The M-theory Archipelago

We combine supersymmetric localization results and the numerical conformal bootstrap technique to study the 3d maximally supersymmetric (${\cal N} = 8$) CFT on $N$ coincident M2-branes (the $U(N)_k \times U(N)_{-k}$ ABJM theory at Chern-Simons level $k=1$). In particular, we perform a mixed correlator bootstrap study of the superconformal primaries of the stress tensor multiplet and of the next possible lowest-dimension half-BPS multiplet that is allowed by 3d ${\cal N} = 8$ superconformal symmetry. Of all known 3d ${\cal N} = 8$ SCFTs, the $k=1$ ABJM theory is the only one that contains both types of multiplets in its operator spectrum. By imposing the values of the short OPE coefficients that can be computed exactly using supersymmetric localization, we are able to derive precise islands in the space of semi-short OPE coefficients for an infinite number of such coefficients. We find that these islands decrease in size with increasing $N$. More generally, we also analyze 3d ${\cal N} = 8$ SCFT that contain both aforementioned multiplets in their operator spectra without inputing any additional information that is specific to ABJM theory. For such theories, we compute upper and lower bounds on the semi-short OPE coefficients as well as upper bounds on the scaling dimension of the lowest unprotected scalar operator. These latter bounds are more constraining than the analogous bounds previously derived from a single correlator bootstrap of the stress tensor multiplet. This leads us to conjecture that the $U(N)_2 \times U(N+1)_{-2}$ ABJ theory, and not the $k=1$ ABJM theory, saturates the single correlator bounds.


Introduction and summary
The conformal bootstrap [1][2][3] is a non-perturbative method that places rigorous numerical bounds [4] on scaling dimensions and operator product expansion (OPE) coefficients that appear in a given four-point function of a conformal field theory (CFT) (for reviews, see [5][6][7][8][9]). In the best scenario, these bounds form precise islands in the space of a small number of CFT data that contain the values of a single known CFT, as was originally shown for the 3d Ising model [10][11][12] and later generalized to the 3d O(N ) models [12,13] and N = 1 Ising model [14,15]. In these cases, the small allowed islands were found in the space of scaling dimensions of the two lowest-lying scalar operators which, in these theories, are the only relevant operators in their corresponding global symmetry charge sectors. For instance, in the case of the Ising model, these operators were the lowest Z 2 -even and Z 2 -odd operators.
Performing a mixed correlator bootstrap study was necessary because it is not possible to access both charge sectors from the four-point function of a single relevant operator. 1 In this paper, we obtain islands and (in principle) infinite amount of CFT data for 3d maximally supersymmetric (N = 8) superconformal field theories (SCFTs) by also studying a mixed correlator system. (Before doing so, we will also review the kinds of islands that can be obtained from a single correlator bootstrap study.) All N = 8 SCFTs may contain 1/2-BPS scalar operators S p with scaling dimension ∆ = p/2 that transform in the [00p0] irreducible representation of the so(8) R-symmetry. Out of them, S 1 must be a free scalar and it thus must decouple from the rest of the SCFT, and S 2 represents the bottom component of the stress tensor multiplet. In general, a given N = 8 SCFT may not have operators belonging to all of these multiplets, or it may have multiple linearly independent 1/2-BPS operators with the same value of p. All local SCFTs, however, do have a unique stress tensor operator and thus a unique S 2 operator. 2 In this paper we will restrict our attention to N = 8 SCFTs that contain an S 3 operator and examine the mixed four-point functions of S 2 and S 3 . As will be explained in more detail shortly, the S 3 operators do not exist in all known 3d N = 8 SCFTs, but they do exist in the theories on N coincident M2-branes in flat space.
Since S 2 and S 3 have fixed scaling dimensions 1 and 3/2, respectively, we cannot find 1 Scaling dimension islands for these and other theories can be found even from single correlators by imposing gaps in multiple operators that appear in the OPE [16], such as spin 1 operators. Islands have also been found for other theories by considering mixed correlators and imposing gaps on both relevant and irrelevant operators [17,18]. However, all these cases involve gaps that, while in many cases plausible, are not rigorously justified. 2 It can be shown by taking OPEs of S 2 with itself any number of times, one can also construct operators S p for all even p.
islands in the space of scaling dimensions as was done in the other cases mentioned above.
Instead, we find small islands in the space of the OPE coefficients of low-lying protected eighth and sixteenth-BPS semishort operators that appear in the S 2 × S 2 , S 2 × S 3 , and S 3 × S 3 OPEs. Both upper and lower bounds can be computed for these quantities because the scaling dimensions of these operators are fixed to distinct values by supersymmetry, unlike the theories discussed above where scaling dimensions are not protected so only upper bounds on OPE coefficients could be computed without additional assumptions. 3 These operators appear in the OPE for an infinite number of spins, so islands for an infinite number of such operators can be derived in principle.
More generally, protected operators appear generically in four point functions of BPS operators in SCFTs with at least four supercharges, and upper and lower bounds on their OPE coefficients can be computed by considering just a single correlator, as was shown originally for the 4d N = 1 bootstrap [19] and later for bootstrap studies in other dimensions with different amounts of supersymmetry [20][21][22][23][24][25][26][27][28][29][30][31][32][33]. 4 In these cases, the bounds are not very constraining, however, and so do not lead to precise islands such as the 3d Ising or O(N ) islands in scaling dimension space mentioned above. For 3d N = 8 theories, on the other hand, the upper and lower bounds that were computed for semishort OPE coefficients previously using only the single correlator S 2 S 2 S 2 S 2 [38] are already very constraining. For instance, the bounds on the OPE coefficients of the spin 0 and 2 eighth-BPS semishort operators for the k = 3 BLG theory that were computed in that study with Λ = 19 have a percent error 5 of 0.5% and 0.3%, respectively, which is comparable to the 0.12% and 0.23% percent error reported for the island in the smallest scaling dimension operator for the O(2) and O(3) models, as computed using Λ = 35 and scanning over OPE coefficients in the best available study in [12]. 6 This infinite number of islands could be computed for infinitely many different 3d N = 8 theories.
While the single correlator OPE coefficient islands are already very precise, the mixed correlator system that is the primary focus of this paper has three advantages: 1. In principle, the free multiplet S 1 is allowed to appear in the mixed OPE S 2 × S 3 , so 3 Lower bounds can be computed for the OPE coefficients of the operators that have been assumed to be relevant, since there is now a gap between them and the continuum of irrelevant operators. Islands in the space of these OPE coefficients are reported in [12,15]. 4 Note that for 4d SCFTs with N ≥ 3, all the protected operators that appear in the four-point function of half-BPS operator are already fixed by the 2d chiral algebra [34], so the bootstrap studies [35][36][37] could not compute any upper bounds on OPE coefficients. 5 Percent error as computed, for instance, in (4.13). 6 Here, Λ is a parameter counting the number of derivatives in the functionals used for numerical bootstrap. See [39] for the precise definition.
we can exclude the presence of a free sector by setting its OPE coefficient to zero.
2. As mentioned above, only certain 3d N = 8 SCFTs contain the operator S 3 , unlike S 2 which appears in all theories with a stress tensor, so we automatically restrict the space of theories by considering correlators of S 3 .
3. Ten protected half and quarter-BPS short scalar operators appear in the mixed system, while only three appear in S 2 S 2 S 2 S 2 . The OPE coefficients of these operators can be computed for a given theory using the 1d topological sector [40] and then imposed on the 3d bootstrap, so having access to more such operators makes the bootstrap much more constraining.
Let us now provide more background and summarize our results. There are only a few known infinite families of N = 8 SCFTs, and they can all be realized (in N = 3 SUSY notation) as Chern-Simons (CS) theories with a product gauge group G 1 × G 2 coupled to two matter hypermultiplets transforming in the bifundamental representation. These families are: the SU (2) k ×SU (2) −k and (SU (2) k ×SU (2) −k )/Z 2 reformulations [41,42] of the theories of Bagger-Lambert-Gustavsson (BLG) [43][44][45][46], 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) [47], which we denote as ABJM N,k for integer N and k = 1, 2; and the U (N + 1) 2 × U (N ) −2 theories [48] of Aharony-Bergman-Jafferis (ABJ) [49], which are labeled by the integer N . The ABJ(M) theories can be interpreted as effective theories on N coincident M2branes placed at a C 4 /Z k singularity in the transverse directions, so that when N → ∞ they contain a sector described by weakly coupled supergravity. 7 When N > 1, ABJM N,1 flows to two decoupled SCFTs in the infrared: a free SCFT with eight massless real scalars and eight Majorana fermions that is isomorphic to ABJM 1,1 , and a strongly coupled interacting SCFT that we denote as ABJM int N,1 . The only known interacting theories that contain an S 3 operator are ABJM int N,1 with N ≥ 3, ABJM 3,2 , and the (SU (2) 3 × SU (2) −3 )/Z 2 BLG theory, where the last two theories are dual to ABJM int 4,1 [53] and ABJM int 3,1 [50], respectively. Thus, we need only consider ABJM int N,1 in this work. In general, 3d N = 8 SCFTs are most conveniently parameterized by c T , which is defined as the coefficient appearing in the two-point function of the canonically-normalized stresstensor, T µν ( x)T ρσ (0) = c T 64 (P µρ P νσ + P νρ P µσ − P µν P ρσ ) 1 16π 2 x 2 , P µν ≡ η µν ∇ 2 − ∂ µ ∂ ν . (1.1) 7 The BLG theories, in contrast, do not have a known M-theory interpretation except when k ≤ 4. For This coefficient can be computed exactly using supersymmetric localization for any N ≥ 2 SCFT with a Lagrangian description (see [54] and [55]). In (1.1), c T is normalized such that it equals 1 for a (non-supersymmetric) free massless real scalar or a free massless Majorana fermion. Thus, c T = 16 for the free N = 8 theory of eight massless real scalars and eight massless Majorana fermions (equivalent to ABJM 1,1 ), and for ABJ(M) theory at large N . We will use N and c T interchangeably to specify ABJM int N,1 . The conformal bootstrap was first applied to 3d N = 8 SCFTs in [39], which used the single correlator S 2 S 2 S 2 S 2 to derive bounds on CFT data as a function of c T , which itself was bounded as c T 15.35 at Λ = 19, so that in the infinite Λ limit this bound is likely saturated by the free theory. Upper bounds were derived on scaling dimensions of unprotected operators, which showed a kink at the value c T ≈ 22.8 that, to our knowledge, does not correspond to any known theory. Upper and lower bounds were also derived for OPE coefficients of the two short scalars with ∆ = 2, both of which also showed kinks at c T ≈ 22.8, and one of which went to zero at this value. Next, in [38] it was shown that these scalar short operators are described by a 1d theory that relates their OPE coefficients, so that there is only one independent short OPE coefficient in terms of c T , and in fact the numerical bounds on the two short scalar operators are exactly related by the 1d theory relation. Upper and lower bounds were also computed for several of infinitely many semishort operators, and these bounds also showed a kink at c T ≈ 22.8. Most recently, in [56] the unique independent short operator OPE coefficient was computed for the ABJ(M) and BLG theories by relating it to derivatives of the mass deformed sphere free energy, which was computed using localization [57] exactly at small N and to all orders in 1/N using the Fermi gas method [58,59]. This analytic formula was found to come close to saturating the numerical bounds for all interacting ABJ(M) theories. For whichever of these theories saturates the bound in the infinite precision limit, all the CFT data in S 2 S 2 S 2 S 2 , both protected and unprotected, could then be numerically determined using the extremal functional method [60][61][62]. To leading order in 1/c T , where all ABJ(M) theories are indistinguishable, the bootstrap predictions were verified by supergravity calculations of lowest twist unprotected operators [63,64] and even more non-trivially subleading twist unprotected operators [64].
Lastly, inputting the values of c T and of the short OPE coefficient that can be computed each k ≤ 4, BLG theory is dual to a theory of ABJ(M) type [48,[50][51][52]. [0020] 2,0 , λ 2 (A,+) [0020] 4,2 (to be defined precisely later) for ABJM int N,1 for N = 2, 3, 4, 20, 50, 100, where orange is allowed. These bounds are derived from the single correlator S 2 S 2 S 2 S 2 with the short OPE coefficients fixed to their ABJM int N,1 values using the 1d theory in Section 3 and [38] for N = 2, 3, 4, and from the all orders in 1/N formulae in [56] for N = 20, 50, 100. The red denotes the N → ∞ GFFT values in Table 4. Zoomed-in plots will be presented in Figure 2.
from the 1d theory, one can obtain very precise islands of allowed regions for semi-short OPE coefficients, as we show in Figure 1. (See also Figure 2 for zoomed-in versions of this plot.) As mentioned already, in this work we study mixed correlators of S 2 and S 3 . We find that the 1d theory for these correlators again relates the OPE coefficients of short scalar operators, so that there are five such independent quantities in terms of c T , instead of just the one that appeared for S 2 S 2 S 2 S 2 . After imposing these relations and setting the free theory OPE coefficient to zero, we numerically bound CFT data as a function of c T for a general interacting 3d N = 8 SCFT. We find that c T itself is now bounded as c T 22.8, which is curiously the same value were we observed a kink in the S 2 S 2 S 2 S 2 bounds. We compute upper and lower bounds for OPE coefficients of short and semishort operators that did not appear in S 2 S 2 S 2 S 2 , as well as upper bounds on the scaling dimension of the lowest dimension unprotected scalar operator that also appeared in S 2 S 2 S 2 S 2 . For the latter, we find that the mixed bound is more restrictive than the S 2 S 2 S 2 S 2 bound, which suggests that the ABJ(M) theory that conjecturally saturated that bound, and could thus be studied using the extremal functional method, must be one of the k = 2 theories that does not appear in our mixed correlator study. This leaves two possibilities: the U (N ) 2 × U (N ) −2 ABJM theory or the U (N + 1) 2 × U (N ) −2 ABJ theory. When N = 2, the former becomes a product of two ABJ with N = 1 theories, and so does cannot lie on the boundary of the allowed region [56], while when N = 4, this theory is dual to ABJM int 4,1 and so also cannot saturate the bound, which suggests that ABJ is the theory saturating the bound. We also found that the mixed correlator scaling dimension bound now goes to ∆ = 1 at c T ∼ 22.8, which suggests that if any theory lives at that point it must include a free sector.
Next, we restrict to ABJM int N,1 by imposing the values of OPE coefficients of short scalar operators. We cannot relate correlators of S 3 to the mass deformed free energy, as we did for S 2 S 2 S 2 S 2 , so instead we compute these OPE coefficients using the 1d theory Lagrangian of ABJM N,1 [40], as was previously done for correlators of S 2 and S 4 in [50]. The calculation involves a number of integrals that grow with N , so we only give results for N = 3, 4.
After imposing these values, we compute precise islands in the space of OPE coefficients of semishort operators, and in several cases are able to determine these values to less than a percent! The rest of this paper is organized as follows. In Section 2, we discuss the constraints of 3d N = 8 superconformal symmetry on four point functions of half-BPS operators, and derive explicit superblocks for correlators of S 2 and S 3 (which we include in an attached Mathematica file). In Section 3, we derive the 1d theory relations for these correlators for general 3d N = 8 SCFTs, and then compute the short scalar OPE coefficients for ABJM int 3,1 and ABJM int 4,1 . In Section 4, we derive the crossing relations for our mixed systems and compute numerical bounds both for general interacting 3d N = 8 SCFTs, and for ABJM int 3,1 and ABJM int 4,1 by imposing the OPE coefficients of the previous section. Finally, in Section 5, we end with a discussion of our results and of future directions.

Half-BPS four-point functions in 3d
We begin by discussing the constraints of the 3d N = 8 superconformal algebra osp(8|4) ⊃ so(5) ⊕ so(8) R on 4-point functions of the bottom component of half-BPS supermultiplets.
The results of this section are are not restricted to ABJM theory, but apply to any N = 8 SCFT that contains these half-BPS operators.

Setup
We consider half-BPS superconformal primaries S k in 3d N = 8 SCFTs that are scalars with ∆ = k 2 and transform in the [00k0] of so(8) R , 8 where k = 1, 2, . . . . We will focus on the lowest three operator S k with k = 1, 2, 3. S 1 is a free scalar with ∆ = 1 2 in the 8 c of so(8) R , which does not exist in an interacting theory. S 2 is a scalar with ∆ = 1 in the 35 c of so(8) R , which is the bottom of the conserved stress tensor multiplet and so must exist in all local 3d N = 8 SCFTs. Lastly, S 3 is a scalar with ∆ = 3 2 in the 112 c of so(8) R , which need not exist in a general 3d N = 8 SCFT.
We are interested in 4-point functions of S k i for i = 1, 2, 3, 4. Since the [00k i 0] irrep of so (8) can be viewed as a rank-k i symmetric traceless product of the 8 c , we can represent the corresponding operators as traceless symmetric tensors S I 1 ... (2.1) Conformal and so(8) R symmetry then imply that four point functions of , and the conformally-invariant cross ratios U, V and so(8) R invariants σ, τ are we give a few examples in Appendix A. We can then expand G k 1 k 2 k 3 k 4 (U, V ; σ, τ ) in terms of this basis as By taking the s-channel OPEs S k 1 × S k 2 and S k 3 × S k 4 in (2.2), we can expand A k 12 ,k 34 is included to match the definition of the conformal blocks in [21], which can be computed recursively as in [10] 9 in terms of the variables r, η in a small r expansion, where r, η are defined as [66] So far, we have imposed the bosonic subgroups of the osp(8|4) algebra. The constraints from the fermionic subgroups are captured by the superconformal Ward identities [67]: where z,z and α,ᾱ are written in terms of U, V and σ, τ , respectively, as These constraints can be satisfied by expanding G k 1 k 2 k 3 k 4 in superconformal blocks as where G k 12 ,k 34 M are superblocks for each supermultiplet M that appears in both OPEs S k 1 ×S k 2 and S k 3 × S k 4 with OPE coefficients λ k 1 k 2 M and λ k 3 k 4 M , respectively. Comparing (2.10) to (2.5) and (2.6), we see that the superblocks are finite linear combination of conformal blocks where O ∆, ,nm are conformal primaries that appear in M. For the four-point functions of In the next subsection, we will derive these superblocks explicitly for four-point functions involving S 2 and S 3 . Before we do that, we will first discuss which M can appear in the OPEs for general S k 1 × S k 2 .
In general, there are twelve different types of osp(8|4) supermultiplets [68] that we list in Table 1. There are two types of shortening conditions denoted by the A and B families. The multiplet denoted by (A, 0) is a long multiplet and does not obey any shortening conditions.
The other multiplets of type A have the property that certain so(2, 1) irreps of spin − 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 ± 1/2 are absent from this product, and consequently, the multiplets of type B are smaller. 10 The half-BPS multiplets that we discussed above with bottom component S k are of (B, +) type and have a 3 = k. The conserved current multiplet appears in the decomposition of the long multiplet at unitarity: ∆ → + 1. This multiplet contains higher-spin conserved currents, and therefore can only appear in the free theory [69]. We will sometimes denote the superconformal multiplets by X , with (∆, ) 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, +)).
Only certain M can appear in the OPE of half-BPS multiplets. Comparing the so (   (2.12) Furthermore, it was shown in [70] that for n = 1 2 (k 1 +k 2 ), 1 2 (k 1 +k 2 )−1 we have the following extra constraints:

Four point functions of S 2 and S 3
We will now restrict our attention to the following four-point functions: 2222 , 3333 , 2233 , 2323 , 3223 , (2.14) which we consider in the s-channel as discussed above, and we use the notation ijkl ≡ S i S j S k S l . The so(8) R tensor products that we must consider for the S 2 × S 2 , S 3 × S 3 , and The supermultiplets M that can then appear in these OPEs according to the rules described above are listed in Tables 2 and 3. Note that 2233 and 2323 contain the same supermultiplets as 2222 and 3223 , respectively.
We can compute the superblocks for each of these M in two steps, following the computation for 2222 in [39]. First, we must determine which conformal primaries appear in ,nm of the superconformal descendents in terms of the superconformal primary OPE coefficients λ k 1 k 2 M λ k 3 k 4 M . For instance, the stress tensor multiplet (B, +) [0020] 1,0 has a superblock that can be written in the A k 12 ,k 34 nm (U, V ) basis of (2.5) in terms of conformal blocks as where note that the relative sign of the odd spin conformal primary is opposite that of the even spins, as expected in general for a four-point function of scalars. The analogous expressions for the other superconformal blocks will have linear combinations of many conformal blocks for each A k 12 ,k 34 nm (U, V ) and are quite unwieldy, so we relegate them to an attached Mathematica file. Note that the superblocks for 2323 and 3223 have coefficients that are related as which follows from the specific form of the so(8) polynomials in Appendix A and from how OPE coefficient of two scalars and a spin operator transforms under exchanging the scalars.
We find that the relative coefficients of the super-descendents obtained by acting on the superconformal primary with αβ Q aα Q bβ an odd number of times vanish, where Q aα is a supercharge with α, β spinor indices and a, b so(8) R indices. These conformal primaries are denoted in red in the tables of Appendix B. This combination of superchargers is odd under parity, so these super-descendents have the opposite parity as the superconformal primary, which motivates a 'bonus' parity for 4-point functions in 3d N = 8 theories, as was originally conjectured in [39] based on the superblocks for 2222 .
We can use conformal Ward identities to relate OPE coefficients of two stress tensor operators, and by supersymmetry any two operators in the stress tensor multiplet, to 1/ √ c T [71]. For the four-point function we consider, this implies that To fix the proportionality constants in our conventions, we can consider a free theory of eight real scalars X I and eight Majorana fermions mentioned in the Introduction, which has c T = 16. The half-BPS operators S 2 and S 3 in this case are given by Performing Wick contractions with the propagator X I ( x)X J (0) = δ IJ | x| , we then find that the 4-point functions in (2.14) in the notation of (2.2) equal: (2.20) By comparing this to the superconformal block expansion (2.10), we can read off the OPE coefficients listed in Table 4, where the scaling dimensions of the long multiplets that appear in each correlator are 2222 free , 2233 free : (A, 0) [0000] ∆, for ∆ = + 1, + 2n , for ∆ = + n , , (A, 0) [0020] ∆, for ∆ = + 2 , + 2n + 1 , for n = 1, 2, 3, . . . . Note that (A, 0) [0000] +1, for > 0 are conserved currents that only exist in a free theory, while (A, 0) and c T along with the general relations (2.18) to find Another limit in which we can compute the correlators (2.14) explicitly is in the generalized free field theory (GFFT) where the operators S 2 and S 3 are treated as generalized free The GFFT describes the c T → ∞ limit of N = 8 theories.
Performing the Wick contractions, we then find that (2.2) equals: (2.23) By comparing this to the superconformal block expansion, we can read off the OPE coefficients listed in Table 4 Table 4: Values of OPE coefficients of the superconformal primaries of short multiplets and the lowest semishort and long multiplets of each type in correlators of S 2 and S 3 in the s-channel of the free and generalized free field theory limits.

Protected 1d topological sector
All N = 8 SCFTs contain 1d topological sectors [34,38,72], which, for theories with Lagrangian descriptions, can be computed using supersymmetric localization [40,73,74]. The supersymmetric localization setup of [40,73,74] applies to any N ≥ 4 theory constructed from N = 4 vector multiplets and hypermultiplets. The theory we are interested in, ABJM N,1 , is, however, not of this type due to the presence of the Chern-Simons interactions. However, as shown in [48], ABJM N,1 does have a dual description as an N = 4 U (N ) gauge theory with a fundamental hyper and an adjoint hyper, for which the results of [40,73,74] do apply.
This is the description we use in this section.
A generic N = 4 theory has two distinct topological sectors: one corresponding to the Higgs branch and one corresponding to the Coulomb branch. Abstractly, these two sectors can be described as follows. The R-symmetry of an N = 4 SCFT is so(4) ∼ = su (2) in the Higgs branch topological sector. Here, r is a dimensionful parameter with units of length. The Coulomb branch sector can be constructed similarly from half-BPS scalar operators O (ȧ 1 ···ȧ 2j C ) with ∆ = j C and j H = 0: The operators (3.1) (or (3.2)) can be argued to have topological correlation functions because they belong to the cohomology of a nilpotent supercharge of Q + rS type [34,38,72].
While for a generic N = 4 SCFT, the two sectors described above are unrelated, for the particular case of N = 8 SCFTs it can be argued that they must be isomorphic. Indeed, it can be shown that the N = 4 operators with ∆ = j H and j C = 0 and those with ∆ = j C and j H = 0 can only originate in pairs from N = 8 operators that are at least quarter-BPS, and N = 8 SUSY relates their correlation functions. Since the description of the Higgs branch sector coming from supersymmetric localization [40] is simpler than that of the Coulomb branch sector [73,74], we will thus focus only on the Higgs branch one.

General features
Before restricting our attention to ABJM N,1 , we first derive some general constraints valid for the 1d sector of any 3d CFT with N = 8 SUSY. In addition to topological invariance, these theories have an su(2) F flavor symmetry, which from the 3d perspective originates as a subalgebra of the so(8) R-symmetry that commutes with the N = 4 R-symmetry su(2) H ⊕su(2) C . 11 We can therefore consider operators O (a 1 ···a 2j F ) (ϕ), a i = 1, 2, transforming in the spin-j F irrep of su(2) F . Their indices can be contracted with polarization vectors y a to form homogeneous polynomials in y of degree 2j F : where y 1 , y 2 = ab y a 1 y b 2 is the su(2) F invariant product of the polarization vectors. Similarly, three-point functions in the theory take the form inequality. For triplets of operators for which this is not the case, the right hand side of (3.5) vanishes. Note that (3.4) and (3.5) depend only on the overall ordering of the operators on S 1 , as expected of any topological theory.
The four-point structures can be used to relate observables in the 1d theory to CFT data in the 3d theory. From (3.4) and (3.5) we can first extract the OPE of a generic 1d theory. 11 What commutes with su(2) H ⊕ su(2) C inside so (8) is an so(4) algebra, so more generally, the 1d topological sector could have so(4) ∼ = su(2) F ⊕ su(2) F flavor symmetry. The 1d theories we focus on here will be invariant under su(2) F , so we will only consider su(2) F as a flavor symmetry.
Assuming that ϕ 1 , ϕ 2 < ϕ, operator products decompose as where we have introduced the symmetrized operator The four-point structures are determined by applying this OPE (3.6) twice. Taking ϕ 1 , ϕ 2 < ϕ 3 , ϕ 4 gives the s-channel expansion, Here, the quantity y 1,2 , y 3,4 2j is an su copies of y 1 , (j − j B + j A ) copies of y 2 , (j − j C + j D ) copies of y 3 , and (j − j D + j C ) copies of y 4 that reduces to y, y 2j if we were to set y 1 = y 2 = y and y 3 = y 4 = y .
Introducing the su(2) F -invariant cross-ratio w ≡ y 1 , y 2 y 3 , y 4 y 1 , y 3 y 2 , y 4 , (3.9) the four-point function (3.8) simplifies to where (3.11) The P (a,b) n (x) are Jacobi polynomials normalized such that With the four-point structures in hand, we can read off the 3d OPE coefficients. We first project the four-point function (2.2) to 1d by placing the operators on a line and implementing the twist in (3.1) (for more details, see [38]). Comparing this with the superconformal block expansion (2.6), a tedious calculation yields [0200] 2,0 [0200] 2,0 (3.14) The quantityλ 2 M is defined as the contribution of a multiplet M ∈ S 2 × S 2 ∩ S 3 × S 3 . More precisely, an orthogonal set of 1d operators O i , all descending from the 3d multiplet Note that the relations (3.14) are derived solely based on representation theory arguments, so they do not rely on the specific details of ABJM theory and of its 1d sector.

1d sector of ABJM N,1
We now specialize to the 1d sector of ABJM N,1 , in the dual presentation where this theory is an N = 4 U (N ) gauge theory with a fundamental hypermultiplet and an adjoint hypermultiplet. To obtain a Lagrangian description of the 1d Higgs branch sector of a general N = 4 gauge theory with hypermultiplet matter fields, Ref [40] first used the stereographic projection to map the 3d theory from flat space to a round S 3 whose radius of curvature r is taken to be the same as the parameter r appearing in (3.1)-(3.2). Since under this mapping a straight line passing through the origin of R 3 gets mapped to a great circle on S 3 , the 1d sector of the S 3 theory will be defined on a great circle. An explicit description was then where α, β = 1, . . . , N , and σ αβ ≡ σ α − σ β . The 1d fields X α β and X β α correspond to the adjoint hypermultiplet, Q α and Q α correspond to the fundamental hypermultiplet, and σ α are the matrix degrees of freedom in the Cartan of U (N ). The 1d operators can be constructed as gauge-invariant products of X, X, Q, and Q. However, the D-term relations of the 3d theory imply that one can always trade the product of Q and Q for products of X and X, so without loss of generality we can construct all 1d operators in terms of X and X.
After integrating out Q and Q, the correlation functions become is the partition function of the Gaussian theory at fixed σ, and the correlation functions in this theory are given by The Gaussian correlators with fixed σ can be computed systematically by performing Wick contractions using the 1d propagator [40] Note that the factorization of ABJM N,1 into ABJM int N,1 and a free theory, as described in the Introduction, is manifest for this presentation of the 1d sector. Indeed, under the identification of the free fields X free , X free as the traces of X, X and the interacting fields X int , X int as the traceless parts, where 1 is the N × N identity matrix, the action (3.17) decomposes into 23) and the partition function immediately factorizes as (3.24)

Operator content
We seek the operator content of the 1d sector of ABJM int N,1 . Under the su(2) F symmetry, X and X T transform as a doublet. To simplify notation, we group X and X T into a quantity X (ϕ, y) = y 1 X(ϕ) + y 2 X T (ϕ) (3.25) that depends on an additional polarization variable y a = (y 1 , y 2 ). The observables of the theory are then given by gauge-invariant products of 2j F X 's.
As explained in [38], respectively, while the internal operators that appear in the OPEs of the operators (3.27) and have representatives in the 1d theory can be part of any of the following multiplets: (B, +) [00k0] Let us now construct these 1d operators. The most general 1d operators one can construct are products tr X n 1 tr X n 2 tr X n 3 · · · . The operator tr X and its various powers are the 1d operators that come from the free sector. The operators in the interacting sector are then constructed from the traceless part of X , namely where as in (3.22), 1 is the N × N identity matrix. We will only focus on 1d operators which descend from superconformal primaries in the interacting sector.

Correlation functions
As mentioned previously, we can compute correlation functions of arbitrary operator insertions through Wick contractions and subsequently performing the matrix integral over σ (3.33) The expressions for the other correlators are quite complicated, so we relegate them to Appendix C.1.
Although we can determine the mixed correlators in the Gaussian theory for arbitrary N , it is difficult to compute the full observables in the 1d sector when the rank of the gauge group is large; indeed, numerical integration over large-dimensional spaces is infeasible when a high degree of precision is required. 13 Consequently, we restrict ourselves to the computation of the 1d data in ABJM int N,1 with N = 3, 4 and list the results in Tables 5 and 6

N,1
Using our results for the two-and three-point functions in Tables 5 and 6, we can use the dictionary (3.13) to extract the 3d OPE coefficients. We list our results for ABJM int N,1 with N = 3, 4 in Table 7. As a consistency check, it is straightforward to verify that these coefficients obey the 1d crossing relations in (3.14).

Numerical bootstrap
We will now use the results of the previous sections to derive the mixed correlator crossing equations for interacting 3d N = 8 SCFTs, taking into account some important redundancies due to supersymmetry. We then use these crossing equations to perform the numerical bootstrap for general N , i.e. c T , as well as N = 3, 4, using the values of the protected OPE coefficients derived in Section 3. We will also show results of the single correlator 2222 bootstrap for N ≥ 2 using the OPE coefficients from Section 3, as well as the all orders in 1/N expression in [56] for the OPE coefficients that appear in the S 2 × S 2 OPE.

Crossing equations
To derive the crossing equations for the four point functions 2222 , 3333 , and 2233 , we equate all the independent channels: (σ, τ ), and Y 1,1 nm (σ, τ ) in Appendix A to write these crossing relations as a 34-dimensional vector V (U, V ) whose entries are linear combinations of We can simplify these crossing equations in four steps. First we restrict to interacting SCFTs by not including the free multiplet (B, +) where the V M in the first couple lines are written as linear combinations of V M for other short multiplets, whose explicit form are given in Appendix D, and | A in the last line denotes that only the semishort and long A-type multiplets are considered here.

Redundant equations
As was first noted for the 2222 crossing equations in [39], where z,z are written in terms of U, V in (2.9), and in the sums we only consider terms are nonzero and independent according to the definition (4.2). We can then truncate these sums to a finite number of terms by imposing that and then consider the finite dimensional matrix V (p,q) i whose rows as labeled by i = 1, . . . 34 are those of V , and whose columns as labeled by (p, q) are the coefficients of the ∂ p z ∂ q z A k 12 ,k 34 nm (U, V )| z=z= 1 2 that appear in each entry of V after expanding like (4.5) using the definition (4.2) of is degenerate, and we find that a linearly independent subspace is given by the entries: where V

Bootstrap algorithms
We now have all the ingredients to perform the numerical bootstrap using the crossing equations (4.4) where the 1d crossing relations have been imposed, the free multiplet has been excluded, and we restrict to the linearly independent set of crossing equations in (4.7).
We will describe the bootstrap algorithms we use following [10,39].
Consider a linear functional α (p,q) i acting on the crossing equations (4.4) expanded in ∂ p z , ∂ q z derivatives around the crossing symmetric point (4.5). The index i refers to the 34 crossing equations in V i , which we have restricted to the linearly independent set in (4.7).
In the following we will suppress the indices on α for simplicity.
To find upper/lower bounds on a given OPE coefficient of a short or semishort multiplet  Tables 2 and 3. If such a functional α exists, then this α applied to (4.4) along with the reality of λ k 1 k 2 M except, possibly, for that of λ k 1 k 2 M * , implies that if s = 1, then for the associated k 1 , k 2 . If we are bounding the OPE coefficient squared of (B, +) [0020] 1,0 itself, which is proportional to 1 c T , then we should remove the second term from the RHS of (4.9). To obtain the most stringent upper/lower bound on λ 2 k 1 k 2 M * , one should then minimize/maximize the RHS of (4.9) under the constraints (4.8). Note that for M * that appear in (4.4) with a matrix constraint, e.g. (A, +) [0020] +2, or (A, 2) [0100] +2, , we cannot obtain any constraints using this algorithm since the LHS of (4.9) must be a scalar. We can also find islands in the space of OPE coefficients (λ M * , λ M * * ) for a given c T , by removing M * * from the second line of (4.8) and replacing the RHS of (4.9) by where we input the value of λ 2 k 1 k 2 M by hand for the appropriate k 1 , k 2 . To find upper bounds on the scaling dimension ∆ * of the lowest dimension operator in a long supermultiplet with spin * that appears in (4.4), we consider linear functionals α In the above algorithms, we fixed the SCFT by inputting the value of c T . We can further fix the theory by also putting in the values of all the short OPE coefficients, that can be fixed using the 1d theory in Section 3. We should then replace the RHS of (4.9) and the first line of (4.11) by ) .

(4.12)
For island plots, we should furthermore add λ 2 k 1 k 2 M α( V M * * ) as in (4.10). The numerical implementation of the algorithms described above requires two truncations: one in the number of derivatives Λ defined in (4.6) that are used to construct α, and one in the spins max that can appear for the long multiplets. in (4.7), and only the operators shown in black in Table 2 appear. Note that only three short operators appear: the stress tensor (B, +) [0020] 1,0 , (B, +) [0040] 2,0 , and (B, 2) [0200] 2,0 . We computed these expressions for ABJM int N,1 with N = 3, 4 in Section 3 (the N = 2 values were already known from [38]), and to all orders in 1/N in [56], which was found to be very accurate even down to N = 2. We will restrict our bootstrap results to ABJM int N,1 by imposing the values of these short OPE coefficients. As discussed in subsection 4.2, we must remove the V   [0020] +2, , computed using the single correlator 2222 with Λ = 43 and the short OPE coefficients fixed to their values for ABJM int N,1 using the 1d theory in Section 3 for N = 2, 3, 4 (N = 2 was already reported in [38]), and the all orders in 1/N results from [56] for N = 20, 50, 100.
In Table 8, we show upper and lower bounds on the semishort (A, +) [0020] +2, that appear for all even spins in the S 2 × S 2 OPE. We compute the percent error as the difference between these lower bounds normalized by the average: In general, the error is extremely small and seems to decrease with increasing spin and N .
In Figures 1 and 2 we show islands in the space of ((A, +) [0020] 2,0 , (A, +) [0020] 4,2 ) for N = 2, 3, 4, 20, 50, 100. For N = 2 we use the exact results reported in [38], for N = 3, 4 we use the exact results from Section 3, while for N = 20, 50, 100 we use the all orders in 1/N expression from [56]. These extremely small islands interpolate between the upper and lower bounds for these values in Table 8. As noted above, these islands get smaller with  for ABJM int N,1 for N = 2, 3, 4, 20, 50, 100, where orange is allowed. These bounds are derived from the single correlator 2222 with Λ = 43 and the short OPE coefficients fixed to their ABJM int N,1 values using the 1d theory in Section 3 and [38] for N = 2, 3, 4, and from the all orders in 1/N formulae in [56] for N = 20, 50, 100.
increasing N .

Mixed correlator bounds for general N
We now move one to the full mixed correlator bootstrap, and present the results of the OPE coefficient extremization (4.8) and scaling dimension upper bound (4.11) algorithms for general N , i.e. c T . Since many supermultiplets appear in (4.4), we will only show results for a representative sample of these multiplets.
We start with the upper bound on the OPE coefficient squared of the stress tensor multiplet (B, +) [0020] 1,0 , which is equivalent to a lower bound on c T . Using Λ = 35 derivatives, we find  [38,39], which was also where λ 2 (B,2) [0200] 2,0 was observed to vanish. This kink had been conjectured to correspond to [0200] 2,0 , but our new upper bound suggests there must be a different explanation.
Next, in Figure 3 we show upper/lower bounds on the OPE coefficients squared of the short (B, +) [0050]  Table 4 at c T → ∞, while the dotted lines show the values of c T for various known SCFTs. As expected, both plots terminate at the upper bound on 16 c T in (4.14). Lastly, in Figure 4, we show upper bounds on the scaling dimension of the lowest dimension long (A, 0) [0000] ∆,0 multiplet as a function of 16 c T , where the orange is the allowed region. The red dot denotes the GFFT value at c T → ∞, the red dotted line shows the 1 c T correction computed in [63], and the dotted lines show the values of c T for various known SCFTs. The gray dashed line shows the upper bounds for the analogous quantity computed using only the 2222 crossing equations in [39], which is slightly less constraining in general. Curiously, the bounds start becoming very different around 16 c T ∼ .4, and the new bound goes to the free theory value at the upper bound on 16 c T in (4.14).  Table 9: Upper and lower bounds on OPE coefficients squared for semishort operators, computed with Λ = 35 using mixed correlators and the short OPE coefficients fixed to their values for ABJM int 3,1 and ABJM int 4,1 using the 1d theory in Section 3.

Mixed correlator bounds for ABJM
We now restrict to the ABJM int N,1 theories for N = 3, 4 by imposing the values of the short OPE coefficients computed in these cases in Section 3. In Table 9, we show upper and lower bounds on various semishort multiplets that appear in (4.4). We compute the percent error as in (4.13). In general, for a given multiplet the error seems to decrease with increasing spin and N . While the error is bigger than the single correlator OPE coefficients in Table 8, it is still quite small.
In Figure 5 we show islands in the space of ((A, +) [0030] [0030] 7 2 ,1 ), which were the two most accurately known OPE coefficients in Table 9, for N = 3, 4. These extremely small islands interpolate between the upper and lower bounds for these values in Table 9. As noted above, these islands get smaller with increasing N .

Discussion
In this paper we bootstrapped a mixed correlator system of S 2 and S 3 operators in 3d N = 8 SCFTs, generalizing the single correlator bootstrap of S 2 S 2 S 2 S 2 [38,39,56]. This  Table 4.
bootstrap study automatically excludes all known 3d N = 8 SCFTs except those that are dual to ABJM int N,1 with N ≥ 3 or the free theory, since these other theories do not contain an S 3 operator. After further restricting to interacting theories by setting the free multiplet OPE coefficients to zero, we computed the lower bound c T 22.8, which is curiously the same value were a kink was observed in all plots of the previous single correlator bootstrap studies. We then computed upper and lower bounds on protected operators and an upper bound on the lowest dimension unprotected scaling dimension as a function of c T in general interacting 3d N = 8 theories. This scaling dimension bound is stronger than the same bound in the single correlator boostrap studies, and so implies that the latter bound, which was conjectured to be saturated by one of the ABJ(M) theories, must in fact be saturated by one of the k = 2 ABJ(M) theories, which is likely the U (N + 1) 2 × U (N ) −2 ABJ theory.
Lastly, we imposed the values of short scalar operator OPE coefficients as computed from the 1d theory for ABJM N,1 with N = 3, 4, and derived precise islands in the space of semishort operator OPE coefficients. Since these semishort operators appear for an infinite number of possible spins, infinite many islands of this type can be computed.
It is instructive to compare these islands against those computed previously in the literature for the O(N ) critical models [10][11][12][13] Looking ahead, there are several ways we could improve both the analytical and numerical calculations in this work. In the single correlator bootstrap study [56], the scalar short operator OPE coefficients were computed to all orders in 1/N , which was found to be very accurate even for N = 2. If similar analytic expressions could be derived for all the short operators in the mixed correlator system, then we could use them to derive precise islands for all values of N in the mixed correlator system (as we did for the single correlator system), not just N = 3, 4. We could also check if the numerics saturate some of these analytic expressions, in which case we could read off all the CFT data that appears in the relevant OPEs, as was done for the single correlator case in [56]. Since the mixed correlator system automatically excludes the k = 2 ABJ(M) theories, 15 there would be no ambiguities as to which theory was saturating the bounds, as was the case in the single correlator study. A more careful study of the spectrum as a function of N could perhaps shed light on the mysterious point with c T ∼ 22.8, which does not correspond to any known theory.
From the numerical perspective, the islands derived in this work come from the single and mixed correlator systems that only have around one and seven crossing equations each, respectively, which makes them almost as simple as the Ising model case, which has one and five crossing equations for the single and mixed correlator systems [10], respectively. These islands also appear to still be shrinking as the derivatives parameter Λ is increased. It should thus be possible to use the rapidly developing SDPB program to drastically increase Λ, so that these islands become small enough that they can be used to constrain the 1/c T expansion of the respective four-point functions, which in the flat space limit can be used to constrain the small momentum expansion of the M-theory S-matrix [75][76][77].
Finally, there are more constraints from localization that could be imposed on the numerical bootstrap. In particular, the squashed sphere free energy F (b, m i ) in ABJ(M) theory can be computed using localization as a function of three masses m i as well as a squashing parameter b. The integrated four point function of S 2 can be related to taking four derivatives of these parameters and setting m i = 0 and b = 1. Indeed, the quantity ∂ m i F | m i =0 was already used in [56] to compute the short OPE coefficient, while ∂ m i ∂ m j F | m i =m j =0 for i = j was used in [76,77] to constrain S 2 S 2 S 2 S 2 in the large c T limit. Imposing new constraints of this type on the numerical bootstrap could further constrain the precise islands presented in this work. = 1, and

B Supermultiplets
The conformal primaries that contribute to a given supermultiplet can be found by decomposing the characters of osp(8|4) into its maximal bosonic subgroup so(5) ⊕ so(8) R . This can be done using the formulae in Appendix B of [39], based on [68]. In the tables below we give the results for the supermultiplets that appear in the S 2 × S 2 , S 2 × S 3 , and S 3 × S 3 OPEs. We denote in red those primaries that are allowed by symmetry, but vanish once we compute the explicit superblocks. As explained in the main text, these primaries have the opposite parity as the superconformal primary.

C 1d correlation functions C.1 Gaussian correlators
In this section, we list the correlation functions of the 1d theory at fixed σ for any value of N . These are used to compute OPE coefficients of short multiplets in ABJM int N,1 , listed in Table 7. All operators O(ϕ i , y i ) are ordered on S 1 in ascending fashion with ϕ 1 < ϕ 2 < · · · < ϕ n . For notational simplicity, we strip off the polarization dependence from all expressions, which can be restored through (3.4) and (3.5), and we work in units where = 1. It follows that all correlators can be subsequently written in terms of ten quantities, where t α,β ≡ tanh(π(σ α − σ β )).
For example, the two-point functions of external operators (3.33) can be rewritten as

(C.2)
We now list the correlation functions according to type. Those related to constructing an orthogonal basis of operators are:

(C.4)
The single two-point function of internal operators with ∆ = j F − 1 is: Along with the 1d crossing equations (3.14), the constraints in (C.10) can be used to rewrite all of the 3d OPE coefficients in terms of λ 1 ≡ λ 2

22(B,+)
[0040] 2,0 , which were computed to all orders in 1/N in [56], as well as three correlators, which we can In terms of these quantities, the OPE coefficients can be written as where 0 a denotes vectors of length a with entries equal to the 2 × 2 zero matrix, and we In the main text, we then used the 1d crossing relations and the conformal Ward identity constraints (2.22) to write all the short OPE coefficients in terms of 1/c T , λ 2