Absence of $D^4 R^4$ in M-Theory From ABJM

Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d ${\cal N} = 8$ SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on $S^d$ for any $d$.


Introduction and Summary
M-theory is a rather mysterious ultraviolet (UV) completion of eleven-dimensional supergravity [1]. It describes the dynamics of massless gravitons and their superpartners, 1 and as such its main observables are the scattering amplitudes of these massless particles. For general momenta, supersymmetry requires that the 4-graviton S-matrix take a factorized form A(η i , s, t) = A SG,tree (η i , s, t)f (s, t) [2]. The first factor is the tree-level scattering amplitude computed in 11d supergravity and depends on the polarizations η i of the four gravitons as well as the Mandelstam invariants. The second factor is an arbitrary symmetric function f of the Mandelstam invariants s, t, and u = −s − t. In the small momentum expansion (or equivalently in the expansion in the 11d Planck length p ), 11d supersymmetry allows the following terms in f : f (s, t) = 1 + 6 p f R 4 (s, t) + 9 p f 1-loop (s, t) + 10 p f D 4 R 4 (s, t) + 12 p f D 6 R 4 (s, t) + · · · (1.1) Terms multiplying n p are homogeneous of degree n in s and t, as required on dimensional grounds. The functions of s and t that constitute the coefficients of n p are given suggestive names: f D 2m R 4 are symmetric polynomials in s, t, u that represent the contributions of the contact Feynman diagrams with D 2m R 4 as well as tree-level exchange diagrams that have the same momentum scaling; f 1-loop is the one-loop supergravity Feynman diagram, etc. All the loop corrections to supergravity can be computed in principle from the 11d supergravity Lagrangian. On the other hand only the lowest few protected f D 2m R 4 corrections to supergravity can be determined by relating them to type II perturbative string theory computations and non-renormalization theorems. 2 These take the form [9][10][11][12][13] f R 4 (s, t) = stu 3 · 2 7 , f D 4 R 4 (s, t) = 0 , f D 6 R 4 (s, t) = (stu) 2 15 with u = −s−t as above. The D 4 R 4 contribution is absent, but would otherwise be consistent with 11d supersymmetry. The goal of this paper is to derive the vanishing of f D 4 R 4 purely from 3d CFT using AdS/CFT.
It was proposed in [14] following earlier work [15][16][17] that an alternative way of determining the 11d 4-graviton S-matrix is from the flat space limit of stress tensor multiplet 4-point correlation functions in the superconformal field theory (SCFT) on N coincident M2-branes.
This theory is part of a family of U (N ) k × U (N ) −k gauge theories coupled to bifundamental matter whose Lagrangian descriptions are due to Aharony, Bergman, Jafferis, and Maldacena (ABJM) [18]. For general N and k, ABJM theory is dual to the AdS 4 ×S 7 /Z k background of M-theory and is the effective theory on N coincident M2-branes placed at a C 4 /Z k singularity in the transverse space. We will only focus on the cases k = 1 or 2 where supersymmetry is enhanced to N = 8 [18][19][20][21][22] from the N = 6 manifestly preserved at all k [18]. Instead of parameterizing these theories by N , we will find it convenient to use the quantity c T ∼ N 3/2 , the coefficient of the canonically-normalized stress-tensor two point function, which has been calculated to all orders in 1/N through supersymmetric localization [23] using the results of [24] and [25].
More concretely, one can consider the four-point function SSSS of the scalar bottom component S of the N = 8 stress tensor multiplet. It was shown in [14] that the N = 8 superconformal Ward identity for the Mellin transform [16,17] M SSSS tree of the tree level SSSS correlator has a finite number of solutions at every order in the 1/c T expansion: where s, t are Mellin space variables that are related to the 11d Mandelstam variables in the flat space limit, M p S are functions of s, t that grow as the pth power at large s, t, and the B's are numerical coefficients unfixed by 3d supersymmetry. If one can determine these B's up to order 1/c n T , then by taking the flat space limit [15] one can reproduce the 4graviton scattering amplitude and read off the function f to order 9(n−1) p . This procedure was carried out in [14] to first non-trivial order: by computing two distinct CFT quantities as a function of N (namely c T as well as a 1/2-BPS OPE coefficient), Ref. [14] was able to fix both B 4 i and thereby reproduce exactly the known value of f R 4 (s, t). (See also [26] for an analogous computation in 6d.) The goal of the present paper is to use ABJM theory to find an additional constraint such that all three B 6 i can be fixed to zero, which implies that f D 4 R 4 = 0.
Developing the momentum expansion up to D 4 R 4 order presents new challenges compared to the computation up to R 4 order performed in [14]. In [14], the computation of the additional CFT quantity besides c T that was needed involved a trick based on the fact that, as any N ≥ 4 3d SCFT, ABJM theory contains a one-dimensional topological sector [27,28].
Such a trick does not seem easily generalizable to the computation of other quantities.
Nevertheless, one may hope to to beyond R 4 order because in 11d all terms up to D 6 R 4 (thus including D 4 R 4 ) preserve some amount of supersymmetry, and thus one may hope to be able to use supersymmetric localization to compute just enough quantities in the field theory in order to recover all these terms in 11d.
In ABJM theory, quite a few BPS quantities can be computed using supersymmetric localization using [24,29]. To go to order D 4 R 4 , it is enough to consider the free energy on a round three-sphere in the presence of real mass deformations. When ABJM theory is viewed as an N = 2 SCFT, it has SU (4) flavor symmetry, and, since the Cartan of SU (4) is three-dimensional, it also admits a three-parameter family of real mass deformations. (See for instance [30] where these deformations were studied at leading order 1/N .) The S 3 free energy F in the presence of two such mass parameters m 1 and m 2 was computed to all orders in 1/N in [31] using the Fermi gas formalism developed in [32]. To make connection with the four-point function of the stress tensor multiplet, we consider the fourth derivatives 3 ∂ 4 F ∂m 4 1 m 1 =m 2 =0 and ∂ 4 F ∂m 2 1 ∂m 2 2 m 1 =m 2 =0 , which, by analogy with the analysis of [25] for two-point functions, can be related to integrated four-point correlators in the SCFT. Calculating these integrated correlators using the solution to the Ward identity at order D 4 R 4 and comparing with the fourth derivatives of F mentioned above, one can fix all three B 6 i = 0 so that f D 4 R 4 (s, t) = 0.
It is worth pointing out that part of the difficulty in performing this computation is that all previous studies [14,23,27,33,34] in 3d N = 8 SCFTs focused on the four-point function of the superconformal primary of the stress tensor multiplet, which is the scalar operator S mentioned above of scaling dimension ∆ S = 1 transforming in the 35 c of the SO(8) R R-symmetry. However, the fourth mass derivatives of the F are more directly related to integrated four-point functions of a linear combination of S and another operator P that belongs to the same superconformal multiplet as S. The operator P is a pseudoscalar of scaling dimension ∆ P = 2 transforming in the 35 s of SO(8) R . As part of our computation, we will therefore derive expressions for the four-point functions SSP P and P P P P in terms of the more easily computable SSSS .
Another notable feature of our computation is a concise expression for integrals over S d of 4-point functions of scalar operators in CFTs. In particular, we find that the integral over 4d variables reduces to an integral over the two conformally-invariant cross ratios U and V of the 4-point function multiplied by a D(U, V ) function, which naturally shows up in tree level calculations in AdS d+1 [35]. (See Eqs. (3.22) and (3.30).) While in this work we only 3 All other fourth derivatives vanish or are linearly dependent on the two mentioned in the main text. apply this result to d = 3 and the specific operators we are interested, this formula applies to any CFT with or without supersymmetry.
The rest of this paper is organized as follows. We start in Section 2 with a brief review of the relevant four-point functions in N = 8 SCFTs and derive the relations between them that were mentioned in the previous paragraph. In Section 3, we discuss the relation between the fourth mass derivatives of the S 3 free energy and integrated correlation functions in the SCFT. In Section 4 we apply these results to ABJM theory, and by taking the flat space limit of the SCFT correlators show that f D 4 R 4 vanishes, as expected. Lastly, we end with a brief discussion of our results in Section 5. Various technical details are relegated to the Appendices as well as to an auxiliary Mathematica file included with this arXiv submission, containing the Ward identities and the large c T expressions for the Mellin amplitudes and position space correlators that we computed.

Stress tensor multiplet 4-point functions
Let us begin by discussing the structure of the stress energy tensor multiplet in N = 8 SCFTs and then derive the relation between the 4-point function of the 35 c scalar S and the 4-point functions involving the 35 s pseudoscalar P that were mentioned in the Introduction.

The N = 8 stress tensor multiplet
In addition to the scaling dimension 1 scalar operator S and the dimension 2 pseudoscalar operator P transforming in the 35 c and 35 s irreps of the SO(8) R R-symmetry, the N = 8 stress tensor multiplet also contains a fermionic operator χ α of dimension 3/2 in the 56 v , the R-symmetry current j µ in the 28, the supercurrent ψ µα in the 8 v , and the stress tensor itself, T µν -See Table 1. if v and w transform in the 8 s ). In addition to the Kronecker delta symbol, SO(8) admits another invariant tensor E aIA (see Appendix A for explicit expressions), which can be used to produce the 8 k from 8 i ⊗ 8 j whenever i, j, and k are distinct. In index free notation, we can represent this as a wedge product ∧ : w transform in the 8 c and 8 s , respectively). 5 Then, we can represent the operators of the stress tensor multiplet as S IJ , P AB , χ AI α , j IJ µ (or j AB µ or j ab µ ), ψ a µα , T µν . The scalars S and P are rank-two traceless symmetric tensors of 8 c and 8 s , respectively, j µ is an anti-symmetric tensor of any eight-dimensional representation, and χ obeys χ AI E AIa = 0 in order to select the 56 v representation from the product 8 s ⊗8 c = 56 v ⊕ 8 v . Including all the SO(8) R indices quickly becomes unwieldy, so instead we will use polarization vectors: we will denote vectors in 8 c by Y , those in 8 s by X, and those in 8 v by Z. Then we can define the operators (2.1) To implement the tracelessness of S IJ and P AB , we demand that Y · Y ≡ Y I Y I = 0 in the definition of S( x, Y ) and similarly for X in the definition of P ( x, X). 6 Likewise, to implement 4 These three representations are all equivalent due to SO(8) triality, and so while we can think of the 8 v as the vector representation while 8 c and 8 s are the two (real, inequivalent) spinor representations, this assignment is arbitrary. 5 The E aIA can be thought of as chiral SO(8) gamma matrices. The Clifford algebra implies By manipulating the Clifford algebra, one can derive other useful relations, for instance 6 Strictly speaking we should view S and P as functions of two distinct auxiliary fields; for instance, the condition E a AI χ AI ( x) = 0, we require X ∧ Y = 0. We can automatically satisfy this condition by choosing X = Y ∧ Z for some Z ∈ 8 v ; this is now a redundant parametrization as there exists a (unique up to normalization) vector Z Y for which Y ∧ Z Y = 0. Lastly, since the R-symmetry current j µ transforms in the adjoint representation 28 ∈ (8 c ⊗ 8 c ) a we can polarize it with two vectors Y 1 and Y 2 , but all expression must be antisymmetric in these two vectors. Alternatively we could polarize it with X 1 and X 2 or Z 1 and Z 2 , depending on whichever is most convenient.
After this long introduction on notation, we can write down how Poincaré supersymmetry generated by the supercharges Q αa relate the operators in the stress tensor multiplet: Here, δ α (Z) represents the action of Q αa Z a on the various operators and σ µ are the 3d gamma matrices, which can be taken to be just Pauli matrices. The supersymmetry variations of j µ , ψ muα , and T µν that were omitted from (2.2) will not be needed in this work.

Ward identities
To derive the relations superconformal symmetry imposes between the four-point functions of the stress tensor multiplet operators, it is enough to first determine the most general form of these four-point functions that is consistent with conformal symmetry, and then require that they be invariant under the Poincaré SUSY transformations in (2.2). (One does not gain any additional information by also imposing invariance under the superconformal generators S αa because invariance under S αa is guaranteed by invariance under Q αa and under the special conformal generators K µ .) For example, conformal symmetry implies that But we can always uniquely reproduce the full SO(8) structures by restricting to Y 1 = Y 2 and it is usually convenient to do so.
the SSSS and P P P P correlators take the form where the S i and the P i are functions of the conformal cross-ratios Note that not all the functions S i and P i are independent. Crossing symmetry implies the relations (2.5) Likewise, SSP P takes the form where the R i are also functions of U and V and we used the product • : 7 For other correlation functions (which are not needed in the rest of this paper), see Appendix B.
For the application presented in this paper, we only need to express the four-point func- 7 Using the identities in Footnote 5, we can derive

Variation
Correlators Used Correlators Obtained δ SSSχ SSSS SSχχ SSSj δ SSP χ SSχχ SP χχ SSP P SSP j δ SP P χ SP χχ SSP P P P χχ SP P j δ P P P χ P P χχ P P P P P P P j Table 2: Taking supersymmetric variations to compute correlators. By setting the variation in the first column to zero, we can use the correlators in the second column to compute the correlators in the third column.
tions SSP P and P P P P in terms of SSSS . These relations can be determined by substituting the general form of the four-point functions into the identities δ SSSχ = 0 , δ SSP χ = 0 , δ SP P χ = 0 , δ P P P χ = 0 . (2.7) In particular, from the first equation in (2.7), we determine SSχχ and SSSj in terms of SSSS , as well as relations on SSSS . Then, from the second line of (2.7), we determine SP χχ , SSP P , and SSP j . Then, from the third line of (2.7), we determine P P χχ and SP P j . Lastly, from the fourth line of (2.7), we determine P P P P and P P P j . See also Table 2.
In practice, plugging (2.3), (2.6), and the analogous equations in Appendix B into (2.7) is an onerous but straightforward task that can be greatly simplified using Mathematica.
Our results are as follows. From the first equation in (2.7), we can show that the S i obey the Ward identities along with other identities which can be derived using the crossing relations (2.5). It can be checked that these equations are equivalent to the Ward identities obtained in [36]. The expressions for the functions R i (U, V ), S i (U, V ), and P i (U, V ) that appear in (2.6) and (2.3) in the SSP P , SSSS , and P P P P correlators, respectively, are related as where the differential operators D R i (U, V, ∂ U , ∂ V ) and D P ij (U, V, ∂ U , ∂ V ) are rational functions U and V and have at most 2 and 4 derivatives, respectively, and are given explicitly in  Table 2 can also be written as differential operators acting on the S i ; their explicit expression can be found in the attached Mathematica notebook.

A check: superconformal blocks
A stringent check on the formulas (2.9) as defined in (B.7)-(B.9) is that they should map superconformal blocks to superconformal blocks. In particular, if we take the S i to correspond to a superconformal block, then these equations determine the corresponding superconformal blocks in the SSP P and P P P P correlators. The fact that these equations produce a finite linear combinations of conformal blocks is nontrivial.
As a simple example, the superconformal block corresponding to the s-channel exchange of the stress tensor multiplet in the SSSS correlator is [23] S i = 1 4 (−g 1,0 + g 3,2 ) 0 0 0 g 1,0 + g 2,1 g 1,0 − g 2,1 , (2.10) where g ∆, are the conformal blocks written in the normalization used in [23]. From (B.7) we find and (B.8)-(B.9) along with their crossed versions imply Superconformal blocks for other multiplets can be worked out in a similar way.
3 Integrated correlators on S 3 Having described the four-point function of the scalar and pseudo-scalar operators in the stress tensor multiplet of an N = 8 SCFT, let us now connect these quantities to the fourth derivatives of the S 3 partition function with respect to various mass parameters. Before delving into the details of these mass deformations, let us note that the formulas (2.3)-(2.6) also hold on a round S 3 , with the only modification that the quantity x ij should undergo the replacement everywhere. Here, r is the radius of the three-sphere, and the three-sphere is taken to have the metric (3. 2) The RHS of (3.1) is just the chordal distance between two points on S 3 . In particular, the replacement (3.1) leaves unchanged the conformally-invariant cross-ratios U and V defined in (2.4).

Three-parameter family of real mass deformations
We are interested in mass deformations on S 3 which preserve sufficient supersymmetry to compute the partition function using supersymmetric localization. This requires at minimum N = 2-preserving mass deformations. Viewed as an N = 2 SCFT, any N = 8 SCFT possesses an su(4) flavor symmetry generated by the subalgebra of so(8) R which commutes with the N = 2 R-symmetry u(1) R . In N = 2 SCFTs, real mass parameters are associated with conserved current multiplets, because they can be thought of as arising from giving supersymmetry-preserving expectation values to the scalars in the background vector multiplets that couple to the conserved current multiplets. In particular, the operators of an N = 2 conserved current multiplet generating a symmetry algebra g with hermitian generators T a are: a ∆ = 1 scalar J = J a T a , a ∆ = 2 pseudo-scalar K = K a T a , and the conserved current j µ = j a µ T a . If we normalize the flat space two-point functions at separated points as for some constant τ , then the real mass deformation on S 3 is given by [25] where m = m a T a is a Lie-algebra valued mass parameter. Here, 'tr' denotes a positivedefinite bilinear form on the Lie algebra, which can be thought of as the trace in a convenientlychosen representation of g. For us, we have the flavor symmetry algebra g = su(4), and we consider a basis of this algebra such that tr(T a T b ) = δ ab . For convenience, we will take 'tr' to be the trace in the fundamental (4) of su (4). Note that τ is related to the stress tensor where our normalization is such that c T = 1 for a free real scalar. For simplicity, let the radius of S 3 be set to r = 1 from now on.
Due to the su(4) symmetry, the S 3 free energy can be expanded in terms of su(4) Casimirs.
For instance, the first few terms in the expansion at small mass are (3.6) We can of course obtain the same information without the need to consider a completely general su(4) mass matrix m, and instead focus on a Cartan subalgebra. Let us order the T a such that the first three (a = 1, 2, 3) correspond to a Cartan subalgebra given explicitly by With tr(T a T b ) = δ ab , the expression (3.6) becomes a sum of polynomials in m a , a = 1, 2, 3 that are invariant under the action of the Weyl group: for any a = 1, 2, 3. We would like to perform a similar calculation in the n = 4 case, first ignoring the O(m 2 ) contributions in (3.4). (We will return to these contributions in Section 3.3.) Because our final goal is to determine the integrated four-point functions of the N = 8 operators S IJ and P AB , we must first relate S IJ and P AB to the N = 2 operators The operators J a , K a , j a µ all arise from the stress tensor multiplet of the N = 8 SCFT, in particular from certain components of the 35 c scalars, 35 s pseudo-scalars, and of the so(8) defined by the decompositions of the fundamental representations 9 the components of the stress tensor multiplet decompose as: with operators in the same su(4) representation belonging to the same N = 2 superconformal multiplet. Therefore, the N = 2 flavor current multiplet consists of those components of S IJ , χ α AI , P AB , and j µ ab that transform in the 15 of su(4) in the decomposition (3.11). We can be more concrete. Given a generator T of su(4) presented as a 4 × 4 hermitian traceless matrix as, for instance, the generators in (3.7), we can ask which linear combination of the S IJ and which linear combination of the P AB correspond to it. Let us first focus on implies that if in the fundamental representation of su(4), then the same generator acting in the 8 c irrep of so (8) can be taken to be equal tõ where ε ≡ iσ 2 . 10 Using the same ingredients, one can also construct a symmetric traceless 9 In our convention, the supercharges of the N = 8 theory transform in 8 v . This fixes our embedding of su(4) ⊕ u(1) ⊂ so(8) if we wish to preserve only N = 2 supersymmetry. The 8 c and 8 s must then decompose as indicated, or the decompositions may be flipped. We choose the convention in which the decompositions are as in (3.10). 10 We can check this claim as follows. Since T is hermitian, then A is an anti-symmetric real matrix and B is a symmetric real matrix, which implies that the generatorT is also hermitian. Then, if [T a , T b ] = if abc T c , matrix T 35c representing the generator T inside the 35 c : , so that T 35c indeed corresponds to states in the 15 of su (4). For the Cartan elements in (3.7), we have This implies that J a ∝ (T a 35c ) IJ S IJ are given by where the normalization constant N J is determined to be such that the normalization (3.3) is obeyed.
A similar procedure can be repeated to give K a , but we have to be careful that the su (4) generators written as 8 × 8 matrices in the 8 s irrep are consistent with the symbols E aIA defined in the previous section. One can check that this is indeed the case for our choice of E aIA , and that the su(4) generators T in (3.12) are also represented by theT in (3.13) in the 8 s representation. 11 Then, by analogy with (3.18), the three K a corresponding to (3.7) we can immediately infer that which can be used to check that theT obey the right commutation relations: where a, b = 1, 2, 3, for now on we set r = 1, and we define the integrated quantity

Integrated four point function
Let us now evaluate more explicitly the quantity I d ∆ A ,∆ B defined in (3.22). While we are mainly interested in the case d = 3 and ∆ A,B = 1 or 2, we nevertheless keep d, ∆ A , and ∆ B completely general in this section. Eq. (3.22) contains 4d integrals as written, but using conformal symmetry one can perform 4d − 2 12 of them, as follows. The first step is to notice 12 For d = 2, 3 these correspond to the (d+1)(d+2) 2 generators of the conformal group SO(d, 2), while for d ≥ 4, after we have used conformal transformation to set one point to the origin and the other to infinity, there is a nontrivial stability group SO(d − 2) so that the number of integrals we can perform is that the integral (3.22) is rotationally-invariant on S d , so one can rotate the point x 4 to any fixed point of our choosing x 4 = x 4 * , using We can then translate x 1 → x 1 + x 3 and x 2 → x 2 + x 3 , and write (3.26) as Note that the x 3 dependence is only in the prefactor now. We can then use the remaining rotational symmetry to set x 1 and x 2 to x 1 * = (r 1 , 0, . . . , 0) and x 2 * = (r 2 cos θ, r 2 sin θ, 0, . . . , 0), respectively: (3.28) Then we can change variables from (r 1 , r 2 , x 3 ) to (z 0 , r, z) defined through r 1 = 2/z 0 , r 2 = 2r/z 0 , x 3 = 2 z/z 0 , after which the integral takes the form where z = (z 0 , z) and G r B∂ (z, x) is the AdS bulk-to-boundary propagator defined in (C.1).
The quantity in square brackets in (3.29) can be written in terms of theD function described in Appendix C, so that we get Note that this formula is symmetric under interchanging ∆ A ↔ ∆ B , because theD functions obey the relationD aabb (U, V ) = U b−aD bbaa (U, V ) [38]. Combined with (3.21), the formula (3.30) allows for an explicit evaluation of the integrated four-point functions provided that we know the functions of U and V appearing in (2.3) and (2.6). We will determine these functions in the 1/c T expansion in the next section.  and similarly for the fermions, so that the free theory action is

Order m 2 terms
In this case we have and indeed from (3.5), and c T = 16. The mass deformation (3.4) implies that the scalars Z i have masses M i given by The real mass deformation on S 3 has both a linear and a quadratic term in M i , namely where is an su(4) singlet. Thus where c = a and c = b. We can then use [37] d 3 x d 3 y g( x) g( y)

38)
13 This expression can also be written as in a manifestly su(4)-invariant form.
as well as the correlators For completeness, let us note that in the free theory, we have R conn Combining this expression with (3.40), we find that This result is indeed correct and serves as a check of our formalism. Indeed, in a free theory, the mass-deformed S 3 free energy can be computed by directly evaluating the required Gaussian integrals. As an alternative, one can use the supersymmetric localization result of [24] that gives with the function (z) defined in (1.3) of [24]. It is straightforward to see that (3.44) implies (3.43) for any a, b = 1, 2, 3.

Order m 2 terms at strong coupling
In a strongly coupled CFT with a holographic dual, as will be the case we study in the next section, one also expects a term of order m 2 similar to (3.35) to supplement (3.4). One difference is that in a generic strongly coupled CFT there is no operator O S of dimension 1 so we can drop the contributions involving O S from (3.37). In addition, the correlators J c ( x)J c ( y) and J a ( x)J b ( y)J c ( z) are c T /16 times larger than in the free theory, so (3.40) becomes will not play any role in the next section.

From ABJM to the 11d S-matrix
We will now apply the machinery of the previous section to fix the coefficients B 4 i and B 6 i of the c , then using [39] we can deduce that the flat space four supergraviton scattering amplitude with momenta restricted to 4 dimensions is (up to an overall ∆ i -independent numerical factor) [14]: We will now derive the c −1 T , c  , and M SSSS (6) , respectively, in M SSSS to take the following form: where M p S are asymptotically degree p polynomials in s, t that we can write in the basis (2.3) as  For p = 1, [40] computed M 1 S as an infinite sum of poles, whose explicit formula we relegate to Appendix E. In the large s, t limit this amplitude is normalized as where recall that u = −s − t in the large s, t limit. The asymptotic form M 1 S,asymp of this AdS 4 Mellin amplitude was derived in [14] from the tree amplitude in 4D ungauged N = 8 supergravity [3,4,41]. M 1 S includes the contribution of the Mellin space conformal blocks for the stress tensor multiplet, which must be proportional to the OPE coefficient λ 2 Stress . On general grounds λ 2 Stress ∝ c −1 T , so in our normalization we have [14,40] For p > 1, M p S are maximum degree p polynomials in s, t, u. These terms were computed explicitly in [14] for p ≤ 10. For instance, for p = 4 we havẽ where the otherS 4 i are given by crossing (4.5). For M 6 S , we relegate the explicit terms to Appendix E. In the large s, t limit these amplitudes are normalized as where M 1 S,asymp was given in (4.6). Note that stu and stu(s 2 + t 2 + u 2 ) are the only crossing symmetric polynomials in s, t, u of their respective degree.
We can also write the Mellin amplitudes for P P P P and SSP P in the large c T expansion as (4.3), where the corresponding polynomial amplitudes M p P and M p R can be written in the same basis as (2.3) and (2.6): whereP p i are related under crossing as P p 2 (s, t) = P p 1 (8 − s − t, t) ,P p 3 (s, t) = P p 1 (t, s) , P p 5 (s, t) = P p 4 (8 − s − t, t) ,P p 6 (s, t) = P p 4 (t, s) .

(4.11)
In Section 2, we showed in position space that the components P i and R i of P P P P and SSP P are related to S i according to the differential operators D R i (U, V, ∂ U , ∂ V ) and D P ij (U, V, ∂ U , ∂ V ) (2.9), which are rational functions of U, V and degree 2 and 4 in ∂ U , ∂ V , respectively. From the definition (D.4) we can relate their Mellin amplitudes as where the hatted operators act onS i (s, t) as (4.13) Using these formulae and the explicit expressions for S p i for p = 1, 4, 6 in (E.1), (4.8), and (E.4), respectively, we can now derive P p i and R p i . For instance, for p = 4 we find where the otherP 4 i are given by crossing (4.11), while forR 4 i we find (4.15) The expressions forP p i andR p i for p = 1, 6 are more complicated and are relegated to Appendix E. The large s, t limit of the p = 1 supergravity term is  Just as we saw with the asymptotic M p S in (4.9), the asymptotic M p S and M p R take the form of a universal polarization factor, which can be read off from the supergravity term, multiplied by the unique crossing symmetric polynomial in s, t of the required degree. This is consistent with the flat space interpretation as an M-theory S-matrix with a universal polarization term multiplied by polynomials in s, t for each order, as shown in (1.1) and (1.2). Note that both M p P and M p R can be use in the flat space formula (4.1), as was originally discussed for M p S in [14]. The numerical factors in (4.17) relative to (4.9) are compensated by the α integral in (4.1) that depends on the dimensions of the external operators.

From Mellin to position space
To compute the integrated correlators, we need the position space expressions S p i (U, V ), P p i (U, V ), and R p I (U, V ) ofS p i (s, t),P p i (s, t), andR p I (s, t) for p = 4, 6. Since the latter expressions are polynomials in s, t, we can rewrite them as finite sums of the function D r 1 ,r 2 ,r 3 ,r 4 (U, V ) described in Appendix C using its Mellin transformD SSSS r 1 ,r 2 ,r 3 ,r 4 (s, t) (D.6), which is a polynomial in s, t after being multiplied by an appropriate power of U . For instance, for p = 4 we find where the other S 4 i and P 4 i are related by crossing (2.5) using (C.7). The analogous expressions for p = 6 are given in Appendix E. Note that the p = 1 Mellin amplitude is an infinite sum of poles in s, t and so would correspond to an infinite sum ofD's, which is why we do not consider this term in the rest of the paper.

Integrals of SSSS , P P P P , and SSP P
We now compute the S 3 integrals of the c T terms in SSSS , P P P P , and SSP P in position space, using the expressions derived in the previous section. Recall that in Section 3.2 we reduced the 12-dimensional integral over each 3-component x i to a 2-dimensional integral over r and θ, where U = 1 + r 2 − 2r cos θ and V = r 2 , and the measure was proportional to U −1D 1,1,1,1 (U, V ), U −2D 2,2,2,2 (U, V ), and U −2D 1,1,2,2 (U, V ) for SSSS , P P P P , and SSP P . The powers of U in the former expressions exactly cancel those in the latter, as given in Section 4.2, so that our integral is over a sum of pairs of D functions. We can then plug in the explicit expressions for theD functions using the algorithm in Appendix C, and then perform the integral numerically to high precision for each component S p i , P p i , and R p I . We find that the results for p = 4 are consistent with the following analytic expressions

Localization for F (m 1 , m 2 ) in ABJM
So far the discussion has applied to any N = 8 SCFT with a large c T expansion. We will now specify to ABJM N,k for k = 1, 2 using the all orders in 1/c T results for F (m 1 , m 2 ) for this theory.
For ABJM N,k , the mass deformed partition function has been computed from localization [29], and takes the form .

(4.23)
Using the Fermi gas technique [31,32], this quantity was computed to all orders in 1/N : where m ± ≡ m 1 ± m 2 , and the function A is given by with derivatives For k = 1, recall that that ABJM N,1 is a product between the free theory ABJM 1,1 and an interacting theory ABJM int N,1 . Recall from Section 3.3 that c free (4.28) Comparing (4.27) to (4.28), we find that the tree level terms of order c −n T (namely c −1 T , c .

(4.29)
This matches the value previously derived in [14] using the 1d sector of ABJM N,k , which was shown to recover the known coefficient of the R 4 term in 11d M-theory. This precise match is a very nontrivial check on our formalism. T , we note that such a coefficient does not appear in the localization results (4.27), so that plugging in the integrals (4.21) and (4.22) into (3.21) and cancelling overall factors we find the two constraints: 0 = −9.638755B 6 6 + 2.368067B 6 4 , 0 = −1.55479B 6 6 + 0.394677B 6 4 . (4.30) The matrix formed by these constraints has determinant 0.14161 = 0, so these constraints are linearly independent, which implies that (4.31) In the flat space limit, this implies that the D 4 R 4 interaction in 11d M-theory is absent, as was previously argued using purely string theory reasoning.

Discussion
The main result of this work is the derivation of the absence of the protected D 4 R 4 term in the 11d M-theory S-matrix, which had previously been derived using duality arguments between M-theory and string theory, by showing that the corresponding c by supersymmetry as well as two constraints coming from the two linearly independent quartic mass terms in the mass deformed S 3 free energy F (m a ), which can be computed to all orders in 1/c T using localization [29] and the Fermi gas formalism [31,32]. A nontrivial check on these constraints was the recovery of the R 4 term, which had previously been computed in [14] and There is one more protected term in the M-theory S-matrix: D 6 R 4 . This term was also computed using duality arguments and string theory, and corresponds to the c ABJM, F (b) has been computed as an N -dimensional integral using localization [42,43].
Unlike F (m a ), however, no all orders in 1/N (and thus 1/c T ) result is known yet for F (b). 14 It would be interesting to derive such a formula using the Fermi gas formalism, which would 14 Except for specific values of b [44]. then allow us to derive D 6 R 4 from ABJM using the methods in this paper.
The idea in this paper of deriving constraints on 4-point functions from the mass deformed sphere free energy has applications to theories in other dimensions and/or with less supersymmetry. In d = 3, the all orders in 1/c T formula for the ABJM S 3 free energy deformed by two masses can also be applied to N = 6 ABJM with gauge group U (N ) k × U (N ) −k for k > 2, which in the large N, k limit is dual to a Type IIA string theory background.
In d = 4, the S 4 free energy deformed by one mass was computed by localization [45] for N = 4 super-Yang-Mills (SYM), whose large N and large 't Hooft coupling limit has a dual in Type IIB string theory. In d = 5, the S 5 free energy deformed by one mass has also been computed [46][47][48] for various 5d SCFTs with holographic duals. In all these cases, we can use the mass deformed free energy to fix a CFT 4-point function to some order in the large N expansion, which could be used to derive the dual quantum gravity S-matrix to the same order.
Ultimately, since the mass deformed free energy is a protected quantity, we can expect that it can be used to derive only the protected terms in the corresponding S-matrix. To explore the unprotected terms, we will need to derive unprotected CFT data in these holographic theories. The only known method of this sort is the numerical conformal bootstrap, which has been applied to the d = 3, 4, 5 holographic theories mentioned above in [23,33,[48][49][50]. We hope that as the precision of the numerical bootstrap studies increases, it will eventually become feasible to derive the full quantum gravity S-matrix from CFT.

A The E invariants
We use the following E aIA symbols, written as 8 × 8 matrices for given a: where I is the 2 × 2 identity matrix and 0 is interpreted as a 2 × 2 matrix with vanishing entries. From these matrices, one can construct the so(8) gamma matrices as B Ward Identities B.1 Structures for χ and j µ We will first expand correlators containing χ in the allowed SO(8) R and conformally invariant structures, which are derived in [51]. Using the notation we normalize χ so that the two point function is given by: We expand the four point function as: along with:

(B.4)
For correlators involving the R-symmetry current j µ , spacetime structures were computed using the embedding space formalism. Since j µ is a conserved current there is actually only one allowed conformal structure; we however did not impose this condition. We could then test whether our final answers satisfied ∂ µ j µ = 0, serving as a non-trivial check on the Ward identities.
Our normalization of j µ is such that: We then expand the four point functions of scalars and currents as:

B.2 Ward Identities for Scalar Correlators
We will now give explicit expressions for the differential operators in (2.9). For the correlator SSP P , the three equations are For P P P P we will give expressions for P 1 and P 4 . The other expressions can be derived from this by applying the crossing relations (2.5).
The expressions (B.7)-(B.9) assume that the operators S( x, Y ) and P ( x, X) are normalized such that their two-point functions are (B. 10) This means that in the small U limit, the functions S 1 , P 1 , R 1 approach 1 as U → 0.

B.3 Ward Identities from SSSχ
By considering the supersymmetric variation δ SSSχ , we can compute SSχχ and SSSj in terms of SSSS . These expressions are first order differential operators and are relatively simple, so we will give them explicitly here as an examples. Other correlators can be found in the attached Mathematica file.
First we will give the SSχχ expressions: For SSSj we find that (B.12)

CD functions
The quartic Witten contact diagram is given by where z are the d + 1 bulk spacetime variables and G r i B∂ is the bulk-to-boundary propagator [52] for an operator of dimension r i . We can then define the conformally invariant function: which is in fact independent of d.
The simplestD r 1 ,r 2 ,r 3 ,r 4 (U, V ) is Φ =D 1,1,1,1 (U, V ), which is just a scalar one-loop box integral in d = 4 and can be written as where we define as usual and Φ has a recursion relation [53] .

D Mellin amplitudes
Holographic correlators take a simpler form in Mellin space. To find the Mellin transform of any 4-point function of the form A 1 A 2 B 1 B 2 of scalar operators with scaling dimensions we first define the conformally invariant function We then separate out the disconnected parts of each correlator, which for the correlators we consider take the form which has the inverse transformation For a 4-point function in a large N expansion, it is convenient to consider the auxiliary Mellin amplitude where s + t + u = 2∆ A + 2∆ B , and the Gamma functions automatically encode the pole contribution of all double-trace operators [54]. The two integration contours in (D.4) then include all poles of the Gamma functions on one side or the other of the contour.
As an example of the simplicity of holographic correlators in Mellin space, recall that tree level correlators are written in term ofD r 1 ,r 2 ,r 3 ,r 4 functions, which in position space are given by a complicated recursive algorithm in terms of Dilogarithm functions as described in Appendix C. In Mellin space, however, theseD r 1 ,r 2 ,r 3 ,r 4 contribute to M A 1 A 2 B 1 B 2 (s, t) as [35]: which for integer ∆ A , ∆ B , r i is a rational function of s, t, u. We can get polynomials in s, t, u by shifting s → s−2 max{∆ A , ∆ B }, which in position space corresponds toŪ max{∆ A ,∆ B } D r 1 ,r 2 ,r 3 ,r 4 .

E Supergravity and D 4 R 4 terms
The degree 1 Mellin amplitudes for SSSS , P P P P , and SSP P in the bases (4.4) and (4.10) have the following crossing-independent coefficients