M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture

We study various aspects of the M-theory uplift of the $A_{N-1}$ series of $(2,0)$ CFTs in 6d, which describe the worldvolume theory of $N$ M5 branes in flat space. We show how knowledge of OPE coefficients and scaling dimensions for this CFT can be directly translated into features of the momentum expansion of M-theory. In particular, we develop the expansion of the four-graviton S-matrix in M-theory via the flat space limit of four-point Mellin amplitudes. This includes correctly reproducing the known contribution of the $R^4$ term from 6d CFT data. Central to the calculation are the OPE coefficients for half-BPS operators not in the stress tensor multiplet, which we obtain for finite $N$ via the previously conjectured relation [arXiv:1404.1079] between the quantum ${\cal W}_N$ algebra and the $A_{N-1}$ $(2,0)$ CFT. We further explain how the $1/N$ expansion of ${\cal W}_N$ structure constants exhibits the structure of protected vertices in the M-theory action. Conversely, our results provide strong evidence for the chiral algebra conjecture.

The goal of this paper is to develop a quantitative study of M-theory by way of its holographic duality to six-dimensional conformal field theory (CFT) with maximal (2,0) supersymmetry, using modern results from the conformal bootstrap and techniques for computing correlation functions in large N CFTs.
The A N −1 (2,0) CFT, on which we will focus, has various descriptions (e.g. [2][3][4][5][6]). Perhaps the most profitable is its realization as the worldvolume theory of N M5 branes in flat space, whose gravitational backreaction generates an AdS 7 × S 4 solution of M-theory. AdS/CFT then provides the usual dictionary for computing various observables in the 1/N expansion [7][8][9]; indeed, the notion of a well-defined 1/N expansion was first made explicit by the existence of the bulk dual. Despite being a non-Lagrangian, non-gauge theory with somewhat On the other hand, given our utter lack of a complete description of M-theory, the bulk is not terribly useful for determining finite N aspects of the dual CFT. However, we can turn this problem around using the modern perspective of the conformal bootstrap, which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work.
An initial implementation of the numerical bootstrap to the (2,0) CFT was performed in [10], which led to the first predictions for finite N data for low-lying non-BPS operators that appear in the stress tensor operator product expansion (OPE). More relevant for us will be the remarkable analytic progress in the BPS sector. The half-BPS supermultiplets in interacting theories have bottom components S k with k = 2, 3, . . . 1 and conformal dimension ∆ k = 2k, which are traceless symmetric tensors of the so(5) R symmetry. The KK reduction on AdS 7 ×S 4 [11][12][13][14] identifies the S k (modulo mixing) with scalar fields φ k in AdS, of squared mass (mL AdS ) 2 = 2k(2k − 6), which uplift to admixtures of the 11d graviton and three-form potential with legs on S 4 . While ∆ k is independent of N , the OPE coefficients λ k 1 k 2 k 3 are not. In [1], it was conjectured that these OPE coefficients sit in one-to-one correspondence with the structure constants C k 1 k 2 k 3 of the well-studied two-dimensional W N chiral algebra, with the auspicious central charge assignment c = 4N 3 − 3N − 1. This algebra is freely generated by an infinite tower of conserved currents W k of spins s = 2, 3, . . . , N , which lie in 1 The k = 1 case only exists for the free theory.
correspondence with the half-BPS operators S k mentioned above.
The W N chiral algebra conjecture is powerful: it determines, in principle, an infinite number of OPE coefficients of the (2,0) CFT. Many of the W N structure constants, which are completely determined by the Jacobi identities, are also explicitly known. 2 This data is highly quantum from the M-theory perspective, as it is known in closed form for finite c and finite N , unlike the currently known analogous results for protected operator algebras in d = 3, 4 maximally-supersymmetric CFTs [17,18] (some of which, however, do admit finite-dimensional integral representations [19]). In [1], it was shown that the W N OPE coefficients with c ≈ 4N 3 in the large N limit correctly reproduce previous computations of tree-level three-point functions in the (2,0) CFT as computed from AdS 7 × S 4 [20,21].
Some further aspects of the conjecture were substantiated in [22] using localization and the W N chiral algebra of A N −1 Toda CFT. One aim of this paper is to test this chiral algebra conjecture beyond leading order in 1/N ; as we explain below, we find strong evidence, both perturbative and non-perturbative, that the conjecture is indeed correct.
Before explaining what exactly we will compute, let us set the target. Even putting aside the deeper non-perturbative aspects of M-theory, the expansion of the 11d four-point superamplitude, A 11 , is not well understood. The 11d amplitude takes the form [23] A 11 (p i ; ζ i ) = f (s, t)A 11 R,tree (p i ; ζ i ) .
(1.1) the non-loop terms, only the R 4 and D 6 R 4 terms are known from previous computations, as reviewed in Appendix F. 3 At loop-level, 1-and 2-loop amplitudes are known from 11d supergravity computations [25,26]. Beginning at D 8 R 4 , the vertices are no longer protected by supersymmetry and their coefficients are not known, although there exist conjectures in the literature [27]. It is of great interest to improve on this state of affairs -specifically, the outstanding problem of determining D 8 R 4 and beyond, and of unveiling the finite N spectrum of M-theory -by computing the CFT four-point functions S k S k S k S k , and uplifting them to M-theory. This would be a remarkable holographic window onto the perturbative structure of M-theory and, by compactification, type IIA string theory.
In this paper, we will articulate a concrete strategy for doing this. As a step toward the longer-term goal of D 8 R 4 , we will explicitly demonstrate this strategy by deriving the R 4 term in (1.3) from CFT, as recently done in a closely related context using AdS 4 ×S 7 and the ABJM CFT [28]. Let us summarize the idea. We first compute S k S k S k S k in Mellin space in the 1/c expansion by solving the 6d superconformal Ward identity and using independent CFT data to fix any free parameters in the solution. This fixing relies crucially on input from W N to fix half-BPS structure constants. We then use the flat space limit formula for Mellin amplitudes [29] to relate this correlator at a given order in 1/c to terms in the 11 1 expansion of A 11 , using the holographic relation where L S 4 = L AdS /2. 4 The direct relation of the 1/c expansion to the 11 expansion follows from dimensional analysis in the reduction on AdS 7 × S 4 , and the absence of a dimensionless coupling in M-theory. One novelty of the 6d case is that (for reasons explained below) in order to fix the parameters necessary to reproduce the R 4 coefficient from presently known (2,0) CFT data, we will need to study the k = 3 correlator, as opposed to the stress tensor multiplet correlator (k = 2). Along the way, we will explain how to uplift S k S k S k S k to 11d for arbitrary k.
In Section 2, we review the basic features of the (2,0) CFT and the implications of superconformal symmetry on the structure of four-point functions of the half-BPS superconformal primaries S k . We then recall the conjectured relation between the W N algebra and 3 The precise tensor appearing at R 4 is t 8 t 8 R 4 , plus 11 terms that do not contribute to the four-graviton amplitude. Further details about R 4 and its superpartners may be found in e.g. [24]. 4 In defining 11 , we use the conventions of [24]. The relation between L AdS and L S 4 is believed to hold to all orders in 11 [30].
A N −1 (2,0) CFT data. Using properties of W N , we show that all half-BPS OPE coefficients λ k 1 k 2 k 3 admit a 1/c expansion of the form where the F i (c) have an expansion in non-negative integer powers of 1/c. This structure is precisely what one expects from M-theory: in particular, it is consistent with the only protected 11d vertices being R, R 4 and D 6 R 4 (hence the subscripts), with the F i (c) encoding bulk loops in the presence of these terms. The fact that λ 2 k 1 k 2 k 3 obeys the form (1.5), for any k i , is strong all-orders evidence for the validity of the identification of W N with central charge c = 4N 3 − 3N − 1 as the chiral algebra of the (2,0) CFT. Conversely, this may be viewed as suggestive evidence of the absence of 10-and 12-derivative terms in 11d (∼ D 2 R 4 and D 4 R 4 + superpartners). 5 In Section 3, we study the four-point functions S k S k S k S k . We work with the corresponding Mellin amplitudes, which we denote M k . After writing their general form, we explore the space of solutions to the 6d superconformal Ward identity, focusing especially (but not exclusively) on k = 2, 3. The solutions are distinguished by whether they are meromorphic or polynomial, and are organized according to their degree in the limit of s, t → ∞.
In Section 4, we give a physical analysis of the solution space in Section 3 and explain how to uplift to M-theory. We first show how to extract the 11d flat-space amplitude A 11 from the 1/c expansion of 6d Mellin amplitudes M k at large s, t, for any k. This involves an adaptation of Penedones' formula to the case of arbitrary KK modes. (See (4.1), (4.3).) A nice feature of this procedure is that, in addition to producing the function f (s, t), the overall kinematic factor K of A 11 can be seen to follow quite directly from the flat space limit of the 6d superconformal Ward identity itself. (See (4.5).) Moreover, the same factor Θ flat 4 (s, t; σ, τ ) appears in the 4d, N = 4 superconformal Ward identity. Therefore, the flat space limit of 4d N = 4 SYM four-point functions implies that type IIB string amplitudes are proportional to the universal K factor to all loop orders. This has sometimes been indirectly argued on general grounds (e.g. [31]), and K is known to appear in type II string theory through three-loop order [32,33]; here we give a rigorous derivation of its appearance to all orders in type IIB.
With this understanding, we explain how the coefficients of the Mellin amplitudes are directly related to CFT data, namely, OPE coefficients and scaling dimensions. The main physical point is that the degree of the solutions is correlated with the order in 1/c at which they first appear in CFT; in particular, a degree-p solution M (p) k has scaling c −(2p+7)/9 to leading order in 1/c. This is shown to follow from the flat-space limit and the absence of a dimensionless coupling in M-theory. This allows us to explain, physically, some features of the Mellin amplitudes found in Section 3. The result may be viewed as an M-theory version of previous arguments relating coefficients of solutions of crossing to powers of the higher spin gap in large N CFTs [34,35]. A related perspective on this c-scaling is given in terms of the dimensional reduction of M-theory on AdS 7 × S 4 . Together with previous knowledge of the M-theory amplitude through 14-derivative order, we can rule out candidate polynomial solutions of crossing symmetry at O(c −17/9 ) and O(c −19/9 ). (A similar argument was made in [28].) This last statement, which uses general features of KK reduction on AdS×M, applies to any CFT with an M-theory dual of this form. We also present a sharp signature of the four-point functions of putative large c CFTs with a hierarchy between the AdS and KK scales, L AdS L M .
In Section 5, we put everything together to develop the precise dictionary between Mtheory and (2,0) CFT. First, we derive the R 4 coefficient via the k = 3 four-point function at ). This is possible because the k = 3 amplitude at O(c −5/3 ) happens to be determined by one free parameter, which we can take to be the OPE coefficient λ 334 . This is in turn fixed by W N . The result perfectly matches the M-theoretic prediction, We note that the O(c −5/3 ) term in λ 334 is extracted from a c −1 N −2 term in W N ; in particular, one does not need to know the sub-leading O(N ) term in c, which descends from the 11d R 4 term in the first place [24]. Turning next to higher order terms ∼ D 2m R 4 , W N does not provide enough constraints on the k = 3 amplitude to completely fix the solutions.
(This is due to the existence of pure polynomial solutions.) Instead, our strategy will be to relate the higher degree Mellin amplitude coefficients to CFT data that is not determined by W N -namely, anomalous dimensions of unprotected double-trace operators and OPE coefficients of protected operators that do not live in W N . Thus, future constraints on this data can be translated into constraints on M-theory amplitudes. Because the number of Mellin amplitudes grows with k, we will focus on the lowest case k = 2. The output of this procedure is given in Table 2.
In Section 6, we conclude with some future directions.
Several Appendices complement the main text. These include technical details on Mellin amplitudes, superconformal Ward identities, superconformal blocks, the OPE of two stress tensor multiplet scalars S 2 , and a review of the derivation of f R 4 (s, t) and f D 6 R 4 (s, t) from the uplift of type IIA string theory. . We can view these operators as the rank-k symmetric traceless products of the 5, so we can denote them as traceless symmetric tensors S I 1 ...I k (x) of so(5), where It is convenient and conventional to contract with an auxiliary polarization vector Y I that is constrained to be null, Y i · Y i = 0, so that We will often perform explicit computations involving the two lowest multiplets. The k = 2 multiplet, D [20], is the stress tensor multiplet, whose bottom component is a scalar with ∆ = 4 in the 14 of so (5), and appears in all local (2, 0) SCFTs. The next lowest half-BPS multiplet, D [30], has a scalar bottom component with ∆ = 6 in the 30 of so(5).
The stress tensor itself T µν has a two-point function where I µνρσ (x) is a fixed tensor structure whose form can be found in [36]. The coefficient c T is proportional to the unique c-type central charge appearing in the (2,0) conformal anomaly [37]. A free (2,0) tensor multiplet may be taken to have c = 1, while a (2, 0) theory labeled by Lie algebra g has central charge c(g) = 4d g h ∨ g + r g , where d g , h ∨ g , and r g are the dimension, dual Coxeter number, and rank of g, respectively. For the A N −1 series of interest here, These results for c were first motivated by R-symmetry anomaly and holographic computations [24,38,39], conjectured in [10], and proven in [22]. (See also [40][41][42][43][44][45] for some recent related results about the c-type anomaly in 6d SCFT.)

Half-BPS Four-Point Functions
Conformal symmetry and so(5) symmetry implies that the four point function of S k (x, Y ) takes the form where U and V are conformally-invariant cross ratios and σ and τ are so(5) invariants formed out of the polarizations: is a degree k polynomial in each Y i separately, the quantity G k (U, V ; σ, τ ) is a degree k polynomial in σ and τ .
So far, we have imposed the bosonic subgroups of the osp(8 * |4) algebra. The constraints from the fermionic subgroups are captured by the superconformal Ward identities [46], which can be expressed as differential operators on all four arguments of G k (U, V ; σ, τ ) whose explicit form we review in Appendix A. For 6d (2, 0) SCFTs, there are two ways of satisfying these constraints.
In the first method, which can actually be used in any dimension, we decompose G k (U, V ; σ, τ ) into superconformal blocks G M by taking the OPE twice in (2.5), which yields where M k runs over all osp(8 * |4) multiplets appearing in the S k × S k OPE, and λ 2 k,M k 6 is the OPE coefficient squared for each such supermultiplet M k . The selection rules for the OPE of half-BPS multiplets have been worked out in [47,48] and were summarized for general k in [10]. The supermultiplets that appear in a four point function of identical S k 's are where the spins j refer to rank-j traceless symmetric irreps of so (6) (2.9) The A multiplets that appear here are unprotected, while the rest are annihilated by some fraction of supercharges, and so have fixed dimension. The D[k0] are the half-BPS multiplets whose bottom component we called S k , and k of these multiplets appear in S k × S k . The lowest such multiplet is always the stress tensor multiplet D [20], whose OPE coefficient is can thus be written as a linear combination of the conformal blocks G ∆ ,j 7 corresponding to 7 We normalize our conformal blocks as the conformal primaries in M k as of maximal degree k. For general k, these can be computed using Appendix D in [49], and we list the explicit forms for k = 2, 3 in Appendix A. The A M k ab∆ j (∆, j) are rational functions of ∆ and j. For k = 2, we work out some of these coefficients in Appendix E. 8 The second way of imposing the constraints from the superconformal Ward identity is special to 6d (2, 0) SCFTs. We can satisfy the Ward identities by writing G k as where Υ is a complicated differential operator whose explicit form is given in Appendix A, and F k (U, V ; σ, τ ) and H k (U, V ; σ, τ ) are degree k and k − 2 in σ, τ , respectively. These functions are defined so that only F k (U, V ; σ, τ ) contributes to the 2d chiral algebra fourpoint function, which we will describe in the next subsection.

The W N Chiral Algebra and AdS 7 × S 4
It was conjectured in [1] that in every (2,0) CFT labeled by g, the OPE data of half-BPS operators and a subset of their protected composites -specifically, among those appearing in S k × S k , the D[2n 0] and B[2n 0] multiplets -are determined by the dynamics of a 2d chiral algebra, W g . In the case g = A N −1 , the algebra is the well-known W N algebra. This algebra is generated by a finite tower of holomorphic currents, W k (z), of integer spins k = 2, 3, . . . , N , and depends on a single free parameter c 2d , the central charge. The conjecture stipulates the so-called "quantum algebra", to be distinguished from the "classical algebra" which is defined as the c → ∞ limit of W ∞ [µ] with fixed µ [50,51]. Henceforth we will refer to the 8 These superconformal blocks have also been derived in different bases in [10,46,47].
central charge simply as c, and write the W N OPE as We employ a unit normalization, C ij0 = δ ij .
Explicit checks of the conjecture require a map between bases of the 6d half-BPS chiral ring and the W N generators. In the so-called "quadratic basis" of W N , the structure constants are conjecturally completely determined [15]. A more physically natural basis for many purposes is the "Virasoro primary basis", in which the currents W k obey the Virasoro primary condition L n>0 |W k = 0. 9 Many low-lying structure constants in the primary basis may be found in [15,16]. In the 1/c expansion, there is a natural map between the single-trace half-BPS operators S k , and the currents W k in the Virasoro primary basis: At subleading orders in 1/c, the 6d spectrum undergoes mixing between single-trace and multi-trace operators: for instance, S 4 mixes with the D [40] projection of :S 2 S 2 :. However, for k = 2, 3, no mixing is possible, due to the absence of k = 1 operators in both the W N algebra and the (2,0) spectrum.
While fundamentally a statement about the spectrum and OPE coefficients, the map to W N has especially powerful implications for 6d four-point functions. As shown in [1], one obtains a holomorphic function of z from the 6d four-point function G k (U, V ; σ, τ ) by twisting σ = z −2 and τ = (1 − z −1 ) 2 . Given the map (2.15), the claim of [1] is that precisely the 6d uplift of the 2d correlator F k (z). To explicitly relate W N structure constants C ijk to 6d OPE coefficients λ ijk , we must determine how the chiral algebra twist relates the 6d blocks to 2d blocks. As derived in Appendix B, where the first term represents the unit operator, and are the SL(2, R) global conformal blocks. The crossing equations for F k (z) imply that only 2k 3 of the infinitely many λ 2 k,M k are independent [53,54]. Of particular use in the following will be the OPE coefficients λ 2 k,D[2n 0] for k = 2, 3. For k = 2, this is nonzero only for n = 1 (cf. (2.10)), and the corresponding W N OPE coefficient is C 222 ∝ 1/c, i.e. the cubic coupling of the stress tensor. On the other hand, k ≥ 3 contain non-trivial information to all orders in 1/c. The 6d squared OPE coefficient [40] is determined by (C 334 ) 2 of W N , with relative coefficient determined by the n = 2 term in the first line of (2.17): In unit normalization, (C 334 ) 2 takes the rational form [16,[55][56][57] ( Let us expand λ 2 3,D [40] in the large c limit, using (2.20) with the necessary identification c = 4N 3 − 3N − 1: Starrting at c −5/3 , all powers of c −1/3 are generated. Moreover, in the primary basis, all W N structure constants C k 1 k 2 k 3 with k i > 2 can be written as rational functions of c and C 334 to some fractional power [15,16,58]. This follows from the structure of the Jacobi identity, as recently proven in full generality in [58]. 10 For example, in our unit normalization, The uplift to 6d then implies that all half-BPS OPE coefficients not involving S 2 , the stress- 123 to a cubic scalar AdS vertex g 123 φ 1 φ 2 φ 3 in the quantum effective action in AdS d+1 , then accounting for factors of 11 , . (2.23) The form of (2.23) follows from dimensional analysis, (L AdS / 11 ) 9 ∝ c, combined with the fact that the reduction on M can only produce powers of L AdS , not 11 . 11 To relate g 123 to the dual CFT OPE coefficient λ k 1 k 2 k 3 , one multiplies (2.23) by a function with an infinite expansion in non-negative integer powers of 1/c ∼ G N , which accounts for bulk loops.
Specializing to the case M = S 4 , we equate the result with λ 2 k 1 k 2 k 3 , thus inferring that g (2n) 123 = 0 for n = 4, 7, 10, . . .. The result is consistent with the minimal form g (2n) 123 = 0 for n = 4, 7. As explained in the introduction, the latter is precisely compatible with the known structure of the M-theory action, thus furnishing compelling evidence for the W N chiral algebra conjecture to all orders in 1/c. Conversely, given the 1/c expansion of λ 2 in (1.5), we have given a holographic argument for the structure of protected vertices in M-theory, in particular, the absence of 10-and 12-derivative terms. 10 Unusual normalization conventions, such as the one in [15,58], can lead to especially simple-looking structure constants. In any convention, 1/c power counting ensures that the normalized OPE coefficients scale as ∼ 1/ √ c to leading order. We also note that [58] proved the uniqueness of the W N algebra, given the list of spectrum-generating currents. 11 Later, we will note an interesting consequence of relaxing the assumption that L AdS ≈ L M .

Mellin Amplitude
We will find it useful to express our four-point function in Mellin space. For this purpose it is useful to separate out the disconnected piece The Mellin transform M k (s, t; σ, τ ) of the connected correlator G conn ≡ G − G disc is then: where the Mellin space variables s, t, and u satisfy the constraint s + t + u = 8k. The two integration contours run parallel to the imaginary axis, such that all poles of the Gamma functions are to one side of the contour.
We can similarly define the Mellin transform M k (s, t; σ, τ ) of the reduced correlator where Θ is a complicated difference operator whose explicit form is given in Appendix A, and should be thought of as the Mellin space version of the differential operator Υ in position space. 12 The Mellin presentation of the superconformal Ward identity will make the physics of the flat space limit especially transparent.

Holographic four-point functions at tree level
Let us now discuss the four-point correlator of the operators S k in the A N −1 (2, 0) theory, with special emphasis on the cases k = 2, 3. We will compute the Mellin amplitudes allowed by the superconformal Ward identity with the constraint that no triple poles appear in the inverse Mellin transform (2.25), which restricts us to tree-level amplitudes [60,61]. We will organize the solutions we find according to their maximal degree in the large s, t, u limit. We will find where the t-and u-channel exchange diagrams are related to the s-channel by crossing In Mellin space, the contact diagrams corresponding to vertices dressed with n derivatives are order-n polynomials in s, t, u. The s-channel exchange for a bulk field φ dual to a boundary conformal primary operator O of dimension ∆ , traceless symmetric spin j , and This is required by the 1/N expansion [60,61].
In our case, the scalar operators S k are dual to the bulk scalars φ k , which descend from linear combinations of the 11d graviton and three-form along S 4 . These are the only elementary scalars in AdS 7 . As shown in 13 According to the standard GKPW dictionary, we expect an exchange diagram for each conformal primary operator in these multiplets. 14 The meromorphic part multiplet dimension spin so (5)   of the s-channel exchange diagram M k,s-exchange for a multiplet D[n0] can then be written (up to overall normalization) as a linear combination of the contributions from its conformal 13 The irreps D[n0] appearing here may be realized, as operators, both by single-trace superconformal primaries S n , as well as multi-trace superconformal primaries. Only the former are elementary fields in AdS, and thus only these are the ingredients of Witten diagrams. 14   nn∆ j can be fixed using the superconformal Ward identity in Appendix A, and are the coefficients of the conformal bock contributions to the superconformal block as defined in (2.11). For example, for k = 2, 3, we give the branching  We will now plug the ansatz (3.6) into the superconformal Ward identities. This will further constrain the solutions, which we organize by the maximal degree p of the polynomial term. The solutions can be divided into those that have a meromorphic term and those that do not.

Meromorphic solutions
We first discuss those solutions that contain a meromorphic term that comes with a polynomial term of maximal degree p. By checking many cases we find that the most general ansatz is where a 2.2; we will explain this further when we relate these solutions to CFT data in Section 4.2.
As we now show, the superconformal Ward identity relates the meromorphic terms to the polynomial piece M (p) k,poly . We begin with p = 1. As we will explain in the next section, these amplitudes descend from the 11d supergravity term, so we denote them by where the polynomial terms are given in Appendix D. A more compact expression for these Mellin amplitudes is given by the reduced form M k defined by (2.27), which for k = 2 is and for k = 3 is . (3.11) Up to an overall normalization, these expressions match those in [59].

Polynomial solutions
We can also find purely polynomial solutions to the Ward identities. Note that the degree of these purely polynomial terms can in general be the same as that of the polynomial amplitudes M (p) k,poly that come with the meromorphic solutions. If we define N k (p) as the number of solutions of maximal degree p for a given k, then the purely polynomial terms M k,pure-poly in M k,tree 15 take the form where a i and b i are non-negative integers. The sum of the number of partitions of all positive integers y ≤ x into 2 and 3 is given by 3,poly do not take such a simple form for all p, but we write the cases p = 5, 6 in Appendix D. In this case we find by checking many solutions that 16 N 3 (p) = n(p − 4) + n(p − 5) + n(p − 6) − 1 . (3.17) For higher k we found no simple pattern for the number of polynomial solutions, but they can be easily computed case-by-case. We do, however, note the following feature: at p = 4, for even k only, there is a unique polynomial solution in addition to the unique meromorphic solution (3.7).

Uplifting to M-theory
Having established the space of solutions to the superconformal Ward identity, we turn to their physical interpretation in the (2,0) CFT and the uplift to M-theory. This relies on the flat space limit of S k S k S k S k , which we perform using an adaptation of Penedones' formula [29]. Our goals are twofold: first, to give a precise dictionary for how to recover 11d amplitudes in the 11 1 expansion from these four-point functions; and second, to show on general grounds how the functional form of the 11d amplitude is reflected in, and may be inferred from, the properties of the 6d CFT correlators. In the next section we apply this technique to derive the R 4 contribution to the 11d graviton amplitude. where (L/2) 4 V 4 = π 2 6 L 4 is the S 4 volume (required by dimensional analysis). We interpret A 11 k as the 11d flat spacetime amplitude of four supergravitons with momenta k µ i restricted to an R 7 AdS 7 | L→∞ , integrated against four supergraviton Kaluza-Klein mode wave functions on S 4 and contracted with so(5) polarization vectors Y i . We can write A 11 k explicitly as

Flat space limit for arbitrary KK modes
2) The ingredients are as follows: A 11 αβγδ (s, t) is an invariant tensor in the supergraviton polarizations α, β, γ, δ; Ψ α I i1 ...I ik (x) is the normalized KK mode wave function for the particle i on a unit S 4 ; and Y are the scalar S 4 harmonics.
To actually extract A 11 k (s, t; σ, τ ) from the integral (4.2) for arbitrary k is not straightforward, nor is it necessary. On general grounds, the uplift to 11d must be proportional to the four-supergraviton amplitude, for any k. This follows from the fact that all operators where P k−2 (σ, τ ) is a crossing-symmetric polynomial of degree-(k−2) in (σ, τ ), and A 11 ⊥ (s, t; σ, τ ) is defined as The orthogonal kinematics Y i · p i = 0 follows from taking the flat space limit of amplitudes in a direct product spacetime like AdS 7 × S 4 . Note that while A 11 (p i ; Y i ) depends in general on the individual momenta, A 11 ⊥ (s, t; σ, τ ) only depends on Mandelstam invariants, as we demonstrate momentarily. where ζ i and p i are the polarization vector and momenta of the i'th 11d graviton, respectively. In general, K is not just a function of (s, t). But in 11d kinematics where ζ i · p j = 0 for all (i, j), one finds Therefore, the universal factor Θ flat 4 (s, t; σ, τ ) that is required by the superconformal Ward identity accounts for the overall momentum/polarization-dependent factor K of the 11d graviton amplitude. 17 It is satisfying that 6d superconformal symmetry generates the K factor in the uplift for any k. As noted in the introduction, the fact that Θ 4 also appears in the 4d N = 4 superconformal Ward identities, combined with (4.8), implies that type IIB string amplitudes are proportional to K to all loop orders. 18 Returning to (4.3), we note that in a ratio of amplitudes at different orders in 11 , P k−2 (σ, τ ) will cancel. Therefore, given the form of (1.3), we may express the tree-level terms f D 2m R 4 (s, t) in terms of the basis of polynomial and meromorphic solutions of fixed degree p = 4 + m: k,poly (s, t; σ, τ ) M k,sugra (s, t; σ, τ ) where the numerical prefactor comes from the β-integral in (4.1). This is one of our main formulas.
An important point is that the sum i in (4.9) is defined to run only over all Mellin amplitudes whose (σ, τ )-dependence is given by P k−2 (σ, τ ). This polynomial is not a function of 11 , so it can be computed once and for all from, say, taking the flat space limit of the supergravity term M k,sugra (s, t; σ, τ ). This places strong constraints on the 6d Mellin amplitudes, not all of which have this factorized form. For instance, for k = 3, the polynomial is unique up to rescaling for some constant b, which can be determined from supergravity [59] to be b = 4. For a given k, there are n(k − 2) independent orbits of crossing, where n(x) was introduced in (3.15), one linear combination of which is picked out by M-theory. 19 We can state the general criterion for which solutions can contribute to f D 2m R 4 (s, t) in terms of the M lmn (cf. (A.9)): they must be crossing-symmetric at large s, t. This discussion makes clear that 11d superPoincare invariance is more constraining that the flat space limit of the 6d superconformal Ward identity.

Explaining the momentum expansion of Mellin amplitudes
Now that we can perform the flat space limit, we return to the previous section's mathematical solutions to the superconformal Ward identities, and interpret them as solutions of the actual (2,0) CFT. The main point is that the degree of the Mellin amplitudes is correlated with powers of 1/c ∼ 9 11 . This is visible from the flat space limit (cf. (4.9)), which determines the corresponding power of momenta, and hence of 11 , in the corresponding 11d S-matrix element. In particular, the Mellin amplitude coefficients B

Dimensional reduction and M-theory constraints on crossing
The scaling (4.13) may also be seen using dimensional reduction of M-theory. 21 On general grounds, the quartic part of the effective Lagrangian in AdS 7 is constrained to take the form where we have used D 2m R 4 to denote all (8 + 2m)-derivative terms in the 7d action, such as To recover the exact coefficient of each amplitude from dimensional reduction would require knowledge of the full supersymmetric completion of the 11d higher derivative terms, which is unknown aside from the R 4 term. The action analysis may be seen as a book-keeping device for the c-scaling of on-shell amplitudes, which are what we actually compute. 22 New tensor structures can also appear after the dimensional reduction that are not present in 11d. For instance, (R µν R µν ) 3 in 11d can generate (R µν R µν ) 2 in AdS, which is different from the t 8 t 8 R 4 and 11 11 R 4 tensors that appear in 11d. rules out some of the low-lying solutions of [66]. 23

M-Theory from CFT Data
With all pieces in place, we now relate M-theory in the 11 1 expansion to CFT data.
We first derive R 4 from CFT. We then lay the groundwork for deriving the tree level higher derivative terms ∼ D 2m R 4 from CFT.
For the case k = 3, the W N algebra gives us a single nontrivial constraint from the OPE coefficient λ 3,D [40] , which is enough to fix M 3,mero . We gave the R 4 contribution to A 11 in (1.6), whose derivation we review in Appendix F. Plugging it into our formula (4.9) with m = 0 and k = 3, we obtain the prediction This precisely matches the OPE coefficients (2.10) and (2.21) as derived from the CFT with help from the W N chiral algebra conjecture. Thus, we have derived the R 4 coefficient from 6d CFT data. We point out that λ 2 3,D [40] c −5/3 ∼ c −1 N −2 , and thus is, fortunately, independent of the O(N ) term in c whose 11d origin is R 4 itself [24]. 23 It has been a long-standing goal in holography to find explicit examples of AdS ×M compactifications with a parametric hierarchy L AdS L M . Such CFTs would have an especially sparse spectrum of light, lowspin single-trace operators. In this case, dimensional reduction will generate positive powers of 11 /L M ≡ c −1/9 M c −1/9 . The quartic effective action in AdS will again take the form (4.15), but where f i (c) → f i (c M ). Relating this to solutions to CFT crossing gives a new diagnostic, using CFT four-point functions, of whether a large c CFT has an M-theory dual with L M L AdS : at a given order in 1/c, the only polynomial solutions to crossing are those of maximal degree p = p max . It would be interesting to use this in an effort to bootstrap the existence of such CFTs.

Higher derivatives from
For higher order terms ∼ D 2m R 4 , W N does not provide enough constraints on the k = 3 amplitude to completely fix the solutions, due to the existence of pure polynomial solutions.
Instead, our strategy will be to relate the higher-degree Mellin amplitude coefficients B (p,i) to CFT data. We will focus on the lowest case k = 2, in which case the higher-derivative Mellin amplitudes are the purely polynomial amplitudes M (p,i) 2,poly of degree p, which are defined using the Mellin space Ward identity (2.27) and the reduced Mellin amplitudes M (p,i) 2,poly in (3.14). We will restrict to p ≤ 10 where we can extract unambiguous information from tree level Mellin amplitudes, without contamination from loop-level data. 24 To extract CFT data from the purely polynomial Mellin amplitudes M for p = 6, 7, 8, 9, 10 (where i = 1 except for p = 10 where i = 1, 2), we will use the algorithm developed for extracting CFT data from Mellin amplitudes in 3d [68].
We use the following normalization for the conformal blocks in the lightcone limit U → 0, This calculation is very similar to that of [28,68], so we will only briefly sketch the derivation.
The amplitudes M (p,i) 2 will contribute to the anomalous dimension of the infinite tower of unprotected double-trace conformal primary operators As discussed in [34,65,66], a purely polynomial Mellin amplitude of maximal degree p, which corresponds to a flat space vertex with 2p derivatives, contributes to the doubletrace operators with spin j ≤ p − 4. We will now show how to fix the N (p) − N (p − 1) coefficients B 2,(p,i) , indexed by i, of each degree p tree level term M (p,i) 2,poly defined by acting with (A.10) on (3.14) by extracting at least N 2 (p) − N 2 (p − 1) different pieces of CFT data from these amplitudes. We will use the OPE coefficients squared a M of the protected multiplets M ∈ {D[04], B[02] j } from (2.8) that are not fixed by W N , as well as the scaling dimension of the lowest twist long multiplet with spin j. The supergravity contribution to 24 The 2-loop 11d amplitude first appears at p = 10, which makes it impossible to fix M (p,i) k for p > 11 purely in terms of tree level data. For p = 10, while the term that scales like c −3 will receive contributions from loop amplitudes, there is also a c −3 log c term that is fixed by the logarithmic divergence of the 2-loop amplitude in 11d supergravity; this should be fixable using tree level CFT data and the techniques of [60,67]. these quantities (∼ c −1 ) was computed in [47,69]. The higher derivative Mellin amplitudes M (p,i) k,poly discussed above will contribute starting at order c − 7+2p 9 , and then will generically include all subleading powers of c −2/9 ∼ 2 11 (see (4.15)). Let's define the quantity γ to the anomalous dimension γ A[00] j+8,j of the leading twist operators [S 2 S 2 ] 0,j . We focus only on leading twist for simplicity, because higher twists are degenerate. We will use the conformal primary (j + 12, j) [40] , because it is the only conformal primary in that R-symmetry channel, so we do not have to worry about mixing with other conformal primaries. To extract these we will need the product of the mean field theory (MFT) OPE coefficient squared a MFT A[00] j+8,j and the coefficient A 22 j+12 j (j + 12, j), which as shown in [47] in our conventions are 22 j+12 j (j + 12, j)a MFT A[00] j+8,j = (j + 1)(j + 2)(j + 9)(j + 10)(j + 5)!(j + 6)! 360(9 + 2j)! . (5.5) Using these quantities and following the algorithm in [28], we find the results listed in the last four rows of Table 2.
For the protected OPE coefficients, we only need to worry about mixing with multiplets that are not in the chiral algebra, because those in the chiral algebra do not receive corrections beyond supergravity. For a D[04] , we can see from the tables of supermultiplets in Appendix E that its superconformal primary does not appear in any other supermultiplets, so we can easily extract its OPE coefficient. For a B[02] j , the superconformal primary now also appears in D[04], so for simplicity we will extract its OPE coefficient from the conformal primary (j + 11, j + 1) [40] . Using the superblock coefficients computed in Appendix E, we find the results listed in the first four rows of Table 2.

Conclusion
This paper developed an explicit program for how to extract the perturbative expansion of the 11d flat space S-matrix from the OPE data of the 6d A N −1 (2, 0) CFT. We mostly focused structure constants in the 1/c expansion were also shown to exhibit the absence of 10-and 12-derivative terms in the 11d effective action. Altogether, the aforementioned matches to 11d physics provide strong support for the chiral algebra conjecture of [1]. Moreover, we provided an explicit roadmap for how the first several low-lying higher-derivative tree level terms ∼ D 2m R 4 in the 11d S-matrix, including unknown terms beyond D 6 R 4 , can be directly recovered from 6d CFT data that is as yet unknown.
Our results give a new motivation for computing 6d CFT data. The only known method at this time of computing unprotected 6d CFT data is the numerical conformal bootstrap.
This program was initiated in [10] for S 2 S 2 S 2 S 2 , but the present bounds are not strong enough to extract the 1/c expansion necessary to determine the M-theory effective action via the method of Section 5. One lesson from our paper is that S 3 S 3 S 3 S 3 can be more constraining (and constrained) than S 2 S 2 S 2 S 2 : the W N chiral algebra contributes terms with a nontrivial expansion in 1/c to the former, but not the latter. By applying the numerical bootstrap to S 3 S 3 S 3 S 3 , one could hope to find strong bounds on CFT data.
For certain protected operators, these bounds could be compared to the nontrivial functions of c determined from the chiral algebra. If these analytic functions were to saturate the numerical bounds, then one could use the extremal functional method [70] to read off the CFT data of all operators that appear in the four-point function, as was initiated in the case of the 3d maximally-supersymmetric ABJM theory in [71]. One could also consider using the exact W N result (2.20) as input to this computation, which would presumably generate stronger bounds for the remaining OPE data.
There are also several details of 6d Mellin amplitudes for S k S k S k S k that we would like to understand better. For instance, while we determined the number of purely polynomial solutions to the superconformal Ward identities for k = 2, 3, we were unable to find a simple pattern for larger k. For k ≥ 4, the operators S k undergo mixing with multi-trace operators in the respective D[2k 0] R-symmetry representations, which might explain the counting in these cases. We would also like to better understand the relation between subleading 1/c corrections to a Mellin amplitude and terms in the 11d effective action. As also noted in [28], these correspond to local higher-point vertices in 11d that involve more than four fields, e.g. R 7 . Thus, the "finite size corrections" to the flat space limit of CFT correlators may be understood in part 25 as suitable soft limits of 11d higher-point amplitudes with four external gravitons. It would be interesting to make this relationship explicit.
Lastly, it would be interesting to extend the methods here and in [28] to CFTs with semiclassical string theory duals, such as the large N 't Hooft limits of N = 6 ABJM in 3d or N = 4 SYM in 4d. The complete string moduli dependence of the D 8 R 4 term in type IIA and IIB is unknown despite many years of sophisticated direct attempts (e.g. [72][73][74][75][76] and references therein). 26 It would be fascinating to determine this using holography. 25 Through 14-derivative order, supersymmetry relates D 2m R 4 to R 4+m . Starting at 16 derivatives, the uplift of the subleading terms that descend from D 2m R 4 in 11d cannot be related in any known way to higher-point terms. 26 We would be remiss not to highlight the recent work [77], which sheds a (negative) light on the status of possible non-renormalization of D 8 R 4 by a direct five-loop supergravity computation.

A Superconformal Ward identity and so(5) harmonics
In position space, the superconformal Ward identity takes the form To implement the Ward identities in Mellin space, we first expand G k (U, V ; σ, τ ) into the R-symmetry polynomials Y ab (σ, τ ) as where the hatted operators act on M ab (s, t) as where u = 8k − s − t and we will have independent constraints on each coefficient in the expansion in powers ofᾱ.
In position space, the Ward identities can be also be solved by writing G k in the form (2.12), where the differential operator Υ acts on H(U, V ; σ, τ ) as The Mellin space version of this differential operator is a difference operator Θ that acts in the following way: , where U, V acts on M k (s, t) as It is straightforward to take the flat space limit of this result directly; the result was given in (4.5).

B Superconformal blocks under chiral algebra twist
In the chiral algebra limit ( where the g ∆,j (z) are SL(2, R) global conformal blocks  27 The prefactors in (B.1) come from the R-symmetry factors Y ab (σ, τ ) that multiply the conformal block G ∆ ,j (U, V ) of each superconformal descendent that appears in the superconformal block G ∆,j (U, V ; σ, τ ) in (2.11). After performing the twist and expanding for smallz, these quantities take the form Since thez dependence must cancel from the superblock after performing the twist, we see that for D[2n 0] only the superconformal primary with Y nn (σ, τ )G 4n,0 (U, V ) survives, while for B[2n 0] j only the superconformal descendent with Y n+1 n+1 (σ, τ )G 4n+6+j,j+2 (U, V ) survives. Putting things together, G k (z)| 2d takes the form (2.17) given in the main text.

(D.2)
Finally, we write the reduced Mellin amplitude in the notation of (3.11) for the lowest few E Supermultiplets and superblocks in S 2 × S 2 In this appendix we discuss the supermultiplets that appear in S 2 × S 2 . First, we list the conformal primaries that appear in each supermultiplet. Following the algorithm in [78], we list these results in Tables 3-8.