The ${\cal N} = 8$ Superconformal Bootstrap in Three Dimensions

We analyze the constraints imposed by unitarity and crossing symmetry on the four-point function of the stress-tensor multiplet of ${\cal N}=8$ superconformal field theories in three dimensions. We first derive the superconformal blocks by analyzing the superconformal Ward identity. Our results imply that the OPE of the primary operator of the stress-tensor multiplet with itself must have parity symmetry. We then analyze the relations between the crossing equations, and we find that these equations are mostly redundant. We implement the independent crossing constraints numerically and find bounds on OPE coefficients and operator dimensions as a function of the stress-tensor central charge. To make contact with known ${\cal N}=8$ superconformal field theories, we compute this central charge in a few particular cases using supersymmetric localization. For limiting values of the central charge, our numerical bounds are nearly saturated by the large $N$ limit of ABJM theory and also by the free $U(1)\times U(1)$ ABJM theory.


D Recurrence Relations 57 1 Introduction
The conformal bootstrap [2][3][4] is an old idea that uses the associativity of the operator algebra to provide an infinite set of constraints on the operator dimensions and the operator product expansion (OPE) coefficients of abstract conformal field theories (CFTs). For twodimensional CFTs, this idea was used to compute the correlation functions of the minimal models [5] and of Liouville theory [6]. In more than two dimensions, conformal symmetry is much less restrictive, and as a consequence it is difficult to extract such detailed information from the bootstrap.
The goal of this paper is to set up and develop the conformal bootstrap program in three-dimensional SCFTs with N = 8 supersymmetry, which is the largest amount of supersymmetry in three dimensions. There are only a few infinite families of such theories that have been constructed explicitly, and they can all be realized as Chern-Simons (CS) theories with a product gauge group G 1 × G 2 , coupled to two matter hypermultiplets transforming in a bifundamental representation. These families are: 2 the SU (2) k × SU (2) −k reformulation [39,40] of the theories of Bagger-Lambert-Gustavsson (BLG) [41][42][43][44], 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) [45], which are labeled by the integer N and k = 1, 2; and the U (N + 1) 2 × U (N ) −2 theories [36] of Aharony-Bergman-Jafferis (ABJ) [46], which 1 See also [8,9] for a different recent method. 2 It is believed that non-trivial N = 8 SCFTs such as some of the ones listed here can be obtained by taking the infrared (IR) limit of N = 8 supersymmetric Yang-Mills (SYM) theories. In particular, the U (N ) and O(2N ) SYM theories are believed to flow in the IR to the U (N ) × U (N ) ABJM theories with levels k = 1 and k = 2, respectively. In addition, SO(2N + 1) SYM is believed to flow to the U (N + 1) 2 × U (N ) −2 ABJ theory. See, for instance, [33][34][35][36][37][38] for evidence of the above relations between ABJ(M) theories and N = 8 SYM. We thank O. Aharony for emphasizing this point to us. are labeled by the integer N . 3 That there should exist SCFTs with osp(8|4) global symmetry had been anticipated from the AdS 4 /CFT 3 correspondence. Indeed, the AdS 4 × S 7 background of eleven-dimensional supergravity was conjectured to be dual to an N = 8 SCFT in three dimensions, describing the infrared limit of the effective theory on N coincident M2-branes in flat space, in the limit of large N . The ABJM theory with CS level k = 1 is an explicit realization of this effective theory 4 that is believed to be correct for any N . When N = 1 the theory becomes free, as the interaction potential between the matter fields vanishes and the gauge interactions are trivial [51]. At large N , ABJM theory is strongly coupled, but it can be studied through its supergravity dual, which is weakly coupled in this limit. The duality with the supergravity description has passed impressive tests, such as a match in the N 3/2 behavior of the number of degrees of freedom [52,53]. At finite N > 1, both ABJM theory and its supergravity dual are strongly interacting and not much detailed information is available. In this paper we aim to uncover such information by using the conformal bootstrap. Indeed, our bootstrap study provides us, indirectly, with non-perturbative information about M-theory.
At some level, our work parallels that of [24], who developed the numerical bootstrap program in four-dimensional theories with N = 4 superconformal symmetry. The authors of [24] studied the implications of unitarity and crossing symmetry on the four-point function of the superconformal primary operator O 20 of the N = 4 stress-tensor multiplet. 5 This superconformal primary is a Lorentz scalar that transforms in the 20 irrep under the so (6) R-symmetry. In the present work, we study the analogous question in three-dimensional N = 8 SCFTs. In particular, we analyze the four-point function of the superconformal primary O 35c of the N = 8 stress-tensor multiplet. This superconformal primary is a Lorentz scalar transforming in the 35 c irrep of the so(8) R-symmetry. 6 Upon using the OPE, the four-point function of O 35c can be written as a sum of contributions, called superconformal blocks, coming from all superconformal multiplets that appear in the OPE of O 35c with itself. In addition, this four-point function can be decomposed into the six R-symmetry channels corresponding to the so (8) irreps that appear in the product 3 The invariance of the ABJM and ABJ theories under the N = 8 superconformal algebra, osp(8|4), is not visible at the classical level, but an enhancement to N = 8 is expected at the quantum level. The arguments for this symmetry enhancement are based partly on M-theory [45] and partly on field theory [35,[47][48][49]. 4 The ABJM and ABJ theories have M-theory interpretation for any N and k. For some special values of k the BLG theories were argued to be isomorphic to ABJM and ABJ theories with N = 2 [36,50]. However, for general k the BLG theories have no known M-theory interpretation. 5 The OPE of the stress-tensor multiplet in N = 4 SYM was first analyzed in [54][55][56][57]. 6 That this operator transforms in the 35 c as opposed to 35 v or 35 s is a choice that we make. See the beginning of Section 2. 35 c ⊗ 35 c . Generically, each superconformal multiplet contributes to all six R-symmetry channels. These superconformal blocks can be determined by analyzing the superconformal Ward identity written down in [1]. Crossing symmetry then implies six possibly independent equations that mix the R-symmetry channels amongst themselves. The situation described here is analogous to the case of 4-d N = 4 theories where one also has six R-symmetry channels and, consequently, six possibly independent crossing equations.
There are a few significant differences between our work and that of [24] that are worth emphasizing: • In the case of 4-d N = 4 theories, the crossing equations contain a closed subset that yields a "mini-bootstrap" program, which allows one to solve for the BPS sector of the theory. In the 3-d N = 8 case, we do not know of any such closed subset of the crossing equations that might allow one to solve for the BPS sector.
• At a more technical level, in the case of 4-d N = 4 theories, the solution to the superconformal Ward identity involves algebraic relations between the six R-symmetry channels. As a consequence, after solving for the BPS sector, it turns out that the six crossing equations reduce algebraically to a single equation. In 3-d, the solution to the superconformal Ward identity can be written formally in terms of non-local operators acting on a single function [1]. As we will show, despite the appearance of these nonlocal operators, the various R-symmetry channels can be related to one another with the help of local second order differential operators. These relations show that the six crossing equations are mostly redundant, but still no single equation implies the others, as was the case in 4-d.
• As in the 4-d case, we will parameterize our abstract 3-d theories by the central charge c T , which is defined as the coefficient of the stress-tensor two-point function (in some normalization). In 4-d, c T is the Weyl anomaly coefficient, which allows one to determine it rather easily in particular realizations of the 4-d N = 4 theory. there is no Weyl anomaly, so in order to connect our bootstrap study to more conventional descriptions in terms of BLG and/or ABJ(M) theory, we calculate c T for several of these theories using the supersymmetric localization results of [58,59].
The remainder of this paper is organized as follows. In Section 2 we set up our conventions and review the constraints on the four-point function of O 35c . In Section 3 we write the crossing equations and describe the differential relations that they satisfy. Section 4 is devoted to the derivation of the superconformal blocks building on the results of [1]. In preparation for our numerical results, in Section 5 we calculate the coefficient c T for several explicit N = 8 SCFTs using supersymmetric localization. In Section 6 we study the crossing equations using the semi-definite programing method introduced in [18] and present our findings. We obtain quite stringent non-perturbative bounds on scaling dimensions of operators belonging to long multiplets and on OPE coefficients. We provide several checks of our results in the free theory (namely the U (1) k × U (1) −k ABJM theory) and in the limit of large c T . We end with a discussion of our results in Section 7. Several technical details are delegated to the Appendices.

Constraints from Global Symmetry
Let us start with a short review of some general properties of the four-point function of the stress-tensor multiplet in an N = 8 SCFT, and of the constraints imposed on it by the osp(8|4) superconformal algebra.
In any N = 8 SCFT, the stress tensor sits in a half-BPS multiplet, whose members are listed in Table 1. These include the spin-3/2 super-current, which in our convention 7 transforms (like the supercharges) in the 8 v of the so(8) R-symmetry; and the spin-1 Rsymmetry current, which transforms in the adjoint (i.e. the 28) of so(8) R . In addition, the multiplet contains a spin-1/2 operator transforming (in our conventions) in the 8 c , and two spin-0 operators with scaling dimension 1 and 2, which transform in the 35 c and 35 s , respectively. in various dimensions were analyzed in detail in [1]. The rest of this section reviews some details of [1] that are relevant for our case of interest.

Constraints from Conformal Symmetry and R-symmetry
Let us start by reviewing the constraints arising from the maximal so(3, 2) ⊕ so(8) R bosonic subalgebra of osp(8|4).
The 35 c of so(8) R can be identified as the rank-two symmetric traceless product of the 8 c .
It is convenient to analyze any such symmetric traceless products by introducing polarization vectors Y i , i = 1, . . . , 8, whose indices we can contract with the 8 c indices to form so(8) R invariants. For instance, for an 8 c vector ψ i we should define ψ = ψ i Y i ; for a rank-two tensor O ij , as is the case for our operator, we should consider and so on.
The Y i should be thought of as a set of auxiliary commuting variables, and they are required to satisfy the null condition Commutativity is related to the fact that the tensor O ij is symmetric, while the null condition is connected to its tracelessness. The advantage of introducing the polarization vectors is that instead of keeping track of the various so(8) R tensor structures that can appear in correlation functions, one can just construct all possible so(8) R invariants out of the polarizations.
Invariance of our SCFT under so (3,2) implies that the four-point function of O 35c evaluated at space-time points x m , with m = 1, . . . , 4, should take the form where G is a quadratic polynomial in 1/U and V /U , U and V being the cross-ratios Being a quadratic polynomial in 1/U and V /U , G(u, v; U, V ) contains six distinct functions of u and v. It is helpful to exhibit explicitly these six functions by writing where the quadratic polynomials Y ab (σ, τ ) are defined as The definition (2.7) could be regarded simply as a convention. It has, however, a more profound meaning in terms of the so(8) R irreps that appear in the s-channel of the fourpoint function (2.4). We have The six polynomials 8 in (2.7) correspond, in order, to the six terms on the right-hand side of (2.8

Constraints from Supersymmetry
The full osp(8|4) superconformal algebra imposes additional constraints on (2.6). For the purpose of writing down these constraints, it is convenient to introduce the following parameterization of the cross-ratios in terms of the variables x,x and α,ᾱ: In this parameterization, the function G(x,x; α,ᾱ) appearing in (2.4) as well as the function A ab (x,x) appearing in (2.6) should be taken to be symmetric under the interchanges x ↔x and α ↔ᾱ. As shown in [1], in terms of the variables (2.9) and (2.10) the superconformal Ward identity takes a particularly neat form.
In [1], the superconformal Ward identity of the four-point function of 1/2-BPS operators was written down for any theory with so(n) R-symmetry in space-time dimension d with 3 ≤ d ≤ 6. It takes the form where ε ≡ (d − 2)/2 is the scaling dimension of a free scalar field in d space-time dimensions.
The solution of the superconformal Ward identity (2.11) depends quite significantly on the parameter ε. In the case ε = 1, which would apply to four-dimensional SCFTs with N = 4 supersymmetry, the solution is very simple. Indeed, in this case, (2.11) reduces to ∂ x G(x,x; 1/x,ᾱ) = ∂xG(x,x; α, 1/x) = 0, as can be seen from using the chain rule. Therefore, G(x,x; 1/x,ᾱ) is independent of x and G(x,x; α, 1/x) is independent ofx. Taking into account the fact that G is symmetric w.r.t. interchanging x withx and α withᾱ, and that G is a quartic polynomial in α andᾱ, as follows from the definition (2.10), one can write the general solution for the four-point function as [61,62] where A(x,x), F (x, α), and C are arbitrary. In other words, apart from the restricted function F (x, α) and the constant C, which can be determined in terms of the anomaly coefficient c [24], the whole four-point function G d=4 (x,x; α,ᾱ) can be written in terms of a single function A(x,x). The six R-symmetry channels in this case are related algebraically.
The solution of the superconformal Ward identity for arbitrary ε, and in particular for ε = 1/2, can be written in terms of powers of the differential operator For non-integer ε, the general solution 9 of (2.11) can be written, formally, in terms of a single arbitrary function a(x,x) as 10 (2.14) The appearance of the operator (D ε ) ε−1 , which is non-local for non-integer ε, makes using (2.14) rather subtle. However, we can demystify the operator D ε and its non-integer powers by interpreting D ε as the Laplacian in d = 2(ε + 1) dimensions. 11 Using conformal transformations we can fix three of the coordinates of the four-point function on a line, such that: x 1 = 0, x 3 = (0, . . . , 0, 1) ≡ẑ and x 4 = ∞. (We denote the unit vector (0, . . . , 0, 1) ∈ R d byẑ because we will eventually be interested in working in three dimensions where we denote the third coordinate by z.) We write the remaining , where θ is the angle between r andẑ, and Ω d−2 parameterizes S d−2 . The four-point function does not depend on Ω d−2 because of the additional rotation symmetry which fixes the line determined by x 1 , x 3 , 9 The solution (2.14) corresponds to the four-point function of 1/2-BPS operators which are rank-2 symmetric traceless tensors of so(n) R . The solution for tensors of arbitrary rank can also be written in a similar way, but it depends on more undetermined functions. The reader is referred to [1] for more details. 10 The function a(x,x) that appears in this equation equals (xx) ε−1 a(x,x) in the notation of [1]. 11 That D ε is the Laplacian in d = 2(ε + 1) dimensions was first observed by Dolan and Osborn in [63]. and x 4 . The cross-ratios in these coordinates are given by In other words, u can be interpreted as the square of the distance to the origin of R d , while v is the square of the distance to the special point (0, . . . , 0, 1).
The operator D ε can then be written as Up to an overall factor of 1/4, D ε is nothing but the d-dimensional Laplacian ∆ acting on functions that are independent of the azimuthal directions Ω d−2 ∈ S d−2 .
In d = 3, the solution (2.14) to the Ward identity can then be written formally as for some undetermined function a( r). Here, both G( r; α,ᾱ) and a( r) should be taken to be invariant under rotations about the z-axis. This expression will become quite useful when we analyze the crossing symmetry in the next section.

Constraints from Crossing Symmetry
In this section we will discuss the constraints of crossing symmetry on the four-point function (2.4).
In terms of G(u, v; U, V ) defined in (2.4), the crossing constraint corresponding to the By expanding (3.1) in U and V one obtains six crossing equations, mixing the different R-symmetry channels (2.6). However, these crossing equations cannot be used in the numerical bootstrap program as they stand, for the following reason. The different R-symmetry channels are related by supersymmetry, so these equations are not independent. Using these dependent equations in a semidefinite program solver like sdpa [64] (as we will discuss in detail in Section 6) results in a numerical instability.
To understand the dependencies between the equations (6.1) we have to study the solution (2.18) of the Ward identity. In terms of (2.14) the crossing equation (3.1) takes the form 12 This expression seems to suggest that there is only one independent crossing equation given Despite the appearance of a non-local operator in the solution of the superconformal Ward identity, we can in fact show that the six R-symmetry channels and, consequently the six crossing equations, satisfy certain differential equations that relate them to one another.
These relations will be crucial for the implementation of the numerical bootstrap program in Section 6.

Relations Between R-Symmetry Channels
The inverse square root of the Laplacian appearing in (2.18) can be defined by its Fourier In expressions of the form ∆ − 1 2 f (r, θ)∆ 1 2 , we can then use the canonical commutation relation of quantum mechanics, [x, p] = i, to commute ∆ 1 2 through f (r, θ). For example, it is straightforward to show that where we defined z ≡ r cos θ.
To proceed, it is convenient to decompose the solution of the Ward identity (2.14) in the 12 In deriving (3.2) we use the fact that under crossing r →ẑ − r and ∆ is invariant. basis These e i are simply related to the different R-symmetry channels A ab by Defining the operators it can be seen from (3.6) that the following relations hold: It is easy to convince oneself that these are the most general relations between the e i that can be obtained by acting with D ± . Moreover, instead of thinking of the solution to the Ward identity as given in terms of a single unconstrained function a(u, v), we can think of it as given in terms of the six constrained functions e i , with the constraints given by (3.9)-(3.12).
The advantage of this formulation of the solution is that the constraints (3.9)-(3.12) only involve local differential operators. Indeed, using (3.4), (3.5), and the coordinate transfor-mation (2.16), we find (3.13) In terms of the x,x coordinates, we have where D 1 2 was defined in (2.13).

Relations Between the Crossing Equations
Defineẽ i to be the same as the e i in (3.6), but with the factors of a(u, v) replaced by . It is clear that theẽ i also satisfy the differential equations (3.9)-(3.12).
The crossing symmetry constraints are simply given byẽ i = 0.
One can solve the differential equations (3.  for some constants a nm , b nm , and c nm that can be determined order by order in the expansion.
We conclude that the maximal set of independent crossing equations can be taken to bẽ e 2 n,m = 0 andẽ 4 n,0 = 0 for all integers n, m ≥ 0.

Superconformal Blocks
In this section we will derive the N = 8 superconformal blocks of the four-point function (2.4 A common approach to deriving superconformal blocks involves analyzing the detailed structure of the three-point function between two O 35c and a third generic superconformal multiplet. In this approach one has to construct the most general superconformal invariants out of the superspace variables appearing in this three-point function (see e.g., [15]). However, it is difficult to implement this method in theories with extended supersymmetry due to complications in using superspace techniques in such theories.
In practice, we will compute the superconformal blocks in our case of interest using two different methods. One method involves expanding the solution of the Ward identity given in (2.18) in conformal blocks. 13 Even though this method is hard to implement due to the appearance of the non-local operator 1/ √ ∆ in (2.14), significant progress was made in [1] and we will build on it in Section 4.2. In the next subsection we will introduce a new strategy for computing the superconformal blocks. This second method relies on the fact that the superconformal Ward identity (2.11) holds separately for each superconformal block. As we will see momentarily in Section 4.1, this approach is simpler and more systematic than working directly with the full solution to the Ward identity.
Before we begin, let us quickly review the unitary irreducible representations of the osp(8|4) superconformal algebra, following [65]. Unitary irreps of osp(8|4) are specified by the scaling dimension ∆, Lorentz spin j, and so(8) R-symmetry irrep [a 1 a 2 a 3 a 4 ] of their bottom component, as well as by various shortening conditions. There are twelve different types of multiplets that we list in Table 2. There are two types of shortening conditions Table 2: Multiplets of osp(8|4) and the quantum numbers of their corresponding superconformal primary operator. The conformal dimension ∆ is written in terms of ∆ 0 ≡ a 1 + a 2 + (a 3 + a 4 )/2. The Lorentz spin can take the values j = 0, 1/2, 1, 3/2, . . .. Representations of the so(8) R-symmetry are given in terms of the four so(8) Dynkin labels, which are non-negative integers.
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. The stress-tensor multiplet that we encountered in Section 2 is of (B, +) type and has a 3 = 2. The conserved current multiplet appears in the decomposition of the long multiplet at unitarity: ∆ → j + 1. This multiplet contains higher-spin conserved currents, and therefore can only appear in the free theory [66].
We will sometimes denote the superconformal multiplets by (∆, j) [a 1 a 2 a 3 a 4 ] X , with (∆, j) and [a 1 a 2 a 3 a 4 ] representing the so(3, 2) and so(8) R quantum numbers of the superconformal primary, and the subscript X denoting the type of shortening condition (for instance, X = (A, 2) or X = (B, +)).

Superconformal Blocks from Ward Identity
Our strategy to compute the superconformal blocks is very simple. Let G (a,b) ∆,j denote the contribution to the four-point function of a multiplet whose primary has dimension and spin This contribution can be written as some linear combination of a finite number of conformal blocks: ∆,j contribution independently. We can expand (4.1) in a Taylor series around x =x = 0 using the known expansions of the conformal blocks (see, for example, [67] or Appendix A). Plugging in this expansion in the suprconformal Ward identity (2.11), we can generate infinitely many equations for the undetermined coefficients λ 2 O . These equations must be consistent if in (4.1) we sum over all the operators O belonging to a given superconformal multiplet.
Before we can apply this strategy outlined above concretely, we need to determine which superconformal multiplets can appear in the OPE. In addition, we should also determine the spectrum of conformal primaries in each of those superconformal multiplets. The first task was preformed in [68], and we list their results in Table 3.
Note that a three-point function of two 1/2-BPS multiplets with a third multiplet of any type is completely determined by the contribution of the superconformal primaries. It then follows that if a superconformal primary has zero OPE coefficient, then so do all its descendants. Consequently, in Table 3, the (B, 2) multiplets in [0100] and [0120] cannot actually appear in the OPE. The reason is that these representations appear in the anti-  symmetric product of the OPE, and can therefore contain only odd spin operators, while the superconformal primaries of the above multiplets have even spin. Similarly, j must be even in the (j +2, j) [0020] (A,+) and the (long) (∆, j) [0000] (A,0) multiplets, and it must be odd for (j +2, j) [0100] (A,2) . Next, we have to identify the conformal primaries belonging to the superconformal multiplets listed in Table 3. For each such superconformal multiplet, we can decompose its corresponding osp(8|4) character [65] into characters of the maximal bosonic sub-algebra so(3, 2) ⊕ so(8) R . This decomposition is rather tedious, and we describe it in Appendix B.
Here, let us list the results. The conformal primaries of the stress-tensor multiplet (1, 0) [0020] (B,+) were already given in Table 1. The conformal primaries of all the other multiplets appearing in Table 3 are given in Tables 4-8. The first column in these tables contains the conformal dimensions and the other columns contain the possible values of the spins in the various Rsymmetry channels. In each table, we only list the operators which could possibly contribute to our OPE, namely only operators with R-symmetry representations in the tensor product (2.8), and only even (odd) integer spins for the representations (a, b) with even (odd) a + b.
Using this information and the Ward identity we can now determine the superconformal blocks. In practice, we expand (4.1) to a high enough order so that we get an overdetermined system of linear equations in the λ 2 O . We can then solve for the OPE coefficients in terms of one overall coefficient. The fact that we can successfully solve an overdetermined system of equations is a strong consistency check on our computation. The final expressions are very complicated, and we collect the results in Appendix C.
As an interesting feature of the superconformal blocks, we find that the OPE coefficients of all the operators which are marked in red in Tables 4-8   -    (long) superconformal multiplet, with j even, ∆ ≥ j +1. The decomposition of this multiplet at unitarity contains a conserved current multiplet, which, in turn, contains higher-spin conserved currents.

Derivation of Superconformal Blocks Using the Results of [1]
The superconformal blocks can also be computed using the solution (2.18) of the Ward identity. 15 One first observes that for all multiplets listed in Tables 4-8 Let us first write (2.18) in terms of the decomposition into so(8) R representations in (2.6), For the long multiplet A 22 is determined (up to an overall coefficient) to be 14 A similar phenomenon occurs in four dimensional N = 4 supersymmetric Yang-Mills theory [62]. There, the operators that decouple are the ones which are not invariant under the "bonus symmetry" discussed in [69,70]. 15 The superconformal blocks of N = 2, 4 SCFTs in d = 4 were derived in this way in [62].
Then, for example, the A 21 channel is given by (4.4) and the other channels are given by similar expressions. This expression can be expanded in conformal blocks by using recurrence relations derived in [1]. We collect these relations 16 in appendix D. The final result matches precisely the long multiplet superconformal block that we found using the method of the previous section.
It turns out that the superconformal blocks of the short multiplets can be derived by taking limits of the long superconformal block. These limits consist of taking ∆ and j in the long block to certain values below unitarity, i.e. ∆ < j + 1. For instance, we can try to obtain the superconformal block of the (2, 0) [0040] (B,+) multiplet (see Table 4) by taking ∆ → −2 and j → 0 in the long superconformal block. In this limit as ∆ → −2 and j → 0 . (4.5) Note that such limits have to be taken with great care for two reasons. The first reason is that some of the conformal blocks g ∆,j are divergent in this limit, but the coefficients multiplying them vanish, so the limit is finite. The divergence arises because the conformal blocks g ∆,j , viewed as functions of ∆, have poles below unitarity. The location and residues of these poles were computed in [20]. For example, there is a "twist-0" pole at ∆ = j given by The second reason why the limits have to be taken with care is that the limits ∆ → 2 and j → 0 do not commute, so the result is ambiguous. We parameterize this ambiguity by taking first ∆ = −2 + cj and later sending j → 0. The constant c is kept arbitrary at this stage. 16 Appendix D also corrects several typos in the equations of [1].
Taking the above considerations into account, for the (2, 0) [0040] (B,+) multiplet we find 17 This result is, in general, inconsistent with unitarity because of the appearance of conformal blocks with negative twists such as g 2,3 . These unphysical blocks can be removed in the limit c → ∞. In this limit, the result matches precisely the (2, 0) [0040] (B,+) superconformal block in (C.4)-(C.9), and we conclude that

Central Charge Computation
For the numerical bootstrap, we need to specify an input that distinguishes different N = 8 SCFTs. As with the 4-d N = 4 case in [24], we use the central charge c T , defined as the 17 We use the identity g ∆,−j−1 = g ∆,j , which can be derived from the conformal Casimir equation.
overall coefficient appearing in the two-point function of the canonically normalized stress tensor [71]: In (5.1), we normalized c T such that it equals one for a real massless scalar or Majorana fermion. For SCFTs preserving N ≥ 2 supersymmetry one can use supersymmetric localization [58,72,73] on the three-sphere to compute c T exactly [59] .
There are two approaches to using supersymmetric localization to compute c T . One way is to compute the squashed sphere partition function [74,75] of the theory with squashing parameter b, where b = 1 corresponds to the round sphere. Taking the derivative with respect to the squashing parameter, the central charge can be computed as [76]. This computation has been carried out in [76][77][78] in a few simple examples.
Another way of obtaining c T makes use of having extended supersymmetry. In our N = 8 SCFTs, the stress tensor sits in the same osp(8|4) multiplet as the so (8)  The extended supersymmetry relates c T to the coefficient appearing in the two-point function of the Abelian flavor currents. In general, the flat-space two-point functions of Abelian flavor currents j µ a , with a being a flavor index, takes the form The normalization in (5.2) is such that for a free chiral superfield (where there is only one flavor current j µ corresponding to multiplication of the superfield by a phase) we have τ = 1 provided that the chiral superfield carries unit charge under the flavor symmetry. As explained in [59], the quantity τ ab can be computed from the S 3 partition function corresponding to a supersymmetry-preserving deformation of the N = 2 SCFT. This deformation can be interpreted as a mixing of the R-symmetry with the flavor symmetry, whereby the matter fields are assigned non-canonical R-charges. The deformed S 3 partition function can be computed exactly using the supersymmetric localization results of [58,72,73].

Setup of the Computation
We will follow the second approach for computing c T exactly in a few N = 8 SCFTs. In N = 2 notation, the matter content of all known N = 8 theories consists of two vector multiplets with gauge group G 1 and G 2 , respectively, and four chiral multiplets that transform in bifundamental representations of G 1 × G 2 . Preserving the marginality of the quartic superpotential, one can consider the most general R-charge assignment parameterized as [79,80] This parameterization is chosen such that τ ab will be diagonal.
F -maximization [79,81,82] tells us that Re As explained in [59], the coefficient τ ab can be computed from the second derivative of Z evaluated at t = 0: As explained above, c T should be proportional to τ ab . The coefficient of proportionality can be fixed through carefully defining representations of the osp(8|4) algebra and then decomposing them into their osp(2|4) sub-algebra representations. A quicker way to fix the proportionality factor is from ABJM theory in the large N limit, where c T is known from supergravity computations [83] and Z was computed as a function of ∆ in [79].
The three-sphere partition function of R-charge deformed ABJ(M) theories with gauge Here, α ranges from 1 to 4 and labels the chiral superfields of our theory.
The only ∆(t) dependence of (5.6) comes from f α (∆). To compute the second derivative required in (5.5), note that At face value, it looks like (5.7) gives something different from (5.8). One can check, however, that the extra term present in (5.7) does not contribute to (5.5) in the cases we study, as required from N = 8 supersymmetry.
Note that sometimes the N = 8 theories that we consider have decoupled sectors. For instance, the U (N ) 1 ×U (N ) −1 ABJM theory has a free N = 8 sector [35], which is not visible at the level of the Lagrangian, but must clearly exist if one identifies this theory with the IR limit of U (N ) SYM. 18 In such cases the theory has more than one stress tensor, and our localization computations are only sensitive to the sum of the central charges corresponding to the different decoupled CFTs.
In particular, we compute the central charges of theories, which are expected to be equivalent to the IR factorizes into two copies of U (2) 2 × U (1) −2 ABJ theories. Indeed we find that the central charges computed below for these product CFTs are given by the appropriate sum of central charges corresponding to the irreducible CFTs (see Table 9). 19 18 On the gravity side this free sector simply corresponds to the center of mass motion of the stack of M2-branes. 19 We are grateful to O. Aharony for pointing this out to us.

Large N Limit
First let's consider the theories with supergravity dual descriptions. For a theory admitting an AdS 4 dual description, the sphere free energy F is proportional to the central charge This relation follows from the fact that the central charge in our normalization is c T = 32L 2 πG 4 [83], and the S 3 free energy is F = πL 2 2G 4 [52,79]. 20 Using localization, the large N limit of F is given by the N 3/2 scaling law [52,79] Combining this expression with (5.9) gives us the central charge of ABJM theories in the large N limit The flavor current two-point function coefficient can be computed using (5.5), so  [84][85][86][87][88] can be generalized to compute c T systematically.
This choice exploits the fact that the product is independent of u + , giving us a delta-function.
Performing the integral, we obtain We see that for ∆ α = ∆ * = 1/2,  For general k, the integral I n can be evaluated by contour integration as explained in [84].
Depending on whether k is even or odd, one can choose a holomorphic function and a contour 21 See Appendix E.1 for details.
that integrates to I n . Summing the residues of the poles gives One can also consider the large k limit, where the theory becomes perturbative. The central charge in the large k limit is The fact that the central charge asymptotes to 64 can be understood from free theory counting. Four chiral multiplets in a single color factor contribute 16 to the central charge.
As chiral multiplets are in the bifundamental representation, there are four copies of them, which sums to 64.

U (2) × U (1) ABJ Theory
We now compute the central charge explicitly for U (2) k × U (1) −k ABJ theory. 22 Carrying out the integral for both Z and its second t-derivatives, we get c T = 32 The central charge for k = 1 is consistent with the ABJ duality [46,89] as

Numerics
All ingredients are now in place for our numerical study of the crossing equations (3.1).
Explicitly, in terms of the functions A ab (u, v) defined in (2.6) and expanded in superconformal blocks in Section 4 (see also Appendix C), these equations are: where we defined Crossing equations such as (6.1) have been used many times recently to rule out the existence of (S)CFTs whose spectrum of operators satisfies certain additional assumptions.
We will perform several such studies with or without additional assumptions besides locality (i.e. existence of a stress tensor), unitarity, and invariance under the N = 8 superconformal algebra osp(8|4). The main observation is that, when expanded in superconformal blocks, the crossing equations (6.1) take the form where M ranges over all the superconformal multiplets that appear in the OPE of O 35c with itself-see Table 3. In (6.4), d i,M should be identified with the middle expression in (6.1) in which one uses only the contributions to the F ± ab coming from the superconformal block of the multiplet M.
There is in fact a superconformal multiplet that appears in the O 35c × O 35c OPE and that was omitted from Table 3. It is a rather trivial multiplet that consists solely of the identity operator in the so(8) R singlet channel. Its superconformal block is given by corresponds to λ Id = 1 by requiring its two-point function to satisfy wherex is taken to zero first. (See also Appendix A.) With the normalization described above, we can relate the OPE coefficient of the stresstensor multiplet (1, 0) [0020] (B,+) (which, for short, will henceforth be referred to as "stress") to the central charge c T discussed in the previous section. We have 23 where, as in the previous section, we normalized c T so that c T = 1 for a theory of a free real scalar field or a free Majorana fermion. In Table 9 we collect the lowest few values of c T that we computed in the previous section for known SCFTs with N = 8 supersymmetry.
. . . . . . where α i,mn are numerical coefficients. In (6.9), we restricted the second sum to run only over m ≥ n because ∂ m∂n d i = ∂ n∂m d i , as follows from the fact that all conformal blocks are chosen to be invariant under x ↔x. Without this restriction, we would be double counting all derivatives with m = n.
Note that still not all the terms in the sum (6.9) are linearly independent. There are two additional sources of linear dependencies between the various terms in (6.9). The first such source can be seen from the definitions (6.1)-(6.2) whereby d 1 and d 2 are even under x → 1 − x andx → 1 −x, while the other d i are odd. Therefore, at the crossing-symmetric point x =x = 1/2, we have ∂ m∂n d i = 0 for i = 1, 2 and m + n odd or i = 3, 4, 5, 6 and m + n even. We should not include these terms that vanish in (6.9).
The second source of dependencies is more subtle and follows from the discussion in We can now attempt to find linear functionals (6.9) that satisfy certain positivity properties in order to obtain bounds on operator dimensions and OPE coefficients.

Obtaining a Lower Bound on c T
In the previous section we have seen that the U The conditions (6.11) and the equation (6.10) imply the bound To obtain the most stringent bound we should minimize −α( d Id ) under the constraints (6.11).
The minimization problem described above needs to be truncated for a numerical implementation. There are two truncations that should be performed: one in the number of derivatives used to construct α and one in the range of multiplets M that we consider.
Instead of (6.9), we can consider the truncated version where the sum over m and n should only contain independent terms. In practice, the cutoff Λ that determines the size of our search space will be taken to be Λ = 15, 17, or 19. We can then minimize −α Λ ( d Id ) under the constraints for all other M with j ≤ j max and ∆ ≥ j + 1 (6.14) Here, ∆ and j refer to the conformal dimension and spin of the superconformal primary, and ∆ ≥ j + 1 is just the unitarity condition. The second equation refers to all multiplets M other than the identity and the stress-tensor multiplet. In practice, we found that taking j max = 20 provides fairly accurate results.
For the long multiplet (∆, j) [0000] (A,0) (henceforth referred to as "long") the quantity α Λ ( d long ) can further be approximated, for each spin, by a positive function times a polynomial in ∆. Such expansion is obtained by expanding the conformal blocks that comprise the long superconformal block in a Taylor series around x =x = 0 using the recursion formula given in [20], and then approximating some of the poles as a function of ∆ that appear in this expansion in terms of a smaller set of poles, as explained in the Appendix of [20].
The minimization of −α Λ ( d Id ) under the constraints (6.14) can then be rephrased as a semidefinite programing problem using the method developed in [18]. This problem can be solved efficiently by freely available software such as sdpa gmp [64]. Implementing it as a dual problem, we obtain λ 2 stress ≤ 17.02, 16

Bounds on Scaling Dimensions of Long Multiplets
A small variation on the method presented in the Section 6.1 yields upper bounds on the lowest scaling dimension ∆ * j of spin-j superconformal primaries in a long multiplet. Such superconformal primaries must all be singlets under the so(8) R-symmetry-see Table 3, where the long multiplet is in the last line. It is worth emphasizing that, as was the case in Section 6.1, these bounds do not depend on any assumptions about our N = 8 SCFTs other than locality and unitarity.
The variation on the method presented in Section 6.1 is as follows. Let us fix c T and look for functionals α satisfying the following conditions: The existence of any such functional α would prove inconsistent all SCFTs with the property that superconformal primaries of spin-j long multiplets all have conformal dimension ∆ ≥ ∆ * j , because if this were the case, then equation (6.10) could not possibly hold. If we cannot find a functional α satisfying (6.15), then we would not be able to conclude anything about the existence of an SCFT for which superconformal primaries of spin-j long multiplets all have conformal dimension ∆ ≥ ∆ * j -such SCFTs may or may not be excluded by other consistency conditions we have not examined. An instance in which a functional α with the properties (6.15) should not exist is if c T is chosen to be that of an ABJ(M) or a BLG theory and if we only impose restrictions coming from unitarity, namely if we take ∆ * j = j + 1 for all j. Indeed, we should not be able to exclude the ABJ(M) and/or BLG theories, assuming that these theories are consistent as is believed to be the case.
As in the previous section, in order to make the problem (6.15) amenable to a numerical study, we should truncate the number of spins used in the second and third lines to  Table 9.
(where in practice we take j max = 20) and replace α by α Λ such that our search space becomes finite-dimensional. We can then use sdpa gmp to look for functionals α Λ satisfying (6.15) for various choices of ∆ * j . In practice, we will take Λ = 15, 17, and 19. We present three numerical studies: 1. We first find an upper bound on the lowest dimension ∆ * 0 of a spin-0 long multiplet assuming that all long multiplets with spin j > 0 are only restricted by the unitarity bound. In other words, we set ∆ * j = j + 1 for all j > 0. This upper bound is plotted as a function of c T in Figure 1 for Λ = 15 (in light brown), Λ = 17 (in black), and Λ = 19 (in orange). As can be seen from Figure 1, there is very good agreement between the latter two values of Λ, especially at large c T .
The upper bound on ∆ * 0 interpolates monotonically between ∆ * 0 < ∼ 1.02 at c T = 16 and ∆ * 0 < ∼ 2.03 as c T → ∞ when Λ = 19. As we will now explain, these bounds are very close to being saturated by the U (1) k × U (1) −k ABJM theory at c T = 16 and by the large N U (N ) k × U (N ) −k ABJM theory (or its supergravity dual) at c T = ∞.  2. Our second numerical study is similar to the first. Instead of obtaining an upper bound on ∆ * 0 , we now obtain an upper bound on ∆ * 2 , which is the lowest scaling dimension 24 For Chern-Simons levels k = 1, 2, the products X i X j must be combined with monopole operators into gauge invariant combinations. 25 Single trace long multiplets are not part of the supergravity spectrum. The only single-trace operators that are dual to supergravity fluctuations around AdS 4 × S 7 are part of the half-BPS multiplets (n/2, 0) [00n0] (B,+) with n ≥ 2 [90]. 26 We thank I. Klebanov for a discussion on this issue.   Table 9.
of a spin-2 long multiplet. We obtain the bound on ∆ * 2 under the assumption that long multiplets of spin j = 2 are only restricted by the unitarity condition. In other words, we set ∆ * j = j + 1 for all j = 2. In Figure 3, we plot the upper bound on ∆ * 2 as a function of c T for Λ = 15 (in light brown), Λ = 17 (in black), and Λ = 19 (in orange). The convergence as a function of Λ is poorer than in the ∆ * 0 case, but it is still reasonably good throughout, especially at large c T .
A main feature of the plot in Figure 3 is that it interpolates monotonically between ∆ * 2 < ∼ 3.11 at c T = 16 and ∆ * 2 < ∼ 4.006 at c T = ∞. It is likely that as one increases Λ, the bound at c T = 16 will become stronger still, since at this value of c T the bound obtained when Λ = 19 is still noticeably different from that obtained when Λ = 17 and convergence has not yet been achieved.
As was the case for the bounds on ∆ * 0 , the bounds on ∆ * 2 are also almost saturated by ABJM theory at c T = 16 and c T = ∞. Indeed, two of the spin-2 so(8) singlets that appear in the O ij × O kl OPE as bottom components of long multiplets are the single trace operator tr X k ∂ µ ∂ ν X k and the double trace (1) ABJM theory, they have scaling dimensions 3 and 4, respectively; in ABJM theory at infinite N , the first has a large anomalous dimension, while the second has scaling dimension 4 because of large N factorization. Therefore, the N = 1 theory has ∆ * 2 = 3, while the large N theory has ∆ * 2 = 4, in agreement with our numerical bounds.
Note that just as in the ∆ * 0 case, our upper bound on ∆ * 2 in Figure 3 also exhibits a kink for c T ≈ 22.8. Within our numerical precision, this kink is in the same location as that in Figure 1. 3. Our last numerical study yields combined upper bounds on ∆ * 0 and ∆ * 2 under the assumption that all long multiplets with spin j > 2 are restricted only by the unitarity bound, i.e. ∆ * j = j + 1 for all j > 2. In Figure 4 we provide such combined upper bounds only for a few values of c T corresponding to the ABJ(M) / BLG theories for which we computed c T in Section 5.
As can be seen from Figure 4, the combined bounds take the form of a rectangle in the ∆ * 0 -∆ * 2 plane, suggesting that these bounds are set by a single N = 8 SCFT, if such an SCFT exists. A similar feature is present for the N = 4 superconformal bootstrap in 4-d [24].

Bounds on OPE Coefficients
We can also obtain upper bounds on various OPE coefficients, just as we did in Section 6.1 for λ 2 stress . In this section we will only do so for the protected multiplets (2, 0) [0040] (B,+) and (2, 0) [0200] (B,2) , which for brevity will henceforth be denoted by (B, +) and (B, 2), respectively.  If such a functional α exists, then (6.10) implies that As a warm-up, let us start with the c T = ∞ limit and see how sensitive the bounds on λ 2 (B,+) and λ 2 (B,2) are on the values of ∆ * j that we choose. Requiring only unitarity means setting ∆ * j = j + 1 for all j. When c T = ∞ we know, however, that there exists an N = 8 SCFT (namely ABJM theory with k = 1, 2 and N = ∞) for which ∆ * j = j + 2. In Table 10 we show the upper bounds on λ 2 (B,+) and λ 2 (B,2) that we obtain under the assumption that ∆ * j = j + 1 for j < J and ∆ * j = j + 2 for j ≥ J as we vary J.  (B,+) and (2, 0) [0200] (B,2) mutliplets. These bounds are computed for c T = ∞ and under the assumption that ∆ * j = j + 1 for j ≥ J and ∆ * j = j + 2 for j < J.
least restrictive and they hold in any SCFT with N = 8 supersymmetry. As we increase J, the bounds converge to In Figures 5 and 6, we show upper bounds on λ 2 (B,+) and λ 2 (B,2) for a wide range of c T . The bounds plotted in blue correspond to ∆ * j = j + 1 for all j and hold for any N = 8 SCFTs. The bounds plotted in orange are more restrictive. They are obtained with ∆ * j = j + 1 for all j > 0 and ∆ * 0 chosen approximately by the bounds given in Figure 1. At large c T , these latter bounds approach approximately the limits in (6.19). At c T = 16, the upper bound for  Table 9.  Table 9.
Lastly, in Figure 7

Discussion
Our conformal bootstrap analysis provides us with true non-perturbative information about N = 8 SCFTs. Generically these theories are strongly coupled, and the conformal bootstrap is possibly the only available method to study them. Indeed, except for the U (1) × U (1) ABJM theory (which is trivial) and BLG theory at large k (which has no known gravity description), all known N = 8 SCFTs are strongly interacting. In addition, while the large N limit of the ABJM theory can be studied through its weakly coupled supergravity dual, it is hard to obtain detailed information directly from the field theory side.
The operator spectrum and OPE coefficients of theories that saturate the bounds provided by the numerical bootstrap can be determined numerically [30]. It is therefore interest- In 4-dimensional N = 4 SCFTs the contributions coming from short multiplets to fourpoint functions of 1/2-BPS operators can be determined analytically. One way to fix these contributions is by proving a non-renormalization theorem [91], showing that they are the same as in the free theory. In interacting 3-dimensional N = 8 theories there are no continuous couplings that one can tune to obtain a free theory. One could then argue that the absence of continuous couplings implies that the short multiplet contributions cannot be determined in the same way as in four dimensions.
However, in 4-d N = 4 SCFTs one can also fix the short multiplet contributions by using 27 Indeed, in other numerical bootstrap studies, kinks were shown to correspond to abrupt changes in the operator spectrum [22]. 28 The solution to the Ward identity is slightly different in those cases (see [1]). 29 In the 4-dimensional N = 4 theory such a study was performed in [26].
only superconformal invariance and crossing symmetry, without ever referring to a free theory or Lagrangian description. 30 It is possible that such contributions to the four-point functions of 1/2-BPS operators in 3-d N = 8 theories could also be fixed in this fashion. In this work we have solved the differential relations between the crossing equations, which were implied by superconformal invariance, in a series expansion around the crossing symmetric point.
While this solution was sufficient for the purpose of implementing the numerical bootstrap, it is possible that with a more thorough analysis of those equations, one would be able to determine the contributions from short operators in three dimensions as well. We hope to return to this interesting question in the future.

Acknowledgments
We A so(d, 2) Conformal Blocks It was shown in [60] that conformal blocks in any dimension can be written as series expansions in two variable Jack polynomials. Jack polynomials 31 can be defined using Gegenbauer polynomials as P (ε) The conformal blocks can then be written as r m,n (∆, j)P The coefficientsr mn can be computed using the recursion relation Our normalization convention is fixed by taking r 00 = 1/4 ∆ . With this convention we have wherex is taken to zero first. This normalization is adapted to the r, η coordinates 32 of [67] which are related to x andx by The normalization (A.5) is equivalent to In practice, to approximate conformal blocks in our numerics we use the recursion relations of [20,67].

B Characters of osp(8|4)
In this section we will review the character formulas of osp(8|4), which were computed in [65], as well as their decomposition under osp(8|4) → so(3, 2) ⊕ so (8). This decomposition was used in Section 4 to determine which conformal primaries reside in each supermultiplet appearing in the O 35c × O 35c OPE, and, in particular, to derive Table 1 and Tables 4-8.
The osp(8|4) characters are defined by where ∆, j, and r = (r 1 , . . . , r 4 ) ∈ 1 2 Z 4 are, respectively, the conformal dimension, spin and so (8)  The characters are most easily computed by first computing the Verma module characters.
Verma modules are infinite (reducible) representations obtained from highest weights by acting unrestrictedly with lowering ladder operators. For instance, the su(2) and so (8) Verma module characters are given by The characters of irreducible representations are obtained from the Verma module characters by Weyl symmetrization, which projects out all the null states in the Verma module.

C Superconformal Blocks
Let us write our results for the superconformal blocks in order of increasing complexity. In all the supermultiplets, we normalize the coefficient of the superconformal primary to one.
The results are presented in terms of the R-symmetry channels A ab (u, v), which were defined in (2.6).

D Recurrence Relations
In this section we collect various recurrence relations that were derived in [1] and used in section 4.2 to derive the superconformal blocks. We also correct various mistakes in Appendix D of [1], some of which lead to inconsistencies with known results in four dimensions.

E Details of Central Charge Computation
Here we lay out details of computations in Section 5. The theories we consider have a natural parity transformation that flips the sign of k, so we choose k > 0 without loss of generality.
(E. 13) Due to w independence we again obtain a delta function integration setting y + z = 0. Since tanh(−x) = − tanh x, one sees that (E.13) vanishes.
(E.18) 35 The computations in this appendix are similar to the ones performed in [95].