The 3d N = 6 Bootstrap: From Higher Spins to Strings to Membranes

We study the space of 3d N = 6 SCFTs by combining numerical bootstrap techniques with exact results derived using supersymmetric localization. First we derive the superconformal block decomposition of the four-point function of the stress tensor multiplet superconformal primary. We then use supersymmetric localization results for the N = 6 U(N)k×U(N +M)−k Chern-Simons-matter theories to determine two protected OPE coefficients for many values of N,M, k. These two exact inputs are combined with the numerical bootstrap to compute precise rigorous islands for a wide range of N, k at M = 0, so that we can non-perturbatively interpolate between SCFTs with M-theory duals at small k and string theory duals at large k. We also present evidence that the localization results for the U(1)2M × U(1 + M)−2M theory, which has a vector-like large-M limit dual to higher spin theory, saturates the bootstrap bounds for certain protected CFT data. The extremal functional allows us to then conjecturally reconstruct low-lying CFT data for this theory. ar X iv :2 01 1. 05 72 8v 1 [ he pth ] 1 1 N ov 2 02 0

We will achieve this by studying the four-point function SSSS of the scalar superconformal primary S of the stress tensor multiplet. Because N = 6 is less than the maximal N = 8 possible superconformal symmetry in three dimensions, the stress tensor multiplet is only 1/3-BPS [27,28]. 3 Three-dimensional N = 6 SCFTs provide a unique window into theories of quantum gravity via the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence [30][31][32]. 4 Firstly, due to the large amount of supersymmetry these theories are amenable to exact computations of various protected quantities, as we will describe in more detail below.
by SU , or where extra U (1) factors with certain Chern-Simons levels appear [38]. 5 This set of theories is very rich, as can be seen from various limits in parameter space. In the limit k N, M , the SCFTs are weakly coupled and one can use perturbation theory. In the limit M, k N with M/k fixed there is a weakly-interacting higher-spin dual description in AdS 4 . In the limit N M, k there is a weakly-interacting M-theory dual description.
Finally, in the limit N, k M with N/k fixed there is a weakly-interacting type IIA string dual description.
Given a known CFT, the ideal outcome of a numerical conformal bootstrap study is to show that under a certain set of assumptions the CFT is unique, or that at least some of the CFT data can be uniquely determined. This has so far been accomplished in one of two ways. The first is to find (small) islands in the space of CFT data surrounding the known CFT. If one can argue that these islands shrink to a point as precision is increased, then one can determine part of the CFT data. For instance, in the case of the 3d Ising model, a numerical bootstrap study of the four-point functions of the only two relevant operators gave allowed regions in the space of scaling dimensions of these operators that look like small islands surrounding the known values of these scaling dimensions for the Ising CFT [39][40][41].
Besides the 3d Ising model, there are very few other examples where similar small islands have been found with minimal physical assumptions-see, for instance, [42][43][44][45]. One can also find islands in the space of OPE coefficients multiplying semishort superconformal blocks which, due to supersymmetry, cannot be deformed into long blocks. Such islands were found for 3d N = 8 SCFTs (which are, of course, a particular case of the 3d N = 6 SCFTs we study here) in [9,12], in which case exactly computable localization results for certain protected OPE coefficients were also inputted to further shrink the islands and identify them with known theories. The theories studied in [9,12] were strongly-interacting gauge theories which, in the limit where the rank of the gauge group is taken to infinity, are dual to weaklycoupled M-theory in eleven dimensions. In the present work we will find similar islands for the much larger class of 3d N = 6 SCFTs.
The other method used to "solve" for part of the CFT data of a known CFT applies to cases where one can argue that this CFT saturates certain bounds in the limit of infinite bootstrap precision. In such a case, it is believed that there is a unique solution to the crossing equation and the CFT data can be extracted using the extremal functional method [46][47][48]. 6 5 We will find evidence from supersymmetric localization that all these more exotic versions have the same correlators we consider as certain U (N ) k × U (N + M ) −k or SO(2) 2k × U Sp(2 + 2M ) −k theories, so we can restrict to these standard theories for simplicity. 6 Ref. [49] showed that it is sometimes possible that there could be several extremal functionals, but in all cases that were studied they produced the same CFT spectrum.
One application of this method has been to the 3d Ising model, which was argued to saturate the lower bound on the coefficient c T which appears in the stress-tensor two-point function [48,50]. In this paper we will find compelling evidence that the U (1) 2M × U (1 + M ) M ABJ theory, which has a higher-spin limit at large M , also saturates certain bootstrap bounds.
This makes the theory amenable to a precision bootstrap study. It is worth noting that the more restrictive bounds on N = 8 SCFTs derived in [10,12] were saturated by the U (N ) 2 × U (N + 1) −2 ABJ theory with N = 8 supersymmetry, which in the large N limit has an M-theory dual description. So the present work shows that one can use the numerical bootstrap to study both SCFTs with supergravity and higher-spin dual descriptions.
As in the N = 8 case studied in [10,12], our bootstrap analyses are aided by supersymmetric localization, allowing us to analytically compute certain protected data that appears in SSSS . Exact computations using supersymmetric localization 7 are possible in any N ≥ 2 SCFT in 3d [52,53], but when N < 4 they can only give local CFT data related to conserved currents, such as c T . For N ≥ 4 SCFTs, however, one can combine localization with the 1d topological sector discovered in [9,54], whose explicit description for Lagrangian theories was determined in [55][56][57], to compute all half-BPS (in N = 4 language) OPE coefficients. For instance, we can use localization in our N = 6 case to compute not only c T but also a certain 1/3-BPS OPE coefficient (that would be half-BPS in N = 4) that appears in SSSS . For the U (N ) k × U (N + M ) −k theory these computations take the form of N dimensional integrals, which can be evaluated exactly for small N , and also to all orders in 1/N using the Fermi gas technique [58,59]. For the U (1) k × U (1 + M ) −k and SO(2) 2k × U Sp(2 + 2M ) −k cases we derive one-dimensional integrals that can be computed exactly for all M and k.
The rest of this paper is organized as follows. In Section 2 we review basic properties of SSSS and derive the superblock decomposition. In Section 3, we compute c T and the squared 1/3-BPS OPE coefficient λ 2 (B,2) 022 2,0 for various choices of M, N, k. In Section 4, we use the numerical bootstrap to compute non-perturbative bounds on CFT data, which we then combine with the localization results to derive precise islands and study the U (1) 2M × U (1 + M ) −2M theory using the extremal functional. Finally, in Section 5, we end with a discussion of our results and of future directions. Various technical details are discussed in the appendices. We also include an attached Mathematica notebook with the explicit SSSS superconformal blocks. 7 For a review, see [51].

Superconformal block expansion of SSSS
In this section we derive the superconformal block expansion for the SSSS four-point correlator. We begin with a brief review of constraints that superconformal symmetry places on SSSS ; a more detailed discussion can be found in Section 2 of [60]. Section 2.2 restricts the supermultiplets which can appear in the S × S OPE, and hence exchanged in SSSS , to a small number of possibilities. Section 2.3 applies the superconformal Casimir equation to each of the allowed supermultiplets to fix the superconformal blocks which contribute to SSSS . Finally, in Section 2.4 we determine the superblock decomposition for SSSS in free fields theories.

Constraints from conformal symmetry and R-symmetry
In any 3d N = 6 SCFT, the stress tensor sits in a 1/3-BPS multiplet [27,28]. The superconformal primary of this multiplet is a dimension 1 scalar operator transforming in the 15 of the so(6) R-symmetry algebra. Using the isomorphism so(6) ∼ = su(4), we will write this operator as a 4 × 4 traceless hermitian matrix S a b ( x), where a = 1, . . . , 4 and b = 1, . . . , 4 are su(4) fundamental and anti-fundamental indices, respectively. To avoiding carrying around indices, we find it convenient to contract them with an auxiliary matrix X, thus defining S( x, X) ≡ X a b S b a ( x) . (2.1) We normalize S( x, X) such that its two-point function is S( x 1 , X 1 )S( x 2 , X 2 ) = tr(X 1 X 2 ) x 2 12 . (2.2) We will not need many details about other operators in the stress tensor multiplet, but for completeness we list the conformal primaries of this multiplet in Table 1. Apart from the superconformal primary, the only operators that will appear in the discussion below are the fermions χ α , F α and F α , which all have dimension 3/2 and transform in the 6, 10, and 10 of so(6) R , as well as the (pseudo)scalar operator P of dimension 2 transforming in the 15 of so(6) R . 8 As discussed in [60], conformal and R-symmetry invariance imply that the four-point function of S( x, X) must take the form where we define the R-symmetry structures it is not the most convenient form to work with for writing a conformal block decomposition because each conformal block will contribute to several different S i . To do better, we can take linear combinations of S i such that each such linear combination corresponds to a specific so (6) irrep being exchanged in the s-channel OPE. The possible such irreps are those appearing in the tensor product We define S r to receive contributions only from operators in the s-channel OPE that belong to so(6) R irrep r, so that [60] Each of the functions S r can then be expanded as a sum of conformal blocks where the sum is taken over all the distinct conformal primary operators O ∆, ,r transforming in the representation r which appear in the S × S OPE. In (2.9), ∆ and are the scaling dimension and spin, respectively, of O ∆, ,r .
Invariance under the full osp(6|4) superconformal algebra relates the various four-point functions of stress-tensor multiplet operators, and furthermore it imposes relations on the S i (U, V ) defined above. As shown in [60], there are two such relations obeyed by the S i and they take the form (2.10)

The S × S OPE
As a first step towards determining the superconformal block decomposition of the SSSS correlator, we turn to the task of determining which N = 6 supermultiplets may appear in the S × S OPE. By using the R-symmetry selection rules and the fact that S( x, X) is a 1/3-BPS operator we can restrict our attention to only a handful of N = 6 supermultiplets.
In Section 2.3 we can then apply the superconformal Casimir equation in order to fully fix the superconformal blocks corresponding to each of these supermultiplets.

N = 6 Supermultiplets
The unitary multiplets of a 3d N = 6 theory are given in [27,28]. Each multiplet can be labeled by the conformal dimension ∆, spin , and so (6)  [0a 2 a 3 ] 1/6 (A, +) [ below the end of the lower continuum in (2.11), also obeying shortening conditions. Multiplets can furthermore be distinguished by their BPSness. For generic representations A-type multiplets are 1/12-BPS and B-type multiplets are 1/6-BPS, but for specific R-symmetry representations the multiplets may be higher BPS. We list all possible multiplets in Table 2.
Note that the stress-tensor multiplet discussed in the previous subsection is a (B, 2) multiplet in the notation of Table 2.
Of course, not all possible multiplets contain operators that can appear in the S ×S OPE due to various selection rules. Note for instance that the operators in the S × S OPE must transform in the irreducible representations of so(6) which appear in (2.7). Due to 1 ↔ 2 crossing symmetry even spin operators must be in the 1, 15, 20 , or 84 while odd spin operators must be in the 15, 45, or 45. A large number of supermultiplets contain operators in at least one of these irreps, so by themselves these conditions are not very restrictive.
We can do better by using the fact that S( x, X) is a 1/3-BPS operator, and as such is annihilated by certain Poincaré supercharges. If Q is a Poincaré supercharge annihilating S( x, X) (for any x but a specific X), then it also annihilates S( x, X)S( y, X). We will explore the consequences of this fact in the next subsection.

Operators in the S × S OPE
Let us begin by writing the generators of osp(6|4) in terms of the so(6) and sp(4) Cartan subalgebras. The Lie algebra so(6) has a three dimensional Cartan subalgebra, spanned by orthogonal operators 9 H 1 , H 2 , and H 3 . The other twelve R-symmetry generators take the form: where for each R the subscripts are correlated and label the weights of each of these generators under the Cartan subalgebra: (2.14) We take the simple roots of so(6) to be the raising operators while their corresponding lowering operators are A highest weight state is one that is annihilated by each element of R + ; the highest weight state of the 15 is then R 1,1,0 .
We perform a similar procedure with the conformal group sp(4). We can take one Cartan element to be the dilatation operator D and the other to be the rotation operator J 0 . The other two rotation operators are the raising and lowering operators J ± . The H i , D, and J 0 span a Cartan subalgebra of osp(6|4).
(The sign in the superscript is uncorrelated with the signs in the subscripts.) Note that the Qs have scaling dimension +1/2 and the Ss have scaling dimension −1/2, so their charges under dilatation operator are also manifest in this notation.
Given an irreducible representation of osp(6|4), the highest weight state |∆, , r is one which is annihilated by the raising operators of osp(6|4): where R + ∈ R + , and is an eigenstate of each of the Cartans: (2.16) Here ∆ and are the conformal dimension and spin of the superconformal primary, and the r = (r 1 , r 2 , r 3 )'s are the highest weight states of the R-symmetry representation of the superconformal primary. These weights are related to the Dynkin label [a 1 a 2 a 3 ] by the equation and always satisfy r 1 ≥ r 2 ≥ r 3 .
We will find it useful to further define along with analogous definitions for the S-supercharges.
Let Φ 2,2,0 ( x) be any operator which appears in the OPEŜ 1,1,0 ×Ŝ 1,1,0 and |Φ = Φ 2,2,0 (0) |0 the associated state. This is the highest weight state of an 84 multiplet which is annihilated by R + and Q + . Without loss of generality we can take this operator to be a conformal primary which is annihilated by J + ; if it is not we can act with the raising operators K µ and J + to construct such an operator. Because any operator S + ∈ S + is of the form [K, Q + ] for some Q + ∈ Q, we find that S + also annihilates |Φ . So in total, we have the conditions (2.21) Our task it to determine which supermultiplets |Φ may belong to.
By acting with operators in S on |Φ we can construct states of lower conformal dimension. Consider first constructing a state |O by acting with all eight supercharges in By assumption |Φ satisfies (2.21), and it is straightforward to see that |O then also satisfies (2.21). Because the S operators anticommute with themselves, we furthermore find that any operator in S 0 ∪ S − annihilates |O . The state |O is therefore annihilated by all of the S and by J + and R + , and so either |O is the highest weight state of the superconformal primary of the supermultiplet, or |O = 0. In either case we conclude that there exists some 0 ≤ k ≤ 8 for which acting with any k + 1 operators from S 0 ∪ S − annihilates |Φ , but for which acting with just k operators does not: for some string of k operators S i ∈ S 0 ∪S − . It is again easy to see that |Φ satisfies (2.21) and is annihilated by the operators in S 0 ∪ S − ; we hence conclude that |O is the highest weight state of the superconformal primary of the multiplet. Note that the different orderings of the operators S i in (2.23) are equivalent, up to an overall minus sign.
Let us denote the so(6) weights of |O by where v i = (v i1 , v i2 , v i3 ) are the so(6) Cartans of the S i we act with in (2.23). Because |O is a highest weight state we must have which provides a useful additional constraint on (2.23).
As discussed in the previous section, |O belongs to one of the three types of unitary representations of osp(6|4). If |O is part of a long multiplet, it is annihilated by all of the raising operators (2.15) but satisfies no other conditions. If instead it belongs to an A-type multiplet it satisfies shortening conditions [27] Q − q 1 ,q 2 ,q 3 − with the specific weights q i depending on the so(6) weights of |O . Finally, if it is part of a B-type multiplet, it is annihilated by both Q + q 1 ,q 2 ,q 3 and Q − q 1 ,q 2 ,q 3 for specific weights q i . Furthermore for B-type multiplets |O is always a scalar.
With this information out of the way, we now simply enumerate all possibilities for (2.23), subject to the constraint (2.25). The simplest case is where |Φ is itself the highest weight primary. Then we have a (B, 2) multiplet in the 84.
Next let us extend this reasoning to the case Let us next consider the cases The next cases to consider are Case (2.29c) violates (2.25) and so is forbidden. For other two cases we find some combination of Q − 1,0,0 and Q + 1,0,0 annihilate |O , so |O must be either an A-type or B-type multiplet. For Finally, we have the case Now |O need not be annihilated by an supercharges, so it can be a long multiplet. The condition (2.25) however forces it to be an so(6) singlet. If |O satisfies any shortening conditions it must be either a conserved current multiplet or the trivial (vacuum) multiplet, but neither of these contain an operator in the 84 so these are both ruled out.
We summarize our results in the first 11 lines of Table 3, where we give the full list of all possible superconformal blocks which contain an operator in the 84.
The shortening conditions on S imply that Q ± 1,0,0 annihilatesŜ 1,±1,0 andŜ 1,0,±1 , and so must annihilate Ψ 2,1,1 ,Ψ 2,1,−1 and Ξ 2,0,0 . We can then repeat the analysis previously performed for Φ 2,2,0 , and recover the same list of multiplets that we found by analyzing the conditions for the operators in the 84. We thus conclude that any supermultiplet appearing in S × S not listed in the first 11 lines of Table 3 can contain non-zero contributions only from operators in the 15 and 1.
Restricting the supermultiplets for which only operators in the 15 and 1 appear in the S × S OPE is more subtle and requires the use of superconformal Ward identities. While we include the details of this analysis in Appendix A, the result is very simple. There are only 3 such supermultiplets: the identity supermultiplet (containing just the identity operator), the stress tensor multiplet itself, as well as a conserved multiplet (A, cons.) whose superconformal primary is an so(6) singlet scalar with scaling dimension 1. Table 3 shows a summary of our analyses containing all the possible supermultiplets which can appear in the S × S OPE. By using the superconformal Casimir equation we shall find that most of these supermultiplets can in fact be exchanged; we mark those that cannot in red.

Superconformal Casimir equation
Just as the s-channel conformal blocks are eigenfunctions of the quadratic conformal Casimir when the Casimir acts only on the first two operators in a four-point function, superconformal blocks are eigenfunctions of the quadratic superconformal Casimir (see for instance [14,61] for similar discussions with less supersymmetry). In the conformal case, this fact implies that the conformal blocks obey a second order differential equation. In the superconformal   To fix conventions, let us denote by M α β , P αβ , K αβ , and D the Lorentz generators, the momentum generators, the special conformal generators, and the dilatation generator.
Here α, β = 1, 2 are spinor indices raised and lowered with the epsilon symbol. The precise normalization of these operators is fixed by our convention for the conformal algebra, which we give in Appendix B. It is straightforward to check that in these conventions, the quadratic conformal Casimir commutes with all conformal generators. The normalization of M α β is such that when acting on an operator of spin placed at x = 0, the first term in this expression evaluates . Similarly, when acting on an operator of scaling dimension ∆ also placed at x = 0, the dilatation operator evaluates to D = ∆. Since a conformal primary O ∆, of dimension ∆ and spin , placed at x = 0, is annihilated by all special conformal generators K αβ , it follows that O ∆, (0) is an eigenstate of C C with eigenvalue ( + 1) + ∆(∆ − 3). By conformal symmetry this implies that for any operator O( x) that belongs to a conformal multiplet whose conformal primary has dimension ∆ and spin , we have The discussion in the previous paragraph can be generalized to the superconformal case for a theory with N -extended superconformal symmetry. (We will of course set N = 6 shortly, but let us keep N arbitrary for now.) The superconformal algebra is generated by the conformal generators M α β , P αβ , K αβ , and D described above, as well as the Poincaré supercharges Q αI , the superconformal charges S αI , and the R-symmetry generators R IJ . Here, I = 1, . . . , N is an so(N ) vector index, and R IJ is anti-symmetric. The normalizations of these operators is fixed by the commutation and anti-commutation relations in Appendix B.
Using these commutation relations, one can check that the quadratic superconformal Casimir commutes with all the conformal generators. Here, the R-symmetry generators are such that when acting on an operator in a representation r of so(N ), we have is the eigenvalue of the quadratic Casimir of so(N ) normalized so that λ R (N ) = N − 1. For the case of so(6) and the various representations we will encounter, we have the quadratic Casimir eigenvalues in Table 4. Eq. (2.33) implies that when acting on the superconformal 45 16 84 20 Table 4: Eigenvalues of C R in the N = 6 case where the R-symmetry algebra is so(6) R .
primary operator O ∆, ,r of spin , dimension ∆, and R-symmetry representation r, placed at x = 0, the superconformal Casimir gives ∆(∆ + N − 3) + ( + 1) − 1 2 λ R (r); this follows because any such an operator is annihilated by S α I . Superconformal symmetry then implies that if O is any operator in a superconformal multiplet whose superconformal primary has dimension ∆, spin and R-symmetry irrep r, we have Let us now use the Casimirs above to obtain an equation for the superconformal blocks.
Suppose we have four superconformal primary scalar operators φ i , i = 1, . . . , 4, of dimension ∆ φ and R-symmetry representation r φ . The four-point function has the conformal block decomposition A superconformal block corresponding to the supermultiplet M r 0 ∆ 0 , 0 whose superconformal primary has quantum numbers (∆ 0 , 0 , r 0 ) consists of the conformal primary operators in the sum on the RHS of (2.35) that belong to the same supermultiplet as O ∆ 0 , 0 ,r 0 : Let us now applying the superconformal Casimir operator (2.33), assuming to act only on the first two operators. To specify which of the four φ's an operator is acting on, let us use a subscript " (12)" if the operator is acting on φ 1 and φ 2 and a superscript "(i)" if the operator acts only on φ i . From (2.34), we see that When we apply this expression to (2.36), we act with the Casimirs with upper index (i) on the LHS of the equation, and with the ones with upper index (12) on the RHS of the gives λ R (r). Thus, we obtain the following relation: The RHS of Eq. (2.38) can be easily evaluated provided we know all the conformal primaries occurring in the multiplet M r 0 ∆ 0 , 0 . To evaluate the LHS, note that and so Eq. (2.38) becomes In general, there are Ward identities relating the LHS of (2.41) to φ 1 φ 2 φ 3 φ 4 , but the relations may not be sufficient to determine the LHS of (2.41) completely in terms of This general discussion can be applied to the case of interest to us, namely the SSSS correlator in 3d N = 6 SCFTs. If we replace φ i ( x i ) by S( x i , X i ), then SSSS can be expanded in R-symmetry channels as in (2.3), and so can all the equations above. In particular, we replace c ∆, ,r → c i ∆, ,r B i in all these equations, with c i ∆, ,r , placed in a row vector, determined in terms of the coefficients a ∆, ,r defined in (2.9) via with α ∆, ,r evaluated in this particular case to The remaining challenge is to evaluate the LHS of (2.43). This can be done by noting that Q αI S( x, X) is a linear combination of the fermions χ, F , and F in the stress tensor multiplet, as given in Appendix D of [60]. Consequently, the LHS of (2.43) can be written in terms of the functions of (U, V ) appearing in the correlators χχSS , χF SS , F F SS , and F F SS . These functions are denoted by C i,a , E i,a , F i,a , and G i,a , respectively, in Appendix D of [60]. Here, the index i runs over the R-symmetry structures and the index a = 1, 2 runs over the two spacetime structures of a fermion-fermion-scalar-scalar correlator. Denoting with the coefficients β i,n given by Thus, Eq. (2.41) reduces to the 6 equations (one for each i): (2.47) To use this equation for finding the coefficients c i ∆, ,r of a given superconformal block, one should also expand the fermion-fermion-scalar-scalar correlators on the LHS in conformal blocks corresponding to operators belonging to the supermultiplet M r 0 ∆ 0 , 0 . Fortunately, we do not have to do this for all 24 functions X n,a because, as explained in Appendix D of [60], C i,a , E i,a , F i,a , and G i,a can be completely determined from S i and F 1,a . Since we have already expanded the S i in conformal blocks, all that is left to do is to also expand F 1,a .
The s-channel conformal block decomposition of a fermion-fermion-scalar-scalar fourpoint function was derived in [62]. For each conformal primary being exchanged, there are two possible blocks appearing with independent coefficients. For F 1,a , if we denote the corresponding coefficients by d ∆, ,r for the first block and e ∆, ,r for the second block, we can then write: where g ∆, are the scalar conformal blocks appearing above and D 1,2 are differential operators: (2.50) (Each doublet of functions (X n,1 , X n,2 ) appearing on the LHS of (2.47) has a similar block decomposition, but as mentioned above, we only need this decomposition for (F 1,1 , F 1,2 ).) Using the relations between X n,a and S i and F 1,a given in Appendix D of [60] together with the decompositions (2.48) and (2.49), we obtain a system of linear equations for c i ∆, ,r , d ∆, ,r , and e ∆, ,r that has to be obeyed for all values of (U, V ). Expanding g ∆, to sufficiently high orders in U is then enough to determine the linearly-independent solutions of this system of equations, and thus determine the coefficients c i ∆, ,r of the superconformal block corresponding to the supermultiplet M r 0 ∆ 0 , 0 . We performed this analysis for all the multiplets described in Table 3. The coefficients c i ∆, ,r for each multiplet are included in the attached Mathematica notebook. The multiplets marked in red in Table 3 did not give solutions to the system of equations that determines the c i ∆, ,r . For each of the remaining multiplets we found between one and three solutions. Since any linear combination of superconformal blocks is a superconformal block, we are free to choose a basis of blocks with specific normalizations. In other words, for the coefficients a ∆, ,r in (2.9) can be written as where I ranges over all superconformal blocks, λ 2 I are theory-dependent coefficients, and a I ∆, ,r represent the solution to the super-Casimir equation for superconformal block I, normalized according to our choosing. In Table 5, we list all the superconformal blocks as well as enough values for a I ∆, ,r in order to determine the normalization of the blocks. 11 A superconformal block G I is simply where the index I = M r 0 ,n ∆ 0 , 0 of the block encodes both the supermultiplet M r 0 ∆ 0 , 0 as well as an integer n = 1, 2, . . . denoting which block this is according to Table 5. (In the cases where there is a single superconformal block per multiplet, we omit the index n.) As discussed in Appendix B.3 of [60], the stress-tensor multiplet forms a representation not only of the superconformal group OSp(6|4), but also of a larger group (Z 2 ×Z 2 ) OSp(6|4) which includes both a parity transformation P and discrete R-symmetry transformation Z.
The parity transformation P extends the spacetime symmetries from Spin(3, 2) ∼ = Sp(4, R) to P in (3,2), while Z extends the R-symmetry group from SO (6) to O(6). In any local CFT the scalar three-point function SSS is non-zero, which implies that in a Z-invariant theory the operator S transforms as a pseudotensor 15 − , while the supercharges transform as O (6) vectors.
Reflection positivity implies that the coefficients a ∆, ,r in (2.9) are non-negative for all r.
Because for each superconformal block in Table 5 there exists an operator that receives contributions only from that block, it follows that the coefficients λ 2 I in (2.51) are nonnegative. This is the reason why we wrote these coefficients in (2.51) manifestly as perfect squares. They are the squares of real OPE coefficients. 12 Let us end this section by describing the unitarity limits of the long blocks obtained by taking ∆ → + 1. For the scalar blocks, we obtain (up to normalization) either a spin-0 conserved block for the parity-even structure or a (B, 1) , ≥ 2 even [011] 1,0 a 1,0,15s = 1 + − Table 5: A summary of the superconformal blocks and their normalizations in terms of a few OPE coefficients. The values a ∆, ,r in this table correspond to a I ∆, ,r in Eq. (2.51)-we omitted the index I for clarity. Note that the (A, ±) are complex conjugates and do not by themselves have well defined Z parity, but together they can be combined into a Z-even and a Z-odd structure.
Lastly, for even ≥ 2 we have three superconformal blocks. The parity even one approaches a spin-conserved block, while the parity odd ones approach the two superconformal blocks for the (A, +) [020] +2, multiplet: Even though the blocks on the RHS of (2.54)-(2.56) involve short or semishort superconformal multiplets, they sit at the bottom of the continuum of long superconformal blocks.
All other short and semishort superconformal blocks are isolated, as they cannot recombine into a long superconformal block. In particular, if the correlator SSSS contains one of these isolated superconformal blocks, any sufficiently small deformation of SSSS also must, while the other blocks can instead disappear by recombining into a long block. This distinction will be important when we consider the numerical bootstrap.

Examples: GFFT and free N = 6 hypermultiplet
There are two theories for which we can determine the superconformal block decomposition and all CFT data. The first is the generalized free field theory (GFFT), where correlators of S are computed using Wick contractions with the propagator S( x 1 , X 1 )S( x 2 , X 2 ) = tr(X 1 X 2 ) | x 12 | . In terms of the functions S i (U, V ), the SSSS correlator is: This theory does not have a stress-energy tensor, and it is thus non-local and therefore not of primary interest here. However, the GFFT four-point function does represent the leading term in the strong-coupling limit of correlators of the local SCFTs that are discussed in the next section. If we think about the S operators as single-trace, then in the superconformal block decomposition of SSSS only double-trace operators appear, with schematic form S∂ µ 1 · · · ∂ µ p S of spin dimension 2 + + 2p with positive integer and p. The GFFT defined as above does not necessarily have N = 6 supersymmetry, but it can be completed into an N = 6 preserving theory by considering similar rules for calculating correlators of any four stress-tensor multiplet operators from Table 1. We can expand (2.57) in superconformal blocks to read off the CFT data given in Table 6.
The second theory that is exactly computable is a free theory. Let us consider four complex scalar φ a , a = 1, . . . , 4 and their complex conjugatesφ a , with the two-point function In this theory, we can consider the operator S( x, The SSSS correlator can then be computed using Wick contractions of the φ andφ's, and in terms of the S i it is given by As was the case with the GFFT, this correlator does not necessarily correspond to an N = 6 SCFT, but it can be embedded in one by considering the φ a as the components of an N = 6 hypermultiplet that also contains 4 complex fermions. The four-point function (2.58) can then be expanded into superconformal blocks to give the CFT data given in Table 6. Note that the free theory has the same spectrum as the GFFT, except that it also contains conserved current multiplets for each spin, has a stress tensor multiplet, and does not have 2,0 multiplet. For both the GFFT theory and the free theory of an N = 6 hypermultiplet, one can alternatively obtain the CFT data listed in Table 6 [38], up to discrete quotients that do not affect correlators of S. 13 In N = 3 SUSY notation, they are Chern-Simons-matter theories with two matter hypermultiplets. There are two possible families of gauge groups 13 See [63] for a conjectured classification that takes into account discrete quotients.
for N, M ≥ 1 where the hypermultiplets are in the bifundamental of SU (M ) × SU (N ), and with M ≤ |k| [34,35], which is the special case of (3.1) where L = 2 with q 1 = q 2 = 1 and with M + 1 ≤ |k| [35,37], is the L = 1, q = 1 case of (3.2). Sending k → −k gives a parity-conjugate theory, so without loss of generality we can focus on k > 0. Seiberg duality 14 The case SU (N ) k × SU (N ) −k describes the BLG theories [64][65][66]. 15 When N = 1, M = 0, the ABJM theory describes a free SCFT equivalent to the theory of eight massless real scalars and eight Majorana fermions described in Section 2.4. For M = 0 and N > 1, ABJM flows to the product of a free SCFT and a strongly-coupled SCFT, while for all other parameters ABJM theory has a unique stress tensor.
imposes additional equivalences between each family:

Exactly calculable CFT data
In the next section, we will derive numerical bounds on the CFT data of 3d N = 6 SCFTs parameterized in terms of c T , which is defined as the coefficient appearing in the two-point function of the canonically-normalized stress-tensor, and, in the normalization (2.2) for the external operator S( x, X), it is inversely related to the square of the OPE coefficient of the stress tensor multiplet: Here c T is defined such that it equals 1 for a (non-supersymmetric) free massless real scalar or a free massless Majorana fermion. Hence c T = 16 for the free N = 6 hypermultiplet described in Section 2.4, which also has N = 8 and is equivalent to ABJM theory with M = 0 and N = 1.
This c T is a particularly useful parameterization of physical theories. It can be computed exactly using supersymmetric localization for any N ≥ 2 SCFT with a Lagrangian description by taking two derivatives of the squashed sphere partition function with respect to the squashing parameter [68,69]. For theories with at least N = 4 supersymmetry, the stress tensor multiplet contains R-symmetry currents that from an N = 2 point of view are flavor currents, and one can argue that c T is proportional to the two-point functions of such flavor currents [11]. Such two-point functions can be computed by taking two derivatives of the round sphere partition function with respect to a mass parameter [70]. For the N = 6 theories that we focus on here, we will define a mass parameter m with a normalization such In these theories, supersymmetric localization [52] implies that the quantity Z(m) can be expressed as an N -dimensional integral for any k, M , 16 and can be evaluated exactly at small N and to all orders in 1/N for M ≤ k ≤ N using the Fermi gas method [10,58,59]. In particular, for N = 1 we can exactly compute the one-dimensional integral for any M, k, such as the large M ∼ k limit that describes the vector like limit, which can also be computed in a large M expansion to any order as described in [71]. For the various quantum gravity theories we discuss the leading order expressions for c T are then [11,72]: M-theory, Type IIA string theory : Higher-spin theory : This short OPE coefficient is the N = 6 analogue of the N = 8 short OPE coefficient computed in [10].
We should point out that in the limit in which log Z and its mass derivatives go to infinity, we have c T → ∞ and λ 2 (B,2) [022] 2,0 → 2. This is expected because in this limit the CFT correlators factorize, and the SSSS correlator is that of the GFFT theory described in Section 2.4. Indeed, as can be seen from Table 6 GFFT has λ 2 (B,2)  Using supersymmetric localization, the mass-deformed U (N ) k × U (N + M ) −k partition function can be reduced to M + 2N integrals [52,75]: up to an overall m-independent normalization factor. Our first task will be to write (3.11) as an N -dimensional integral that we can then evaluate more easily. For the massless case m = 0 such a reduction was achieved in [76]. In Appendix E.1, we extended their methods to the massive partition function (3.11) and show that where Z 0 is again an overall factor which is independent of the mass parameter m. Since our interest is ultimately in computing derivatives of log Z with respect to m, the value of Z 0 is unimportant. Let us now discuss various limits in which we can evaluate (3.12) exactly or approximately.

Small M, N, k
When M, N, k are small integers, we can evaluate (3.12) as contour integrals. Let us begin with the case N = 1. We must computê where we define π(x + i(l + 1/2)) k . (3.14) All poles of F M,k (x) are located at x = i 2 + iK for K ∈ Z . Furthermore F M,k (x) is periodic in the complex plane, with if a concise expression exists.
The above analysis can be generalized to the N > 1 case by repeatedly integrating over z k . When N = 2, for instance, we must evaluate We evaluate this by first integrating over z 1 while fixing |Im(z 2 )| < k 2 . We can perform this integral by closing the contour in the upper half complex plane and then summing over the poles, which occur at where K is a positive integer. Because both K(z) and tanh z are periodic in the complex plane, we need only sum the poles with imaginary part less than k; the rest can be resummed as a geometric series. Having integrated over z 1 , we perform the z 2 integral in a similar fashion. For general N we must repeat this process for each of the N integration variables.
We list results in Table 8 for the U (2) k × U (M + 2) −k theory, and in Table 9

Supergravity limit
We will also be interested in the large N expansion. Taking N large while keeping M and k fixed (with M ≤ k) corresponds to the M-theory limit, while taking N large while keeping M and N/k fixed describes the Type IIA string theory limit. Results for the M-theory limit were already computed in [10] to all orders in 1/N , which we now briefly review. Using the Fermi gas method [58], the mass deformed partition function was computed to all orders in where the constant map function A is given by [77].
We can now simply take derivatives of these exact functions as in We will use this expansion in the numerics section specifically for N = 10 and a range of M, k, which we summarize in Table 10.

SO(2) 2k × U Sp(2 + 2M ) −k theory
We now discuss the mass-deformed sphere partition function for the SO(2) 2k ×U Sp(2+2M ) −k theory. Using supersymmetric localization, this quantity can be written as an (M + 1)dimensional integral [52,78]: up to an overall m-independent factor. In [79] it was shown that the massless partition function m = 0 could be further simplified to a single integral. Generalizing their results to non-zero m is straightforward, as we outline in Appendix E.2. We show that where Z 0 is an overall constant which is independent of m. We will now compute mass derivatives of this quantity, first at finite M, k, and then in the large M and fixed λ = M/k expansion.

Finite M, k
For computing c T and λ 2 (B,2) [022] 2,0 let us use the first expression (3.30). We are thus led to evaluate where we define (3.32) The function G M,k (x) has poles at for integer N , and, as with F M,k (x) in the previous section, is periodic in the complex plane: . Unlike in the previous section, however, G M,k (x) does not vanish at infinity and so we cannot directly evaluate (3.31) by closing the contour in the upper-half plane. To fix this issue we instead compute the regularized integral in the limit ξ → 0 + . We then find that [022] 2,0 using (3.8) and (3.10). We list results for various M and k in Table 11.

Higher-spin limit
Finally, we study the large M limit of the SO(2) 2k × U Sp(2 + 2M ) −k theory, keeping λ = M k fixed. To this we shall find it useful to perform a change of variables   Expanding the other term in (3.35) at large k is straightforward. Performing a change of variables ξ = k −1/2 x, we then find that where the suppressed terms are higher order in m and/or k −1 . We can thus compute derivatives of Z M,k (m) at each order in k −1 as Gaussian integrals to get c T = 32k sin(πλ) π + 8(1 + 2 cos(πλ) + cos(2πλ)) + π sin(πλ) [  Comparing to the exact results in Table 11, we see that already for k = 2 the approximations (3.44) Together with (3.26), this expression will be useful in the next section.

Numerical bootstrap
We

Crossing equations
The position space crossing equations were written in (2.6). For the s-channel superblock expansion the nontrivial constraint is the one given by (x 1 , X 1 ) ↔ (x 3 , X 3 ). In terms of the S r (U, V ) basis in (2.8), the crossing equations (2.6) can be written in using a 6-component where we define Combining the crossing equations with the superconformal block decomposition, we can then define a d i I for each superconformal block I listed in Table 2 by replacing each S r in d i by G r I defined in (2.52). The crossing equations in terms of these d i I can then be written as [011] 1,0 [011] 1,0 where we normalized the squared OPE coefficient of the identity multiplet to λ 2 Id = 1, and parameterized our theories by the value of λ 2 (B,2) These six crossing equations are in fact redundant due to N = 6 superconformal symmetry, similar to the N = 8 case in [11,12]. It is important to remove these redundancies, since otherwise they cause numerical instabilities in the bootstrap algorithm. As in [12], we can do this using the explicit expressions for the crossing equations in (4.3) in terms of superblocks, where each S r (U, V ) is a linear combination of conformal blocks for each supermultiplet. We then expand in z,z derivatives as where z,z are written in terms of U, V as In the sums in Eqs. (4.4) we only consider terms that are nonzero and independent according to the definition (4.2). We then truncate these sums to a finite number of terms by imposing that p + q ≤ Λ , (4.6) and then consider the finite dimensional matrix d (p,q) i whose rows as labeled by i = 1, . . . 6 are those of d i , and whose columns as labeled by (p, q) are the coefficients of the ∂ p z ∂ q z S r (U, V )| z=z= 1 that appear in each entry of d i after expanding like (4.4) using the definition (4.2) of F ±,r (U, V ) in terms of S r (U, V ). Finally, we check numerically to see which crossing equations are linearly independent for each value of Λ, and find that a linearly independent subspace for any Λ is given by where we include all nonzero z,z derivatives for the crossing equations listed. 17 We now have all the ingredients to perform the numerical bootstrap using the crossing equations (4.3), where we restrict to the linearly independent set of crossing equations (4.7).
We can now derive numerical bounds on both OPE coefficients and caling dimensions using numerical algorithms that are by now standard (see for instance [12,39]) and can be implemented using SDPB [41,80]. In each case, the numerical algorithms involve finding functionals α that act on the vector of functions d i (U, V ) and return a linear combination of derivatives of these functions evaluated at the crossing-symmetric point U = V = 1/4. In all the numerical studies presented below, we will restrict the total derivative order Λ (see (4.6)) to be Λ = 39, and we will only consider acting with α on blocks that have spin up to max = 50.

Bounds on short OPE coefficients and the extremal functional conjecture
We begin by deriving numerical bootstrap bounds on the squared OPE coefficients λ 2 (B,2)    The existence of such an α implies that if s = 1, then in Eqs. (3.26) and (3.44), respectively, and we see that these values do lie outside the N = 8 region and come close to saturating the N = 6 bounds. We chose to plot the exact results for these particular theories because, of all the exact results that we computed, these ones are the SCFTs in their respective families that come closest to saturating the lower bounds in Figure 1. The red dots are slightly closer to the lower bound than the orange ones, as can also be seen analytically at large c T by comparing the 1/c 2 T terms in (3.26) and (3.44). We hence conjecture that, in the infinite Λ limit, it is the U (1) 2M × U (1 + M ) −2M theory that saturates the numerical lower bound. This is reminiscent of the N = 8 case in [10], where the U (N ) 2 × U (N + 1) −2 theory was found to saturate the corresponding lower bound.
To orient the reader about the spectrum of the superconformal block decomposition of the SSSS correlator in the higher-spin limit, we list the low-lying CFT data for a paritypreserving theory such as U (1) 2M × U (1 + M ) −2M in Table 12. 19 The spectrum is similar to that in Table 6, except that the conserved multiplets combine with double trace multiplets according to (2.54)-(2.56) to form single trace long multiplets whose scaling dimensions are close to unitarity. Since these multiplets are single trace, their OPE coefficients are O(c −1 T ), as opposed to all the other double trace operators whose OPE coefficients are O(c 0 T ). For N = 6 SCFTs that do not preserve parity, all the scaling dimensions for any given spin should appear in all structures.

Bounds on semishort OPE coefficients
Let us now discuss upper and lower bounds on OPE coefficients for isolated superconformal blocks that appear in the SSSS . (Recall that the isolated superconformal blocks are the short and semishort superblocks in Table 5 which do not appear on the RHS of (2.54)-(2.56).) Using the algorithm presented in (4.11)-(4.12), we determined such bounds as shows in Figure 2. In these plots, our Λ = 39 N = 6 upper/lower bounds are shown in black, and they can be compared to the Λ = 43 N = 8 bounds computed in [10], which in these figures are shown in blue. As in Figure 1 discussed above, in all these plots, the N = 6 and N = 8 lower bounds meet at around 16 c T ∼ .71. Note that the N = 6 upper/lower bounds do not converge at the GFFT and free theory points yet, whose exactly known values where listed in Table 6 and are denoted by gray dots, which is evidence that they are not fully parity-preserving higher-spin limit converged. The exception is the bound on the OPE coefficient for (A, +) [002] +2, , which is our most constraining plot.
In addition to the upper and lower bounds, in Figure 2 we also plotted in dashed red the values of the OPE coefficients as extracted from the extremal functional under the assumption that the lower bound of Figure 1 is saturated. As we can see, the extremal functional values for the OPE coefficients come close to saturating several of the bounds in this figure, but not all.

Bounds on long scaling dimensions
Lastly, let us show bounds on the scaling dimensions of the long multiplets. To find upper bounds on the scaling dimension ∆ * of the lowest dimension operator in a long supermultiplet with spin * that appears in (4.3), we consider linear functionals α satisfying for all short and semi-short I / ∈ {Id, (B, 2) [011] where we set all ∆ I to their unitarity values except for ∆ I * . If such a functional α exists, then this α applied to (4.3) along with the reality of λ I would lead to a contradiction. By running this algorithm for many values of (c T , ∆ I * ) we can find an upper bound on ∆ I * in this plane.
Since for the long multiplets Long For general N = 6 SCFTs, the bounds for different n need not be the same, but we do expect that a long multiplet Long in a generic N = 6 SCFTs will contribute to all superconformal structures and, if this is the case, the lowest dimension long multiplet must obey all the bounds obtained separately from each superconformal structure. Since the superconformal structures are distinguished by they parity P and Z charges (see Table 5), in an SCFT that preserves these symmetries, ∆ ( ,n) represents the upper bound on the lowest long multiplet with the P and Z charges that correspond to the structure Long [000],n ∆, as given in Table 5. superconformal structure, which for parity preserving theories has the same parity as the superprimary. The orange shaded region is allowed, and the plot ranges from the GFFT limit c T → ∞ to the free theory c T = 16. The black line denotes the N = 6 upper bound computed in this work with Λ = 39, the blue line denotes the N = 8 upper bound computed in [10] with Λ = 43. The red dashed line denotes the spectrum read off from the functional saturating the lower bound in Figure 1, which we identify with the U (1) 2M × U (1 + M ) −2M theory. The gray dots denote the GFFT and free theory values from Table 6.
Our bounds on the scaling dimensions of long multiplets of spin = 0, 1, 2 are presented in Figures 3-7. The bounds on the first superconformal structure, namely ∆ (0,1) for = 0, ∆ 1 for = 1, and ∆ (2,1) for = 2, shown in Figures 3, 4, and 5, respectively, are relatively straightforward to understand: they interpolate smoothly between the values of the corresponding scaling dimensions at the free N = 6 hypermultiplet theory at 16 c T = 1 and the GFFT at 16 c T = 0. This is unlike in the N = 8 case where the upper bounds on the scaling dimension exhibit kinks at 16 c T ∼ .71. The bounds on the other structures, namely ∆ (0,2) for = 0 and ∆ (2,2) and ∆ (2,3) for = 2, are more subtle. Let us start discussing ∆ (0,2) first. Recall that, as per (2.54), the unitarity limit of the Long superconformal structure, which for parity preserving theories has the same parity as the superprimary. The orange shaded region is allowed, and the plot ranges from the GFFT limit c T → ∞ to the free theory c T = 16. The black line denotes the N = 6 upper bound computed in this work with Λ = 39, the blue line denote the N = 8 upper bound computed in [10] with Λ = 43. The gray dots denote the GFFT and free theory values from Table 6. about the possibility of having a (B, 1) [200] 2,0 multiplet appearing in the S × S OPE. If we assume that there are no (B, 1) 2,0 operators that appear in the S × S OPE, then we obtain the bound in Figure 6. As we can see from this figure, the bound ∆ 0,2 smoothly goes from the GFFT value 1 at 16 c T = 0 to the free theory value 3 at 16 c T = 1. This suggests that it is possible for N = 6 SCFTs to not contain (B, 1) [200] 2,0 multiplets, and indeed the U (1) 2M ×U (1+M ) −2M theory is an example of an N = 6 SCFT with this property.
For ∆ (0,2) , the extremal functional result that we identify with the U (1) 2M ×U (1+M ) −2M theory, shown in red, is close to the unitarity value for large c T . This is suggestive of an approximately broken higher spin current, as one generically expects for such vector-like theories. For ∆ 1 and ∆ (2,1) we do not yet have sufficient precision to accurately show the extremal functional results. We would also expect to see approximately conserved currents in these sectors, but since they are single trace their OPE coefficients start at O(c −1 T ), which make them especially difficult to see from numerics.
(A, 1) [100] 7/2,3/2 N = 6 multiplet as per (D.4). Next, we can derive revised bounds ∆ (2,n) , with n = 2, 3, under the assumption that the S × S OPE contains the (A, 1) [100],n−1 7/2,3/2 superblocks. As we can see from Figure 7, we found that the bounds ∆ (2,n) are slightly above 5 for all c T , with little dependence on c T . This is consistent with the value at both GFFT and free theory. For comparison, we also showed the second lowest operator for N = 8 theories, which corresponds to the lowest long spin 2 N = 8 operator. 21 We do not show any extremal functional results for these plots, because we do not yet have sufficient numerical precision.

Islands for semishort OPE coefficients
In the previous subsection we discussed numerical bounds that apply to all 3d N = 6 SCFTs.
In particular, we noticed that the upper/lower bounds on (A, +) [022] 2,0 ) fixed to their values in each theory using the exact localization results of [9] for N = 2, 3, 4 as shown in Table 9 and the all orders in 1/N formulae in [10] for N = 10 as shown in Table 10.
extremely constraining. This implies that for a given value of c T , we could find a small island in the space of (λ 2 (A,+) [002] 5/2,1/2 , λ 2 (A,+) [002] 9/2,5/2 ) using the OPE island algorithm described in the previous subsection. We can make this island even smaller, and correlate it to a physical theory, by imposing values of c T and λ 2 (B,2) [022] 2,0 computed using localization in Section 3. We show the results of these plots for U (N ) k × U (N ) −k for a variety of N, k in the Figure 8.
Note that the islands are small enough that we can distinguish each value of N and k, which allows us to non-perturbatively interpolate between M-theory at small k and Type IIA at large k.
One difficulty with trying to fix a physical theory by imposing two exactly computed quantities, c T and λ 2 (B,2) [022] 2,0 fixed to their values in each theory using the all orders in 1/N localization formulae in [10] for N = 10. Note that the axes describe a very narrow range in parameter space.
theories with any real value of these parameters, so we are effectively trying to parameterize a 3-dimensional space of theories. Since we are only imposing two quantities, these islands are expected to have a finite area even at high numerical precision corresponding to the third direction in "theory space". Thankfully, this third direction appears to be very small. We can quantify this by fixing N = k = 10 and computing islands for several different values of M ≤ k/2 = 5. As shown in Figure 9, the island is not very sensitive to the value of M < N , which explains why we were able to get such small islands in a 3-dimensional conformal manifold by just imposing two values of the parameters.

Conclusion
In this paper, we have three main results. which extends the all orders in 1/N results for the M ≤ k ≤ N case [10]. Finally, we combined all these ingredients to non-perturbatively study the ABJM theories. In particular, by inputting the exact values of c T and λ 2 (B,2) [022] 2,0 for a given k, N and M = 0, we found precise rigorous islands in the space of semishort OPE coefficients that interpolate between M-theory at small k and type IIA string theory at k ∼ N . We also conjectured that in the infinite precision limit, the numerical lower bound on λ 2 (B,2) [022] 2,0 is saturated by the family of theories, which allowed us to non-rigorously read off all CFT data in SSSS using the extremal functional method. Interestingly, in the regime of large c T , we found a spin zero long multiplet whose scaling dimension approaches a zero spin conserved current multiplet at large c T , as expected from weakly broken higher spin symmetry.
There are several ways we can improve upon our 3d N = 6 bootstrap study. From the numerical perspective, it will be useful to improve the precision of our study. This is parameterized by the parameter Λ defined in the main text. While we used Λ = 39 in this work, which is close to the Λ = 43 values used in the analogous N = 8 studies [10,12], for N = 6 this value has not led to complete convergence. For instance, we found the lower bound c T ≥ 15.5, compared to the N = 8 result c T ≥ 15.9; both are expected to converge to the free theory c T = 16. More physically, we expect that approximately conserved currents should appear in the extremal functional that conjecturally describes the U (1) 2M × U (1 + M ) −2M theory. We found such an operator in the zero spin sector as shown in Figure 6, but for higher spin these operators are hard to see from the numerics at large c T , since their OPE coefficients start at O(c −1 T ). The main obstacle to increasing Λ at the moment is not SDPB, which due to the recent upgrade [80] can easily handle four crossing equations at very high Λ, but simply the difficulty in computing numerical approximations to the superblocks at large Λ. In particular it would be extremely useful to have an efficient code for approximations of linear combinations of conformal blocks with ∆ dependent coefficients around the crossing symmetric point. Currently the code scalar blocks code, found on the bootstrap collaboration website, 22 is only able to efficiently compute single conformal blocks.
From the analytical perspective, it would be interesting to derive the 1/c T corrections to SSSS for ABJ theories in the vector-like limit, such as for U (N ) k × U (N + M ) −k at finite N and large M, k with fixed M/k. This would complement the order 1/c T correlator that was computed in the supergravity limit in [81] and was successfully matched to numerical bootstrap results in [82]. We expect that the vector-like correlator can be computed by generalizing [83,84] to N = 6, and hope to report on these results soon in a future publication.
We could also make further use of localization to improve our results. In this study we were only able to impose two analytic constraints, c T and λ 2 (B,2) [022] 2,0 , while ABJM is parameterized by three parameters N, M, k. For this reason there are not enough constraints to uniquely pick out a single ABJM theory and so we should not expect our islands to shrink indefinitely as we increase Λ. We think this is the reason why the islands shown in Figure 8, while small, are still much bigger than the N = 8 islands computed in [12]. Localization conveniently provides us a third quantity given by four mixed mass derivatives of the mass deformed free energy, which as shown [60] is related to an integral of SSSS . While this integrated constraint can be imposed analytically in a large N expansion as in [60,74], it is not yet known how to impose it on the numerical bootstrap in our case. Perhaps the method used in [85], where a similar similar integrated constraint was successfully imposed on the numerical bootstrap of a certain supersymmetric 2d theory, could be applied to our case.
Another option would be to look at a larger system of correlators involving fermions. This would allow us to impose parity, which would restrict the set of known N = 6 SCFTs to a few families such as U (N ) k × U (N ) −k parameterized by only two parameters each.
Once we can fully fix the three-parameters ABJM theory, it would be interesting to see if we can match integrability results computed for the lowest dimension singlet scaling dimension in the leading large N 't Hooft limit at fixed λ 't Hooft = N/k and M = 0. On the integrability side some results are available, for instance, in [86,87]. On the localization side, we would need to compute the derivatives of the mass deformed free energy in the 1/N expansion at finite λ 't Hooft . In fact, the zero mass free energy has already been computed in this limit in [88] by applying topological recursion to the Lens space L(2, 1) matrix model, so computing c T and λ 2 (B,2) [022] 2,0 should correspond to just computing two-and four-body operators in this matrix model. This could potentially lead to the first precise comparison between integrability and the numerical conformal bootstrap.
Finally, it would be interesting to consider the superconformal block decomposition of other correlators involving operators which are less than half-BPS but still have scalar superprimaries, which makes them feasible to numerical bootstrap. For instance, in N = 4 theories the stress tensor multiplet is only 1/4-BPS but still has a scalar superprimary. Similarly, while conserved current multiplets are half-BPS for N = 4 theories (and were studied using the numerical bootstrap in [18]), for N = 3 theories they are 1/3-BPS and for N = 2 theories are only 1/4-BPS. Many localization results exist which could be applied to these cases, including N = 4 results computed in [89].
To make further progress we can consider the correlator where B i are defined as in (2.4). The Ward identities relating P i (U, V ) to S i (U, V ) have been computed in [60]. Note that because both S and P transform in the 15 of so (6) Combining this with (A.2) and the Ward identities derived in [60], we find that where D is the differential operator Our next step is to rewrite the cross-ratios (U, V ) using radial coordinates (r, η) Conformal blocks have a relatively simple form in radial coordinates: where each p ∆, ,k (η) is polynomial in η [90]. In particular, the leading term is given by where P n (x) is the n th Legendre polynomial. Since S for some polynomial q r (η).
Let us translate (A.5) into radial coordinates: Substituting (A.10) into this equation we find that q r (η) satisfies Legendre's equation Hence q r (η) is a polynomial if and only if ∆ ∈ Z, in which case q r (η) = aP ∆+1 (η) for some arbitrary constant a. Since unitarity implies that ∆ ≥ 0, we conclude that S Any operator in a superconformal multiplet has twist greater than or equal to the twist of the superconformal primary. Thus if M is not the trivial supermultiplet then its superconformal primary must have twist one. Examining Table 2, we see that aside from the stress-tensor multiplet the only other such multiplets are conserved currents: A-type multiplets whose superprimary is an R-symmetry singlet with conformal dimension ∆ = + 1.
We conclude that any superblock in which the only exchanged operators transform in the 1 or 15 must correspond to the exchange of the trivial, stress-tensor, or a conserved current multiplet.

C Characters of osp(6|4)
In this section we will review the character formulas of osp(6|4), which were computed in [27], as well as their decomposition under osp(6|4) → so(3, 2) ⊕ so (6). This decomposition was used in Section 2 to determine which conformal primaries reside in each supermultiplet appearing in the S × S OPE.
The N = 8 stress tensor correlator was written in [11] in the basis 3) 24 In [11], the long multiplet was denoted as (A, 0).

E Simplifying the S 3 partition function
In this appendix we describe how to simplify the S 3 partition function for various N = 6 theories. In Section E.1 we consider the U (N ) k × U (N + M ) −k , and in Section E.2 we consider the SO(2) 2k × U Sp(2 + 2M ) −2k theory. We close by demonstrating that additional U (1) factors do not change the S 3 partition function
To achieve this we will follow the methods of [76] which considered the special case m + = m − = 0. We are ultimately only interested in computing Z up to an overall normalization constant Z 0 which is independent of m ± .
Our first step is to use the determinant formula: . (E.5) The µ and ν integrals then become Gaussian and can be easily performed. We thus find that In this section we shall reduce the SO(2) 2k × U Sp(2 + 2M ) −k sphere partition function (3.29) down to a single integral. As in the previous section, we will work with a slightly more general partition function where we have two mass deformations m ± . We follow the derivation in [79], which considered the special case m + = m − = 0. They however consider the general SO(2N )×U Sp(2N +2M ) theory, while here we only focus on the N = 1 case, for which the manifest N = 5 SUSY is enhanced to N = 6.
We begin by rewriting (E.10): (E.12) We can simplify this expression by introducing canonical position and momentum operatorŝ q andp which satisfy [q,p] = 2πik. We denote theq eigenstates by |ν , and introduce states We then find that our version of equation ( To this end we note that the mass-deformed S 3 partition function for the G × U (1) L can be generically written as Z G×U (1) L (m + , m − ) = dχ 1 . . . dχ N e iπ ab K ab χaχ b Z G (m + + 2q · χ, m − + 2q · χ) , (E. 17) where K ab is the matrix of Chern-Simons levels for the U (1)s q = (q 1 , . . . , q L ) are the charges of the (bi)fundamentals under each U (1), and Z G (m + , m − ) is the S 3 partition function for the theory without any U (1) factors. In order for G × U (1) N to have N = 6 supersymmetry, K ab and q a must satisfy the condition a,b K ab q a q b = 1 k G (E. 18) for some G dependent constant k G , where K ab is the inverse of K ab [38].
Because K ab is symmetric, we can then perform a second change of basis to χ 2 , . . . , χ L , so that K ab take the form We can now integrate over χ 2 , χ 3 , . . . leaving us with Z G×U (1) L (m + , m − ) ∝ dχ 1 e iπ(K 11 −K 2 12 )χ 2 Z G (m + + 2χ 1 , m − + 2χ 1 ) , (E.20) We then note that, in this basis, the condition (E.18) becomes: and so We now simply recognize the right-hand side of this equation is the partition function for the G × U (1) theory, and have hence shown what we set out to prove.