Solving M-theory with the Conformal Bootstrap

We use the conformal bootstrap to perform a precision study of 3d maximally supersymmetric ($\mathcal{N}=8$) SCFTs that describe the IR physics on $N$ coincident M2-branes placed either in flat space or at a $\C^4/\Z_2$ singularity. First, using the explicit Lagrangians of ABJ(M) \cite{Aharony:2008ug,Aharony:2008gk} and recent supersymmetric localization results, we calculate certain half and quarter-BPS OPE coefficients, both exactly at small $N$, and approximately in a large $N$ expansion that we perform to all orders in $1/N$. Comparing these values with the numerical bootstrap bounds leads us to conjecture that some of these theories obey an OPE coefficient minimization principle. We then use this conjecture as well as the extremal functional method to reconstruct the first few low-lying scaling dimensions and OPE coefficients for both protected and unprotected multiplets that appear in the OPE of two stress tensor multiplets for all values of $N$. We also calculate the half and quarter-BPS operator OPE coefficients in the $SU(2)_k \times SU(2)_{-k}$ BLG theory for all values of the Chern-Simons coupling $k$, and show that generically they do not obey the same OPE coefficient minimization principle.


Introduction and summary
The conformal bootstrap [3][4][5] can be used to place rigorous bounds on scaling dimensions and operator product expansion (OPE) coefficients of operators that appear in the conformal block decomposition of a given four-point function or of a given system of four-point functions [6]. Generally, these bounds get more and more stringent as one explores a larger and larger set of crossing symmetry constraints. 1 Quite remarkably, in certain cases, the bounds include small islands of allowed regions in theory space that contain well-known CFTs [7][8][9]. 2 For a generic point within the allowed region 3 formed by these bounds, there are generally many different solutions to the crossing equations obeying unitarity constraints. At the boundary of the allowed region, however, there is believed to be a unique such solution, which can then be used to read off the CFT data (scaling dimensions and OPE coefficients) that enters the conformal block decomposition of the given four-point function(s). This solution can be found, for instance, using the extremal functional method of [11,36,37]. If we have reasons to believe that a known CFT lives on this boundary, we can therefore potentially determine at least part of its CFT data.
A notable application of this method is to the 3d Ising model. In [37], it was argued that the critical Ising model has the minimal value of the stress tensor coefficient c T (to be defined more precisely shortly) in the space of possible 3d CFTs with Z 2 symmetry, and thus it is believed to sit at the boundary of the region of allowed values of c T . Reconstructing the corresponding unique solution of the crossing equations using the extremal functional method, one can then read off all low-lying CFT data in the critical Ising model. See [38][39][40][41] for other cases where this method was applied.
In this paper, we will apply the extremal functional method to maximally supersymmetric (N = 8) superconformal field theories (SCFTs) in 3d. To do so, we would first need to argue that the 3d N = 8 SCFTs of interest to us saturate some numerical bounds on OPE coefficients or scaling dimensions. Indeed, using supersymmetric localization we will be able to calculate the values of certain OPE coefficients that we can also bound using the numerical bootstrap technique. As we will show, for N = 8 SCFTs with holographic duals, these OPE coefficients come very close to saturating the bootstrap bounds obtained numerically. We conjecture that these OPE coefficients for some of these theories precisely saturate the bootstrap bounds in the limit of very precise numerics.
Let us now provide more background and summarize our results. There are only a few known infinite families of N = 8 SCFTs, and they can all be realized (in N = 3 SUSY notation) as Chern-Simons (CS) theories with a product gauge group G 1 × G 2 coupled to two matter hypermultiplets transforming in the bifundamental representation. These families are: the SU (2) k × SU (2) −k reformulation [42,43] of the theories of Bagger-Lambert-Gustavsson (BLG) [44][45][46][47], which are indexed by an arbitrary integer Chern-Simons level k; the U (N ) k × U (N ) −k theories of Aharony-Bergman-Jafferis-Maldacena (ABJM) [1], which are labeled by the integer N and k = 1, 2; and the U (N + 1) 2 × U (N ) −2 theories [48] of Aharony-Bergman-Jafferis (ABJ) [2], which are labeled by the integer N . We will denote these theories as BLG k , 4 ABJM N,k , and ABJ N , respectively. When N = 1, the ABJM 1,1 theory describes a free SCFT equivalent to the theory of eight massless real scalars and eight Majorana fermions. When N > 1, ABJM N,1 flows to two decoupled SCFTs in the infrared: a free SCFT isomorphic to ABJM 1,1 , and a strongly coupled interacting SCFT.
The ABJ(M) theories can be interpreted as effective theories on N coincident M2-branes placed at a C 4 /Z k singularity in the transverse direction, so that when N → ∞ they contain a sector described by weakly coupled supergravity. The BLG theories, in contrast, do not have a known M-theory interpretation except when k ≤ 4, in which case they are dual to an ABJ(M) theory [48][49][50][51]. N = 8 SCFTs were first studied using the conformal bootstrap in [52,53], in which upper and lower bounds were placed on the CFT data that enters the conformal block expansion of the four-point function of the stress-tensor multiplet. These bounds were computed as a function of c T , which is defined as the coefficient appearing in the two-point function of the canonically-normalized stress-tensor, T µν ( x)T ρσ (0) = c T 64 (P µρ P νσ + P νρ P µσ − P µν P ρσ ) 1 16π 2 x 2 , P µν ≡ η µν ∇ 2 − ∂ µ ∂ ν . (1) This coefficient can be computed exactly using supersymmetric localization for any N ≥ 2 SCFT with a Lagrangian description (see [54] and [55]). In (1), c T is normalized such that it equals 1 for a (non-supersymmetric) free massless real scalar or a free massless Majorana fermion. Thus, c T = 16 for the free N = 8 theory of eight massless real scalars and eight massless Majorana fermions (equivalent to ABJM 1,1 ), and for ABJ(M) theory at large N . By varying c T over the range c T ∈ [16, ∞), one could obtain non-perturbative information about M-theory, albeit only in the form of bounds.
In [49], the OPE coefficients of the half and quarter-BPS operators that appear in the stress-tensor four-point function were computed for BLG 3 and the interacting sector of ABJM 3,1 . 5 This calculation was made possible due to the observation that N = 4 SCFTs (of which N = 8 SCFTs are a particular class) contain 1d topological sectors [53,56,57] that can be accessed explicitly using supersymmetric localization [58]. The OPE coefficients were found to saturate the lower bounds obtained numerically in [53], just as c T saturated the lower bounds in the Ising model, which suggests that the extremal functional approach can be used in this case.
In this work, we generalize the computation of [49] to all N = 8 SCFTs mentioned above. Instead of working directly in the 1d topological theory obtained from supersymmetric localization [58], we argue that one can relate certain integrated correlators in the 1d theory to derivatives of the partition function of an N = 4-preserving mass deformation of the SCFT on S 3 . For each theory, this mass-deformed S 3 partition function can be expressed as a matrix integral using the results of Kapustin, Willet, and Yaakov [59]. For BLG k , the matrix integral can be computed exactly for all k. For ABJ(M) theory, the corresponding integrals can be computed either exactly at small N , or to all orders in the 1/N expansion using the Fermi gas methods in [60]. From the mass-deformed partition function, one can then determine the integrated four-point function in the 1d theory, and from it, as well as from crossing symmetry, one can extract the OPE coefficients of interest.
The analytic expressions for the OPE coefficients can then be compared to the numerical bootstrap bounds. We find that the lower bound is close to being saturated by these OPE coefficients in ABJ N and the interacting sector of ABJM N,1 for all N , in ABJM N,2 for all N = 2, and in BLG k only when k = 3, 4. These are exactly the cases when these theories have a unique stress tensor and have M-theory duals. 6 The formulae for these OPE coefficients become k-dependent beyond order 1/c T , so only one of these curves could saturate the numerical bounds at infinite precision, but the analytic expressions are too similar to distinguish between them at the current level of numerical precision. This motivates our conjecture that all N = 8 theories with holographic duals saturate the bootstrap bounds at large c T , and that at least one of them saturates the bounds for all c T , so that we can apply the extremal functional method to it. Using this technique, we read off all the low-lying CFT data in the stress-tensor four-point function, including that of unprotected operators, for theories that are dual to M-theory. This provides a complete numerical non-perturbative 5 These theories are conjectured to be dual [49], so the OPE coefficients are in fact identical for both theories.
6 (SU (2) × SU (2))/Z 2 BLG is dual to ABJM 2,1 , which is a product of a free and interacting theory. ABJM 2,2 is dual to SU (2) × SU (2) BLG and (SU (2) × SU (2))/Z 2 BLG is dual to two copies of ABJ 1 . Both BLG theories have the same stress tensor four-point function, which makes ABJM 2,2 a product theory in this sector. For k = 3, 4, BLG k is dual to the interacting sector ABJM 3,1 and ABJ 2 , respectively. description of M-theory for the operators in this sector.
The rest of this paper is organized as follows. In Section 2 we review the conformal block decomposition of the four-point function of the scalar operator at the bottom of the N = 8 stress tensor multiplet. In Section 3, we explain our method for computing the OPE coefficients, and we perform this computation for BLG and ABJ(M) theory. In Section 4, we present our evidence for the conjecture that holographic theories saturate the bootstrap bounds on the OPE coefficients we computed in Section 3. In Section 5, we use the extremal functional method to read off all the low-lying CFT data for theories that saturate the bootstrap bounds. Finally, in Section 6, we end with a discussion of our results and of future directions.

Four-point function of stress-tensor
Let us begin by reviewing some general properties of the four-point function of the stresstensor multiplet in an N = 8 SCFT, and of the constraints imposed by the osp(8|4) superconformal algebra (for more details, the reader is referred to e.g. [61][62][63]).
Unitary irreps of osp(8|4) are specified by the quantum numbers of their bottom component, namely by its scaling dimension ∆, Lorentz spin j, and so(8) R-symmetry irrep with Dynkin labels [a 1 a 2 a 3 a 4 ], as well as by various shortening conditions. There are twelve different types of multiplets that we list in Table 1. 7 There are two types of shortening conditions denoted by the A and B families. The multiplet denoted by (A, 0) is a long multiplet and does not obey any shortening conditions. The other multiplets of type A have the property that certain so(2, 1) irreps of spin j − 1/2 are absent from the product between the supercharges and the superconformal primary. The multiplets of type B have the property that certain so(2, 1) irreps of spin j ± 1/2 are absent from this product, and consequently, the multiplets of type B are smaller. 8 The stress-tensor multiplet is of (B, +) type, 9 with its superconformal primary being a is a rank-two traceless symmetric tensor.) In order to not carry around the so(8) indices, it 7 The convention we use in defining these multiplets is that the supercharges transform in the 8 v = [1000] irrep of so(8) R . 8 This description is correct only when j > 0. When j = 0, the definition of the multiplets also requires various conditions when acting on the primary with two supercharges. 9 Whether it is (B, +) or (B, −) is a matter of convention. The two choices are related by the triality of is convenient to contract them with an auxiliary polarization vector Y I that is constrained In the rest of this paper we will only consider the four-point function of O Stress ( x, Y ).

Superconformal invariance implies that it takes the form
where  are the only multiplets we will consider in this paper, we denote the short multiplets other than the stress-tensor as (B, +) and (B, 2), the semi short multiplets as (A, 2) j and (A, +) j where j is the spin, and the long multiplet as (A, 0) j,n , where n = 0, 1, . . . denotes the nth lowest multiplet with that spin-See the last column of Table 2. For explicit expressions for the functions G M , see [52].
Of particular importance will be the OPE coefficient which the stress tensor multiplet enters in the four-point function (4). In the conventions of [52], if we normalize O Stress such that the OPE coefficient of the identity operator is λ Id = 1, then where c T is the coefficient appearing in the two-point function (1) of the canonically normalized stress tensor. The c T coefficient was computed using supersymmetric localization in [52] for the known N = 8 SCFTs with Lagrangian descriptions, and in the next section we will reproduce this result from a different calculation.
It is worth pointing out two limits in which the four-point function (4) is known exactly and one can extract all OPE coefficients. The first limit is the free theory of eight real scalars X I and eight Majorana fermions mentioned in the Introduction. The scalar O Stress,IJ in this case is given by Performing Wick contractions with the propagator X I ( x)X J (0) = δ IJ 4π| x| , we then find that (4) equals: By comparing this to the conformal block expansion, we can read off the OPE coefficients listed in Table 3, where the scaling dimensions of the long multiplet are given by ∆ free (A,0) j,n = j + δ n,0 + 2n , with n = 0, 1, 2, . . ..
Another limit in which we can compute (4) explicitly is in the generalized free field theory (GFFT) where the dimension one operator O GFFT Stress,IJ ( x) is treated as a generalized free field with two-point function The GFFT describes the c T → ∞, i.e. λ 2 Stress = 0, limit of N = 8 theories. Performing the Wick contractions, we then find By comparing this to (4), we can read off the OPE coefficients listed in Table 3, where the scaling dimensions of the long multiplet are given by with n = 0, 1, 2, . . ..  Table 3: Values of OPE coefficients in the free and generalized free field theory limits for the (B, 2) and (B, +) multiplets, the (A, 2) j multiplet for spin j = 1, 3, 5, the (A, +) j multiplet for j = 0, 2, 4, and the (A, 0) j,n multiplet for j = 0, 2, 4 and n = 0, which is the lowest multiplet with that spin.

Computation of OPE coefficients
A crucial input into the numerical bootstrap analysis, which we will use to isolate the N = 8 SCFTs with holographic duals, is that we can compute the squared OPE coefficients λ 2 Stress , λ 2 (B,+) , and λ 2 (B,2) in all Lagrangian N = 8 SCFTs using supersymmetric localization. Conceptually, this computation can be split into several parts, each of which we discuss in separate subsections. In Section 3.1, we explain how, in N = 4 SCFTs with flavor symmetries, one can relate the fourth derivative of the mass-deformed S 3 partition function with respect to the mass parameter to certain OPE coefficients. In Section 3.2, we apply this analysis to N = 8 SCFTs. In Section 3.3, we use the existing results for the mass-deformed partition function of ABJM and BLG theories in order to extract λ 2 Stress , λ 2 (B,+) , and λ 2 from the results of the previous two sections.

Topological sector of N = 4 SCFTs from mass-deformed S 3 partition function
In this subsection, let us discuss some general results for N = 4 SCFTs. N = 4 SCFTs are invariant under the superconfomal algebra osp(4|4), which contains, as its bosonic subalgebra, the conformal algebra so(3, 2) as well as an so(4) R symmetry which we write as su(2) H ⊕ su(2) C . We will use a, b, c, etc. for the su(2) H fundamental indices andȧ,ḃ,ċ, etc. for the su(2) C fundamental indices. These indices are raised and lowered with the anti-symmetric symbol, for instance φ a = ab φ b or φ a = ab φ b , with 12 = − 12 = 1.
It was shown in [53,56] that each 3d N = 4 SCFT contains 1d topological sectors that capture information on the half-BPS spectrum of the 3d theory. There are two such sectors whose abstract constructions mirror each other by simply flipping the roles played by su(2) H and su(2) C . For our final goal of computing λ 2 Stress , λ 2 (B,+) , and λ 2 (B,2) in a given N = 8 theory, it does not matter which of the two 1d sectors of the 3d N = 4 SCFT we focus on, so let us focus on the one built from su(2) C -invariant operators. (This sector is associated with the Higgs branch in the terminology of [53,56].) The construction of the 1d operators that descend from 3d local operators is quite simple and proceeds as follows. The Here, r is a parameter with dimensions of length that was introduced in order for the expression for O(x) to be dimensionally correct. That the operators (12) are topological follows abstractly from properties of the superconformal algebra: these operators are in the cohomology of a nilpotent supercharge with respect to which translations in x are exact.
While (12) was written for the case of an SCFT defined on R 3 , one can perform a similar construction on any conformally flat space. In particular, using the stereographic projection, the 3d SCFT can also be placed on a round three-sphere of radius r such that the 1d line gets mapped to a great circle parameterized by ϕ = 2 arctan x 2r . In this case the 1d operators O(ϕ) are periodic on this circle if j H is an integer and anti-periodic if j H is a half oddinteger. Defining the 1d theory on a great circle of S 3 as opposed to a line in R 3 has the benefit that when the 3d SCFT has a Lagrangian description, then it is possible to perform supersymmetric localization on S 3 in order to obtain an explicit Lagrangian description of the 1d sector itself. In the case where the Lagrangian of the 3d theory involves only hypermultiplets and vector multiplets, the 1d theory Lagrangian was derived in [58].
Regardless of whether the 1d theory has a Lagrangian description or not, let us describe a procedure for calculating certain integrated correlation functions in the 1d theory. We will be interested in the case where the 1d operator J(ϕ) comes from a 3d operator J ab ( x) with ∆ = j H = 1. Such a 3d operator is the bottom component of a superconformal multiplet (J ab , K˙a˙b, j µ , χ aȧ ) that in addition to J ab also contains the following conformal primaries: a pseudoscalar K˙a˙b of scaling dimension 2, a fermion χ aȧ of scaling dimension 3/2, and a conserved current j µ . Thus, this is a conserved flavor current multiplet, and all its operators transform in the adjoint representation of the flavor symmetry. To exhibit the adjoint indices, , where A runs from 1 to the dimension of the flavor symmetry Lie algebra.
Let us choose a basis for this Lie algebra where the two-point function of the current multiplets is diagonal in the adjoint indices: Let us also normalize the current j A µ canonically, meaning that for an operator O transforming in a representation R of the flavor symmetry, we have 10 At the linearized level, such a current multiplet couples to a background N = 4 vector (Quadratic terms in the background vector multiplet are also required in order to preserve gauge invariance and supersymmetry.) Let us provide a prescription for computing correlation functions of the integrated operator dϕ J A (ϕ). To obtain this prescription, first place the SCFT on a round S 3 , then introduce an N = 4-preserving (adjoint valued) real mass parameter m = m A T A . Introducing such a parameter requires the following background vector multiplet fields: Here, we follow the conventions of [58] for the hypermultiplet and vector multiplet fields and their SUSY variations. The quantities h andh are constant matrices, normalized such that h ab h ab =h˙a˙bh˙a˙b = −2. The mass-deformed theory is invariant under the superalgebra su(2|1) ⊕ su(2|1), and the mass parameter m appears as a central charge in this algebra. Up 10 For example, a free hypermultiplet has su(2) flavor symmetry and a current multiplet as described.
to linear order in m, the mass deformation amounts to adding (16) to the conformal action on S 3 . The S 3 partition function Z(m) can be computed using supersymmetric localization [59] even for N = 4 theories for which the localization to the 1d sector performed in [58] does not apply.
However, Z(m) also computes the partition function of the 1d theory deformed by Such a statement can be proven explicitly 11 in the case where the results of [58] apply, but it should hold more generally. This statement should simply follow from the supersymmetric Ward identities, as was shown in similar 4d examples in [65,66]; it would be nice to investigate this more precisely in the future. In other words, we claim that the supersymmetric Ward identity must imply that the expressions (16) and (17) are equal up to Q-exact terms.
Consequently, we have that This is the main result of this subsection.
As a particular case, we can consider n = 2. From (13), we see that on a line in R 3 , we 11 Let us consider N hypermultiplets (q ai , ψȧ i ), where i = 1, . . . , 2N , charged under a vector multiplet with gauge group G and generators t α and flavor symmetry G F with generators T A , respectively. Both G and G F are embedded into the flavor symmetry U Sp(2N ) of N ungauged free hypers. We then have where T Aij is a symmetric matrix in the ij indices. Consequently, using (12), In [58], it was shown that the partition function of the 1d topological theory, defined on a circle, is described by the partition function So deforming the 3d theory by a mass parameter m is equivalent to deforming the 1d theory by (17). For a single free hyper, we have N = 1, Ω ij = ij and T Aij = 1 2 σ Aij for the SU (2) flavor symmetry-see Footnote 10.
Comparing to (19), we deduce (with no summation over A), where we defined the S 3 free energy F S 3 = − log Z. This formula agrees with the result of [67]. (For an N = 4 mass-deformed SCFT on S 3 , the free energy is real, so one does not have to take its real part as in [67].)

Application to N = 8 SCFTs
In order to apply the above results to N = 8 SCFTs, one would first have to go through the exercise of decomposing the various representations of the N = 8 superconformal algebra into representations of the N = 4 superconformal algebra in order to establish which of the N = 8 irreps contain Higgs branch scalar operators with ∆ = j H and j C = 0. In the notation introduced in Table 1, it can be checked that these irreps are of (B The precise projection to 1d was performed in [53]. Converting to the notation here, 13 we have The crossing symmetry of this four-point function implies Let us now use the results of the previous section in order to extract the OPE coefficients λ 2 Stress , λ 2 (B,+) , and λ 2 (B,2) . Eq (23) gives a way to compute λ 2 Stress . From λ 2 Stress = 256/c T and (23) we obtain λ 2 Stress = 8/τ , and from (21) we further obtain The other OPE coefficients can be calculated by specializing the four-point function (24) to A = B = C = D and integrating over ϕ: 12 In N = 4 notation, an N = 8 free theory is a product between a theory of a free hypermultiplet and a free twisted hypermultiplet. The su(2) F acts on the hypermultiplet only, so τ has the same value as in the free hypermultiplet theory, namely τ = 1/2, as explained in Footnote 10. 13 Up to an overall constant, Eq. (3.31) of [53] gives the 4-point function of J A σ A ijȳ iȳj , whereȳ i are auxiliary polarization variables, and σ A ij ≡ (−iσ 2 τ A ) ij , where τ A are the standard Pauli matrices.

ABJ(M) theory
For ABJ(M) theory, one can use (30) and (26)- (29) to evaluate λ 2 Stress , λ 2 (B,+) , and λ 2 (B,2) . The number of integrals increases with N , however, and unlike the BLG case where analytical formulas were possible for the entire family of theories, in the ABJ(M) case we can perform these integrals analytically only for small values of N -See Table 4.
One can also perform a 1/N expansion, where M = N or M = N + 1. There are several approaches for developing a 1/N expansion: one can either work more generally in the 't Hooft limit where N is taken to be large while N/k is held fixed [68], and then take N/k large; or one can work at fixed k while taking N large [69,70]. We will follow the approach originating in [70], where for m = 0 it was noticed in [70] that the S 3 partition function for ABJM theory can be rewritten as a partition function of a non-interacting Fermi gas of N particles with kinetic energy T (p) = log cosh(πp) and potential energy U (q) = log cosh(πq).
Phase space quantization and statistical physics techniques allow one to calculate the S 3 partition function to all orders in 1/N , and this expansion resums into an Airy function.
The S 3 partition function in the presence of a mass deformation was computed using the same method in [60].
where k > 0, M ≥ N , and the function A is given by In order to plug this expression into (26)-(29), one needs the following derivatives of A: 14 The result of [60] is only for N = M , but here we generalize it to N = M using the results of [71]. 15 In the notation of [60], we can take ζ λ = im and ζ µ = 0.
Using (26), we then find which are expressions valid to all orders in 1/N . The analogous expressions for λ 2 (B,2) are rather complicated, so we delegate them to Appendix B. The formulas for λ 2 (B,+) can then be determined from (25).
A comment is in order for the ABJM N,1 theory. This theory is a direct product between a free sector, identified with ABJM 1,1 , and an interacting sector. Since the free sector has c T = 16, we have that the interacting sector of the ABJM N,1 theory has Extracting the value of λ 2 (B,2) of just the interacting sector knowing λ 2 (B,2) for the full theory requires more thought. In the free theory one has λ 2 (B,2) = 0, as such a multiplet does not exist as can be checked explicitly by decomposing the 4-point function of O Stress in superconformal blocks [52]. Using this fact, and the general formulas for how squared OPE coefficients combine when taking product CFTs [53], one has where λ 2 (B,2),GFFT = 32/3 is the generalized free theory value of the λ 2 (B,2) OPE coefficient as given in (3).
In Table 5 we compare λ 2 (B,2) , λ 2 (B,+) and c T to the exact values for N = 2 and find excellent agreement.
We can also expand λ 2 (B,2) and λ 2 (B,+) directly in terms of c T , by comparing the large N expansions. When expanding the Airy functions in (39) and (53) for large N , one should be careful to expand in terms of the entire argument of the Airy function, and not just N , which is how these functions were originally defined in [70]. We find that the large c T expansion is where note that the leading order correction is independent of k.

Bootstrap bound saturation 4.1 Numerical bootstrap setup
We will now briefly review how the numerical bootstrap can be applied to the stress-tensor multiplet four-point function in N = 8 theories. For further details, see [52]. Invariance where M ranges over all the superconformal multiplets listed in Table 2, d M are functions of superconformal blocks, and λ 2 M are squares of OPE coefficients that must be positive by unitarity. As in [52], we normalize the OPE coefficient of the identity multiplet to λ Id = 1, and parameterize our theories by the value of λ Stress , which is related to c T through (6).
for all n < n , and fixed ∆ (A,0) j ,n , In either case, if such a functional α exists, then this α applied to (43) along with the positivity of all λ 2 M except, possibly, for that of λ 2 M implies that if s = 1, then Note that the final condition ∆ (A,0) j ,n +1 > ∆ (A,0) j ,n when M is a long multiplet (A, 0) j ,n is so that M is isolated from the continuum of possible long multiplets. To obtain the most stringent upper/lower bound on λ 2 M , one should then minimize/maximize the RHS of (46) under the constraints (44).
The numerical implementation of the minimization/maximization problem described above requires two truncations: one in the number of derivatives used to construct α and one in the range of multiplets M that we consider. We used the same parameters as in [38], namely spins in {0, . . . , 64} ∪ {67, 68,71,72,75,76,79,80,83,84,87, 88} and derivatives parameter Λ = 43. The truncated minimization/maximization problem can now be rephrased as a semidefinite programing problem using the method developed in [13]. This problem can be solved efficiently using SDPB [72].

Bounds on OPE coefficients
Let us now compare the analytical values of the OPE coefficients λ 2 (B,2) , λ 2 (B,+) , and λ 2 Stress = 256/c T found in Section 2 to the numerical bootstrap bounds obtained using the method outlined in the previous subsection [52]. As noted in [52], the numerical bounds on these OPE coefficients exactly satisfy the constraint (25), so it suffices to discuss the bounds on just two of them, which for simplicity we choose to be λ 2 (B,2) and λ 2 Stress . The main lesson from this comparison will be that λ 2 (B,2) saturates the lower bounds for all N = 8 theories with holographic duals at large c T , so we can use the extremal functional method to read off the spectrum of all operators in the OPE O Stress × O Stress in this regime. At smaller values of c T , we expect that one of these holographic theories saturates the bounds, so the results for the extremal functional hold for that theory.
In Figure 1 Table 4 for the interacting sector of ABJM N,1 , ABJM N,2 , BLG k , and ABJ N , respectively, where in all cases the dots go right to left for increasing k, N . We also list the free theory ABJM 1,1 as a magenta dot 16 . The red, gray, and green dotted lines show the large N values for these theories for N ≥ 2 as given in (39) and (53).
There are several features of the BLG k plot that we would like to emphasize. For k = 1, 2, which are the values where BLG k theory is dual to a product theory (see footnote 6), the OPE coefficients lie in bulk of the allowed region. This is expected, because as described in [52], all product theories generically lie in the bulk region. On the other hand, for k = 3, 4, which are the values where BLG k theory is dual to the interacting sector of ABJM 3,1 and ABJ 2 , respectively, the OPE coefficients are close to saturating the lower bound. Lastly, for k > 4, where it is not known whether the BLG k theories have an M-theory interpretation, the OPE coefficients of the BLG k theories interpolate between the lower and upper bounds. 16 ABJM 1,2 is not a free theory, but has the same stress tensor four-point function as a free theory.  Table 4 in BLG k for k ≥ 1. The magenta dot denotes the free ABJM 1,1 theory, the gray and green dots denote the exact values in Table 4 for ABJM N,2 and ABJ N , respectively, for N = 1, 2, ∞, and the red dots denote ABJM int N,1 for N = 2, 3, ∞. The red, gray, and green dotted lines show the large N formulae (39) and (53)  The k → ∞ value is a little off from the upper bound, which is likely explained by the fact that the upper bound numerics are not fully converged.
The ABJ(M) plot also has two interesting features. We first note that the ABJ 1 ∼ = ABJM int 2,1 theory is close to the numerical point where λ 2 (B,2) = 0. In fact, this is the only known interacting theory with λ 2 (B,2) = 0 [53], so we suspect that with infinite accuracy the numerics would converge to this theory. We next note that all the ABJ(M) values seem to saturate the lower bound up to numerical error, with the exception of ABJM 2,2 , which as explained before has the same stress tensor four-point function as a product theory and so must lie in the bulk.
The fact that λ 2 (B,2) for all unique ABJ(M) theories is close to saturating its lower bound c T ≈ 0.340. The orange shaded region is the allowed island, while the red dotted line shows the exactly known value given in Table 4 for λ 2 (B,2) in this theory. These bounds were computed with Λ = 43. may at first suggest that inputing any value of this OPE coefficient (within the bounds in Figure 1) into the numerical bootstrap code could uniquely specify that theory. To test this idea, in Figure 2 we plot upper/lower bounds of λ 2 (B,2) as a function of λ 2 (A,+) 0 at the ABJM 3,1 point with 16 c T ≈ 0.340 as given in Table 4. While the allowed region is a small island, it does not shrink to a point. On the other hand, as the zoomed in plot shows, when λ 2 (B,2) is at its extremal values then λ 2 (A,+) 0 is uniquely fixed. This matches the general numerical bootstrap expectation that all CFT data in the relevant four-point function is fixed at the boundary of an allowed region. Since the extremal value is very close to the exactly known value, as shown by the red dotted line, if we assume that it would exactly saturate the bound at infinite precision, then we can read off the spectrum of ABJM 3,1 by looking at the functional α that extremizes λ 2 (B,2) . Similar plots can be made for all the other unique ABJ(M) theories, so that λ 2 (B,2) minimization gives the spectra of all theories with holographic duals that saturate the lower bound.

Operator spectrum from numerical bootstrap
We now report our numerical results for the scaling dimensions and OPE coefficients of low-lying operators that appear in the OPE of O Stress with itself. We are interested in theories with holographic duals, and the lowest such known theories are ABJM int 2,1 and ABJ 1 with 16 c T = .75 and λ 2 (B,2) = 0. As we see from Figure 1, our numerics are not completely converged in that region, so we find that λ 2 (B,2) = 0 at the numerical point 16 c T ≈ .71. As such, in the following plots we will show results for 16 c T > .71. Let us describe the (A, 0) unprotected operators that we expect to see in the spectrum.
At the c T → ∞ generalized free field value we have the dimension j + 2 + 2n double trace where n = 0, 1, 2, . . . and µ i are space-time indices. The OPE coefficients of these operators are given in Table 3. At infinite N , these are the only operators with nonzero OPE coefficients. At large but finite N , there are also m-trace operators [O Stress ] m n,j , with m > 1, whose scaling dimension ∆ m n,j and OPE coefficients λ m n,j scale as [73] ∆ m n,j =j + m + 2n as well as single trace operators whose scaling dimension scales with N . For all ABJ(M) theories, c T ∼ N 3/2 [68,74]  n,j , this suppression is extremely strong for even N ∼ 10. As a result, we do not expect the numerical bootstrap bounds to be sensitive to these higher trace operators at the currently feasible levels of precision. The situation is similar to high spin operators, which also have OPE coefficients that are highly suppressed [75][76][77], and so one can restrict to a finite number of operators with spin below some cutoff without affecting the numerics. It is the ability to ignore higher spin operator which in fact makes the numerical bootstrap possible at all.

(A, 0) scaling dimensions
We can read off the scaling dimensions by looking at the zeros of the functional α(∆ (A,0) j,n ) that minimizes λ 2 (B,2) . We trust those scaling dimensions that remain stable as we increase the number of derivatives Λ in the bootstrap numerics. We observed that ∆ (A,0) j,n for j = 0, 2, 4 and n = 0, 1 are stable, and have values that in fact coincide with the upper bounds that we can independently compute for these quantities.  Figure 4: The λ 2 (A,2) j and λ 2 (A,+) j OPE coefficients with spins j = 1, 3, 5 and j = 0, 2, 4, respectively, in terms of the stress-tensor coefficient c T , where the plot ranges from the generalized free field theory limit c T → ∞ to the numerical point 16 c T ≈ .71 where λ 2 (B,2) = 0. The red dots denotes denote the known values at the generalized free field theory points given in Table 3, while the red dotted lines show the linear fit for large c T given in (50). These bounds were computed with Λ = 43.
In Figure 3 we show our numerical results for ∆ (A,0) j,n for n = 0, 1 and j = 0, 2, 4. All three of these plots show the same qualitative features. As described above, we only observe double trace operators, whose OPE coefficients are not suppressed at large N , i.e. small c T .
We can gauge how accurate these plots are by comparing to the c T → ∞ generalized free field values given in (9). The plots seem to match the generalized free field theory values quite accurately. For large c T , we find the following best fits As we see from Figure 3, these linear fits are only accurate for large c T .  Table 3, while the red dotted lines show the linear fit for large c T given in (51). These bounds were computed with Λ = 43. the n = 1 value, so that we can only extract long multiplet OPE coefficients with n = 0.
We find that the upper and lower bounds coincide, which matches our expectation that the extremal functional fixes these values. Note that in principle we could have extracted the OPE coefficients directly from α following the algorithm of [36,38], but we found that this algorithm was very numerically unstable in our case.
In Figure 4 we show our numerical results for λ 2 (A,2) j and λ 2 (A,+) j with j = 1, 3, 5 and j = 0, 2, 4, respectively. Just as with the ∆ (A,0) j,n plots, these plots accurately match the generalized free field theory values listed in Table 3. For large c T , we find the following best As we see from Figure 4, these linear fits seem to be accurate for all values of c T .
In Figure 5 we show our numerical results for λ 2 (A,0) j,n with j = 0, 2, 4 and n = 0. Just as with the ∆ (A,0) j,n plots, these plots accurately match the generalized free field theory values listed in Table 3. For large c T , we find the following best fits As we see from Figure 5, these linear fits are only accurate for very large c T .

Discussion
There are two primary results in this work. One notable feature of our numerical results is that we do not observe unprotected triple (or higher) trace operators in the spectrum, even though they should appear in O Stress ×O Stress for finite c T , i.e. finite N . We do not expect our numerics to be sensitive to these operators, because their OPE coefficients are highly suppressed, e.g. the triple trace operator with dimension ≈ 3 at large N has an OPE coefficient squared that is suppressed as N −9/2 . These operators are thus analogous to high spin operators, which also do not effect the numerical bootstrap because their OPE coefficients are highly suppressed [75][76][77]. However, with more precise numerics, we expect these operators to become visible, and it would be interesting to explore this point more in the future. It would also be interesting to observe whether the analogous triple trace operators also do not appear in the numerical bootstrap of other theories with large N duals, such as the N = 4 superconfromal bootstrap in d = 4 [78,79].
Another notable feature of our numerical results is that the values of the squared OPE coefficients of the double trace semi-short multiplets plotted in Figure 4 are approximately linear functions in 1/c T for the entire range of c T . It would be interesting to find an explanation for this almost linearity. From the bulk point of view, it implies that the only significant corrections to the c T = ∞ values come from a tree level computation in the bulk.
There are several new directions that could strengthen our conjecture that only certain N = 8 theories with holographic duals saturate the numerical lower bounds. From the numerical perspective, it would be useful to impose additional assumptions that would automatically exclude the theories that do not saturate the lower bounds. For instance, in order to exclude the BLG k theories with k > 4, one can apply the bootstrap to a mixed correlator between O Stress and the half BPS multiplet in so(8) irrep [0030]. As one can check from the superconformal index, this latter operator does not exist for BLG k with k > 4, while it does for generic ABJ(M) theories. Another feature of this mixed correlator is that the free multiplet appears in it, so by setting its OPE coefficient to zero one could also exclude the free theory.
From the analytic perspective, it would be useful to include the non-perturbative corrections in 1/N to the results presented in this paper. These non-perturbative corrections have already been calculated for the S 3 free energy in many cases [60,68,70,[80][81][82][83][84][85][86][87][88][89], but to extract the OPE coefficients from these results we would need an expression for these corrections as a smooth function of m. With these non-perturbative corrections included, we would have exact values of these OPE coefficients also for N = 6 ABJ(M) theories with gauge group U (N ) k × U (M ) −k , so that we could see how these quantities interpolate between the N, M → ∞ and fixed k M-theory limit, the N, M, k → ∞ and fixed N/k Type IIA string theory limit, and the N, k → ∞ and fixed M higher spin theory [90,91] limit.
Lastly, it would also be interesting to calculate more BPS OPE coefficients in ABJ(M) theory in a large N expansion using the Fermi gas approach [70].  18 Thus, by taking 6 derivatives of the mass deformed S 3 partition function we can compute the integrated 6-point function in the 1d theory, and thereby determine all these OPE coefficients. For BPS operators that do not appear in any n-point functions of the stress tensor, such as operators in the [00a0] irrep for odd a, we could still express their OPE coefficients as matrix integrals using the 1d methods of [49,58,92]. These matrix integrals could then be computed as expectation values of n-body operators in the Fermi gas, along the lines of [93]. 17 λ Stress (B,+),(B,+) also appears, but this OPE coefficient is related to c T . 18 We thank Ran Yacoby for pointing this out to us.