AdS4/CFT3 from weak to strong string coupling

We consider the four-point function of operators in the stress tensor multiplet of the U(N)k× U(N)−k ABJM theory, in the limit where N is taken to infinity while N/k5 is held fixed. In this limit, ABJM theory is holographically dual to type IIA string theory on AdS4× ℂℙ3 at finite string coupling gs∼ (N/k5)1/4. While at leading order in 1/N, the stress tensor multiplet four-point function can be computed from type IIA supergravity, in this work we focus on the first subleading correction, which comes from tree level Witten diagrams with an R4 interaction vertex. Using superconformal Ward identities, bulk locality, and the mass deformed sphere free energy previously computed to all orders in 1/N from supersymmetric localization, we determine this R4 correction as a function of N/k5. Taking its flat space limit, we recover the known R4 contribution to the type IIA S-matrix and reproduce the fact that it only receives perturbative contributions in gs from genus zero and genus one string worldsheets. This is the first check of AdS/CFT at finite gs for local operators. Our result for the four-point correlator interpolates between the large N, large ’t Hooft coupling limit and the large N finite k limit. From the bulk perspective, this is an interpolation between type IIA string theory on AdS4× ℂℙ3 at small string coupling and M-theory on AdS4× S7/ℤk.


Introduction and summary
Even though holographic correlators have been a subject of study since the early days of the AdS/CFT correspondence [1][2][3] (see for example [4][5][6][7][8][9][10][11][12] for early work on four-point functions), they are in many cases hard or even impossible to compute directly. For instance, in the case of higher derivative contact interactions in string theory or M-theory, where the full supersymmetric completion of the first correction to the supergravity action is not completely known (see however [13][14][15][16]), one cannot even write down the full set of relevant Witten diagrams. In the past few years, however, it has become clear that in certain cases one can essentially 'bootstrap' the answer using various consistency conditions [17][18][19][20][21][22][23][24][25]. These consistency conditions include crossing symmetry, the analytic properties of the correlators in Mellin space, and supersymmetry. In particular, for tree level Witten diagrams with supergravity and/or higher derivative vertices in 2d [26][27][28], 3d [22][23][24], 4d [19][20][21], 5d [29], and 6d [18,25] maximally supersymmetric theories, these consistency conditions determine the Witten diagrams contributing to the 4-point functions 1 of 1/2-BPS operators up to a finite number of coefficients. For low orders in the derivative expansion, one can further determine these coefficients using other methods, such as supersymmetric localization [31,32] or the relation between the Mellin amplitudes and flat space scattering amplitudes in 10d or 11d [33][34][35][36][37][38]. In particular, refs. [21,23,25] showed that the tree-level Witten diagram corresponding to an R 4 contact interaction, which is the first correction to supergravity in both 10d and 11d, can be completely determined using either supersymmetric localization or the flat space scattering amplitudes. The agreement between the two methods of fixing the undetermined coefficients in this case provides a precision test of AdS/CFT beyond supergravity.

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The goal of this work is to move away both from maximal supersymmetry and from 1/2-BPS multiplets and to study the stress tensor multiplet tree level Witten diagrams in the 3d N = 6 U(N ) k ×U(N ) −k gauge theory of Aharony, Bergman, Jafferis, and Maldacena (ABJM theory) [39], at large N . 2 The reason for pursuing this generalization is that it offers the possibility of an unprecedented test of AdS/CFT at finite string coupling g s . Indeed, if in ABJM theory we take N to be large and of the same order as k 5 , then the holographic dual is a weakly curved AdS 4 × CP 3 background of type IIA string theory with finite g s [39]. Using the consistency conditions mentioned above supplemented by supersymmetric localization results, we will be able to fully determine the contribution of the R 4 contact diagrams to the four-point function of the lowest dimension operator in the same super-multiplet as the stress tensor. The flat space limit of the Mellin amplitude then reproduces precisely the R 4 contribution to the four-point scattering of super-gravitons in type IIA string theory as a function of g s . This function receives contributions from genus zero and genus one string worldsheets [41]. The reason why such a finite g s test of AdS/CFT was not available in the maximally supersymmetric cases is that in 3d and 6d the bulk dual was an M-theory as opposed to string theory background, while in the 4d case, whose dual is type IIB string theory on AdS 5 × S 5 , the required supersymmetric localization result in the limit of large N and finite g s ∝ g 2 YM is hard to evaluate due to the contribution of instantons in the localized S 4 partition function [31,[42][43][44][45].
In more detail, in this work we consider the four-point function of the scalar superconformal primary of the N = 6 stress tensor multiplet, which is a 1/3-BPS operator that can be represented as a traceless tensor S a b , with a, b = 1, . . . , 4, transforming in the 15 of the SU(4) R R-symmetry [46][47][48]. In addition to the large N , fixed N/k 5 limit mentioned above where ABJM theory is dual to type IIA string theory at finite g s , we will also consider the M-theory limit where N is taken to infinity while k is kept fixed, as well as the 't Hooft strong coupling limit where N is taken to infinity while N/k is fixed and large and where ABJM theory is dual to weakly coupled type IIA strings on AdS 4 × CP 3 . The latter two limits can be obtained from the first: for small values of N/k 5 , one recovers the weakly coupled type IIA limit, while for large N/k 5 one recovers the M-theory limit. In all these limits, we focus on the first few tree-level Witten diagrams that compute the SSSS correlator. Our results will be expressed in terms of the following Mellin amplitudes (whose definition will be made precise in the next section): Each of these Mellin amplitudes gives rise to correlation functions that are crossinginvariant and solve the superconformal Ward identities. The first one, M SG (s, t) corresponds to the sum of the contact and exchange diagrams using supergravity vertices. The other two correspond to six-derivative and eight-derivative interaction vertices, respectively.

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With these ingredients and the definitions µ ≡ N/k 5 and λ ≈ N/k (see eq.  where we expanded the Mellin amplitudes in 1/c T instead of 1/N , with c T being the theorydependent constant that appears in the two-point function of the canonically-normalized stress tensor T µν : (1.3) (As shown in [49], c T is exactly calculable in ABJM theory using the supersymmetric localization results of [50] and [51]. It behaves as c T ∝ k 1/2 N 3/2 at large N .) The Mellin amplitudes in (1.2) can then be related to the 4-point scattering amplitudes of supergravitons in 11d and 10d flat space using the relation proposed in [36]: M-theory: A 11 = A 11 SG 1 + 6 1 3 · 2 7 stu + O( 9 11 ) , type IIA, small g s : A 10 = A 10 SG 1 + where A 11 SG and A 10 SG are the scattering amplitudes in 11d and 10d supergravity, respectively, 11 is the 11d Planck length, s is the 10d string length, and s, t, u = −s − t are the Mandelstam invariants.
Eqs. (1.4) are the well-known formulas describing the scattering of massless states in M-theory and string theory in the small momentum expansion. They have the following structure. The leading term in each equation is the supergravity scattering amplitude,

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and it contains information about the polarization and the type of massless particles being scattered. In each case, the corrections to the supergravity amplitude are captured by a single function of s and t that can be expanded at small s and t. Besides the supergravity terms, the only other terms written down in (1.4) are proportional to stu and correspond to an R 4 correction. 3 The various supergravity and R 4 terms in the three equations are not independent. Indeed, if in the first equation, one makes the replacement 11 = s (2πg s ) 1/3 , then the supergravity term in the first equation matches the supergravity terms in the other two, and the R 4 term in the first equation matches the g 2 s stu terms in the other two. Consequently, the first term in the first equation of (1.2) is identical to the first term in the second equation of (1.2) and the second term in the first equation of (1.2) is identical to the first term on the second line of the second equation of (1.2).
The terms given in (1.2) are derived solely using supersymmetric localization [32,50], as was originally done in [24] for k = 1, 2 when the theory has enhanced N = 8 supersymmetry. Supersymmetric localization can be used to compute the S 3 free energy in the presence of real mass deformations of Lagrangian theories with at least N = 2 supersymmetry. When viewed as an N = 2 SCFT, ABJM theory has an SO(4)×U(1) flavor symmetry, 4 and it can be deformed by three real mass parameters corresponding to the Cartan of SO(4) × U(1). We will focus on two of the three masses, which we denote by m + and m − . The S 3 free energy F (m + , m − ) was computed to all orders in 1/N for any k ≤ N in [67] using the Fermi gas formalism developed in [68]. The two independent choices of four derivatives can be related to integrated four-point functions of the stress tensor multiplet, which can in turn be related to SSSS using Ward identities to fix all the coefficients shown in (1.2). In the m ± → 0 limit, the non-perturbative corrections to F (m + , m − ) are expected to take the form e − √ N k and e − √ N/k , so this expansion also holds to all order in the finite 't Hooft coupling λ ∼ N/k and finite µ = N/k 5 expansions, with no non-perturbative in µ terms.
The rest of this paper is organized as follows. In section 2, we set up the computation of the SSSS correlator in terms of tree-level Mellin amplitudes. In particular, we determine M 3 and M 4 using the consistency conditions mentioned above. Implementing these constraints is much trickier than in the maximal SUSY, 1/2-BPS case, and we get guidance from solving a similar problem for flat space scattering amplitudes. Section 3 contains a 3 It would be interesting to study the next few terms not written down in (1.4) in future work. In particular, in M-theory the next correction not written down in (1.4) is at order 9 11 , and it comes from the 11d supergravity amplitude that can be found in [52,53]. The term after that, at order 12 11 , comes from a D 6 R 4 interaction, it is protected, and it can be related to the D 6 R 4 term in the type IIA string theory amplitude at order g 4 s [54][55][56]. In the string theory case, all terms at order g 0 s can be resummed into an expression involving Gamma functions that can be found, for instance, in [57,58]. Starting at order g 2 s , the scattering amplitude contains both analytic and non-analytic terms that can be derived from the tree level terms using unitarity [59]. While the type II string theory S-matrix is known to order g 4 s for finite s [60,61], the lowest few protected terms in the small s expansion are also known to order g 6 s [62]. For work on the Mellin amplitudes corresponding to the non-analytic terms, see [63][64][65]. 4 The U(1) is a flavor symmetry whose current lies in the N = 6 stress tensor multiplet, and so exists for all N = 6 SCFTs [66]. If the theory has N = 8 supersymmetry, then SO(6)R × U(1) is enhanced to SO(8)R.

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derivation of the supersymmetric localization constraints in ABJM theory. In section 4, we combine the localization constraints with the general setup developed in section 2. We end with a discussion of our results in section 5. Many technical details are relegated to the appendices. 2 The SSSS correlator at strong coupling We will begin by discussing the SSSS four-point function at strong coupling. In any of the strong coupling limits mentioned in the Introduction, the correlator SSSS can be written in terms of tree-level and loop Witten diagrams, although in this paper we focus only on the tree-level contributions. The leading tree-level contribution comes from supergravity exchange diagrams. These are corrected by higher derivative contact interactions, suppressed by the ratio p /L in 11d or s /L in 10d, depending on the limit being taken. Beyond the supergravity term, the tree-level Witten diagrams take a particularly simple form in Mellin space: at each order in the perturbative expansion only a finite number of Mellin amplitudes M i (s, t) contribute, each of which is polynomial in s, t. In this section our task is to determine the first few such amplitudes, using the flat space limit, crossing symmetry, the supersymmetric Ward identities, and locality.

Setup
As mentioned in the Introduction, the S operator is the superconformal primary of the stress tensor multiplet, and transforms in the 15 of the so(6) R-symmetry. In index notation we write the operator as S b a ( x), where the raised index a = 1, . . . , 4 transforms in the 4 of su(4) ∼ = so(6) and the lowered indices in the 4. We will find it more convenient however to use an index-free notation by defining where X is an arbitrary traceless 4 ⊗ 4 matrix. We normalize this operator so that Using both conformal and so(6) symmetry, we can expand where we define the R-symmetry structures

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For future reference, we note that it is sometimes useful to write the four-point function in a conformal block expansion, which reads 5 where G ∆, (U, V ) are the 3d conformal blocks normalized as in [69], T R (X i ) are the SU(4) invariants corresponding to the s-channel exchange of an operator in the irrep R, and λ 2 ∆, ,R are squared OPE coefficients. The SU(4) irreps R of the operators that appear in S × S are where s/a denotes the symmetric/antisymmetric product. As explained in appendix A.1, we find We can distinguish between 15 s and 15 a by (anti)symmetrizing appropriately, and we should only consider the real combination 45 ⊕ 45.
Holographic correlators are simpler in Mellin space. To compute the Mellin transform of S i (U, V ), we first compute the connected correlator by subtracting the disconnected part and then we define M i (s, t) through where u = 4 − s − t. The Mellin transform (2.10) is defined such that a bulk contact Witten diagrams coming from a vertex with n = 2m derivatives gives rise to a polynomial M i (s, t) of degree m [36]. (This property holds both for scalars and for operators with spin, provided that the Mellin amplitudes for operators with spin are defined appropriately.) The two integration contours in (2.10) are chosen such that 6 11) 5 We could reorganize this block expansion into superconformal blocks (as opposed to conformal blocks) for each supermultiplet, but it is unnecessary to do so for our purposes. 6 This is the correct choice of contour provided that M i (s, t) does not have any poles with (s) < 2 or (t) < 2 or (u) < 2. If this is not the case (such as for the supergravity Mellin amplitude), the integration contour will have to be modified in such a way that the extra poles are on the same side of the contour as the other poles in s, t, u, respectively.

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which include all poles of the Gamma functions on one side or the other of the contour. These poles naturally incorporate the effect of double trace operators [70].
In this paper we focus on tree-level Witten diagrams, and in the rest of this section we aim to determine a basis of Mellin amplitudes that can be used to write the contribution from contact Witten diagrams with small numbers of derivatives. These Mellin amplitudes obey three constraints: 1. They obey the crossing symmetry requirements coming from the crossing symmetry of the full SSSS correlator.
2. They obey the SUSY Ward identities following from N = 6 superconformal symmetry. The SUSY Ward identities not only constrain M i (s, t), but they also allow us to determine the Mellin amplitudes corresponding to correlators of other operators in the stress-tensor multiplet.
3. The M i (s, t) and all other Mellin amplitudes related to them by SUSY are polynomials in s, t. We call the collection of Mellin amplitudes corresponding to four-point functions of operators in the same super-multiplet a super-Mellin amplitude, and we define the degree of a polynomial super-Mellin amplitude n to be the highest degree of any component Mellin amplitude.
For fixed m, we will label the Mellin amplitudes obeying these requirements as M i m (s, t) in case there is a unique such amplitude for a given m or by M i m,k (s, t) in the case that there are multiple such amplitudes indexed by k. These Mellin amplitudes represent a basis for contact Witten diagrams, with the number of derivatives in the interaction vertex being bounded from below by 2m. In section 4, we will use these Mellin amplitudes and the constraints coming from supersymmetric localization explored in the next section in order to determine the first few terms in the strong coupling expansion of the SSSS correlator.
Note that, in general, supersymmetry relates the contact interactions for bulk fields with various spins, and in flat space SUSY preserves the number of derivatives of the interaction vertices it relates. In AdS however, the number of derivatives within a given super-vertex may vary, with the change in the number of derivatives being compensated by an appropriate power of the AdS radius L. Thus, it may happen that a four-scalar vertex with a given number of derivatives is part of a supervertex containing other vertices with more derivatives. The corresponding Mellin amplitudes M i (s, t) will then have lower degree than those of some four-point function of superconformal descendants of S, and so M i n (s, t) may have degree less than n. This fact will be very important in the analysis that follows.

The flat-space limit and a toy problem
Finding the Mellin amplitudes M i n (s, t) that obey the conditions listed above is a difficult task, as satisfying the third condition requires us to calculate Ward identities for many JHEP01(2020)034 different correlators and then examine the locality properties of their Mellin amplitudes. We can simplify matters by first solving an analogous problem for flat space scattering amplitudes.
At large AdS radius, we can recover flat space scattering amplitudes for scalars using the Penedones formula [37]. Applied to the superconformal primary S the relationship is (up to an overall normalization N (L)) Here, κ > 0, and A i (s, t) is the corresponding 4d flat space scattering amplitude of graviscalars (or more precisely a scattering amplitude in 10d string theory or 11d M-theory with the momenta restricted to lie within 4d and polarizations transverse to this 4d space), computed in the limit where the AdS radius L is taken to infinity while keeping some other dimensionful length scale UV fixed. For string or M-theory duals we can take UV to be either the 10d string length or 11d Planck length, as we will do in section 4. From (2.13) we expect that each Mellin amplitude M i m,k (s, t) must give rise to a local N = 6 scattering amplitude A i m,k (s, t). This mapping should furthermore be one-to-one, since if two amplitudes M i m,k 1 and M i m,k 2 have the same large s, t limit, then their difference M i m,k 1 − M i m,k 2 will be a local Mellin amplitude with degree at most m − 1. Thus, if we can find all of the number of local scattering amplitudes of a given degree in s, t, then this will also tell us the number of Mellin amplitudes which occur at this degree: 7 # of degree m scattering amplitudes in 4d SUGRA = # of degree m Mellin amplitudes in 3d SCFT . (2.14) Because the flat space scattering amplitudes are obtained as the large s, t limits of Mellin amplitudes, finding all crossing-invariant, supersymmetric, and local N = 6 flat space scattering amplitudes is a strictly simpler problem than finding all Mellin amplitudes with the same properties.

Counterterms in N = 6 supergravity
The toy problem described in the previous section is that of finding four-point scattering amplitudes corresponding to counterterms in 4d N = 6 supergravity. Spinor helicity and on-shell supersymmetric methods provide an efficient means to classify allowed counterterms in a theory. They were first applied to 4d N = 8 in [71,72], and have subsequently been generalized to other maximally supersymmetric theories in [73,74]. In the context of N = 6 supergravity these methods have been applied to study amplitudes involving bulk graviton exchange [75,76]. 7 At a more abstract level, we can justify the correspondence (2.14) as follows. Local Mellin amplitudes correspond to bulk contact Witten diagrams, which are themselves in one-to-one correspondence with local counterterms in AdS. But since AdS is maximally symmetric, local counterterms in AdS are equivalent to local counterterms in flat-space. Since local counterterms in flat-space correspond exactly to scattering amplitudes, we find that Mellin amplitudes and scattering amplitudes are in one-to-one correspondence. Let us begin with a quick review of on-shell superspace (see also appendix B.1); for a detailed textbook treatment of the subject we recommend [77]. In N = 6 supergravity, the massless particles split into two supermultiplets: a multiplet we denote by Φ that contains the positive helicity graviton h + , and its CPT conjugate multiplet we denote by Ψ that contains the negative helicity graviton h − . In addition to the graviton h ± , these multiplets also contain the gravitino ψ ± , the gauginos g ± , fermions F ± , scalars φ, and the graviphoton a ± . Table 1 lists the particles in these multiplets, along with their transformation properties under the SU(6) R-symmetry of N = 6 supergravity. In the on-shell superspace formalism, the Φ and Ψ superfields are polynomials in the Grassmann variables η I , with I = 1, . . . 6 transforming in the 6 of SU(6): 8

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In a four-point superamplitude, such as A[ΦΦΨΨ], each particle i = 1, . . . , 4 is associated to some Grassmannian variable η I i . To compute a component scattering amplitude we simply differentiate with respect to some of the Grassmannian variables while setting all others to zero. For instance: .
In this way a superamplitude A encodes all the amplitudes of its component particles.

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Up to crossing there are five possible 4 particle superamplitudes we can construct from Φ and Ψ. However, under CPT the two supermultiplets Φ and Ψ are conjugates, and their scattering amplitudes are related by complex conjugation (see appendix B.2 for a description of how discrete space-time symmetries act on the scattering amplitudes): Our task now is to constrain the forms of these superamplitudes, beginning with invariance under supersymmetry. As explained in [77], for a given particle i the supermomentum is defined to be and it satisfies the on-shell SUSY algebra by construction. For a given amplitude the total supermomentum is thus: where the first factor is the Grassmann delta function which is annihilated by both Q I andQ I , and f i (s, t) are arbitrary functions 10 of s and t. The delta function δ 12 (Q) is automatically invariant under SU(6) R , even if the full theory does not preserve SU(6) R [72]. 11 Note that every term in each superamplitude contains JHEP01(2020)034
us the relations  Having determined the allowed forms of f i (s, t), we can now determine the number of derivatives in each interaction vertex. To this count each angle and square bracket contribute 1, δ 12 (Q) contributes 6, and each power of s, t, u contributes 2. For instance, if we set f 2 (s, t) = s k and consider the amplitude A[ΦΦΨΨ] = s k δ 12 (Q) [12] 4 34 2 , it follows that this amplitude comes from an interaction vertex with 8 + 2k derivatives, namely from an D 2k R 4 term.
With this in mind, we can now systematically find all local counterterms up to a certain number of derivatives. In table 3 we list all local counterterms up to 15 derivatives, corresponding to Mellin amplitudes up to degree 7.5. 12 In particular, the first local counterterm has 6 derivatives, is unique, and contributes only to A[ΦΦΦΦ]. The next local counterterm has 8 derivatives and is also unique and contributes only to A[ΦΦΨΨ]. There are two 10 derivative counterterms, one contributing to A[ΦΦΦΦ] and one to A[ΦΦΦΦ], and so on. The counterterm with the lowest number of derivatives that contributes to A[ΦΦΦΨ] has 15 derivatives and will not be important in this work.

Implications for N = 6 SCFts
Having systematically computed the local amplitudes in N = 6 supergravity, we will now discuss the implications for holographic N = 6 SCFTs. First, we can deduce that there are five independent superconformal invariants in the four point function of four stress tensor multiplets. This counting follows from the number of unknown real functions needed to JHEP01(2020)034 fully determine the scattering amplitudes of supergravitons, one for f 1 (s, t) and two each for f 2 (s, t) and f 3 (s, t), as these latter two functions are in general complex.
Second, from table 3 we can immediately deduce how many polynomial Mellin superamplitudes exist for a given degree in s, t. For instance, at third degree we have a single polynomial super-Mellin amplitude with scalar component M i 3 (s, t), and at fourth degree we additionally have another polynomial super-Mellin amplitude with scalar component M i 4 (s, t). Here, by third and fourth degree we mean that the super-amplitudes that M i 3 (s, t) and M i 4 (s, t) have degree 3 or 4 for some of the components of the amplitude, but not necessarily for the scalar components M i 3 (s, t) and M i 4 (s, t) themselves. These scalar components may be of less than third and fourth degree, respectively.
In fact, it can be argued that while the scalar component M , and then must relate φ and φ to the superconformal primary S. We perform both computations in appendix C and find that (2.28)

Exchange amplitudes
So far we have considered local contact amplitudes. The only other tree-level diagrams which appear in four point functions consist of exchange diagrams. These can be built up from the on-shell three point amplitudes using on-shell recursion relations (see for instance chapter 3 of [77]), and so our first task is to find the allowed three point amplitudes. Three point amplitudes are subtle due to special kinematics; conservation of momentum implies that either For real momenta [ij] * = ji so this would seem to rule out any interesting amplitudes. This issue is however resolved by analytically continuing to complex momenta. Locality and little-group scaling then uniquely fix three-point functions to take the form: where c is an arbitrary constant [77,78]. Superamplitudes must furthermore satisfy the supersymmetric Ward identities, and this uniquely fixes them to take the form: (2.33) The g 1 term in the A[ΦΦΨ] superamplitude corresponds to the usual supergravity threepoint function, and in particular gives rise to a graviton scattering amplitude (2.34) The g 2 and g 3 terms both vanish due to crossing symmetry; if we exchange 1 ↔ 2 then A[ΦΦΦ] and A[ΦΦΨ] must be even, but this is only possible if g 2 = g 3 = 0.

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Operator ∆ Spin so(6) R irrep Flat space Table 4. The conformal primary operators in the N = 6 stress tensor multiplet. For each such operator, we list the scaling dimension, spin, so(6) R representation, and the particle whose scattering amplitudes it is related to in the flat space limit of the AdS 4 dual.
Since there is only one supergravity three-point function, we can now determine the corresponding unique four point exchange amplitude. Because the tree-level graviton amplitudes in pure supergravity are identical to those in pure gravity [77], we can simply use the pure gravity result to deduce that We can then substitute this into (2.27) to find that the A[φφφφ] amplitude at large s, t is expected to be

Supersymmetric Ward identities
Our task now is to determine M i 3 (s, t) and M i 4 (s, t). In order to do so we will need to compute the superconformal Ward identities relating the S i (U, V ) both to one another and to the correlators of the superconformal descendants of S a b .
The operators in the N = 6 stress tensor multiplet are shown in table 4. There are three fermions with dimension 3/2, the χ α , the F α , and its Hermitian conjugate the F α . In addition to the pseudoscalar P , at dimension 2 there are two conserved currents; the Rsymmetry current J µ in the 15, and the U(1) flavour current j µ which is an SO(6) singlet. Completing the multiplet are the supercurrent ψ µα in the 6 and finally the stress tensor itself, T µν . In table 4, we also list which particles these operators correspond to in the flat space limit.
To impose superconformal invariance on a correlator, it is sufficient to impose conformal invariance, R-symmetry invariance, and invariance under the Poincaré supercharge Q αI . We have already seen how to impose the first two symmetries on the SSSS , and it JHEP01(2020)034 is straightforward to expand other correlators in the multiplet as a sum of conformal and R-symmetry invariants. Explicit expressions for these can be found in appendix D. The supersymmetric Ward identities then follow by imposing that the Q variations vanish: (2.38) We can use (2.37) as well as other similar SUSY Ward identities in order to determine the relations between SSSS and other four-point functions of operators in the stress tensor multiplet. Note, however, that we will not be able to determine the four-point function of the stress tensor multiplet completely. This should already be clear from the flat space limit, where we can ask the analogous question for the flat space scattering amplitudes: given A[φφφφ], can we determine all the other component amplitudes? The answer is no, because it is only the superamplitude A[ΦΦΨΨ] that contributes to A[φφφφ]. Therefore, knowing A[φφφφ] allows us to determine the function f 1 (s, t) in (2.21) via (2.28) and leaves the complex functions f 2 (s, t) and f 3 (s, t) undetermined. In other words, A[φφφφ] determines only one out of five super-amplitudes.
The situation is better for N = 6 SCFTs where from SSSS we can determine more than just one out of five superconformal invariants. The reason for this improvement is that although some of the superconformal invariants do not contribute to SSSS in the flat space limit, they do contribute at subleading orders in 1/L. It can be argued that SSSS is related to two out of the five super-invariants as follows. While the stress tensor multiplet forms a representation of the superconformal group OSp(6|4), it also forms a representation of a larger group that includes two Z 2 transformations: a parity transformation P and discrete R-symmetry transformation Z whose precise definitions are given in appendix B.3. Moreover, the superconformal Ward identity relates four-point structures that have the same P and Z charges. Because the SSSS correlator is P-even and Z-even, and only one other structure has this property, it follows that from SSSS we can determine at most two out of the five superconformal structures. Explicit computations show that we can indeed determine two superconformal invariants. In etc. Table 5. Examples of CFT four-point correlators that contribute to the five superconformal invariants. Each superconformal invariant can be labeled by its transformation properties under the discrete symmetries P and Z. For every CFT correlator in the first column, we list how it contributes to the superconformal invariants in Mellin space: either at leading order, in which case we list the scattering amplitude it should match at this order; either at subleading order, in which case we write "Subleading"; or it does not contribute, in which case we write "None". that contribute to each superconformal structure. The correlator SSSS allows us to determine the conformal structures in the second and fifth columns of this table.
In the next section, we will need to know the relation between SSP P and SSSS . To derive this relation, we need to consider one more variation, δ SSP χ . Using the results of (2.37) and the variation δ SSP χ , we can compute SSP P , along with SP χχ and SP χF . More details can be found in table 6 and in appendix D. Note that while, as discussed above, the superconformal Ward identities fall short of making it possible to determine the all five superconformal invariants (for instance, we cannot determine SSF F fully), we will be able to completely determine the correlators SSP P and, if we wish, P P P P in terms of SSSS . We will now use these Ward identities to find the degree m polynomial Mellin amplitudes M i m (s, t) with m = 3, 4.

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where we defined 14 The Mellin transforms of S i corresponding to contact interactions were found in [23].
With our definition (2.10) (with S i conn replaced by S i conn and M i replaced by M i ), the result in [23] for the quartic amplitude is To relate (2.41) to M i 4 (s, t) we should relate the so(8) R structures (2.40) to the su(4) R ones defined in (2.4). Under the decomposition so(8) → su(4), we have 8 c → 4 + 4, which 13 The fact that this representation is the 35c as opposed to one of the other two 35-dimensional irreducible representations of so(8)R assumes a choice of the triality frame.
14 Despite the use of matrix so(8) polarizations here, the S i (U, V ) here are equal to the Si(U, V ) in [23].

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implies 35 c → 10 + 10 + 15. To select the 15, we should restrict the 8 × 8 matrices X to take the form where X is a 4 × 4 traceless hermitian matrix, I 2 is the 2 × 2 identity matrix, and σ 2 is the second Pauli matrix. (See eq. (3.16) of [23]. 15 ) Then it is straightforward to check that This implies that S i = S i for i = 1, 2, 3 and S i = 1 4 S i for i = 4, 5, 6 and analogously for the Mellin amplitudes. Thus, where the other M i 4 are given by crossing (2.12). The Melin amplitudes M i 4 are normalized so that at large s, t they obey (2.29).
The degree 3 polynomial Mellin amplitude M i 3 is not allowed by N = 8 supersymmetry, and so we must compute it using the N = 6 Ward identities derived in the previous section. In particular, we impose the following constraints to find M 3 : 1. M i 3 must satisfy crossing symmetry (2.12).
2. M i 3 must be a degree 2 polynomial solution of the SSSS Ward identities given in position space (2.38), which can be translated into Mellin space using the rules (E.1). The ansatz for M 3 is only degree 2, since in the previous section we showed that A 3 does not appear in the scattering of four scalars, so M 3 must vanish in the flat space limit.
3. M 3 must remain a polynomial when expressed as correlator of other operators in the stress tensor multiplet using the Ward identities in the previous section. 16 The degree of these polynomials is at most 2 if the corresponding flat space amplitude vanishes, and 3 otherwise.
Condition 3 was trivially satisfied in maximally supersymmetric cases considered before in various dimensions [21,24], where polynomial Mellin amplitudes for SSSS remained 15 The factor of 1/ √ 2 is just a choice of normalization. 16 Instead of imposing this requirement, we could alternatively impose the condition that certain operators in the S × S OPE do not acquire anomalous dimensions. For instance, we can uniquely determine M3 if we impose this requirement for the spin 0 operators of dimension 2 in the 84, 20 , and 15s irreps of SO(6)R, as well as for the spin 1 operator of dimension 3 in the 45 ⊕ 45, all of which belong to protected multiplets and do not mix with unprotected operators.

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polynomials for all other stress tensor multiplets correlators. In our case though, we find that just imposing conditions 1 and 2 leads to five linearly independent solutions: a degree 0, a degree 1, and three degree 2: degree 0: 1st degree 2: 2nd degree 2: 3rd degree 2: (2.45) To reduce these to a unique amplitude, we must consider the other Ward identities SSχχ , SSχF , SSF F , and SSF F given in appendix D, which we can transform into Mellin space as in E.2. We can write SSχχ in terms of the structures C a,I (U, V ) defined in (D.7), where the indices a = 1, 2, 3 and I = 1, 2 refer to the various R-symmetry and conformal structures, respectively. The Mellin transform M SSχχ where the SSχχ Ward identity D C ai,1 is given in position space in (D.9), we express derivatives and powers of U and V in Mellin space using the rules (E.1), and s-dependent prefactors come from the difference in the definition of the scalar and fermion Mellin amplitudes in (2.10) and (E.3). We find that degree 0 amplitude in (2.45) gives which contain poles, and so must be discarded.
When we apply this method to the Ward identities for SSF F and SSF F , a new subtlety is that these Ward identities (D.17), (D. 19), and (D.21) depend on both SSSS and SSF F , and in particular can be written in terms of S 1 (U, V ) and S 4 (U, V ), as well as the first conformal structure F a,1 (U, V ) for SSF F as defined in (D.7), where here a = 1, 2 for the two R-symmetry structures. So to get the constraints from these Ward identities up to degree 2, we must also consider a degree 2 polynomial ansatz for the Mellin JHEP01(2020)034 , which satisfies the crossing relations where the s-dependent prefactors come from the difference in the definition of the fermion Mellin amplitudes in (E.3) for the two different conformal structures. After imposing the SSχF , SSF F , and SSF F Ward identities, just as we did for SSχχ above, and demanding that all poles vanish, we find that M SSF F a,1 (s, t) is completely fixed in terms of M i (s, t) up to degree 2, and that only a single degree 2 solution for M i (s, t) survives: which in fact corresponds to the degree 3 Mellin amplitude M 3 (s, t) as discussed before.

Supergravity exchange Mellin amplitude
We will also use the supergravity amplitude M i SG (s, t), which contains an infinite series of poles that correspond to the stress tensor multiplet operators (or the exchange of the graviton multiplet in the bulk) and their descendants. This amplitude is unique and can be derived using the method we used above for determining M i 4 by translating the N = 8 SCFT results into N = 6 language. For the case of 3d N = 8 CFTs, M i SG was derived in [22]. From eqs. (E.1) and (4.8) of [24], and converting to N = 6 notation as we did before in subsection 2.7.1, we find that where the other M i SG are given by crossing (2.12). We normalize M i SG so that at large s, t they obey (2.36) with g 1 = 1.

Constraints from supersymmetric localization
In order to determine the coefficients of the Mellin amplitudes M 3 and M 4 derived in the previous section in the case of ABJM theory, we will use information from supersymmetric localization. Similarly to [21,23,24], we will focus on the mass-deformed partition function of ABJM theory on a round S 3 . While it would be interesting to also obtain constraints coming from the partition function on a squashed S 3 [79], in this work we will use the round sphere simply because the mass-deformed partition function can be computed [67] using the Fermi gas formalism developed in [68] to all orders in the 1/N expansion. A similar result for the squashed sphere partition function is not currently available.

Integrated correlators on S 3
To set the stage, let us begin with the result for the S 3 partition function in the presence of a mass deformation. On S 3 , there are two classes of mass deformations of ABJM theory that one can consider: in N = 2 notation, there are superpotential mass deformations and real mass deformations. The S 3 partition function has no dependence on the superpotential mass parameters, so we will focus on the real mass parameters. These real masses are associated with global symmetries, because they can be constructed by coupling the conserved currents of the N = 2 theory to background vector multiplets and giving expectation values proportional to the mass parameters to the scalars in the vector multiplets. Since ABJM theory has N = 6 SUSY for arbitrary k, it has an SO(6) R R-symmetry as well as an U(1) F global symmetry, with both the SO(6) R and U(1) F conserved currents belonging to the same multiplet as the stress-energy tensor, as discussed in the previous section. When passing to N = 2 notation, a U(1) R subgroup of SO(6) R can be viewed as the N = 2 R-symmetry, and in SO(6) R × U(1) F there are three other U(1)'s that commute with one another and with U(1) R . (They are the Cartans of an SO(4) × U(1) flavor symmetry from the N = 2 point of view.) Each of these U(1)'s can be coupled to an Abelian background vector multiplet, so for each of them one may consider introducing a real mass parameter. There are thus three distinct real mass parameters.
For simplicity, in this work we will focus on only two of the three real mass parameters of ABJM theory. 17 Recall that ABJM theory in N = 2 notation is a theory of two U(N ) vector multiplets coupled to bifundamental chiral multiplets The two mass parameters we consider, denoted m + and m − , correspond to giving masses (m + /2, m − /2, −m + /2, −m − /2) to W 1 , W 2 , Z 1 , Z 2 , respectively. The partition function can be written as [32,50]: (3.1) The purpose of this section is to relate the mixed derivatives all evaluated at m + = m − = 0, to the correlation functions of the S a b operators introduced in the previous section.
In the ABJM Lagrangian on a unit radius S 3 , the parameters m + and m − appear at linear order as where J ± are linear combinations of the S's and K ± are linear combinations of the P 's. In terms of the Lagrangian fields, the J ± are scalar bilinears which are quadratic in the JHEP01(2020)034 bottom components of the chiral multiplets W i = (W i , χ i ) and Z i = (Z i , ψ i ), while the K ± are fermion mass terms quadratic in the fermions in the same chiral multiplets: The mixed derivatives (3.2) are given in terms of connected correlation functions as conn +(2-and 3-pt functions) , conn +(2-and 3-pt functions) , where the 2-and 3-point function terms not written in (3.5) come from the O(m 2 ) terms not written in (3.3). We will not write down these 2-and 3-point function contributions because they will be automatically taken into account in the final formulas, by analogy with the similar situation encountered in [21].
To determine how J ± and K ± are related to S and P , let us first note that 18 so J ± and K ± can be written as where we defined Because tr C † a C b and tr Ψ † a Ψ b transform in the 15 of SU(4) R , they must be proportional to S a b and P a b , respectively. Eq. (3.6) then implies that where N J and N K are normalization constants. On general grounds, the two-point functions of J ± and K ± must be proportional to the coefficient c T appearing in the two-point function of the canonically normalized stress-energy tensor. Because the two-point functions of S and P are both normalized as in (2.1), knowing that N 2 J and N 2 K are proportional to c T allows us to determine them in a free theory, such as the k → ∞ limit of the U(1) k × U(1) −k ABJM theory. In this 18 The reason why the components of the chiral multiplets do not appear in the same order in this expression is that we require the U(1)R symmetry to be generated by the su(4)R matrix diag{1/2, −1/2, 1/2, −1/2}. JHEP01(2020)034 limit, the W i and Z i chiral multiplets are free, and C a ( . From the definition (3.6), we then have free theory: These expressions should be compared with what we obtain from (3.8) and (2.1), which is free theory: Thus, for a free theory, we have N 2 J = 1/(64π 2 ) and N 2 K = 2N 2 J . In conventions in which a free massless real scalar or a free real Majorana fermion has c T = 1, as in (1.3), the free theory has c T = 16. From this, and the fact that N 2 J and N 2 K should be proportional to c T , we conclude that we must have Using (3.10) and (3.11) and explicitly evaluating the integrals gives [51] c Having determined the normalization factors in (3.8), we can then evaluate (3.5). The result will be given in terms of the functions S i that appear in the SSSS correlator in eq. (2.3) as well as analogous functions that appear in SSP P and P P P P . While this is certainly a valid procedure, 19 it is possible to obtain simpler formulas by making use of the fact that all N ≥ 4 SCFTs in 3d have a 1d topological sector [80][81][82][83][84][85].
In general, a 3d N = 4 SCFT has SU(2) H × SU(2) C R-symmetry, and one can consider 1/2-BPS operators that have scaling dimension ∆ = j H , where j H is the SU(2) H spin, and are invariant under SU(2) C . Such operators can be written as rank-2j H symmetric tensors 19 The result is where S i are the functions appearing in (2.3), R i are the functions appearing in the SSP P correlator given in (D.5), P i are the six functions appearing in the P P P P correlator defined as in (2.3) but with S → P and S i → P i , and 14) The powers of Ω in (3.14) appear because the operators are integrated over S 3 as opposed to R 3 .
where we can take 20 If we want to express the topological operator in terms of the operator O a 1 a 2 ...a 2j H when the theory is placed on S 3 , we havẽ where the extra factor accounts for the fact that the operators on R 3 and those on S 3 differ by a Weyl factor. In this case, the 1d topological theory lives on a circle parameterized by x, with the point at x = +∞ being identified with the point at x = −∞.
To connect this discussion to the N = 6 ABJM theory, let us embed the N = 4 SU(2) H × SU(2) C R-symmetry into SU(4) R such that SU(2) H corresponds to the top left 2 × 2 block of an SU(4) R matrix written in the fundamental representation and SU(2) C corresponds to the bottom right 2 × 2 block. Raising and lowering indices with the epsilon symbol, eqs. (3.15) and (3.17) applied to S givẽ It is straightforward to check that the superconformal Ward identities (2.38) imply that the four-point function ofS R 3 , namely , is piece-wise constant. The advantage of the topological sector is that we can replace the integrated operator √ g(iJ + + K + ) by a different operator that is integrated only along the circle. Such a replacement can be rigorously justified in the class of N = 4 theories studied in [83][84][85] where it was shown how one can obtain a 1d action for the topological sector by using supersymmetric localization in the 3d N = 4 theory. Unfortunately, ABJM theory with JHEP01(2020)034 k > 1 falls outside the range of theories studied in [83][84][85]. Nevertheless, as explained in section 3.1 of [86], one expects that such a replacement should be possible in ABJM theory as well. In particular, one expects [86] 4π dx where 21J Thus, instead of (3.5), we may write Because the correlation function JJJJ is topological, we can place the four operators at any four locations of our choosing and multiply the answer by (2π) 4 . Using (3.18), we have where where the −6 comes from subtracting the disconnected part. After relating N J to c T using (3.11), we obtain The quantity I ++ [S i ] is independent of z. It can be simplified significantly using the conformal block expansion introduced in eq. (2.6). Indeed, (3.24) can be written as

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Each S R must be expanded in conformal blocks, which as z → 0 behave as (z/4) ∆ where ∆ is the scaling dimension of the corresponding conformal primary. Since I ++ is independent of z, it follows that the only conformal primaries that can contribute must have either ∆ = 0 in the 1, 15 s , 20 channels, ∆ = 1 in the 15 a and 45 ⊕ 45 channels, or ∆ = 2 in the 84 channel. The only ∆ = 0 operator is the identity operator and it appears in the 1 channel with squared OPE coefficient λ 2 0,0,1 = 1 by convention. The 15 a and 45 ⊕ 45 channels contain only odd spin operators, and for them ∆ = 1 would violate the unitarity bound. Thus, there are no ∆ = 1 operators contributing to (3.26). Consequently, the only operators that can contribute to (3.26) are the identity operator and any ∆ = 2 operators in the 84. Such operators must be scalars because these are the operators that are non-trivial in the 1d theory [81]. Using G 2,0 (U, V ) ≈ U/16 at small U , we have As explained in more detail in appendix A.2, the OPE coefficient λ 2 2,0,84 can be written in terms of the Mellin amplitude corresponding to the 84 channel, which is defined as 28) with the contour in the t integral obeying 0 < t < 1. For a derivation, see appendix A.2.

Large c T expansion
We will now show how integrated correlators can be expanded to all orders in 1/c T . Using the Fermi gas method [68], the localization formula (3.1) for the mass deformed partition function was computed to all orders in 1/N [67]: where the constant map function A is given by

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and in the second line we wrote A in the large k expansion [87]. We will be interested in derivatives of Z(m ± ) at m ± = 0, in which case we expect the non-perturbative corrections to take the form e − √ N k and e − √ N/k , which is known for Z(0) that has been computed exactly for all N and k in [68,[88][89][90][91][92][93][94]. The large N expansion is then expected to apply to the finite k, the strong coupling 't Hooft limit with 't Hooft coupling 40) and the finite µ ≡ N/k 5 limit discussed in the Introduction, which interpolates between finite k as µ → ∞ and the strong coupling 't Hooft limit as µ → 0. In particular, the non-perturbative corrections e − where we have only shown the lowest couple terms in 1/c T for simplicity. We can evaluate A (4) (k) and A (k) using the definition in the first line of (3.39), which holds for finite k, in which case the ζ(3) term is cancelled by the integral term. 22 For the strong coupling 't Hooft limit, we find the all orders in 1/λ and 1/c T result 't Hooft: where we used the large k formula for A(k) in the second line of (3.39), so ζ(3) terms appear. In fact, ζ(3) and π are the only transcendental numbers that appear to any order in 1/λ and 1/c T . 22 For instance, for k = 1, 2 these values are [86] A (1) = 1 6 + π 2 32 , A (2) = 1 24 ,

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Finally, for the finite µ limit we find finite µ: where we again used the large k formula for A(k).
From the finite µ limit we can derive both the 't Hooft limit and the finite k limit by taking µ → 0 and µ → ∞ respectively. To reproduce the 't Hooft limit (3.43) we first solve for µ in terms of λ and c T using (3.12) and (3.40), which at leading order in 1/c T gives µ = 8192λ 4 9c 2 T π 2 + . . . .

(3.45)
We then take the large c T limit followed by the large λ limit. The ζ (3) We then take the large c T limit. In this limit, the ratio c 2 T µ −3 is finite, so we must sum infinitely many terms in the finite µ limit to recover the finite k limit. This infinite sum cancels all the ζ(3) terms which appear at finite µ. The µ

N = 6 ABJM correlators at large c T
We will now combine the results of the previous to sections and determine the first few terms in the large N expansion of the SSSS correlator in ABJM theory. We will do this for the finite k, finite µ, and strong coupling 't Hooft limits, which correspond to M-theory on AdS 4 × S 7 /Z k for the first limit, or to type IIA string theory on AdS 4 × CP 3 in the second and third limits.
In each of these limits, we can use the Penedones formula (2.13) to relate the SSSS Mellin amplitude to the four-point scattering amplitudes of gravitons and their superpartners in 11d (in the M-theory case) or 10d (in the type IIA case) flat space, with momenta restricted to lie within a four-dimensional subspace. Of course, the flat space limit of the SSSS correlator in ABJM theory cannot give the four-point scattering amplitude of all massless particles in 11d or 10d. Indeed, in either 11d M-theory or in 10d type IIA string theory, the massless particle spectrum consists of 128 bosons and 128 fermions that are related by maximal SUSY. The flat space limit of the SSSS correlator must match the four-point scattering amplitude of only 15 of the 128 bosons, which all have the property JHEP01(2020)034 that after restricting their momenta to lie within 4d, they can be thought of as scalars from the 4d point of view. 23 Note that when using eq. (2.13), we should keep either the 11d Planck length 11 or the 10d string length s fixed as we send L → ∞. In other words, we should more precisely send L/ 11 or L/ s to infinity.
As explained in section 2, the ingredients we will use to construct the first few terms in the large N expansion of the SSSS correlator are the Mellin amplitudes

(4.2)
Here, the normalization constant N (L) appearing in (2.13) depends on our precise choice of normalization for the SSSS correlator. If we normalize this correlator such that the disconnected piece scales as c 0 T , then we should take N (L) = N 0 L D , where D = 7 for the case of an 11d dual and D = 6 for the case of a 10d dual.
In addition to (4.1), we will also consider the contact Mellin amplitudes which are part of degree-5 super-Mellin amplitudes corresponding to D 2 R 4 and D 4 F 2 R 2 interaction vertices, respectively. While in section 2 we did not determine the forms of M i

5,1
and M i 5,2 , we know that such Mellin amplitudes must exist because they must reproduce the scattering amplitudes in the 3rd line of table 3 in the flat space limit. Upon a convenient choice of normalization, the flat space limits of the Mellin amplitudes can be taken to be 1 where A l i,j are k-dependent numerical coefficients. In the flat space limit only the maximal degree Mellin amplitudes contribute at each order in 1/c T , and so from (4.2) and (4.4) we find that Note that neither A i 3 nor A i 5,2 give rise to scalar scattering amplitudes in flat space, which is why they do not appear in (4.7). Comparing (4.7) to the known M-theory four-point scattering amplitude [54] A 11 = A 11 SG 1 + 6 11 1 3 · 2 7 stu + O( 9 11 ) , (4.8) where A 11 SG is the 11d supergravity scattering amplitude, we can immediately deduce that Although M i 3 and M i 5,2 do not give rise to scattering amplitudes for the 11d supergravitons that are scalars from the 4d point of view, they do contribute to the scattering of other particles in the same multiplet. The M-theory amplitude (4.8) however encodes the scattering amplitudes for all such particles, and it does not contain any terms of order 13 11 or 17 11 . From this we conclude that A 3 3 = A 5 5,2 = 0 . (4.10) As a final aside, note that the O(c −2 T ) term (4.6) is not a local Mellin amplitude. It instead corresponds to the one-loop supergravity term, which is not analytic in s and t. We will not study this term further.

't Hooft strong coupling limit
We next consider the strong coupling 't Hooft limit of ABJM theory, whereby we first take N → ∞ with fixed λ (see (3.40) for the definition of λ), and then take λ → ∞. In this double limit, ABJM theory is dual to weakly coupled type IIA string theory on AdS 4 × CP 3 [39]. The leading order AdS/CFT relations are [23,39] where both s /L and the string coupling g s are small in this double expansion. The ellipses in (4.11) stand for terms that are suppressed at large c T in both expressions. Similarly to the M-theory limit discussed above, we can expand M i (s, t) in powers of s /L, with the appropriate powers of s /L being such that after taking the flat space limit, the string theory scattering amptliude has an expansion in s times momentum. Unlike M-theory however, type IIA string theory has an additional dimensionless parameter, the string coupling constant g s , that governs the strength of string interactions. Simultaneously expanding in both, we find that where B l i,j and B l i,j are numerical coefficients. The leading order 1/c T behavior corresponds to tree-level string theory, and the higher order terms are loop corrections. At fixed order in 1/c T and 1/λ only the maximal degree Mellin amplitudes contribute in the flat space limit, and so we find that (4.13) Although the 1/c 2 T terms are one-loop corrections, non-analytic Mellin amplitudes will occur first at λ 0 /c 2 T corresponding to the one-loop correction in supergravity. Comparing this to the IIA S-matrix at weak coupling [

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Like the M-theory amplitude, the type IIA super-amplitude does not contain any terms which could correspond to M i 3 or M i 5,2 , which in 10d contribute at 12 s and 16 s . We hence conclude that these terms do not contribute at leading order: Finally, we consider the large c T expansion of ABJM at finite µ ≡ N/k 5 . Like the 't Hooft strong coupling limit, ABJM theory in this limit is dual to type IIA string theory on AdS 4 ×CP 3 , except now the string coupling g s is finite. The AdS/CFT relations are [23,39] (4.17) with corrections suppressed at large c T . The relation (4.17) implies that M i (s, t) can be expanded at large c T in terms of M i n (s, t) as This expression can be compared with the type IIA scattering amplitude at fixed g s , which is given by [59] A 10 IIA = A 10 SG 1 + 6 s stu Note that the 6 s term only receives contributions from tree-level and one-loop, and it does not have any other perturbative or non-perturbative corrections.
From comparing (4.20) and (4.19), we conclude that We can recover both the finite k and strong coupling 't Hooft limit expansions from (4.18) by taking the µ → ∞ and µ → 0 limits respectively, as we explain at the end of section 3.2. Using the relations (3.45) and (3.46), we find that the c

Fixing the SUGRA coefficients
Our goal is now to fix the coefficients A l i,j , B l i,j , B l i,j , and C l i,j in each expansion considered above, purely using CFT data. We will begin with the supergravity coefficients A l SG , B l SG , B l SG , and C l SG , which we fix by determining how the various Mellin amplitudes contribute to the squared OPE coefficient λ 2 1,0,15s with which the S operator appears in the S × S OPE. As we will explain, this OPE coefficient is proportional to 1/c T , and this fact will allow us to determine all A l SG , B l SG , B l SG , and C l SG exactly. Our starting point is the expression (A.8) for λ 2 1,0,15s in terms of the Mellin amplitude M 15s = 1 6 M 2 + M 3 − M 4 + 1 2 (M 5 + M 6 ) corresponding to the 15 s channel in the S × S OPE. For the reader's convenience, we reproduce it here 22) and refer the reader to appendix A.2 for a derivation. As can be seen from (4.22), it is only the pole as s → 1 in M 15s that contributes to λ 2 1,0,15s . Therefore local Mellin amplitudes cannot contribute to λ 2 1,0,15s , so the only contribution will come from the supergravity exchange Mellin amplitude. Indeed, the supergravity exchange amplitude M i SG (s, t) does have a pole at s = 1 with a residue independent of t: and thus M SG in each of the expansions presented above contributes to λ 2 1,0,15s an amount equal to Note that although we have not yet discussed Mellin amplitudes for loop corrections, by suitably adding to them an appropriate multiple of M SG we can always define them such that they do not contribute to the √ U term, so that λ 2 1,0,15s is purely fixed by the coefficient of M SG . Furthermore, because the three-point function of three stress tensor multiplets is uniquely determined up to an overall coefficient [47], λ 2 1,0,15s must be proportional to the stress-tensor three-point function, which itself is proportional to 1/c T according to the conformal Ward identity [96]. We hence determine that

(4.25)
Our final step is to determine the relationship between λ 2 1,0,15s and c −1 T . We can do so by considering the free N = 6 theory of four complex scalars and four 2-component complex fermions, where the scalars φ a (φ a ) transform in the 4 (4) of SU(4) R . (This is the same as the U(1) k × U(1) −k ABJM theory in the limit k → ∞ considered in the previous section.) We write S in this case as  (4.27) so that by computing S 15s (U, 1) and comparing to (A.5), we find that λ 2 1,0,15s = 4. This free theory has 8 real scalars and 8 Majorana fermions, so c T = 16 according to (1.3). Because the relationship between λ 2 1,0,15s and c −1 T is fixed by the superconformal Ward identity, we conclude that in general

3) with the crossing independent coefficients
Combining (4.28) with (4.25) we conclude that which is the same coefficient that was found for the N = 8 case in [22]. This is the same coefficient we would obtain if we decomposed the known N = 8 answer from [22] into N = 6 language as we did in section 2.7.1. Indeed, the supergravity term does not depend on k when written in terms of c T , because it is proportional to the effective 4d Newton constant G 4 ∝ 1/c T .

Constraints from supersymmetric localization
Let us now explore the constraints on the coefficients A l i,j , B l i,j , B l i,j , and C l i,j coming from the supersymmetric localization constraints of section 3. To do so, we can compute the integrated constraints I ++ [S i ] in (3.24)   , we can obtain the following results. First, without using the constraints from the flat space limit or the constraints (4.25) coming from the superconformal block expansion, the supersymmetric localization constraints (3.25) and (3.37) reproduce the coefficients in the first line of (4.25). This is a stringent consistency check on the accuracy of our computations.
Second, using the constraints (4.25) coming from the superconformal block expansion as an input, the supersymmetric localization constraints allow us to fix the coefficients at JHEP01(2020)034 the next two orders in each of the expansions (4.6), (4.12), and (4.18). The result is finite k:  These equations agree with the constraints from the flat space limit, thus providing a very non-trivial precision test of AdS/CFT. Third, using both the constraints (4.25) as well as the constraints coming from the flat space limit as input, the constraints from supersymmetric localization allow us to conclude that We can then plug these values back into (4.6), (4.12), and (4.18) to get the final answers (1.2) as advertised in the Introduction.

Discussion
In this paper we used superconformal symmetry, the flat space limit, and most importantly supersymmetric localization results for the mass deformed sphere free energy to compute the R 4 correction to the stress tensor multiplet bottom component four point function SSSS in N = 6 U(N ) k × U(N ) −k ABJM theory in the large N finite µ = N/k 5 limit. After taking the flat space limit we matched the known type IIA string theory S-matrix for finite g s , which is the first check of AdS/CFT of this type for local operators. This finite µ result interpolates between the large N finite k limit at µ → ∞ and the large 't Hooft coupling λ ∼ N/k limit at µ → 0, which in the flat space limit are related to the S-matrix of M-theory and weakly coupled type IIA string theory, respectively.
There were several technical innovations in this work relative to similar studies of N = 8 ABJM theory in [24] and N = 4 SYM in [21], which all stem from the fact that our theory is not maximally supersymmetric like these other theories. One implication is that the stress tensor multiplet is 1 3 -BPS, not 1 2 -BPS as in the other cases, so the Ward identities that we derived for various four point functions in this multiplet are the first such derivation for operators annihilated by less than half the supercharges. Another novelty of this calculation was that demanding bulk locality, i.e. that higher derivative corrections to supergravity correspond to polynomial Mellin amplitudes, in stress tensor correlators other than SSSS gave additional constraints, unlike the maximally supersymmetric cases were only SSSS gave such constraints. Finally, in the flat space limit, stress tensor multiplet correlators in holographic theories are dual to supergraviton multiplet amplitudes JHEP01(2020)034 in one more dimension. For maximally supersymmetric supergravity there is just one such amplitude supermultiplet, but for our sub-maximal case two amplitudes exist, which is related to the fact that we found an extra subleading term in the large N expansion of SSSS relative to the analogous expressions in N = 8 ABJM and N = 4 SYM.
A crucial ingredient in our finite g s check of AdS/CFT was the conjecture that the all orders in large N localization expression for derivatives ∂ 4 m ± F m ± =0 and ∂ 2 m + ∂ 2 m − F m ± =0 of the mass deformed sphere partition function F (m ± ) in [67] only receive non-perturbative corrections of form e − √ N/k and e − √ N k . When m ± = 0, these corrections can be interpreted as instanton effects in string theory, and it was proven in [68,[88][89][90][91][92][93][94] that for F (0) they do take the form mentioned above. Since a small mass deformation changes the geometry only slightly, we expect that for sufficiently small masses these instanton effects have the same N and k scaling as for m ± = 0. It would be interesting to find a more rigorous justification of this fact in the future. Looking ahead, there are more localization constraints that can be used to fix SSSS . As discussed in section 3, the N = 6 ABJM free energy can be computed using localization as a function of not only the two masses m ± considered in this work, but also of a third mass m. The reason why there are three mass parameters is that, as an N = 2 SCFT, any N = 6 SCFT has SU(2) × SU(2) × U(1) flavor symmetry, and the Cartan of the flavor symmetry algebra is three-dimensional. In addition to the three mass parameters, one can also consider placing the theory on a squashed sphere parameterized by squashing parameter b [79] (with b = 1 corresponding to the round case). There are then seven potentially independent combinations of four derivatives of these parameters that can be related to integrated 4-point functions of the stress tensor multiplet: all evaluated at m ± = m = 0 and b = 1. Only the first two were considered in this work. In section 2, we showed that there are seven polynomial Mellin amplitudes of maximal degree six, 24 as well as the supergravity Mellin amplitude that is already fixed by the conformal Ward identity. This means that we could potentially use localization to fix the coefficients of all these Mellin amplitudes, which would thus allow us to determine the D 4 R 4 term in the large N finite µ limit, that could be checked in the flat space limit against the known [97] finite g s term in the type IIA S-matrix. These are the highest order terms we would expect to be able to fix with N = 6 supersymmetry. For N = 8 ABJM theory, the U(1) flavor symmetry combines with SU(4) R to form the larger R-symmetry SO(8) R , so the dependence on m is now related to that on m ± . As discussed in [24], there are only two quartic Casimir invariants for SO (8), so only the first four constraints in (5.1) would be linearly independent. On the other hand, for N = 8 there are only three polynomial Mellin amplitudes of maximal degree 7, so we could fix the tree level D 6 R 4 term, which is the highest order term that is protected by supersymmetry. In fact, there is only one additional Mellin amplitude at maximal degree 8, so four constraints would seem sufficient to fix tree level D 8 R 4 , but this term is not expected to be fixed by 24 As shown in table 3, we have one degree 3, one degree 4, two degree 5, and three degree 6.

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supersymmetry, so it is likely that one of these constraints becomes redundant for N = 8 ABJM when we take the large N limit.
To go beyond these protected coefficients, we need a more general method such as the numerical conformal bootstrap. Our computation of the N = 6 Ward identities for SSSS opens the door to a numerical bootstrap study of N = 6 ABJM theory, which would generalize the N = 8 studies of [49,81,86]. In the N = 8 case, the bootstrap bounds were found to be conjecturally saturated by CFT data in ABJM theory, so that all lowlying CFT data, both protected and unprotected, could be read off up to numerical error. If a similar thing occurs for N = 6 ABJM theory, then we can use this unprotected CFT data to extend the derivation in this work to higher order, and perhaps even interpolate between M-theory at finite k and type IIA string theory at weak and strong coupling in the 't Hooft limit of ABJM theory.

Acknowledgments
We thank Yifan Wang for collaboration at early stages of this project, and we also thank him as well as Ofer Aharony and Igor Klebanov for useful discussions. DJB and SSP are supported in part by the Simons Foundation Grant No. 488653, and by the US NSF under Grant No. 1820651. DJB is also supported in part by the General Sir John Monash Foundation. SMC is supported by the Zuckerman STEM Leadership Fellowship. SSP is also supported in part by an Alfred P. Sloan Research Fellowship.

A.1 Derivation of the SU(4) invariants
The SU(4) invariants presented in (2.8) can be derived as follows. The T R (X i ) are eigenfunctions of the SU(4) quadratic Casimir C 2 acting on X 1 and X 2 , namely

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In terms of S i , the functions of (U, V ) corresponding to the various representations are

A.2 Extracting OPE coefficients
We will be interested in extracting 25 two OPE coefficients of protected (1/3-BPS) scalar operators in the S × S OPE: the OPE coefficient of an operator with ∆ = 1 in the 15 s irrep of SU(4) (this is the same as the external operator S a b ), and that of an operator with ∆ = 2 in the 84. In the theories of interest to us, both of these operators are the lowest dimension operators in their corresponding R-symmetry channels. Let us start with λ 2 1,0,15s , and let us take U → 0 while setting V = 1. In this limit, Thus, in order to extract λ 2 1,0,15s , all we need to do is extract the coefficient of √ U in the small U expansion of S 15s (U, 1). Note that the disconnected piece S disc,15s (U, 1) = O(U ) in this limit, so the √ U term in the small U expansion of S 15s (U, 1) must come from a pole at s = 1 in the Mellin amplitude M 15s (s, t) corresponding to S 15s (U, V ), namely (see (A.4)). Performing the s integral in (2.10) and picking up the residue at s = 1, we obtain where the integration contour can be chosen such that t < 2. Comparing with (A.5), we have 25 See [22,98] for similar calculations in N = 8 SCFTs.

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Next, let us consider extracting λ 2 2,0,84 by considering S 84 (U, 1) in the limit U → 0. Because G 2,0 (U, 1) = U 16 + · · · in this limit, we have The integration contour here must be such that t is smaller than the minimum between 2 and the pole in t of M 84 (s, t) with the smallest real part, and such that 2 − t is smaller than the minimum between 2 and the pole in u of M 84 (s, t) with the smallest real part. Such a condition is obeyed by 0 < t < 2 for polynomial M 84 (s, t), but it is tricky to impose it when M 84 (s, t) has both a pole at t = 1 and a pole at u = 1, as is the case for the SUGRA amplitude. In the case that both of these poles are present, let us use 0 < t < 1. Because if we closed the t contour on the right we would pick up both the pole at t = 1 and that at u = 1, we should subtract by hand the contribution from the pole at u = 1. Thus, the correct formula is (A.13) Comparing with (A.9), we extract (A.14) with the t contour obeying 0 < t < 1.

JHEP01(2020)034 B Discrete symmetries of N = 6 theories
Both N = 6 SCFTs and flat space scattering amplitudes may posses various discrete symmetries that can be used to impose selection rules. The symmetries we will focus on here are parity P, time reversal combined with charge conjugation, CT , and a discrete R-symmetry we will call Z. Even for theories that break these symmetries, organizing the SCFT correlators and scattering amplitudes in terms of them will prove very useful.

B.1 Review of spinor helicity formalism
For massless fermions, the Dirac equation for the wavefunction of 4-component spinors implies Here ± indicated the helicity h = ± 1 2 of the wavefunction. If we take our Dirac matrices to be in the Weyl basis, namely where 1 stands for the 2 × 2 identity matrix and σ i , i = 1, 2, 3 are the standard Pauli matrices, then the top two components of the Dirac spinor transform in the (1/2, 0) and bottom two in the (0, 1/2) of SO (3,1). For a given momentum p µ = (E, E sin θ cos φ, E sin θ sin φ, E cos θ), we can then define the angle and square brackets as are solutions to (B.1). Let us consider the scattering of massless particles b ± i for i = 1, 2, . . .. We define the scattering amplitude to be: where a ± i (p) is the annihilation operator of the i th particle, annihilating a particle of helicity ± and momentum p i .

B.2 Discrete symmetries for scattering amplitudes
We will begin by discussing the discrete symmetries of the 4d amplitudes, motivated by two reasons: 1) given that in N = 6 supergravity, we have two CPT conjugate multiplets, we should understand how CPT relates the scattering amplitudes; and 2) we can use discrete symmetries in order to classify the structures that appear in the super-amplitude. As mentioned above, we will discuss parity P, the product CT , as well as a discrete Rsymmetry we denote by Z.
Under parity P, we reverse the spatial components of the momentum of a particle, while leaving the spin unchanged. Flipping the direction of p is equivalent to sending θ → π − θ and φ → φ ± π in (B. Hence the effect of parity is to swap all angle brackets with square brackets and vice versa, while leaving all coefficients unchanged. For instance, P(c 12 ) = c [12] for any constant c.
The second discrete symmetry we consider is CT . Under CT , the spatial components of momentum also flip sign, just like for P, but in addition CT also implements complex conjugation. Thus, from (B.3), we see that CT acts as either (CT )ȧ˙b or (CT ) ab as follows: Thus, the effect of CT is to flip all the brackets and perform complex conjugation on the coefficients -for instance CT (c 12 ) = c * 21 for any constant c. The combined transformation of the two symmetries above, CPT , is a symmetry of all unitary QFTs. On amplitudes, it acts by exchanging angle brackets with flipped square brackets and vice versa, and it complex conjugates the coefficients. For instance, CPT (c 12 ) = c * [21]. Using CPT , we can relate a given amplitude to the amplitude of the CPT conjugate particles. For particles b 1 , b 2 , etc. with CPT conjugate particles b 1 , b 2 , etc., we have Because CT does not change the helicity of the particles, it relates a given amplitude to itself. Thus, we can classify the various scattering amplitudes based on whether they are CT -even or CT -odd. (In a CT -preserving theory, such as pure N = 6 supergravity, all 26 In terms of the four-component spinors (B.4), the action of parity takes the usual form:

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amplitudes should be CT -even. But the CT symmetry may be broken by higher derivative corrections.) For instance, if we consider the amplitude we see that and so the amplitude (B.9) is CT even / odd if f 3 (s, t) is real / pure imaginary. From this we conclude that A[ΦΦΦΦ] can be thought of as containing two distinct superstructures, one of which is CT even and the other CT odd. Similar manipulations show that A[ΦΦΦΨ] also contains a CT even and CT odd structure, corresponding to f 2 (s, t) purely real and purely imaginary respectively. On the other hand, one can show that A[ΦΦΨΨ] is always CT even, even in a theory in which CT is not a symmetry. We can see this by considering the graviton scattering amplitude: But the two amplitudes in (B.11) are also related by CPT , and from comparing this expression with (B.11) we conclude that f 1 (s, t) must be real. Then Let us now consider all possible discrete R-symmetries of N = 6 supergravity and its higher derivative corrections. Before doing so, let us recall that, as discussed in the main text, the various particles in the Φ and Ψ multiplets transform under an SU(6) R Rsymmetry that is a symmetry of pure supergravity and of the higher derivative corrections considered here. Under SU(6) R , the supercharges transform contravariantly where M I J is a unitary matrix with determinant 1. The supergraviton fields h ± , ψ ± I , g ± IJ , . . . transform covariantly, so that overall the superfields Φ and Ψ are invariant. To see what discrete R-symmetries might be possible, let us first focus on the pure supergravity case and consider relaxing the condition that M I J has determinant 1. Instead, let us consider a more general element of U (6). Without loss of generality let us consider a transformation: η I → e iθ η I (B.15) JHEP01(2020)034 in the center of U (6). We can also allow the superfields Φ and Ψ to pick up an overall phase: The supergravitons will then transform as: We cannot however choose α , β, and θ arbitrarily. The graviton and gauge fields are real, and so we can only transform them by a factor of ±1. This restricts us to the cases e iθ = ±i or e iθ = ±1, as well as e iα = ±1 and e iβ = ±1. The case e iθ = ±1 is already in SU (6), so let us focus on the possibility e iθ = ±i.
To determine e iα and e iβ , let us make use of the CTinvariance of supergravity in order to write the scattering amplitudes for three gravitons as with real g. Since the right-hand sides of these equations are invariant under the transformation considered above, we deduce that e iα = 1 and e iβ = −1.
We can now check that the transformation: is in fact a symmetry of pure supergravity, as is the symmetry −Z which sends η I → −iη I . Under both Z and −Z the gauge fields flip sign while the gravitons h ± and the graviscalar φ are left invariant. The fermions will transform with additional factors of i: The full symmetry group is now (Z 4 × SU(6))/Z 2 , the subgroup of U(6) of matrices with determinant ±1. Note however that only fermion bilinears are physical. As a result, the transformation η I → −η I acts trivially on all amplitudes. After quotienting the SU(6) by this Z 2 symmetry, we find that the symmetry group acting on the amplitudes is Z 2 ×(SU(6)/Z 2 ), with Z 2 = I.
While Z is a discrete R-symmetry of pure supergravity, it may or may not be a symmetry of the corrections to supergravity, so we can classify the various amplitude structures as Z-even or Z-odd. Since δ (12) Table 7. Four particle scattering in N = 6 supergravity. The dimension is the mass dimension of the lowest bulk counterterm contributing to the amplitude, and CT and Z are the discrete symmetries defined in the main text.
We can alternatively deduce this from (2.23), since A[ΦΦΦΨ] contains an amplitude with an odd number of gauge fields, while the other two amplitudes contain an even number. We can summarize these results in table 7. In total, the scattering of four supergravitons is fixed up to five arbitrary functions of s and t. To determine the Z and CT even part of the amplitude there are two functions, while for each of the other combinations there is a single function. Because the only superamplitude contribution to scalar scattering, A[ΦΦΨΨ], is automatically Z and CT invariant, it is impossible to know whether these symmetries are present or not in the full theory just by considering scalar scattering, without any additional information.

B.3 Discrete symmetries for N = 6 SCFTs
Analogous P, CT , and Z symmetries exist for N = 6 superconformal theories, with CT P always being a symmetry. Individually, P, CT , and Z may not be symmetries of a given theory, as we will see, but they are symmetries of the free theory (or more generally of the U(1) k × U(1) −k ABJM theory for all k) and of the leading order large c T holographic correlators.
Under P and CT , the ∆ = 1 operators S are even, while the ∆ = 2 operators P are odd. Just as for amplitudes, we expect that three out of the five superconformal structures given in table 5 are P or CT even, while the other two are P and CT odd.
The Z R-symmetry is trickier in the case of SCFTs than for scattering amplitudes, because while in the case of scattering amplitudes it commutes with the SU(6) R R-symmetry, for SCFTs it does not commute with the SO(6) R R-symmetry. Instead, it extends SO(6) R to O(6) R . Let us define the Z generator so that it corresponds to the O(6) matrix that is not part of SO (6). The group O(6) has two 6-dimensional representations: the vector representation 6 + under which a vector v I transforms as v I → Z IJ v J , and the pseudovector representation under which v I → −Z IJ v J . By convention, we take the supercharges to transform as the 6 + . 27 The representations of O(6) appearing in the stress tensor multiplet are all antisymmetric products of the 6 + , because we can start with the stress-energy tensor, which is a singlet, and obtain all the other operators by acting with JHEP01(2020)034 15 10 + 10 15 6 1 Table 8. O(6) and SO(6) assignments for operators in the stress-tensor multiplet.
anti-symmetric products of the superconformal generators. Thus: the rank-0 tensor is the singlet 1 + that is invariant under Z; the rank-1 anti-symmetric tensor is the 6 + ; the rank-2 anti-symmetric tensor is the adjoint representation 15 + ; the rank-3 anti-symmetric tensor, the 20 is irreducible under O(6) but would've been reducible to 10 + 10 under SO (6); the rank-4 anti-symmetric tensor is the 15 − and can also be represented as a rank-2 antisymmetric tensor with the same SO(6) transformation properties as the 15 + except for an additional minus sign under Z; the rank-5 anti-symmetric tensor is the 6 − and can also be represented as a pseudovector; and lastly, the rank-6 anti-symmetric tensor 1 − is invariant under SO (6)  To gain intuition about the Z transformation, let us describe how it acts in the free N = 6 theory of 4 complex fields C a and 4 complex two-component fermions ψ a where it is actually a symmetry. Both φ a and ψ a transform in the 4 of SU(4) R , and their conjugates φ † a and ψ † a transform in the 4 of SU(4) R . In this case, one can show that the Z symmetry acts as charge conjugation and similarly on ψ a and ψ † a . Indeed, from φ a and ψ a , we can construct the various operators in the stress-tensor multiplet. For example, etc.

(B.24)
It is easy to see that under (B.23), j µ acquires a −1 factor, as implied by table 8. To see whether S a b , P a b , (J µ ) a b transform in the expected way, we should represent these operators as rank-two anti-symmetric tensors of SO (6). This is done by defining

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and similarly for P and J µ , with the C matrices given in (D.1) and C being their complex conjugates. (The C and C matrices are the Clebsch-Gordan coefficients for the 6 of SU (4) in the products 4 ⊗ 4 and 4 ⊗ 4, respectively.) One can check that (B.23) implies as expected from table 8. One can make similar checks for the other operators in the stress tensor multiplet. One can ask whether Z is a symmetry in ABJM theory as well, where the scalars φ a and fermions ψ a are bifundamental fields transforming in the (N, N) of the U(N ) k × U(N ) −k gauge symmetry. If the two gauge fields corresponding to the U(N ) factors are A 1µ and A 2µ , the action is invariant under Z provided that A 1µ −A 2µ change sign under Z. In the N = 1 case, this can be accomplished by requiring A 1µ → −A 1µ and A 2µ → −A 2µ under Z, and one can check that the action (including the Chern-Simons terms) is invariant under this transformation. Thus, Z is a symmetry of the U(1) k × U(1) −k ABJM theory. Such a transformation of A 1µ and A 2µ does not leave the action invariant in the non-Abelian case due to the cubic terms in the Chern-Simons action. In the non-Abelian case, however, one can consider sending A 1µ ↔ A 2µ under Z, which also has the effect of flipping the sign of A 1µ −A 2µ . Under this transformation, the action stays unchanged with the only exception that k → −k. Thus, the Z transformation is not a symmetry of the U(N ) k ×U(N ) −k for N > 1.
Note that CT and P are not separately symmetries either of ABJM theory with k > 1, because they also send k → −k. However, the combination PZ where Z is assumed to interchange the two gauge fields in addition to acting as in (B.23) becomes a new parity symmetry of ABJM theory [39]. To summarize, ABJM theory with k = 1 preserves CT , P, Z separately, while ABJM theory with k > 1 preserves only CPT and PZ (or CT Z).
Having discussed CT , P, and Z, let us now argue that the 4-point superconformal invariants (i.e. invariants under OSp(6|4)) can be classified as even or odd under P (or CT ) and Z. This fact may not be immediately obvious, because it may happen that the superconformal Ward identities mix together Z-odd with Z-even structures. The argument that this mixing does not occur is as follows. As we have seen, the stress tensor multiplet naturally forms a representation of (Z 2 × Z 2 ) OSp(6|4), that is, the superconformal group extended by the action of parity P (or CT ) and also Z. As we shall see, this means that we can classify correlation functions of the stress tensor multiplet by their P (or CT ) and Z transformation properties. It is a consequence of the following proposition: (B.27)

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Next we prove that for any h ∈ H, then hV ∈ V. To do so, let us check that hV satisfies (B.27): where we have used the fact that G is a normal subgroup of H to write h −1 gh = g for some g ∈ G. Hence V is a representation of H for which G acts trivially, and so we conclude that V is a representation of H/G.
To apply this to OSp(6|4), take H to be the group (Z 2 × Z 2 ) OSp(6|4) group. We can think of the stress tensor multiplet as a superfield S a b (x µ , θ α c ) which forms a representation t of the group H. The OSp(6|4) invariant structures in . . S an bn x ν n , θ αnAn n are then maps from t ⊗n → 1, and so by proposition 1 can be classified by their represen- Analogous to the amplitudes case, the correlator SSSS is always invariant under P, CT , and Z separately. This is also true for SSP P (and also P P P P ), so if we are only interested in the Ward identities relating these correlators, we can restrict to structures that are even under all of these transformations without loss of generality. This also means it is impossible to check whether a theory is P or Z invariant from just SSSS without having more information about the theory. If, however, we had some information about the spectrum of the theory, then we could potentially determine whether a theory is paritypreserving or not based on the conformal block expansion of SSSS . Without such extra assumptions, in order to see whether a theory is invariant under P or Z, we would need to see whether the P-odd or Z-odd part of a correlator such as SSSJ µ vanishes. The amplitudes calculation furthermore suggests that together SSSS and SSSJ µ should suffice to fix all four-point functions of the stress tensor multiplet operators.

C Relating A[ΦΦΨΨ] to SSSS
In this appendix, we shall explain how to relate the superamplitude A[ΦΦΨΨ] to the large s, t behaviour of the Mellin amplitudes in SSSS . The calculation proceeds in two steps. First we compute the amplitude A[φ ABCD φ EF GH φ IJ φ KL ], where we have made explicit the SU(6) indices on φ and φ. We then relate φ and φ to the CFT operator S a b , requiring us to convert the SU(6) structures to SO(6) structures.
To compute A[φ ABCD φ EF GH φ IJ φ KL ], we must differentiate A[ΦΦΨΨ] with respect to the Grassmannian variables:

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To simplify the process of differentiating δ (12) (Q), we can use SU (6) invariance to expand for some functions F i (s, t). We can then choose specific numbers for each index A through L to isolate each structure, and hence to find that 3) Now we must relate A[φφφφ] to SSSS . To do so, we can rewrite S b a as an antisymmetric 6 × 6 matrix: where C I ac are SO(6) gamma matrices. Explicit expressions for these matrices are given in appendix D. Up to normalization, we then find thať This expression forŠ IJ breaks the SU(6) symmetry down to SO(6) due to the presence of the δ IA symbol. Applying this to the four-point function, we find that We must now expand our final answer in terms of the SO(6) structures appearing in (2.3).
To do so we choose a series of polarization matrices (X i ) a b and then defině Contracting both sides of (C.6) with matrices X IJ i , on the left-hand side we find that We then Mellin transform and take the flat-space limit (2.13) to find that A 1 (s, t)A 12 A 34 +· · ·+A 6 (s, t)B 1342 (C.9) for some overall normalization constant N . Computing the right-hand side of (C.6) is more straightforward; we simply contract the X I i J i i matrices with the various permutations of A[φφφφ]. By imposing (C.6) for many differents matrices (X i ) a b we can completely determine A i (s, t) in terms of f 1 (s, t), and upon choosing a suitable value for N we can reproduce (2.27).

D.1 Stress tensor multiplet four-point functions
To describe the supersymmetric variations which relate operators in the stress tensor multiplet, it will be convenient to introduce index-free notation to encode the so(6) ≈ su (4) representations which appear. We will use indices I, J, . . . for the 6; and raised and lowered a, b, . . . indices for the 4 and 4 as in section 2. The gamma matrices C I ab and C Iab convert antisymmetric tensors of the 4 and 4 into the 6; a convenient basis for these matrices is: where σ i are the Pauli matrices.
We can now describe operators in index-free notation as: with analogous notation for other operators in the stress tensor multiplet. To implement tracelessness of S a b we impose the condition X a a = 0, and similarly we impose that the matrix Y ab is symmetric. We can alternative think of the matrix X b a as an antisymmetric tensorX IJ via the mappingX Similarly, the Z I can also be written as antisymmetric tensors / Z ab = C I ab Z I and / Z ab = C ab I Z I . We can normalize our operators by defining their two-point functions, as we did for S in (2.2): We can expand four point correlators as a sum over conformally invariant and so(6) invariant structures. As explained in section 2.6 we restrict to those structures which are parity preserving and C invariant. For instance,

D.2 Ward identities
As discussed in section 2.6, to compute the supersymmetric Ward identities we need only the action of the Poincaré supercharges Q αI on the operators in the stress tensor multiplet.

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Using the index free notation of the prevoius section, these variations can be written as δ α (Z)S( x, X) = 1 4 F α ( x, X · / Z) + F α ( x, / Z · X) + 1 4 χ α ( x,X · Z) , etc. (D.8) Here, δ α (Z) represents the action of Z I Q αI on the various operators, and σ µ are the 3d gamma matrices, which we can take to be the Pauli matrices. We have omitted the supersymmetric variations of J, j, ψ, and T as they are not needed in this work.
We will now give the Ward identities for two scalars and two fermions derived in section 2.7. We will begin with SSχχ and SSχF , which can be derived from δ SSSχ . We will omit those functions of the cross-ratios that are related to these under crossing.
The expressions for SSχχ are:

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The expressions for SSF χ are: , (D.14) Next we shall give expressions for SSF F and SSF F , which can be computed from δ SSSF . Unlike the previous correlators, we cannot completely fix these in terms of SSSS . We will instead also leave F 1,1 (U, V ) and F 2,1 (U, V ) undetermined. We then find that the other components of SSF F are: Furthermore, by imposing conservation on SSSJ , we find that F 1,1 (U, V ) and F 2,1 (U, V ) are constrained by the Ward identities: (D.20) We also find the following expressions for SSF F : Finally, in section 3.1 we need Ward identities relating SSP P to SSSS . These expressions can be derived by considering the supersymmetric variation δ SSP χ :

E Mellin amplitudes
In this appendix we will first review how to convert supersymmetric Ward identities to position space. We will then describe the Mellin space formulation for four point functions of two scalars and two fermions, following [99].