1) four-point function.To understand the partition function Z N = 1 N ! (cid:90) d N σ (cid:89) α<β ( πσ αβ ) Z σ , (7.22)we will change to the variable x α = σ α √ N (7.23)which should be treated as continuously indexed. Sums over the eigenvalues then be-come integrals over x where the measure is N ρ ( x )d x . Correspondingly, (7.22) becomes afunctional integral over all of the possible choices for this measure. In the saddle-pointapproximation, this partition function is Z N ∼ e − F [ ρ ∗ ] (7.24)where − F [ ρ ] is the log of the integrand in (7.22) and ρ ∗ is the density that extremizes it.To proceed, we will need the approximationslog 2 cosh (cid:16) π √ N x (cid:17) ∼ π √ N | x | log tanh (cid:16) π √ N x (cid:17) ∼ π √ N δ ( x ) (7.25)where the latter coefficient follows from Taylor expanding log (cid:16) ± e π √ Nx (cid:17) and integratingterm-by-term. This allows us to write the free energy as F = N (cid:88) α =1 log 2 cosh (cid:16) π √ N x α (cid:17) − N (cid:88) α,β =1 log tanh (cid:16) π √ N x αβ (cid:17) ∼ πN (cid:90) d x (cid:20) | x | ρ ( x ) + 14 ρ ( x ) (cid:21) . (7.26)Importantly, the power of N in (7.23) has been chosen self-consistently so that (7.26) canindeed be extremized. Following [80, 81], the solution is ρ ∗ ( x ) = max (cid:16) √ − | x | , (cid:17) (7.27)which leads to the well known result Z N ∼ e π √ N . (7.28)Although the partition function itself will cancel out from correlation functions at leadingorder, the remaining calculations will depend crucially on the density (7.27).– 39 – igure 1 : A planar diagram representing the B k two-point function with k contractionsof double-lines. The shaded areas, from which the lines emanate, represent the single-traceoperators O k . To translate between the diagram and (7.31), we explicitly show the α j indices and the β τ ( j )+1 indices to which they are being set equal.For the correlators at fixed σ , we will set ϕ > ϕ and express the leading term using G σ ( ϕ ) = e − σϕ [1 + tanh( πσ )] (7.29)which appears in (7.18). When tracing over a product of k of these functions (and taking x k +1 ≡ x for brevity), it is a simple exercise to show that N (cid:88) α ,...,α k =1 G − σ α α ( ϕ ) . . . G − σ αkα ( ϕ ) = k (cid:89) j =1 (cid:104) π √ N ( x j − x j +1 ) (cid:105) ∼ N − k Γ (cid:0) k (cid:1) √ π Γ (cid:0) k +12 (cid:1) k − (cid:89) j =1 δ ( x j − x j +1 ) . (7.30)The dominant contribution to (cid:104)O k O k (cid:105) will indeed have this form. This can be seen bypicking an arbitrary τ ∈ S k and considering k (cid:89) j =1 (cid:68) ˆ X α j α j +1 ( ϕ , y ) ˆ X β τ ( j ) β τ ( j )+1 ( ϕ , y ) (cid:69) = (cid:16) y πr (cid:17) k N (cid:88) α ,...,α k =1 G − σ α α ( ϕ ) . . . G − σ αkα ( ϕ ) N (cid:88) β ,...,β k =1 k (cid:89) j =1 δ α j β τ ( j )+1 δ β τ ( j ) α j +1 . (7.31)The most important permutation is the one that allows every term to survive in the innersum – namely τ ( j ) = τ ( j + 1) + 1 = ⇒ τ ( j ) + j ∈ { , . . . , k } (7.32)which follows from examining the Kronecker deltas. Clearly, all k of these permutationscontribute equally. Figure 1 shows the result of (7.32) diagrammatically. As expected in Unsurprisingly, this corresponds to the planar diagram when we use double-line notation as a book-keeping device. The standard genus expansion would then suggest the scaling N k since a planar diagramwith 2 vertices and k edges also has k faces. Instead of this, the matrix integral causes N to appear with anon-trivial power. – 40 – large N approximation, this pattern of contracting indices leads to the largest possiblenumber of closed loops.Now that we know the most important term in a two-point function, we can integrateit to get B k = k Γ (cid:0) k (cid:1) √ π Γ (cid:0) k +12 (cid:1) N k +12 (8 πr ) k (cid:90) d x ρ ( x ) k = Γ (cid:0) k +22 (cid:1) √ π Γ (cid:0) k +32 (cid:1) (2 N ) k +12 (8 πr ) k (7.33)at leading order. The single-trace three-point function will only survive if α , α and α from (3.14) are all non-negative integers. When this holds, the leading contraction can bededuced in a similar manner yielding C k k k = k k k N (cid:18) N πr (cid:19) k k k (cid:90) d x d x ρ ( x ) ρ ( x ) cosh (cid:16) π √ N x (cid:17) (7.34) (cid:81) α − j =1 (cid:82) d s j ρ ( s j )cosh (cid:104) π √ N ( x − s ) (cid:105) . . . cosh (cid:104) π √ N ( s α − − x ) (cid:105)(cid:81) α − j =1 (cid:82) d t j ρ ( t j )cosh (cid:104) π √ N ( x − t ) (cid:105) . . . cosh (cid:104) π √ N ( t α − − x ) (cid:105)(cid:81) α − j =1 (cid:82) d u j ρ ( u j )cosh (cid:104) π √ N ( x − u ) (cid:105) . . . cosh (cid:104) π √ N ( u α − − x ) (cid:105) = 16 k + k + k Γ (cid:16) k +22 (cid:17) Γ (cid:16) k +22 (cid:17) Γ (cid:16) k +22 (cid:17) √ π Γ (cid:0) α +12 (cid:1) Γ (cid:0) α +12 (cid:1) Γ (cid:0) α +12 (cid:1) Γ (cid:16) k + k + k (cid:17) (cid:32) √ N πr (cid:33) k k k . The evaluation of this integral requires another formula like (7.30) which is proven in [38].Putting (7.33) and (7.34) together, we have the squared OPE coefficient λ k k k = 2 k α !( α + 1) α C k k k B k B k B k (7.35)= 2 − k − k α ! α !( α + α )! (cid:18) π β (cid:19) ( k + 1)!( k + 1)!( k + 1)!Γ (cid:0) α +12 (cid:1) Γ (cid:0) α +12 (cid:1) Γ (cid:0) α +12 (cid:1) Γ (cid:16) β (cid:17) (2 N ) . This only differs from (3.13) by a normalization. The possibility of more R-symmetry representations opens up when we move onto double-trace operators. Let us keep track of them in an index-free way with[ O E O k ] [0 ,j,k + E− j, ( ϕ, y ) = ( k − j )!( E − j )! k ! E ! (cid:10) ∂ y , ∂ y (cid:48) (cid:11) j O E ( ϕ, y ) O k ( ϕ, y (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) y (cid:48) = y (7.36)where we have defined (cid:10) ∂ y , ∂ y (cid:48) (cid:11) ≡ ∂ ay ∂ by (cid:48) (cid:15) ab . A straightforward calculation shows that thethree-point functions we are interested in are essentially squares of two-point functions.– 41 –his means that, at leading order, O E ( ϕ , y ) is always contracted with O E ( ϕ , y ) and O k ( ϕ , y ) is always contracted with O k ( ϕ , y (cid:48) ). OPEs with E = k also allow the oppositetype of contraction at the cost of a ( − j sign. As a result, C E k [ O E O k ] [0 ,j,k + E− j, = 1 + ( − j δ k E π Γ (cid:0) k +22 (cid:1) Γ (cid:0) E +22 (cid:1) Γ (cid:0) k +32 (cid:1) Γ (cid:0) E +32 (cid:1) (2 N ) k + E +22 (8 πr ) k + E . (7.37) (a) C O O ] (b) B [ O O ] Figure 2 : Diagrams showing the correlation functions that effectively reduce to products oftwo-point functions of the type shown in Figure 1. Single-trace operators are shaded in redwhile double-trace operators are shaded in yellow. We have not attempted to distinguishbetween the different R-symmetry representations that [ O O ] can have in the diagram.The main technical challenge of this subsection is that each one leads to derivatives beingdistributed in different ways.This time, it is the calculation of the norm that requires more work. The key piece ofalgebra is (cid:15) a b ∂ a y . . . (cid:15) a j b j ∂ a j y (cid:104) y , y (cid:105) k − j (cid:10) y , y (cid:48) (cid:11) p y (cid:48) b . . . y (cid:48) b p y b p +1 . . . y b j (cid:12)(cid:12)(cid:12)(cid:12) y (cid:48) = y = (cid:15) a p +1 b p +1 ∂ a p +1 y . . . (cid:15) a j b j ∂ a j y (cid:104) y , y (cid:105) k − j p (cid:88) q =0 (cid:32) pq (cid:33) q !( j − p )!( j − p + q )! (cid:10) y , y (cid:48) (cid:11) p − q (cid:10) y (cid:48) , y (cid:48) (cid:11) q y (cid:48) b p +1 . . . y (cid:48) b p − q y b p − q +1 . . . y b j (cid:12)(cid:12)(cid:12)(cid:12) y (cid:48) = y = (cid:104) y , y (cid:105) k − j + p p (cid:88) q =0 (cid:32) pq (cid:33) p !( j − p )!( k − p + 1)!( j − p + q )!( k − j + p − q + 1)! . (7.38) Clearly, the permutation (7.32) should be used in both cases. – 42 –his in turn allows us to compute (cid:68) ∂ y , ∂ y (cid:48) (cid:69) j (cid:68) ∂ y , ∂ y (cid:48) (cid:69) j (cid:104) y , y (cid:105) k (cid:10) y (cid:48) , y (cid:48) (cid:11) E (cid:12)(cid:12)(cid:12)(cid:12) y (cid:48) = y = k ! E ! (cid:68) ∂ y , ∂ y (cid:48) (cid:69) j ( k − j )!( E − j )! (cid:104) y , y (cid:105) k − j (cid:10) y (cid:48) , y (cid:48) (cid:11) E− j (cid:10) y , y (cid:48) (cid:11) j (cid:12)(cid:12)(cid:12)(cid:12) y (cid:48) = y = k ! E !( k − j )! (cid:15) a b ∂ a y . . . (cid:15) a j b j ∂ a j y (cid:104) y , y (cid:105) k − j j (cid:88) p =0 (cid:32) pj (cid:33) j ! (cid:104) y (cid:48) , y (cid:48) (cid:105) E− j − p p !( E − j − p )! (cid:10) y , y (cid:48) (cid:11) p y (cid:48) b . . . y (cid:48) b p y b p +1 . . . y b j (cid:12)(cid:12)(cid:12)(cid:12) y (cid:48) = y = k ! E !( k − j )! (cid:104) y , y (cid:105) k + E− j j (cid:88) p =0 (cid:32) jp (cid:33) j !( k + 1)!( E − j − p )!( k + p − j + 1)! . (7.39)With (7.38) and (7.39) in hand, the desired two-point function is B [ O E O k ] [0 ,j,k + E− j, = 1 + δ k E π ( k − j )!( E − j )! k ! E ! j !( k + E − j + 1)!( k + E − j + 1)! Γ (cid:0) k +22 (cid:1) Γ (cid:0) E +22 (cid:1) Γ (cid:0) k +32 (cid:1) Γ (cid:0) E +32 (cid:1) (2 N ) k + E +22 (8 πr ) k + E (7.40)which means λ E k [ O E O k ] [0 ,j,k + E− j, = 2 k + E ( E − j )!( k − j + 1) E− j C E k [ O E O k ] [0 ,j,k + E− j, B E B k B [ O E O k ] [0 ,j,k + E− j, = 2 k + E (cid:2) − j δ k E (cid:3) k ! E ! j ! k + E − j + 1( k + E − j + 1)! . (7.41)These OPE coefficients, which satisfy (7.13) look like they would have been quite hard toderive otherwise. By demanding agreement with the standard methods for generalized freetheories, we predict the identity σ E = E (cid:88) j =0 k ! E ! j ! k + E − j + 1( k + E − j + 1)! Y ,k −EE , E− j ( σ, τ ) + O ( σ p τ q ) , p + q < E (7.42)which holds for all of the harmonic polynomials we have checked. The matrix model cannotdetermine the lower degree terms because they describe double-trace operators that live inA-type multiplets.Due to a remarkable correspondence with a Fermi gas, all of the perturbative terms inthe partition function have been computed and resummed into an Airy function [52, 53].This is the origin of the expression (2.9). It would be very interesting to see if thesehigher order techniques could be generalized to cases with operator insertions. Due tothe connection between stress tensor correlators and the partition function on a squashedsphere, some initial progress in this direction was made in [32]. Among the results of the last section are tree-level single-trace OPE coefficients, namely(7.21) and (7.35). Although the limited set of matrix model techniques we have reviewed These Fermi gas methods also have applications to the protected subsector in four dimensions [82]. More recently, a framework for expanding more general Higgs and Coulomb branch correlators wasdeveloped in [83]. We thank Jihwan Oh for bringing this paper to our attention. – 43 –s not powerful enough to determine tree-level double-trace OPE coefficients outright, westill have (7.9) and (7.11) as highly non-trivial constraints. In this section, we will extractthe necessary single-particle and double-particle OPE coefficients using Mellin space andfind perfect agreement. To solve for generic CFT data encoded in a Mellin amplitude one needs detailed knowledgeof the superconformal blocks. A question for us is whether the situation is more favorablefor the short multiplets of B-type. As we will explain shortly, three necessary steps in thiscalculation are: projecting onto the R-symmetry polynomial for [0 , j I − j II , j II , s to set ∆ − (cid:96) = j I and taking V → (cid:96) = 0. Calculations atlow weight, such as those in [84], have demonstrated that this is sufficient to describe aunique conformal primary. We will now show that the same conclusion continues to holdfor arbitrary external weights.First, we must show that B [0] [0 ,j I − j II , j II , j I cannot overlap with a super descendant ofany other B [0] [0 ,j (cid:48) I − j (cid:48) II , j (cid:48) II , j (cid:48) I . To achieve degeneracy in the scaling dimension (which can onlyhappen for j (cid:48) I < j I ), we start off with the ∆ = j (cid:48) I primary and act with 2( j I − j (cid:48) I ) supercharges.To see that the so (8) representations thus produced stay away from [0 , j I − j II , j II , Q [1 , , , α , Q [ − , , , α , Q [0 , − , , α , Q [0 , , , − α Q [0 , , − , α , Q [0 , , − , − α , Q [1 , − , , α , Q [ − , , , α (8.1)where we have left the su (2) weights implicit. The only supercharge in this list that hasa positive first Cartan is Q [1 , , , α . According to table 1, this is precisely the one thatannihilates the primary. We can therefore be certain that the first Cartan of a genericrepresentation will not increase if we constantly use the algebra to move Q [1 , , , α to theleft in the states that we generate.There are also non-generic representations to consider which have at least one vanishingDynkin label. If one of the supercharges in (8.1) lowers this Dynkin label again, theresulting weight vector λ must be transformed according to λ (cid:55)→ ω ( λ + ρ ) − ρ, ω ∈ Weyl( so (8)) (8.2)for some ω which makes all of the Dynkin labels non-negative. The Weyl vector of so ( )is ρ = [3 , , , 0] and the Weyl group of so (8) is S (cid:110) ( S ) . The S acts by permutingthe Cartan eigenvalues and an S can be taken to flip the sign of any two of them. Wetherefore find that under (8.2), h (cid:55)→ ± ( h i + 4 − i ) − , i ∈ { , , , } . (8.3)To verify that (8.3) can never be larger than the original h , we simply go through all eightpossibilities and apply the inequalities that follow from h = r + r + r + r , h = r + r + r , h = r + r , h = r − r . (8.4)– 44 –n the highest-weight case, (8.4) is what ensures h i − h i +1 ≥ λ considered here is slightly negative, we must weaken this conditionto h i − h i +1 ≥ − (cid:96) + 1 for A [2 (cid:96) ] [0 ,j (cid:48) I − j (cid:48) II , j (cid:48) II , j (cid:48) I + (cid:96) +1 and greater than (cid:96) + 1 for a long multiplet. Clearly, Q [1 , , , α is the only superchargewhich raises h faster than it raises ∆. We can therefore imagine that one catches upto the other after we descend from the primary twice. However, this leads to Dynkinlabels of [2 , j (cid:48) I − j (cid:48) II , j (cid:48) II , 0] which cannot appear in (6.3). To keep us in an admissible so (8)representation, each appearance of Q [1 , , , α must be accompanied by either Q [ − , , , α or Q [ − , , , α . This has the net effect of ensuring that h never increases by 1 unless ∆ increasesby 1 as well. We will now fill in the technical steps of taking a four-point function in Mellin space andextracting a particular set of quantum numbers using (3.2).If we wanted to solve for the position space correlator in the 0 < U < s variable. In practice, we only need to look at one pole for each term in the tensor product(6.3). These come in two distinct types. When j I < min (cid:16) k + k , k + k (cid:17) , all of the gammafunctions in (3.2) stay finite when s is set equal to the twist of B [0] [0 ,j I − j II , j II , j I . It is onlythe single-trace function M ( s ) which can possibly diverge at this value of s . This leads toan especially simple t integral once the s = j I residue is isolated. Setting V = 1 brings itinto a form that yields immediately to the first Barnes lemma (cid:90) i ∞− i ∞ d t πi Γ( a − t )Γ( a − t )Γ( b + t )Γ( b + t ) = Γ( a + b )Γ( a + b )Γ( a + b )Γ( a + b )Γ( a + a + b + b ) . (8.5)It is worth pointing out that adding up the residues in the t variable is not enough to derive(8.5) as this would miss a contribution from the arc at infinity. One way to see this is tostart off with V as a parameter and only take V → V , which is neglected in the naive approach, generically multipliesa sum that diverges as V → j I to its maximum possible value is howwe first reach a vanishing gamma function argument. As explained in footnote 7, s = j I is strictly a double-particle pole. In fact, a slightly closer look at the amplitudes revealsthat only M ( t ) and M ( u ) contribute in this case. The contour of the resulting t integralencloses the single-particle poles of the former and avoids the single-particle poles of the This is because j I = min (cid:0) k + k , k + k (cid:1) is associated with the harmonic polynomials of maximal degreebut all of the monomials σ E , . . . , τ E are absent from M ( s ) ( s, t ; σ, τ ). – 45 –atter. We must therefore proceed according to two slightly different methods. Both ofthem are based on F ( a , a ; b ; z ) = Γ( b )Γ( a )Γ( a )Γ( b − a )Γ( b − a ) (cid:90) i ∞− i ∞ d t πi Γ( a − t )Γ( a − t )Γ( b − a − a + t )Γ( t )(1 − z ) − t (8.6)which generalizes (8.5). Notably, (8.6) was part of the original justification for [89]. Wehave not set z = 0 above because we want to use it to accommodate simple poles in t thatcome from the Mellin amplitudes. This is possible thanks to the integral representations1 t − m − δ = − (cid:90) (1 − z ) m − δ − t d z, t + 2 m + δ = 12 (cid:90) (1 − z ) m − δ + t d z. (8.7)Importantly, we have arranged the exponents so that only the first integral can possiblydiverge when the t contour is closed to the right. This reflects the fact that single-particlepoles of M ( t ) must be counted while those of M ( u ) must be skipped. We can now derivetwo very powerful inverse Mellin transformations from (8.6) and (8.7). The first is (cid:90) i ∞− i ∞ d t πi Γ (cid:0) a − t (cid:1) Γ (cid:0) a − t (cid:1) Γ (cid:0) b + t (cid:1) Γ (cid:0) b + t (cid:1) t − m − δ (8.8)= − Γ( a + b )Γ( a + b )Γ( a + b )Γ( a + b )Γ( a + a + b + b ) (cid:2) b + m + δ (cid:3) F (cid:34) a + b , a + b a + a + b + b , 1 + b + m + δ (cid:35) while the second is (cid:90) i ∞− i ∞ d t πi Γ (cid:0) a − t (cid:1) Γ (cid:0) a − t (cid:1) Γ (cid:0) b + t (cid:1) Γ (cid:0) b + t (cid:1) t + 2 m + δ (8.9)= Γ( a + b )Γ( a + b )Γ( a + b )Γ( a + b )Γ( a + a + b + b ) (cid:2) a + m + δ (cid:3) F (cid:34) a + b , a + b a + a + b + b , 1 + a + m + δ (cid:35) . To see that the right hand sides of (8.8) and (8.9) are both separately invariant under a ↔ a and b ↔ b , we need to use the Thomae relations.There is still work to do since these hypergeometric functions depend on m . For this,we refer to Appendix B where it is shown that the sum over descendants (an infinite sumfor AdS × S ) can always be written as a finite sum of integrals of the form W = (cid:90) i ∞− i ∞ d t πi Γ( − t )Γ(1 + a − b − b − b − t ) ( b ) t ( b ) t ( b ) t (1 + a − c − c ) t (1 + a − c ) t (1 + a − c ) t . (8.10)This integral has attracted a great deal of interest since it appears in the crossing kernel forcollinear blocks [90, 91]. One of its interesting mathematical properties is a representationin terms of a single very well-poised hypergeometric function.Γ(1 + a − b )Γ(1 + a − b )Γ(1 + a − b )Γ(1 + a )Γ(1 + a − b − b )Γ(1 + a − b − b )Γ(1 + a − b − b ) W (8.11)= F (cid:34) a , 1 + a , b , b , b , c , c a , 1 + a − b , 1 + a − b , 1 + a − b , 1 + a − c , 1 + a − c (cid:35) . – 46 –ortunately, we have been able to find closed form expressions for all instances of (8.10)that appear in the rest of this section. As we will see, these evaluations make use of (8.11)and also more elementary contour manipulations. For reasons that are not yet clear to us,it appears that one method leads to simple expressions if and only if the other leads tocomplicated expressions. We are now in a good position to return to the E = 2 correlators that involve squaredOPE coefficients. The first tree-level result we can derive is very simple. Since the Mellinamplitudes (3.4) are regular at s = k − , we must have δλ kB [0] [0 , ,k − , k − = 0 . (8.12)Moving onto the harmonic polynomials of degree 1, M [0 , ,k − , k k ( s, t ) = − πc T Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 (2 s + k + 2)(2 s + 2 kt − k − k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) (cid:0) s − m − k (cid:1) + . . . M [0 , ,k, k k ( s, t ) = 8 πc T Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 (2 s + k + 2)(2 s + k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) (cid:0) s − m − k (cid:1) + . . . (8.13)where the omitted terms only have poles in t and u . Clearly, we must take m = 0 in (8.13)to get a pole at s = k . Performing the t integral with (8.5) leads to δλ kB [0] [0 , ,k − , k = 0 , δλ kB [0] [0 , ,k, k = 2 k kc T . (8.14)Another way to derive this result would have been to use the Ward identity with Bosesymmetry. We now come to the more challenging calculation which is the one where theR-symmetry polynomials have maximal degree. M [0 , ,k − , k k ( s, t ) = 8 π ( k + 1) c T (cid:34) Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 (2 s − k − s + 2 t + 2 kt + k + k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) (cid:0) t − m − k (cid:1) + k − √ π ∞ (cid:88) m =0 (2 s − k − s + 2 t + 2 kt − k − k − (cid:0) + m (cid:1) − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) ( s + t + 2 m − k − (cid:35) (8.15a) M [0 , ,k, k k ( s, t ) = − π ( k + 2) c T (cid:34) Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 (2 s − k − s + 4 t + 2 kt + k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) (cid:0) t − m − k (cid:1) − √ π ∞ (cid:88) m =0 (2 s − k − s − ks + 2 t + kt − (cid:0) + m (cid:1) − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) ( s + t + 2 m − k − (cid:35) (8.15b) M [0 , ,k +2 , k k ( s, t ) = 16 kπ ( k + 1)( k + 2) c T (cid:34) Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 (2 s − k − s − k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) (cid:0) t − m − k (cid:1) − √ π ∞ (cid:88) m =0 (2 s − k − s − k − (cid:0) + m (cid:1) − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) ( s + t + 2 m − k − (cid:35) (8.15c)– 47 –he common zero in all of these projected amplitudes is no accident. Despite the squaredgamma function, s = k +22 must be only a simple pole since the short multiplets belowthreshold are protected by recombination rules. After extracting this pole, we can manuallycompute the t integral with (8.8) and (8.9). Note that this requires a partial fractiondecomposition which reintroduces explicit contact terms (to be treated with (8.5)). Theresulting expressions are δλ kB [0] [0 , ,k − , k +22 = 2 k +9 π c T k − k − k +8 π ( k + 1) c T Γ (cid:0) k (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 m + 2 mk + k + 2 k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) F 1, 1, 1 k m m + 1+ k − √ π Γ (cid:0) k (cid:1) Γ (cid:0) k +22 (cid:1) ∞ (cid:88) m =0 m + 2 mk + k + 3 m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) Γ (cid:0) + m (cid:1) F 1, 1, k k m m + 1 (8.16a) δλ kB [0] [0 , ,k, k +22 = 2 k +8 π ( k + 2) c T Γ (cid:0) k (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 m + 2 mk + k + 3 k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) F 1, 1, 1 k m m + 1 − √ π Γ (cid:0) k (cid:1) Γ (cid:0) k +22 (cid:1) ∞ (cid:88) m =0 m + 2 mk − k + 6 m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) Γ (cid:0) + m (cid:1) F 1, 1, k k m m + 1 (8.16b) δλ kB [0] [0 , ,k +2 , k +22 = 2 k +10 kπc T Γ (cid:0) k (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 ( k + 1) − ( k + 2) − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) F 1, 1, 1 k m m + 1+ 1 √ π Γ (cid:0) k (cid:1) Γ (cid:0) k +22 (cid:1) ∞ (cid:88) m =0 ( k + 1) − ( k + 2) − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) Γ (cid:0) + m (cid:1) F 1, 1, k k m m + 1 . (8.16c)The middle line vanishes when k = 2 which makes sense because of Bose symmetry.As a warm-up to computing (8.16) in closed form, let us first verify that crossingsymmetry holds at tree-level in the topological subsector. This is somewhat simpler dueto cancellations that take place when (8.16a), (8.16b) and (8.16c) are added with the right– 48 –oefficients. Defining S = ∞ (cid:88) m =0 (cid:0) (cid:1) m m ! (cid:0) k +32 (cid:1) m (cid:0) m + (cid:1) F (cid:34) 1, 1, 1 k + 1, + m (cid:35) (8.17) S = ∞ (cid:88) m =0 (cid:0) (cid:1) m (cid:0) − k (cid:1) m m ! (cid:0) (cid:1) m (cid:0) m + (cid:1) F (cid:34) 1, 1, k k + 1, + m (cid:35) , the crossing equation (7.9) takes the form k k − kπ − k Γ (cid:0) k (cid:1) π Γ (cid:0) k − (cid:1) Γ (cid:0) k +32 (cid:1) S − k Γ (cid:0) k (cid:1) π Γ (cid:0) k − (cid:1) Γ (cid:0) k +22 (cid:1) S = 0 . (8.18)Our task is now to evaluate (8.17).Starting with S , Appendix B instructs us to write it as the following Mellin-Barnesintegral. S = Γ (cid:0) k +32 (cid:1) √ π Γ (cid:0) k (cid:1) (cid:90) i ∞− i ∞ d t πi Γ( − t )Γ (cid:0) − − t (cid:1) Γ(1 + t ) Γ (cid:0) k + t (cid:1) Γ (cid:0) k +32 + t (cid:1) Γ (cid:0) k +22 + t (cid:1) Γ (cid:0) k +42 + t (cid:1) (8.19)= Γ (cid:0) k +32 (cid:1) Γ (cid:0) k − (cid:1) Γ( k + 2)Γ (cid:0) k +42 (cid:1) Γ (cid:0) k +12 (cid:1) Γ( k + 1) F (cid:34) k + 1, k +32 , k , k , k +22 , k +32 , 1 k +12 , k +42 , k +42 , k +22 , k +12 , k + 1 (cid:35) . We have turned it into a single hypergeometric function using (8.11). Even though thereare three ways to do this, the other two are much less favorable. The important propertyof (8.19) is that many of the numerator parameters can be paired with denominator pa-rameters that are smaller by non-negative integers. After reducing the order four times,(8.19) becomes S = Γ (cid:0) k +32 (cid:1) Γ (cid:0) k − (cid:1) Γ( k + 2)Γ (cid:0) k +42 (cid:1) Γ (cid:0) k +12 (cid:1) Γ( k + 1) (cid:34) k ( k + 2) ( k + 1) ( k + 4) ( k + 6) F (cid:34) k +42 , k +42 , 3 k +82 , k +82 (cid:35) + 4 k ( k + 2)( k + 1) ( k + 4) F (cid:34) k +22 , k +22 , 2 k +62 , k +62 (cid:35) + F (cid:34) k , k , 1 k +42 , k +42 (cid:35)(cid:35) = k − 116 Γ (cid:0) k − (cid:1) Γ (cid:0) k +22 (cid:1) (cid:20) k − 1) + k ψ (1) (cid:18) k (cid:19)(cid:21) (8.20)where we have used some of the tricks in [92].The same technique does not appear to work for S . The approach we will take insteadstarts with the substitution s = t + k − . Equivalently, the iterated Mellin-Barnes integralsthat appear in Appendix B should be evaluated in the opposite order. This leads to S = √ π Γ (cid:0) k +22 (cid:1) (cid:0) k (cid:1) Γ (cid:0) − k (cid:1) (cid:90) i ∞− i ∞ d s πi Γ( − s )Γ (cid:0) k − − s (cid:1) Γ (cid:0) − k + s (cid:1) Γ (cid:0) + s (cid:1) Γ (1 + s )Γ (cid:0) + s (cid:1) (8.21)after we choose the right ordering for the numerator parameters in (8.17). Crucially, thetwo gamma function arguments in (8.21) that depend on k also add up to unity. This– 49 –llows us to remove the k dependence from the integrand entirely after using the reflectionformula. The important caveat to keep in mind is that, after we do this, the naturalcontour for the resulting integral will live in the critical strip − < (cid:60) ( s ) < 0. Since thisis not the contour for (8.21), we need to go between them by subtracting the residues at , , . . . , k − . An arbitrary residue from this set is given bylim s → k − − j Γ( − s )Γ (cid:0) k − − s (cid:1) Γ (cid:0) − k + s (cid:1) Γ (cid:0) + s (cid:1) Γ (1 + s )( s + − k + j ) − Γ (cid:0) + s (cid:1) = Γ (cid:0) − k (cid:1) Γ (cid:0) k − (cid:1)(cid:0) − k + j (cid:1) (8.22)which is a nice object to have in a finite sum. Putting the finite sum and the massagedintegral together, S = Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) √ π Γ (cid:0) k (cid:1) (cid:90) i ∞− i ∞ d s πi Γ( − s )Γ (cid:0) − s (cid:1) Γ (cid:0) + s (cid:1) Γ (1 + s )Γ (cid:0) + s (cid:1) + k − (cid:88) j =0 π (cid:0) − k + j (cid:1) = Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) √ π Γ (cid:0) k (cid:1) (cid:20) π π (cid:18) π − ψ (1) (cid:18) k (cid:19)(cid:19)(cid:21) . (8.23)From (8.20) and (8.23), we can now see that crossing symmetry holds exactly in the 1dtopological sector for all k . Figure 3 : The complex s -plane showing the poles of Γ (cid:0) k − − s (cid:1) Γ (cid:0) − k + s (cid:1) offset fromthe real axis for clarity. The red contour encloses the sequence that increases from s = k − (crosses) and avoids the sequence that decreases from s = k − (dots). It is thereforethe Mellin-Barnes contour for (8.21). It must be deformed until it reaches the imag-inary axis which is the Mellin-Barnes contour for the integral in (8.23) that involvesΓ (cid:0) − s (cid:1) Γ (cid:0) + s (cid:1) . The blue contour therefore includes residues at a discrete set of points.Note that we have suppressed the poles of other gamma functions that are irrelevant tothis discussion. – 50 –xtending this analysis to all of (8.16) is now conceptually straightforward – it leads tothree sums similar to (8.20) and three sums similar to (8.23). Performing this calculation,we arrive at δλ kB [0] [0 , ,k − , k +22 = 2 k +6 ( k − k ( k + 1) π c T (cid:20) k + k + 2 k + 4) − k ( k + 2) π − k ( k + 2) ψ (1) (cid:18) k (cid:19)(cid:21) δλ kB [0] [0 , ,k, k +22 = 2 k +8 k ( k + 2) π c T (cid:20) k − k − − k π + k ( k + 2 k − ψ (1) (cid:18) k (cid:19)(cid:21) δλ kB [0] [0 , ,k +2 , k +22 = 2 k +7 k ( k + 1)( k + 2) π c T (cid:20) k − k + 2) + k π + 2 k ψ (1) (cid:18) k (cid:19)(cid:21) . (8.24) To learn even more about the E = 2 correlators, we can look at the channel that does notproduce squared OPE coefficients. There is one protected operator whose contribution wecan extract from M [0020]22 kk ( s, t ) = 16 kπ c T ∞ (cid:88) m =0 s ( s + 2) m !Γ (cid:0) − m (cid:1) Γ (cid:0) k − − m (cid:1) Γ (cid:0) + m (cid:1) ( s − m − 1) + . . . (8.25)by taking the residue at s = 1. Again, the omitted terms only have poles in t and u .Setting m = 0 and applying the familiar steps with (8.5) leads to δλ B [0] [0020]1 λ kkB [0] [0020]1 = 128 kc T . (8.26)Comparing to (7.11), this is exactly what it should be. There are two more checks to be doneand these involve the Mellin amplitudes projected onto degree 2 harmonic polynomials. M [0200]22 kk ( s, t ) = 8 k πc T Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 s − km !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1)(cid:34) s + 6 t − k − t − m − k + 2 s + 6 t − k − s + t + 2 m − − k (cid:35) M [0040]22 kk ( s, t ) = 16 k πc T Γ (cid:0) k +22 (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 ( s − k )( s − k − m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1)(cid:34) t − m − k − s + t + 2 m − − k (cid:35) (8.27)The analogue of (8.16), found by taking the residue at s = 2, is much simpler now becausethe hypergeometric functions immediately reduce to polygamma functions. δλ B [0] [0200]2 λ kkB [0] [0200]2 = 1024( k − π c T − k πc T Γ (cid:0) k (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 (6 m + k + 3) ψ (1) (cid:0) + m (cid:1) m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) δλ B [0] [0040]2 λ kkB [0] [0040]2 = 256 k πc T Γ (cid:0) k (cid:1) Γ (cid:0) k − (cid:1) ∞ (cid:88) m =0 ψ (1) (cid:0) + m (cid:1) m !Γ (cid:0) − m (cid:1) Γ (cid:0) k +32 + m (cid:1) (8.28)– 51 –ecause of this, we do not need the full power of Appendix B to get a closed form. Thefastest way is to write ψ (1) (cid:18) 12 + m (cid:19) = lim η,ζ → (cid:34) ∂ ∂η Γ( η + m )Γ (cid:0) + m (cid:1) − ∂ ∂η∂ζ Γ( η + m )Γ( ζ + m )Γ (cid:0) + m (cid:1) (cid:35) (8.29)and use Gauss’ theorem on the sums that have η and ζ as free parameters. We arrive at δλ B [0] [0200]2 λ kkB [0] [0200]2 = 2563 kπ c T (cid:20) k − k + 4) − k ( k + 2) π + 2 k ( k + 2) ψ (1) (cid:18) k (cid:19)(cid:21) δλ B [0] [0040]2 λ kkB [0] [0040]2 = 256( k − kπ c T (cid:20) k π − k ψ (1) (cid:18) k (cid:19)(cid:21) (8.30)which is the same as what comes out of (7.11). In this paper, we performed a detailed study of the structures of tree-level four-point holo-graphic correlators in maximally superconformal theories. We made essential use of twoimportant ingredients, namely the MRV limit and residue symmetrization procedure intro-duced in [1, 2]. The former was demonstrated in [1, 2] to lead to remarkable simplifications,and in this paper we find further use of it in applying the superconformal twist. The latterallows the holographic correlators to be written purely in terms of exchange amplitudes,and makes it easier to recognize underlying structures. In the first of our main results, thesymmetrization prescription led us to a tantalizing formula (3.33) which exhibits a remark-able dimensional reduction structure. It is also interesting that the differential operatorappearing there acts on a scalar exchange Witten diagram, thereby avoiding the contactterm ambiguity associated with spinning vertices. Note this structure would be spoiled ifadditional contact terms in the amplitudes were present. Our second main result wasthat the chiral algebra four-point functions predicted by [20, 21] can all be extracted fromMellin amplitudes of the parent theory. This calculation exploited the fact that one of thecross-ratios drops out which means we can access the full chiral algebra correlator withoutever leaving the MRV limit. This insight failed in an interesting way for the topologicalsector of ABJM theory which had to be treated by other means. We will now recap thesefacets of the paper and comment on future directions.In section 3, we showed that the basic ingredients of holographic Mellin amplitudestake the form of a differential operator acting on a simple linear combination of scalarexchange Witten diagrams. This turned out to be related to the linear combination oneneeds in order to repackage bosonic conformal blocks into superblocks of the Parisi-Sourlassuperconformal algebra. As shown in [15, 19], correlation functions in d dimensions withthis supersymmetric property can be interpreted as those of a theory in d − AdS d +1 × S d − amplitudes exhibited features of AdS d − × S d − . To better understand this Similar absence of intrinsic contact interactions was also observed for the stress tensor multiplet five-point function for AdS × S IIB supergravity [93]. – 52 –henomenon, one must confront the fact that the dimensional reduction is rather formal.In particular, negative space-time dimensions can appear. Nevertheless, this is not thefirst time that a seed theory in a fictitious number of dimensions has been proposed in theholographic context. By a very different argument, [16–18] showed that SO ( d, × SO ( d )enhances to the higher dimensional conformal group SO ( d + d , 2) for the case of AdS × S and AdS × S . We also note that our application of Parisi-Sourlas supersymmetry wassomewhat indirect as it required a differential operator to be extracted first. This opera-tor, which is different for each multiplet, led to expressions that only involve the simplestrepresentations (scalar primaries of SO ( d, 2) and symmetric traceless tensors of SO ( d )). Ittherefore has some similarities to weight-shifting operators [94–96] which have been intro-duced as a tool to compute spinning conformal blocks and spinning Witten diagrams fromknowledge of their scalar counterparts. In the future, it would also be very important tocheck whether the dimensional reduction structure (3.33) is a coincidence for four-pointfunctions, or a universal feature shared by all n -point functions ( e.g. in the AdS × S five-point function computed in [93]). If the latter scenario were true, then the computation oftree-level higher-point holographic correlators would be greatly facilitated by uplifting themuch simpler correlators of the underlying scalar seed theory.In sections 4 and 5, we studied four-point functions in the chiral algebras that havebeen conjectured for the 6d N = (2 , 0) theory and 4d N = 4 SYM respectively. Our large N expressions (4.14) and (5.10), which are new for all but a handful of choices for the externalweights, were derived by two methods. First, by directly applying the singular OPE in twodimensions and second, by writing down the corresponding half-BPS holographic correlatorand going to the twisted configuration. The fact that these two methods agree is a featherin the cap of AdS/CFT. Our results also reveal another property of the MRV limit which,as explained in [1, 2], projects out the exchanges of long multiplets with low twist. Whatwe have seen is that the MRV limit is, at the same time, a refined enough tool that crucialshort multiplets ( e.g. all Schur operators) are still there.Of considerable interest to us is that chiral algebra four-point functions provide a richframework for constraining holographic correlators beyond tree-level. The simplest demon-stration of this is that the OPE data in (cid:104)O O O k O k (cid:105) that survives the superconformal twistis 1 /c d -exact. Let us now raise the stakes with a loop-level prediction for a correlator thathas not been computed yet. Consider J (3) × J (3) in the super W -algebra for AdS × S .There are two chiral operators missing from (5.6) which are allowed by symmetry and lowenough in dimension to provide a singular term.Θ( z ) = (cid:10) ∂ y , ∂ y (cid:48) (cid:11) √ J (2) ( z ; y ) J (2) ( z ; y (cid:48) ) : (cid:12)(cid:12)(cid:12)(cid:12) y = y (cid:48) , Ω( z ; y ) = 1 √ J (2) ( z ; y ) J (2) ( z ; y ) : (9.1)The overall coefficients ensure that these are unit normalized. Knowing the four-pointfunctions of strong generators, three-point functions involving (9.1) follow by taking acoincident limit. Since Ω and J (4) are degenerate, we only need the one for the Sugawarastress tensor. (cid:68) Θ( z ) J (3) ( z ; y ) J (3) ( z ; y ) (cid:69) = 45 √ c d y z z z (9.2)– 53 –dding this piece to the OPE that defines (cid:10) J (3) J (3) J (3) J (3) (cid:11) , we find that the four-pointfunction so obtained is only crossing symmetric if the corrections to existing OPE coeffi-cients satisfy C + C = − c d − c d . (9.3)Putting everything together, we have bootstrapped F ( χ ; α ) = 1 + ( αχ ) + χ (cid:18) α − − χ (cid:19) + 27 χ (1 + αχ )(1 − α + αχ )(1 − χ + αχ ) c d (1 − χ ) − χ α ( α − c d (1 − χ ) + O (cid:18) c d (cid:19) . (9.4)It will be important to undertake a systematic extension of these results as the AdS uni-tarity method [97, 98] continues to progress.Finally, sections 7 and 8 focused on 3d N = 8 ABJM theory whose half-BPS cor-relators, under the superconformal twist, are topological rather than meromorphic. Thisconstitutes a significant obstacle to the goal of making loop-level predictions along the linesof (9.4). Indeed, even at tree-level, only single-trace OPE coefficients were within reachof the matrix model techniques we used. Nevertheless, we have the double-trace OPEcoefficients (8.24) and (8.30) from Mellin space, and there is a strong indication that theTQFT results will match these once they are computed. This is because the 1d spectrumis sparse enough that all-order crossing equations like (7.9) are finite and therefore easy tocheck. In order to pursue subleading calculations in matrix quantum mechanics, the sta-tistical methods of [52, 53, 83] appear promising and it is also worth noting that bootstrapapproaches have recently been developed for these theories in [99, 100].The last property of these special OPE coefficients that should be highlighted is theirconnection to the crossing kernel. The Lorentzian inversion formula [101] establishes thatthe crossing kernel determines CFT data along double-twist trajectories and the nicest ver-sion of this should occur when there is enough supersymmetry to have exact double-twistoperators in the spectrum. It is all the more pleasing that we see the one for sl (2) as all ofthe operators discussed here contribute to a lower dimensional subsector. Although the in-tegrals used to derive this are quite general, they are especially useful for Mellin amplitudeswith infinitely many poles. In the maximally supersymmetric case, this means AdS × S .Existing techniques, which we have generalized, appear to have started with [12, 84] afterthe stress tensor four-point function first became available. With an expanded toolkit,and more correlators at our disposal, it will be worthwhile to see if some of the interest-ing patterns observed for AdS × S [102–104] have analogues in other dimensions. Thiscould suggest interplay between the superconformal twist and other organizing principlesfor these theories. Acknowledgments We thank Fernando Alday for discussions and collaboration in the initial stages of this work.This project has received funding from the European Research Council (ERC) under the– 54 –uropean Union’s Horizon 2020 research and innovation programme (grant agreement No787185). The work of X.Z. is supported in part by Simons Foundation Grant No. 488653. A An alternate derivation In general, the transformation of a tree-level Mellin amplitude back to position space doesnot yield a closed form result. Conversely, we have seen that the integral (3.2) does lead tosimple expressions for the special case of a four-point function in the twisted configuration.We carried out this calculation in subsections 4.2 and 5.2 in order to recover the chiralalgebra correlators (4.14) and (5.10) respectively.In both cases, it was enough to extract a finite number of singularities in the cross-ratio χ and discard the rest of the integral. As is familiar in 2d CFT, the infinitely many regularterms from one channel are instead captured by singular terms in the other channels.However, one might like to check that the full integral can indeed be evaluated and doesnot lead to surprises. We will show that this is the case for the (cid:104)O O O k O k (cid:105) correlatorsin this appendix. While our original derivation appears to be more powerful, a brute forceapproach along these lines could be helpful for studying more general observables that donot obey crossing. A.1 Four dimensions We are interested in reproducing the following correlator. (cid:68) J (2) ( z , y ) J (2) ( z , y ) J ( k ) ( z , y ) J ( k ) ( z , y ) (cid:69) = (cid:18) y z (cid:19) (cid:18) y z (cid:19) k F kk ( χ ; α ) (A.1) F kk ( χ ; α ) = 1 + kk d (cid:20) χ − χ − χ (2 − kχ )1 − χ α + ( k − χ − χ α (cid:21) This is a special case of (5.10), which we have expressed in terms of k d = − N − . Noticethat (A.1) is also k d times the canonically normalized (cid:104) J J OO(cid:105) which is often computedusing the standard Ward identity J a a ( z ) O b ...b k ( w ) = (cid:15) a b O a ...b k ( w ) + · · · + (cid:15) a b k O b ...a ( w ) + ( a ↔ a ) − z − w ) + . . . (A.2)for an affine su (2) current. This makes it easy to see that (A.1) is more than just atree-level correlator in the chiral algebra. It is exact to all orders since there is no way forloop corrections to modify the universal singular term in (A.2). If one prefers to work with adjoint indices, these can be re-introduced by taking J ab = (cid:15) ac ( T A ) cb J A = 12 [ (cid:15) ac ( T A ) cb + (cid:15) bc ( T A ) ca ] J A (A.3)where T A = √ σ A are the generators, defined to have structure constants f ABC = √ (cid:15) ABC . – 55 –et us now write down the associated Mellin amplitude in the parent theory. From(3.11), we find M kk ( s, t ; σ, τ ) = − kN ( k − × (cid:20) ( t − k − u − k − 2) + ( s + 2)( t − k − σ + ( s + 2)( u − k − τs − 2+ ( s − k )( u − k − τ + ( s − k )( t + k ) στ + ( t + k )( u − k − τt − k + ( s − k )( t − k − σ + ( t − k − u + k ) σ + ( s − k )( u + k ) στu − k (cid:21) . As in subsection 5.2, we will implement the superconformal twist by taking α (cid:48)− = χ (cid:48) → s -channel. Going through the three lines of (A.4),we can now list the poles in s which contribute in this limit. • For the first line, there is an increasing sequence of poles starting at s = 2. Only thefirst (the single-particle one) contributes. This is because α (cid:48) enters linearly whichmeans that χ (cid:48) s − is the smallest power of χ (cid:48) that we get. • For the other two lines, we only have the sequence of double-particle poles which startsat s = 4. Again, only the first contributes. This is because α (cid:48) enters quadraticallywhich leads to χ (cid:48) s − as the lowest power.After picking up these residues, the remaining integral to compute is F kk ( χ ; α ) = kN (cid:90) i ∞− i ∞ d t πi (1 − χ ) t − k − Γ (cid:20) k + 2 − t (cid:21) (A.5) (cid:34) χα ( k + 2 − t )Γ (cid:20) t − k (cid:21) + χ (1 + α )( k − t )Γ (cid:20) t − k (cid:21) + χ (1 − α ) t − k + 2 t − k Γ (cid:20) t − k + 22 (cid:21) + χ α (1 − α ) t + kt − k Γ (cid:20) t − k + 22 (cid:21) + χ α t − k − t − k Γ (cid:20) t − k + 22 (cid:21) + χ α (1 − α ) t − kt − k Γ (cid:20) t − k + 22 (cid:21) (cid:35) . To simplify the integrand, an obvious approach is to absorb the explicit factors of ( t − k ) − into the gamma functions. Even though there are two choices for how to do this, onlyone respects our prescription of encircling single-trace poles in the t -channel but not the u -channel. If the t = k pole is kept (not kept), it should be absorbed into the gammafunction that has t appearing negatively (positively) so that the result can be treated witha standard Mellin-Barnes contour. In other words, the last line of (A.5) goes with the lastline of 2 k − t Γ (cid:18) k + 2 − t (cid:19) = Γ (cid:18) k − t (cid:19) t − k Γ (cid:18) t − k + 22 (cid:19) = Γ (cid:18) t − k (cid:19) (A.6)– 56 –nd vice versa . In view of (8.6), the integral becomes a sum of Gaussian hypergeometricfunctions after we make this simplification. All of their parameters can be seen to take thevalues 1, 2, 3 or 4. Analytic expressions for them follow from standard contiguous relationsafter using F (cid:18) , 1; 1; χχ − (cid:19) = 1 − χ, F (cid:18) , 1; 2; χχ − (cid:19) = χ − χ log(1 − χ ) (A.7)as a seed. The cancellation of all logarithmic terms generated in this way is a non-trivialcheck of our calculation. By adding up all of the non-logarithmic terms, we find F kk ( χ ; α ) = 2 kN χχ − (cid:2) α ( kχ − − α χ ( k − (cid:3) (A.8)which is nothing but the tree-level piece of (A.1). A.2 Six dimensions The correlator to verify from Mellin space is now (cid:68) W (2) ( z ) W (2) ( z ) W ( k ) ( z ) W ( k ) ( z ) (cid:69) = F kk ( χ ) z z k (A.9) F kk ( χ ) = 1 + 2 kc d χ ( kχ − χ + 2)(1 − χ ) . Once again, this is very special. Up to a c d factor, it takes the form (cid:104) T T OO(cid:105) which iseasily computed using the Ward identity for the stress tensor. T ( z ) O ( w ) = h O ( w )( z − w ) + ∂ O ( w ) z − w + . . . (A.10)Clearly, this means that (A.9) is 1 /c d -exact. It is only the more singular Ward identities of(4.5) where there is room to have loop corrections in the form of normal ordered products.The right Mellin amplitude to use is again obtained from (3.11). To perform thesuperconformal twist and go to position space, we take α − = α (cid:48)− = χ (cid:48) → s . We can again list them in two steps. • The part for the s -channel has poles starting at s = 4. Only the first one (the first oftwo single-particle poles) contributes because of σ and τ entering linearly. This leadsto χ (cid:48) s − and higher powers of χ (cid:48) . • The part for the other two channels has poles starting at s = 8 all of which are double-particle. Only the first contributes because these channels have σ and τ appearingwith total degree 2. Hence, χ (cid:48) s − is the smallest power of χ (cid:48) produced.This time, we will only write the part of the Mellin amplitude that survives after theunimportant terms are removed. M kk ( s, t ; χ (cid:48) ) ∼ kχ (cid:48)− N (2 k − s ( s + 2) s − k χ (cid:48)− ( s − k )( s − k − N (2 k − (cid:20) t − k + 1 t − k − u − k + 1 u − k − (cid:21) – 57 –fter picking up the residues just described, we are left with F kk ( χ ) = k N (cid:90) i ∞− i ∞ d t πi (1 − χ ) t − k − Γ (cid:20) k + 4 − t (cid:21) (A.12) (cid:34) χ Γ (cid:20) t − k (cid:21) − kχ t − k Γ (cid:20) t − k + 42 (cid:21) − χ t − k − (cid:20) t − k + 42 (cid:21) + 2 kχ t − k Γ (cid:20) t − k + 42 (cid:21) + χ t − k + 2 Γ (cid:20) t − k + 42 (cid:21) (cid:35) . Two of the terms look like they cancel but they do not. They localize to a single t = 2 k residue because the t = 2 k pole is kept by one contour but not the other. The other termscan also be evaluated by paying attention to the contour. This leads to F kk ( χ ) = k N (cid:18) χ − χ (cid:19) (cid:20) kχ + 2 F (cid:18) , 2; 4; χχ − (cid:19)(cid:21) + k N (cid:18) χ − χ (cid:19) (cid:20) F (cid:18) , 4; 7; χχ − (cid:19) + 2 F (cid:18) , 4; 7; χχ − (cid:19)(cid:21) (A.13)which, upon using (A.7), matches (A.9). B The infinite sum over descendants In a myriad of applications, one starts with (3.2) and extracts a pole in s corresponding toa double-particle operator. Even though the remaining t integral can always be evaluatedexactly by (8.8) and (8.9), these do not tell us what to do with the sum over m whichlabels the level of the descendant being exchanged. Recalling the explicit form of theMellin amplitude (3.11), this sum is finite for AdS × S and AdS × S . However, forthe case of AdS × S , the sum can be either finite or infinite depending on the externalweights. It is therefore necessary to develop some technology for the calculation being donein section 8.It is explained in footnote 31 that the B [0] [0 ,j I − j II , j II , j I multiplets with maximal j I are associated with the residue of an M ( t ) or M ( u ) Mellin amplitude at s = s min ≡ min(∆ + ∆ , ∆ + ∆ ). We therefore have two channels to analyze as far as the infinitesum is concerned. To start, we will take a closer look at the truncation conditions in eachchannel to see what they tell us about the parameters in our hypergeometric functions.After this, we will explain the appearance of (8.10) in the protected OPE coefficients. B.1 Integers vs half-integers While it is clear that the parameters in (8.8) and (8.9) are either integers or half-integers,we are going to need a slightly more detailed statement. It is useful to define two constantsthat have a discrete choice available. a ∈ (cid:26) ∆ + ∆ , ∆ + ∆ (cid:27) , b ∈ (cid:26) s min − ∆ − ∆ , s min − ∆ − ∆ (cid:27) . (B.1) It is useful to notice from selection rules that ∆ + ∆ and ∆ + ∆ are either both integers or bothhalf-integers. – 58 –n the t -channel, the condition for the sum (3.4) to be infinite is∆ + ∆ − p , ∆ + ∆ − p ∈ Z + 1 . (B.2)The fact that p ≡ | k | , | k | (mod 2) only tells us that the above combinations are integers.Next, the hypergeometric function in (8.8) is T (∆ i ; m, p ) = F (cid:34) (∆ + ∆ ) + b , (∆ + ∆ ) + bs min , 1 + b + m + p (cid:35) . (B.3)We can now quote a basic property of (B.3) which will be important in the next subsection. The denominator parameter that depends on m differs from the non-trivial numeratorparameters by a half-integer . This follows immediately from (B.2) regardless of the valueof b .Proceeding identically in the u -channel, having an infinite sum in (3.4) requires∆ + ∆ − p , ∆ + ∆ − p ∈ Z + 1 . (B.4)This is a refinement of p ≡ | k | , | k | (mod 2). From (8.9), we now have a hypergeometricfunction that looks like U (∆ i ; m, p ) = F (cid:34) a + ( s min − ∆ − ∆ ), a + ( s min − ∆ − ∆ ) s min , 1 + a + m − s max + p (cid:35) (B.5)with the obvious definition of s max . We observe that the previously discussed property alsoholds for (B.5). The denominator parameter that depends on m differs from the non-trivialnumerator parameters by a half-integer . This follows immediately from (B.4) regardless ofthe value of a . B.2 Translation to a finite sum Let us now consider the sum I k = ∞ (cid:88) m =0 (cid:18) mk (cid:19) ( a ) m ( a ) m m !( b ) m ( c + m − F (cid:34) a , a b , c + m (cid:35) (B.6)subject to the properties discussed above. Only I will concern us since the other casescan all be obtained by shifting the parameters. The most useful way to proceed is to writethe hypergeometric function as an Euler integral, essentially undoing the last step in thederivation of (8.8) and (8.9). This leads to I = ∞ (cid:88) m =0 ( a ) m ( a ) m m !( b ) m (cid:90) 10 2 F ( a , a ; b ; z )(1 − z ) c + m − d z = (cid:90) 10 2 F ( a , a ; b ; 1 − z ) F ( a , a ; b ; z )(1 − z ) c − d z (B.7)– 59 –hich can be computed with the methods of [91]. Performing two Pfaff transformationsand passing to the Mellin-Barnes representation, we arrive at I = Γ( b )Γ( b )Γ( a )Γ( b − a )Γ( a )Γ( b − a ) (cid:90) i ∞− i ∞ d s d t (2 πi ) (cid:90) ∞ y t − s − a (1 + y ) a + a − c d y Γ( − s )Γ( a + s )Γ( b − a + s )Γ( b + s ) Γ( − t )Γ( a + t )Γ( b − a + t )Γ( b + t ) . (B.8)This is where our proof that it is always possible to make a − c a half-integer will payoff. Since a is also a half-integer, we get to use the binomial theorem with finitely manyterms. Following [91] again, I = ( a ) b − a ( a ) b − a Γ( b − a )Γ( b − a ) a + a − c (cid:88) n =0 (cid:18) a + a − cn (cid:19) (cid:90) i ∞− i ∞ d t πi Γ( − t )Γ( a + t )Γ( b − a + t )Γ( b + t )Γ( a − n − − t )Γ(1 + n + t )Γ(1 + n + b − a − a + t )Γ(1 + n + b − a + t ) . (B.9)The Mellin-Barnes integral derived here bears an unmistakable resemblance to thecrossing kernel for collinear blocks. As emphasized in [91], the crossing kernel has the specialproperty that it is very well poised – the gamma function arguments in the numerator anddenominator have the same sum. This will only be true for (B.9) if a − a + a − a = 0 . (B.10)Remarkably, the amplitudes in [2] are such that a − a = (cid:40) − (∆ − ∆ − ∆ + ∆ ) , M = M ( t ) − (∆ − ∆ + ∆ − ∆ ) , M = M ( u ) (B.11)which means that | a − a | = | a − a | after we compare to (B.3) and (B.5). Since theoriginal sum (B.6) is symmetric under a ↔ a and a ↔ a separately, we are free tochoose the signs of the parameter differences and thereby ensure that (B.10) holds. References [1] L. F. Alday and X. Zhou, “All tree-level correlators for M-theory on AdS × S ,” Phys.Rev. Lett. (2020) 131604, .[2] L. F. Alday and X. Zhou, “All holographic four-point functions in all maximallysupersymmetric CFTs,” .[3] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Gravitonexchange and complete four point functions in the AdS / CFT correspondence,” Nucl.Phys. B562 (1999) 353–394, arXiv:hep-th/9903196 [hep-th] .[4] G. Arutyunov and S. Frolov, “Four point functions of lowest weight CPOs in N=4 SYM(4)in supergravity approximation,” Phys. Rev. D62 (2000) 064016, arXiv:hep-th/0002170[hep-th] . – 60 – 5] G. Arutyunov and E. Sokatchev, “Implications of superconformal symmetry for interacting(2,0) tensor multiplets,” Nucl. Phys. B635 (2002) 3–32, arXiv:hep-th/0201145 [hep-th] .[6] G. Arutyunov and E. Sokatchev, “On a large N degeneracy in N=4 SYM and the AdS /CFT correspondence,” Nucl. Phys. B663 (2003) 163–196, arXiv:hep-th/0301058[hep-th] .[7] G. Arutyunov, F. Dolan, H. Osborn, and E. Sokatchev, “Correlation functions and massiveKaluza-Klein modes in the AdS / CFT correspondence,” Nucl. Phys. B (2003)273–324, arXiv:hep-th/0212116 .[8] G. Arutyunov and S. Frolov, “Scalar quartic couplings in type IIB supergravity on AdS(5)x S**5,” Nucl. Phys. B579 (2000) 117–176, arXiv:hep-th/9912210 [hep-th] .[9] L. Rastelli and X. Zhou, “Mellin amplitudes for AdS × S ,” Phys. Rev. Lett. (2017)091602, .[10] L. Rastelli and X. Zhou, “How to succeed at holographic correlators without really trying,” JHEP (2018) 014, .[11] L. Rastelli and X. Zhou, “Holographic four-point functions in the (2, 0) theory,” JHEP (2018) 067, .[12] X. Zhou, “On superconformal four-point Mellin amplitudes in dimension d > JHEP (2018) 187, .[13] X. Zhou, “On Mellin Amplitudes in SCFTs with Eight Supercharges,” JHEP (2018)147, arXiv:1804.02397 [hep-th] .[14] G. Parisi and N. Sourlas, “Random Magnetic Fields, Supersymmetry and NegativeDimensions,” Phys. Rev. Lett. (1979) 744.[15] A. Kaviraj, S. Rychkov, and E. Trevisani, “Random Field Ising Model and Parisi-Sourlassupersymmetry. Part I. Supersymmetric CFT,” JHEP (2020) 090, arXiv:1912.01617[hep-th] .[16] S. Caron-Huot and A.-K. Trinh, “All tree-level correlators in AdS × S supergravity:Hidden ten-dimensional conformal symmetry,” JHEP (2019) 196, .[17] L. Rastelli, K. Roumpedakis, and X. Zhou, “ AdS × S tree-level correlators: Hiddensix-dimensional conformal symmetry,” JHEP (2019) 140, .[18] S. Giusto, R. Russo, A. Tyukov, and C. Wen, “The CF T origin of all tree-level 4-pointcorrelators in AdS × S ,” Eur. Phys. J. (2020) 736, .[19] X. Zhou, “How to Succeed at Witten Diagram Recursions without Really Trying,” JHEP (2020) 077, arXiv:2005.03031 [hep-th] .[20] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli, and B. C. van Rees, “Infinite chiralsymmetry in four dimensions,” Commun. Math. Phys. (2015) 1359–1433, .[21] C. Beem, L. Rastelli, and B. C. van Rees, “W symmetry in six dimensions,” JHEP (2017) 017, .[22] S. M. Chester, J. Lee, S. S. Pufu, and R. Yacoby, “Exact correlators of BPS operators fromthe 3d superconformal bootstrap,” JHEP (2015) 130, .[23] C. Beem, W. Peelaers, and L. Rastelli, “Deformation quantization and superconformalsymmetry in three dimensions,” Commun. Math. Phys. (2017) 345–392, .[24] P. Liendo, I. Ramirez, and J. Seo, “Stress tensor OPE in N = 2 superconformal theories,” – 61 – HEP (2016) 019, .[25] M. Lemos and P. Liendo, “ N = 2 central charge bounds from 2d chiral algebras,” JHEP (2016) 004, .[26] C. Beem, “Flavor symmetries and unitarity bounds in N = 2 SCFTs,” Phys. Rev. Lett. (2019) 241603, .[27] C. Beem, L. Rastelli, and B. C. van Rees, “The N = 4 superconformal bootstrap,” Phys.Rev. Lett (2013) 071601, .[28] C. Beem, M. Lemos, P. Liendo, L. Rastelli, and B. C. van Rees, “The N = 2superconformal bootstrap,” JHEP (2016) 183, .[29] C. Beem, M. Lemos, L. Rastelli, and B. C. van Rees, “The (2, 0) superconformalbootstrap,” Phys. Rev. D93 (2016) 025016, .[30] M. Lemos, P. Liendo, C. Meneghelli, and V. Mitev, “Bootstrapping N = 3 superconformaltheories,” JHEP (2017) 032, .[31] C. Beem, L. Rastelli, and B. C. van Rees, “More N = 4 superconformal bootstrap,” Phys.Rev. D96 (2017) 046014, .[32] N. B. Agmon, S. M. Chester, and S. S. Pufu, “Solving M-theory with the conformalbootstrap,” JHEP (2018) 159, .[33] N. B. Agmon, S. M. Chester, and S. S. Pufu, “The M-theory archipelago,” JHEP (2020)010, .[34] A. Gimenez-Grau and P. Liendo, “Bootstrapping Coulomb and Higgs branch operators,” .[35] A. Bissi, A. Manenti, and A. Vichi, “Bootstrapping mixed correlators in N = 4 SuperYang-Mills,” .[36] D. J. Binder, S. M. Chester, M. Jerdee, and S. S. Pufu, “The 3d N = 6 bootstrap: Fromhigher spins to strings to membranes,” .[37] F. Bonetti and L. Rastelli, “Supersymmetric localization in AdS and the protected chiralalgebra,” JHEP (2018) 098, .[38] M. Mezei, S. S. Pufu, and Y. Wang, “A 2d/1d holographic duality,” .[39] A. Feldman, “On a gravity dual to flavored topological quantum mechanics,” JHEP (2020) 113, .[40] C. Beem, W. Peelaers, L. Rastelli, and B. C. van Rees, “Chiral algebras of class S,” JHEP (2015) 020, .[41] M. Lemos and W. Peelaers, “Chiral algebras for trinion theories,” JHEP (2015) 113, .[42] F. Bonetti, C. Meneghelli, and L. Rastelli, “VOAs labelled by complex reflection groupsand 4d SCFTs,” JHEP (2019) 155, .[43] M. Dedushenko, S. S. Pufu, and R. Yacoby, “A one-dimensional theory for Higgs branchoperators,” JHEP (2018) 138, .[44] M. Dedushenko, Y. Fan, S. S. Pufu, and R. Yacoby, “Coulomb branch operators and mirrorsymmetry in three dimensions,” JHEP (2018) 037, .[45] M. Dedushenko, Y. Fan, S. S. Pufu, and R. Yacoby, “Coulomb branch quantization and – 62 – belianized monopole bubbling,” JHEP (2019) 179, .[46] D. Bashkirov and A. Kapustin, “Supersymmetry enhancement by monopole operators,” JHEP (2011) 015, .[47] D. Bashkirov and A. Kapustin, “Dualities between N = 8 superconformal field theories inthree dimensions,” JHEP (2011) 074, .[48] M. Headrick, A. Maloney, E. Perlmutter, and I. G. Zadeh, “Renyi entropies, the analyticbootstrap and 3D quantum gravity at higher genus,” JHEP (2015) 059, .[49] M. Nirschl and H. Osborn, “Superconformal Ward identities and their solution,” Nucl.Phys. B711 (2005) 409–479, hep-th/0407060 .[50] 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] .[51] M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. (1994)1387–1404, hep-th/9308075 .[52] M. Mari˜no and P. Putrov, “ABJM theory as a Fermi gas,” J. Stat. Mech: Theory Exp. (2012) P03001, .[53] Y. Hatsuda, “Spectral zeta function and non-perturbative effects in ABJM Fermi-gas,” JHEP (2015) 086, .[54] H. Osborn and A. Petkos, “Implications of conformal invariance in field theories in generaldimensions,” Ann. Phys. (1994) 311–362, hep-th/9307010 .[55] M. Cornagliotto, M. Lemos, and V. Schomerus, “Long multiplet bootstrap,” JHEP (2017) 119, .[56] Y. Oshima and M. Yamazaki, “Determinant formula for parabolic Verma modules of Liesuperalgebras,” J. Algebra (2018) 51–80, .[57] K. Sen and M. Yamazaki, “Polology of superconformal blocks,” Commun. Math. Phys. (2020) 785–821, .[58] P. Bowcock, “Quasi-primary fields and associativity of chiral algebras,” Nucl. Phys. B356 (1991) 367–386.[59] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning conformal correlators,” JHEP (2011) 071, .[60] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, “Spinning conformal blocks,” JHEP (2011) 154, .[61] J.-F. Fortin and W. Skiba, “New methods for conformal correlation functions,” JHEP (2020) 028, .[62] G. Mack, “D-independent representation of conforal field theories in D dimensions viatransformation to auxiliary dual resonance models. Scalar amplitudes,” .[63] J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,” JHEP (2011) 025, .[64] A. L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju, and B. C. van Rees, “A naturallanguage for AdS / CFT correlators,” JHEP (2011) 095, .[65] F. Aprile, J. Drummond, P. Heslop, and H. Paul, “One-loop amplitudes in AdS × S supergravity from N = 4 SYM at strong coupling,” JHEP (2020) 190, – 63 – rXiv:1912.01047 [hep-th] .[66] L. F. Alday and X. Zhou, “Simplicity of AdS supergravity at one loop,” JHEP (2020)008, .[67] E. D’Hoker, D. Z. Freedman, and L. Rastelli, “AdS / CFT 4-point functions: How tosucceed at z-integrals without really trying,” Nucl. Phys. B562 (1999) 395–411, hep-th/9905049 .[68] J. Penedones, J. A. Silva, and A. Zhiboedov, “Nonperturbative Mellin amplitudes:Existence, properties, applications,” JHEP (2020) 031, .[69] S. Lee, S. Minwalla, M. Rangamani, and N. Seiberg, “Three point functions of chiraloperators in d = 4, N = 4 SYM at large N,” Adv. Theor. Math. Phys. (1998) 697–718, hep-th/9806074 .[70] R. Corrado, B. Florea, and R. McNees, “Correlation functions of operators and Wilsonsurfaces in the d = 6, (0, 2) theory in the large N limit,” Phys. Rev. D60 (1999) 085011, hep-th/9902153 .[71] F. Bastianelli and R. Zucchini, “Three point functions of chiral primary operators in d = 3,N = 8 and d = 6, N = (2, 0) SCFT at large N,” Phys. Lett. B467 (1999) 61–66, hep-th/9907047 .[72] S. Giombi, H. Khanchandani, and X. Zhou, “Aspects of CFTs on Real Projective Space,” arXiv:2009.03290 [hep-th] .[73] L. F. Alday, C. Behan, P. Ferrero, and X. Zhou, “in progress,”.[74] L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions fromFour-dimensional Gauge Theories,” Lett. Math. Phys. (2010) 167–197, arXiv:0906.3219 [hep-th] .[75] A. Campoleoni, S. Fredenhagen, S. Pfenninger, and S. Theisen, “Asymptotic symmetries ofthree-dimensional gravity coupled to higher-spin fields,” JHEP (2010) 007, arXiv:1008.4744 [hep-th] .[76] A. Campoleoni, S. Fredenhagen, and S. Pfenninger, “Asymptotic W-symmetries inthree-dimensional higher-spin gauge theories,” JHEP (2011) 113, arXiv:1107.0290[hep-th] .[77] P. H. F. Aprile, J. M. Drummond and H. Paul, “The double-trace spectrum of N = 4 SYMat strong coupling,” Phys. Rev. D98 (2018) 126008, .[78] C. Cordova, T. T. Dumitrescu, and K. Intriligator, “Multiplets of superconformal symmetryin diverse dimensions,” JHEP (2019) 163, .[79] M. Hogervorst and B. C. van Rees, “Crossing symmetry in alpha space,” JHEP (2017)192, .[80] C. P. Herzog, I. R. Klebanov, S. S. Pufu, and T. Tesileanu, “Multi-matrix models andtri-Sasaki Einstein spaces,” Phys. Rev. D83 (2011) 046001, .[81] M. Mezei and S. S. Pufu, “Three-sphere free energy for classical gauge groups,” JHEP (2014) 037, .[82] J. Bourdier, N. Drukker, and J. Felix, “The exact Schur index of N = 4 SYM,” JHEP (2015) 210, .[83] D. Gaiotto and J. Abajian, “Twisted M2 brane holography and sphere correlation – 64 – unctions,” .[84] S. M. Chester, “ AdS /CF T for unprotected operators,” JHEP (2018) 030, .[85] F. A. Dolan and H. Osborn, “On short and semi-short representations for four dimensionalsuperconformal symmetry,” Annals Phys. (2003) 41–89, hep-th/0209056 .[86] M. Buican, J. Hayling, and C. Papageorgakis, “Aspects of superconformal multiplets inD¿4,” JHEP (2016) 091, .[87] S. Lee and S. Lee, “Notes on superconformal representations in two dimensions,” Nucl.Phys. B956 (2020) 115033, .[88] N. B. Agmon and Y. Wang, “Classifying superconformal defects in diverse dimensions partI: Superconformal lines,” .[89] R. Gopakumar, A. Kaviraj, K. Sen, and A. Sinha, “A Mellin space approach to theconformal bootstrap,” JHEP (2017) 027, .[90] J. Liu, E. Perlmutter, V. Rosenhaus, and D. Simmons-Duffin, “ d -dimensional SYK, AdSloops, and 6 j symbols,” JHEP (2019) 052, .[91] R. Gopakumar and A. Sinha, “On the Polyakov-Mellin bootstrap,” JHEP (2018) 040, .[92] M. S. Milgram, “447 instances of hypergeometric F (1),” .[93] V. Gon¸calves, R. Pereira, and X. Zhou, “20 (cid:48) Five-Point Function from AdS × S Supergravity,” JHEP (2019) 247, arXiv:1906.05305 [hep-th] .[94] D. Karateev, P. Kravchuk, and D. Simmons-Duffin, “Weight shifting operators andconformal blocks,” JHEP (2018) 081, .[95] M. S. Costa and T. Hansen, “AdS weight shifting operators,” JHEP (2018) 040, .[96] D. Baumann, C. D. Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The cosmologicalbootstrap: Weight-shifting operators and scalar seeds,” .[97] O. Aharony, L. F. Alday, A. Bissi, and E. Perlmutter, “Loops in AdS from conformal fieldtheory,” JHEP (2017) 036, .[98] L. F. Alday and S. Caron-Huot, “Gravitational S-matrix from CFT dispersion relations,” JHEP (2018) 017, .[99] H. W. Lin, “Bootstraps to strings: Solving random matrix models with positivity,” JHEP (2020) 090, .[100] X. Han, S. A. Hartnoll, and J. Kruthoff, “Bootstrapping matrix quantum mechanics,” Phys. Rev. Lett. (2020) 041601, .[101] S. Caron-Huot, “Analyticity in spin in conformal theories,” JHEP (2017) 078, .[102] L. F. Alday and A. Bissi, “Loop corrections to supergravity on AdS × S ,” Phys. Rev.Lett. (2017) 171601, .[103] P. H. F. Aprile, J. M. Drummond and H. Paul, “Unmixing supergravity,” JHEP (2018)133, .[104] F. Aprile, J. Drummond, P. Heslop, and H. Paul, “The double-trace spectrum of N = 4SYM at strong coupling,” arXiv:1802.06889 [hep-th] ..