Exact Correlators of BPS Operators from the 3d Superconformal Bootstrap

We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with ${\cal N} \geq 4$ supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In ${\cal N} = 4$ SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite number of OPE coefficients in the interacting SCFT ($U(2)_2 \times U(1)_{-2}$ ABJ theory) that constitutes the IR limit of $O(3)$ ${\cal N} = 8$ super-Yang-Mills theory. More generally, in ${\cal N} = 8$ SCFTs with a unique stress tensor, we are led to conjecture that many superconformal multiplets allowed by group theory must actually be absent from the spectrum, and we test this conjecture in known ${\cal N} = 8$ SCFTs using the superconformal index. For generic ${\cal N} = 8$ SCFTs, we also improve on numerical bootstrap bounds on OPE coefficients of short and semi-short multiplets and discuss their relation to the exact relations between OPE coefficients we derived. In particular, we show that the kink previously observed in these bounds arises from the disappearance of a certain quarter-BPS multiplet, and that the location of the kink is likely tied to the existence of the $U(2)_2 \times U(1)_{-2}$ ABJ theory, which can be argued to not possess this multiplet.


Introduction
In conformal field theories (CFTs), correlation functions of local operators are highly constrained by conformal invariance. For supersymmetric CFTs, conformal invariance is enhanced to superconformal invariance, which leads to even more powerful constraints on the theory. In this case, the most tightly constrained correlation functions are those of 1 2 -BPS operators, because, of all non-trivial local operators, these operators preserve the largest possible amount of supersymmetry. Indeed, it has been known for a long time that such correlation functions have special properties. For instance, in N = 4 (maximally) supersymmetric Yang-Mills (SYM) theory in four dimensions, the three-point functions of 1 2 -BPS operators were found in [1] to be independent of the Yang-Mills coupling constant. 1 In contrast, not much is known about such three-point correlators in 3d CFTs with N = 8 (maximal) supersymmetry. 2 Indeed, these 3d theories are generally strongly coupled isolated superconformal field theories (SCFTs), which makes them more difficult to study than their four-dimensional maximally supersymmetric analogs. In particular, a result such as the non-renormalization theorem quoted above for 4d N = 4 SYM would not be possible for 3d N = 8 SCFTs, as these theories have no continuous deformation parameters that preserve the N = 8 superconformal symmetry. Nevertheless, we will see in this paper that three-point functions of 1 2 -BPS operators in 3d N = 8 SCFTs also have special properties. For example, at least in some cases, these three-point functions are exactly calculable. As we will discuss below, even though in this work we focus on N = 8 SCFTs as our main example, the methods we use apply to any 3d SCFTs with N ≥ 4 supersymmetry.
The main method we use is the conformal bootstrap [8][9][10][11], which has recently emerged as a powerful tool for obtaining non-perturbative information on the operator spectrum and operator product expansion (OPE) coefficients of conformal field theories in more than two space-time dimensions [12]. Its main ingredient is crossing symmetry, which is a symmetry of correlation functions that follows from the associativity of the operator algebra. In most examples, crossing symmetry is combined with unitarity and is implemented numerically on various four-point functions (see, for example, ). This method then yields numerical bounds on scaling dimensions of operators and on OPE coefficients in various CFTs, where the CFTs are considered abstractly as defined by the CFT data. The application of the conformal bootstrap to the study of higher dimensional CFTs has been primarily numerical, and exact analytical results have been somewhat scarce. (See, however, [3,[44][45][46][47][48].) In this 1 See [2] for a recent proof and more references, and also [3] for generalizations. 2 Some large N results derived through the AdS/CFT correspondence are available-see [4][5][6][7].
light, one of our goals in this paper is precisely to complement the numerical studies with new exact analytical results derived from the conformal bootstrap.
Generically, any given four-point function of an (S)CFT can be expanded in (super)conformal blocks using the OPE, and this expansion depends on an infinite number of OPE coefficients.
In N ≥ 2 SCFTs in 4d and N ≥ 4 SCFTs in 3d, the latter being the focus of our work, it was noticed in [3,28] that it is possible "twist" the external operators (after restricting them to lie on a plane in 4d or on a line in 3d) by contracting their R-symmetry indices with their position vectors. 3 The four-point functions of the twisted operators simplify drastically, as they involve expansions that depend only on a restricted set of OPE coefficients. When applied to these twisted four-point functions, crossing symmetry implies tractable relations within this restricted set of OPE coefficients.
The 3d construction starts with the observation that the superconformal algebra of an N = 4 SCFT in three dimensions contains an su(2|2) sub-algebra. This su(2|2) is the superconformal algebra of a one-dimensional SCFT with 8 real supercharges; its bosonic part consists of an sl(2) representing dilatations, translations, and special conformal transformations along, say, the x 1 -axis, as well as an su(2) R R-symmetry. From the odd generators of su(2|2) one can construct a supercharge Q that squares to zero and that has the property that certain linear combinations of the generators of sl (2) and su(2) R are Q-exact. These linear combinations generate a "twisted" 1d conformal algebra sl(2) whose embedding into su(2|2) depends on Q. 4 If an operator O(0) located at the origin of R 3 is Q-invariant, then so is the operator O(x) obtained by translating O(0) to the point (0, x, 0) (that lies on the x 1 -axis) using the twisted translation in sl (2). A standard argument shows that the correlation functions then the correlation function (1.1) vanishes. Indeed, we can obtain non-trivial correlation functions only if all O i (x i ) are non-trivial in the cohomology of Q. 6 We will prove that the cohomology of Q is in one-to-one correspondence with certain 1 2 -BPS superconformal primary operators 7 in the 3d N = 4 theory. Applying crossing symmetry on correlation functions like (1.1), one can then derive relations between the OPE coefficients of the 1 2 -BPS multiplets of an N = 4 SCFT.
In this paper, we only apply the above construction explicitly to the case of 3d N = 8 SCFTs, postponing a further analysis of 3d SCFTs with 4 ≤ N < 8 supersymmetry to future work. Three-dimensional N = 8 SCFTs are of interest partly because of their relation with quantum gravity in AdS 4 via the AdS/CFT duality, and partly because one might hope to classify all such theories as they have the largest amount of rigid supersymmetry and are therefore potentially very constrained. Explicit examples of (inequivalent) known N = 8 SCFTs are the U (N ) k ×U (N ) −k ABJM theory when the Chern-Simons level is k = 1 or 2 [52], the U (N + 1) 2 × U (N ) −2 ABJ theory [53], and the SU (2) k × SU (2) −k BLG theories [54][55][56][57][58][59].
Since any N = 8 SCFT is, in particular, an N = 4 SCFT, one can decompose the N = 8 multiplets into N = 4 multiplets. From the N = 8 point of view, the local operators that represent non-trivial Q-cohomology classes are Lorentz-scalar superconformal primaries that belong to certain 1 4 , 3 8 , or 1 2 -BPS multiplets of the N = 8 superconformal algebra-it is these N = 8 multiplets that contain 1 2 -BPS multiplets in the decomposition under the N = 4 superconformal algebra. 8 An example of an operator non-trivial in Q-cohomology that is present in any local N = 8 SCFT is the superconformal primary O Stress of the N = 8 stress-tensor multiplet.
This multiplet is 1 2 -BPS from the N = 8 point of view. The OPE of O stress with itself contains only three operators that are non-trivial in Q-cohomology (in addition to the identity): O Stress itself, the superconformal primary of a 1 2 -BPS multiplet we will refer to as "(B, +)", and the superconformal primary of a 1 4 -BPS multiplet we will refer to as "(B, 2)". Using crossing symmetry of the four-point function of O Stress , one can derive the following relation between the corresponding OPE coefficients: Stress − 5λ 2 (B,+) + λ 2 (B,2) + 16 = 0 . (1. 2) The normalization of these OPE coefficients is as in [41] and will also be explained in Section 3. In this normalization, one can identify λ 2 Stress = 256/c T , where c T is the coefficient appearing in the two-point function of the canonically-normalized stress tensor T µν : with P µν ≡ η µν ∇ 2 − ∂ µ ∂ ν . (With this definition, c T = 1 for a theory of a free real scalar field or for a theory of a free Majorana fermion.) Note that in the large N limit of the U (N ) k × U (N ) −k ABJM theory at Chern-Simons level k = 1, 2 (or, more generally, in any mean-field theory) the dominant contribution to λ (B,+) and λ (B,2) comes from double-trace operators and is not suppressed by powers of N as is the contribution from single-trace operators.
Eq. (1.2) is the simplest example of an exact relation between OPE coefficients in an N = 8 SCFT. In Section 3 we explain how to derive, at least in principle, many other exact relations that each relate finitely many OPE coefficients in N = 8 SCFTs. In doing so, we provide a simple prescription for computing any correlation functions in the 1d topological theory that arise from 1 2 -BPS operators in the 3d N = 8 theory. There are three applications of our analysis that are worth emphasizing. The first application is that one can use relations like (1.2) to solve for some of the OPE coefficients in certain N = 8 SCFTs. A non-trivial example of such an SCFT is the U (2) 2 × U (1) −2 ABJ theory, which can be thought of as the IR limit of O(3) supersymmetric Yang-Mills theory in 3d. This ABJ theory is interacting and has c T = 64/3 ≈ 21.33. As far as we know, no detailed information on OPE coefficients is currently available for it. In Appendix D we compute the superconformal index of this U (2) 2 × U (1) −2 ABJ theory and show that it contains no (B, 2) multiplets that could contribute to (1.2), so in this case λ (B,2) = 0. We conclude from (1.2) that λ 2 (B,+) = 64/5, which is the first computation of a non-trivial OPE coefficient in this theory. In Section 3 we compute many more OPE coefficients corresponding to three-point functions of 1 2 -BPS operators in this theory, and we believe that all of these coefficients can be computed using our method.
As a second application of our analysis, we conjecture that in any N = 8 SCFT with a unique stress tensor, there are infinitely many superconformal multiplets that are absent, even though they would be allowed by group theory considerations. 9 If we think of the N = 8 SCFT as an N = 4 SCFT, the theory has an su(2) 1 ⊕ su(2) 2 global flavor symmetry.
Assuming the existence of only one stress tensor, we show that there are no local operators in the N = 8 SCFT that, when restricted to the Q-cohomology, generate su(2) 2 . We therefore conjecture that all the operators in the 1d topological theory are invariant under su(2) 2 . It then follows that the N = 8 SCFT does not contain any multiplets that would correspond to 1d operators that transform non-trivially under su(2) 2 . There is an infinite number of such multiplets that could in principle exist. In Appendix D we give more details on our conjecture, and we show that it is satisfied in all known N = 8 SCFTs. Our conjecture therefore applies to any N = 8 SCFTs that are currently unknown.
The third application of our analysis relates to the numerical superconformal bootstrap study in N = 8 SCFTs that was started in [41]. Notably, the bounds on scaling dimensions of long multiplets that appear in the OPE of O Stress with itself exhibit a kink as a function of c T at c T ≈ 22.8. By contrast, the upper bounds on λ 2 (B,+) and λ 2 (B,2) shown in [41] did not exhibit any such kinks. Here, we aim to shed some light onto the origin of these kinks first by noticing that one can also obtain lower bounds on λ 2 (B,+) and λ 2 (B,2) , and those do exhibit kinks at the same value c T ≈ 22.8. Our analysis suggests that these kinks are likely to be related to the potential disappearance of the (B, 2) multiplet, and in particular to the existence of the U (2) 2 × U (1) −2 ABJ theory that has no such (B, 2) multiplet. Remarkably, the exact relation (1.2) maps the allowed ranges of λ 2 (B,+) and λ 2 (B,2) onto each other within our numerical precision. While the allowed regions for these multiples are extremely narrow, the existence of the U (2) 2 × U (1) −2 ABJ theory combined with the construction of products CFTs that we describe in Section 4.3, shows that these regions must have nonzero area.
The rest of this paper is organized as follows. In Section 2 we explain the construction of the 1d topological QFT from the 3d N ≥ 4 SCFT. In Section 3 we use this construction as well as crossing symmetry to derive exact relations between OPE coefficients in N = 8 SCFTs. Section 4 contains numerical bootstrap results for N = 8 SCFTs. We end with a discussion of our results in Section 5. Several technical details such as conventions and superconformal index computations are delegated to the Appendices.

Topological Quantum Mechanics from 3d SCFTs
In this section we construct the cohomology announced in [3] in the case of three-dimensional SCFTs with N ≥ 4 supersymmetry. 10 We start in Section 2.1 with a review of the strategy of [3]. In Section 2.2 we identify a sub-algebra of the 3d superconformal algebra in which we exhibit a nilpotent supercharge Q as well as Q-exact generators. In Sections 2.3 and 2.4 we construct the cohomology of Q and characterize useful representatives of the non-trivial cohomology classes.

General Strategy
One way of phrasing our goal is that we want to find a sub-sector of the full operator algebra of our SCFT that is closed under the OPE, because in such a sub-sector correlation functions and the crossing symmetry constraints might be easier to analyze. In general, one way of obtaining such a sub-sector is to restrict our attention to operators that are invariant under a symmetry of the theory. In a supersymmetric theory, a particularly useful restriction is to operators invariant under a given supercharge or set of supercharges.
A well-known restriction of this sort is the chiral ring in N = 1 field theories in four dimensions. The chiral ring consists of operators that are annihilated by half of the Poincaré supercharges: [Q α , O( x)] = 0, where α = 1, 2 is a spinor index. These operators are closed under the OPE, and their correlation functions are independent of position. Indeed, the translation generators are Q α -exact because they satisfy {Q α ,Qα} = P αα . Combined with the Jacobi identity, the Q α -exactness of the translation generators implies that the derivative of a chiral operator, [P αα , O( x)] = {Q α ,Õα( x)} is also Q α -exact. These facts imply that correlators of chiral operators are independent of position, because where in the third equality we used the supersymmetric Ward identity.
In fact, in unitary SCFTs correlation functions of chiral primaries are completely trivial.
Indeed, in an SCFT, the conformal dimension ∆ of chiral primaries is proportional to their U (1) R charge. Since all non-trivial operators have ∆ > 0 in unitary theories, all chiral 10 We were informed by L. Rastelli that a general treatment of this cohomological structure will appear in [61].
primaries have non-vanishing U (1) R charges of equal signs, and, as a consequence, their correlation functions must vanish. Therefore, the truncation of the operator algebra provided by the chiral ring in a unitary SCFT is not very useful for our purposes.
One way to evade having zero correlation functions for operators in the cohomology of some fermionic symmetry Q satisfying Q 2 = 0 (or of a set of several such symmetries) is to take Q to be a certain linear combination of Poincaré and conformal supercharges.
Because Q contains conformal supercharges, at least some of the translation generators do not commute with Q now. Nevertheless, there might still exist a Q-exact "R-twisted" translation P µ ∼ P µ + R a , where R a is an R-symmetry generator. Let P be the set of Qexact R-twisted translations, and let P ⊂ {P µ } d µ=1 be the subset of translation generators which are Q-closed but not Q-exact, if any. It follows that if O( 0) is Q-closed, so that O( 0) represents an equivalence class in Q-cohomology, then represents the same cohomology class as O( 0), given that P i ∈ P and P a ∈ P. Here, the R-symmetry indices of O are suppressed for simplicity. In addition, a very similar argument to that leading to (2.1) implies that the correlators of O(x;x) satisfy for separated points (x i ,x i ). Now these correlators do not have to vanish since the Rsymmetry orientation of each of the inserted operators is locked to the coordinatesx i . 11 The correlation functions (2.3) could be interpreted as correlation functions of a lower dimensional theory. In particular, in [3] it was shown that in 4d N = 2 theories one can choose Q such that P and P consist of translations in a 2d plane C ⊂ R 4 . More specifically, holomorphic translations by z ∈ C are contained in P, while anti-holomorphic translations byz ∈ C are contained in P. The resulting correlation functions of operators in that cohomology are meromorphic in z and have the structure of a 2d chiral algebra. In the following section we will construct the cohomology of a supercharge Q in 3d N = 4 SCFTs such that the set P is empty and P contains a single twisted translation. The correlation 11 We stress that (2.3) is valid only at separatedx i points. If this were not the case then we could set x 1 = · · · =x n = 0 in (2.3) and argue that f (x 1 , . . . ,x n ) = 0, since due to (2.2) the R-symmetry weights of the O(x i ; 0) cannot combine to form a singlet. We will later see in examples that the limit of coincidentx i is singular. From the point of view of the proof around (2.1), these singularities are related to contact terms. Such contact terms are absent in the case of the chiral ring construction, but do appear in the case of our cohomology. functions (2.3) evaluate to (generally non-zero) constants, and this underlying structure can therefore be identified with a topological quantum mechanics.

An su(2|2) Subalgebra and Q-exact Generators
We now proceed to an explicit construction in 3d N = 4 SCFTs. We first identify an su(2|2) sub-algebra of the osp(4|4) superconformal algebra. This su(2|2) sub-algebra represents the symmetry of a superconformal field theory in one dimension and will be the basis for the topological twisting prescription that we utilize in this work.
Let us start by describing the generators of osp(4|4) in order to set up our conventions.
The bosonic sub-algebra of osp(4|4) consists of the 3d conformal algebra, sp(4) so (3,2), and of the so (4) su(2) L ⊕ su(2) R R-symmetry algebra. 12 The 3d conformal algebra is generated by M µν , P µ , K µ , and D, representing the generators of Lorentz transformations, translations, special conformal transformations, and dilatations, respectively. Here, µ, ν = 0, 1, 2 are space-time indices. The generators of the su(2) L and su(2) R R-symmetries can be represented as traceless 2 × 2 matrices R b a andR˙a˙b respectively, where a, b = 1, 2 are su(2) L spinor indices andȧ,ḃ = 1, 2 are su(2) R spinor indices. In terms of the more conventional The odd generators of osp(4|4) consist of the Poincaré supercharges Q αaȧ and conformal supercharges S β aȧ , which transform in the 4 of so(4) R , and as Majorana spinors of the 3d Lorentz algebra so(1, 2) ⊂ sp(4) with the spinor indices α, β = 1, 2. The commutation relations of the generators of the superconformal algebra and more details on our conventions are collected in Appendix B.
The embedding of su(2|2) into osp(4|4) can be described as follows. Since the bosonic sub-algebra of su(2|2) consists of the 1d conformal algebra sl(2) and an su(2) R-symmetry, we can start by embedding the latter two algebras into osp(4|4). The sl(2) algebra is embedded into the 3d conformal algebra sp(4), and without loss of generality we can require the sl (2) generators to stabilize the line x 0 = x 2 = 0. This requirement identifies the sl(2) generators with the translation P ≡ P 1 , special conformal transformation K ≡ K 1 , and the dilatation generator D. We choose to identify the su(2) R-symmetry of su(2|2) with the su(2) L R- 12 In this paper we will always take our algebras to be over the field of complex numbers. symmetry of osp(4|4). Using the commutation relations in Appendix B one can verify that, up to an su(2) R rotation, the fermionic generators of su(2|2) can be taken to be Q 1a2 , Q 2a1 , S 1 a1 , and S 2 a2 . The result is an su(2|2) algebra generated by with a central extension given by From the results of Appendix B, it is not hard to see that the inner product obtained from radial quantization imposes the following conjugation relations on these generators: where ε 12 = −ε 21 = 1.
Within the su(2|2) algebra there are several nilpotent supercharges that can be used to define our cohomology. We will focus our attention on two of them, which we denote by Q 1 and Q 2 , as well as their complex conjugates: (2.8) With respect to either of the two nilpotent supercharges Q 1,2 , the central element Z is exact, In addition, the following generators are also exact: 11) These generators form an sl(2) triplet: [ L 0 , L ± ] = ± L ± , [ L + , L − ] = −2 L 0 . We will refer to the algebra generated by them as "twisted," and we will denote it by sl (2). Note that L − is a twisted translation generator. Since it is Q-exact (with Q being either Q 1 or Q 2 ), L − preserves the Q-cohomology classes and can be used to translate operators in the cohomology along the line parameterized by x 1 .

The Cohomology of the Nilpotent Supercharge
Let Q be either of the nilpotent supercharges Q 1 or Q 2 defined in (2.8), and let Q † be its conjugate. Let us now describe more explicitly the cohomology of Q. The results of this section will be independent of whether we choose Q = Q 1 or Q = Q 2 . where ∆ is the scaling dimension (eigenvalue of the operator D appearing in (2.10)), and m L is the su(2) weight associated with the spin-j L (j L ∈ 1 2 N) irrep of the su(2) L R-symmetry (eigenvalue of the operator R 1 1 appearing in (2.10)).
A superconformal primary operator of a unitary N = 4 SCFT in three dimensions must satisfy ∆ ≥ j L + j R . (See Table 1 for a list of multiplets of osp(4|4) and Appendix A for a review of the representation theory of osp(N |4).) It then follows from (2.13) and unitarity that superconformal primaries that are non-trivial in the Q-cohomology must have dimension ∆ = j L and they must be Lorentz scalars transforming in the spin (j L , 0) irrep of the su(2) L ⊕ su(2) R R-symmetry. In addition, they must occupy their su(2) L highest weight state, m L = j L , when inserted at the origin. Such superconformal primaries correspond to the 1 2 -BPS multiplets of the osp(4|4) superconformal algebra that are denoted by (B, +) in Table 1. 13 In Appendix C we show that these superconformal primaries are in fact all the operators of an N = 4 SCFT satisfying (2.13). 13 The (B, −) type 1 2 -BPS multiplets are defined in the same way, but transform in the spin-(0, j R ) representation of the so(4) R symmetry. We could obtain a cohomology based on (B, −) multiplets by exchanging the roles of su(2) L and su(2) R in our construction, but we will not consider this possibility here. Representations of the so(4) ∼ = su(2) L ⊕ su(2) R R-symmetry are given in terms of the su(2) L and su(2) R spins denoted j L and j R , which are non-negative half-integers.

Operators in the 1d Topological Theory and Their OPE
We can now study the 1d operators defined by the twisting procedure in (2. where u a (x) ≡ (1, x). The translated operator O k (x) represents the same cohomology class as O 11···1 ( 0). The index k serves as a reminder that the operator O k (x) comes from a superconformal primary in the 3d theory transforming in the spin-j L = k/2 irrep of su(2) L .
From the 1d point of view, k is simply a label.
The arguments that led to (2.3) tell us correlation functions O k 1 (x 1 ) · · · O kn (x n ) are independent of x i ∈ R for separated points, but could depend on the ordering of these points on the real line. Therefore, they can be interpreted as the correlation functions of a topological theory in 1d.

Correlation Functions and 1d Bosons vs. Fermions
As a simple check, let us see explicitly that the two and three-point functions of O k i (x i ) depend only on the ordering of the x i on the real line. Such a check is easy to perform because superconformal invariance fixes the two and three-point functions of O a 1 ···a k ( x) up to an overall factor. Indeed, let us denote where we introduced a set of auxiliary variables y a in order to simplify the expressions below.
The two-point function of O k (x, y) is: Indeed, this two-point function only depends on the ordering of the two points x 1 and x 2 .
It changes sign under interchanging x 1 and x 2 if k is odd, and it stays invariant if k is even.
Therefore, the one-dimensional operators O k (x) behave as fermions if k is odd and as bosons if k is even.
To perform a similar check for the three-point function, we can start with the expression required by the superconformal invariance of the 3d N = 4 theory. This expression may be non-zero only if (2.18) is a polynomial in the y i . This condition is equivalent to the requirement that k 1 , k 2 , and k 3 satisfy the triangle inequality and that they add up to an even integer. Setting y a i = u a i = (1, x i ), we obtain We can choose to interpret this expression as meaning that operators with even (odd) k behave as bosons (fermions) under cyclic permutations, as we did above. Equivalently, we can choose to interpret it as meaning that upon mapping from R to S 1 we must insert a twist operator at x = ±∞; the twist operator commutes (anti-commutes) with O k if k is even (odd). The effect of (2.20) on correlation functions is that under cyclic permutations we have where we chose the ordering of the points to be x 1 < x 2 < . . . < x n . Eqs. (2.17) and (2.19) above obey this property.

The 1d OPE
with the ordering of points x 1 < x 2 < x 3 < x 4 , one can use the OPE to expand the product . Using (2.21), one can also use the OPE to expand . Equating the two expressions as required by (2.21), one may then obtain non-trivial relations between the OPE coefficients.

Application to N = 8 Superconformal Theories
The topological twisting procedure derived in the previous section for N = 4 SCFTs can be applied to any SCFT with N ≥ 4 supersymmetry, and in this section we apply it to N = 8 SCFTs. We start in Section 3.1 by determining how the operators chosen in the previous section as representatives of non-trivial Q-cohomology classes sit within N = 8 multiplets; we find that they are certain superconformal primaries of 1 4 , 3 8 , or 1 2 -BPS multiplets. We then focus on the twisted correlation functions of 1 2 -BPS multiplets, because these multiplets exist in all local N = 8 SCFTs. For instance, the stress-tensor multiplet is of this type.
More specifically, in Section 3.

The Q-Cohomology in N = 8 Theories
In order to understand how the representatives of the Q-cohomology classes sit within N = 8 super-multiplets, let us first discuss how to embed the N = 4 superconformal algebra, osp(4|4), into the N = 8 one, osp(8|4). Focusing on bosonic subgroups first, note that the so(8) R symmetry of N = 8 theories has a maximal sub-algebra (3.1) The so(4) R and so(4) F factors in (3.1) can be identified with an R-symmetry and a flavor symmetry, respectively, from the N = 4 point of view. In our conventions, the embedding of su(2) 4 into so(8) R is such that the following decompositions hold: The first line in (3. It is now straightforward to describe which N = 8 multiplets can contribute to the Qcohomology of Section 2.3. 14 A list of all possible N = 8 multiplets is given in Table 2. Since in N = 4 notation, the N = 8 theory has an so(4) F flavor symmetry, we should be more explicit about which so(4) F representation a (B, +) multiplet of the N = 4 the- 14 Here Q can be chosen to be either Q 1 or Q 2 , just as in Section 2.3.
Note that the (B, +) multiplets in (3.4)-(3.7) have j R = 0, as they should, and that they transform in irreps of the flavor symmetry with (j 1 , j 2 ) = a 3 2 , a 4 2 , which in general are non-trivial. The operators in the topological quantum mechanics introduced in the previous section will therefore also carry these flavor quantum numbers. We will see below, however, that in the examples we study we will have only operators with j 2 = 0.

Twisted (B, +) Multiplets
In this section we will construct explicitly the twisted version of N = 8 superconformal primaries of (B, +) type. This construction will be used in the following sections to compute correlation functions of these operators in the 1d topological theory. Let us start by recalling some of the basic properties of these operators in the full three-dimensional theory. A (B, +) superconformal primary transforming in the [00k0] irrep will be denoted by O n 1 ···n k ( x), where the indices n i = 1, . . . , 8 label basis states in the 8 c = [0010] irrep. This operator is symmetric and traceless in the n i , and it is a Lorentz scalar of scaling dimension ∆ = k/2-see Table 2.
As is customary when dealing with symmetric traceless tensors, we introduce the polar- where the normlization convention for our operators is fixed by (3.9). The coefficient λ in (3.10) may be non-zero only if k 1 , k 2 , and k 3 are such that the 3-point function is a polynomial in the Y i .
The topologically twisted version of the (B, +) operators O k ( x, Y ) can be constructed as follows. According to (3.6) We can project O k ( x, Y ) onto this irrep by choosing the polarizations Y n appropriately. In particular, Y n transforms in the 8 c of so(8) R ; as given in (3.2), this irrep decomposes into irreps of the four su(2)'s as We can choose to organize the polarizations Y n such that transforms as a fundamental of so(4) R,2 and is invariant under so(4) L,1 .
Since the k-th symmetric product of the (2, 1, 2, 1) irrep in (3.11) is given precisely by the Explicitly, we set where σ i aȧ for i = 1, . . . , 4, are defined in terms of the usual Pauli matrices as σ i aȧ ≡ (1, iσ 1 , iσ 2 , iσ 3 ), and we introduced the variables y a andȳ˙a that play the role of su(2) L and su(2) 1 polarizations, respectively. It is easy to verify that the ansatz (3.12) respects the condition Y · Y = 0 that the so(8) polarizations Y n must satisfy. We conclude that the N = 4 superconformal primary that contributes to the cohomology is obtained from

by plugging in the projection (3.12). It is given by
. (3.14) Note that the twisted operator O k (x,ȳ) represents a collection of k + 1 operators like the ones defined in Section 2.4, packaged together into a single expression with the help of the The components O k,a 1 ···a k+1 (x) transform as a spin-k/2 irrep of su(2) 1 .
By applying the projection (3.12) and (3.14) to the two-point and three-point functions in (3.9) and (3.10), we find that the corresponding correlators in the 1d theory are where the angle brackets are defined by The correlators (3.16)

Twisted Four Point Functions
As we discussed in Section 2. In addition, we will see that applying crossing symmetry to these 4-point functions leads to a tractable set of constraints. These constraints allow us to derive simple relations between OPE coefficients that hold in any N = 8 theory.
The simplicity of the crossing constraints in the 1d theory is easy to understand from its 3d origin. In general the OPE between two (B, +) operators in the 3d theory contains only a finite number of operators non-trivial in Q-cohomology. 15 Indeed, there is a finite number of R-symmetry irreps in the tensor product [00m0]⊗[00n0], and multiplets of B-type are completely specified by their R-symmetry irrep. 16 A given correlator in the 1d theory therefore depends only on a finite number of OPE coefficients, and the resulting crossing constraints therefore also involve only a finite number of OPE coefficients of the 3d theory.
Let us discuss the representations in the OPE of two (B, +) operators that transform as [00n0] and [00m0] of so(8) R in more detail. 17 The possible R-symmetry representations in this OPE are (assuming m ≥ n) , (3.19) where in the second line we have indicated the N = 8 multiplets that may be non-trivial in Q-cohomology in each of the so(8) R irreps appearing in the product (see Table 2). There is an additional kinematical restriction on the OPE when m = n. In this case the tensor product decomposes into a symmetric and anti-symmetric piece corresponding to terms in (3.19) with even and odd q, respectively. Operators that appear in the anti-symmetric part of the OPE must have odd spin, and therefore cannot be of B-type (whose superconformal primary has zero spin). Passing to the cohomology, every term on the right-hand side of (3.19) represents a type of multiplet that is non-trivial in the Q-cohomology and that A few case studies are now in order.

The Free Multiplet
The simplest possible case to consider involves the OPE of O 1 (x,ȳ), which arises from twisting the superconformal primary O 1 ( x, Y ) of the free N = 8 multiplet consisting of 8 free real scalars and fermions. While it is trivial to write down the full correlation functions in this theory, it will serve as a good example for the general 1d twisting procedure.
According to (3.19) and the discussion following it, the relevant so(8) R irreps in the O 1 × O 1 OPE appear in the symmetric tensor product: The contribution to the cohomology in the 35 c = [0020] irrep comes from the superconformal primary of the stress-tensor (B, +) multiplet that we will simply denote here by O 2 , and the only contribution from the [0000] multiplet is the identity operator 1. After the twisting, task is simpler here, since we are just interested in contributions that are non-trivial in cohomology.
the O 1 × O 1 OPE can therefore be written as where the factor √ 2 was chosen for later convenience. One can check that the twisted 2-point and 3-point functions in (3.16) and (3.17) are reproduced from this OPE.
Note that the OPE coefficient λ is fixed by the conformal Ward identity in terms of the coefficient c T of the 2-point function of the canonically normalized stress-tensor. In particular, in the conventions of [41] λ = 8/ √ c T and a free real boson or fermion contributes one unit to c T . 18 A free N = 8 multiplet therefore has c T = 16, and as we will now see, this can be derived from the crossing symmetry constraints.
Using the invariance under the global su(2) symmetry, and assuming the 4-point function of O 1 can be written as The variablew should be thought of as the single su(2) 1 -invariant cross-ratio, and is defined in terms of the polarizations asw Applying the OPE (3.21) in the s-channel (i.e., (12) (34)) gives The only other OPE channel that does not change the cyclic ordering of the operators is the t-channel (i.e., (41) (23) ). In computing it we should be careful to include an overall minus sign from exchanging the fermionic like O 1 (x 4 ,ȳ 4 ) three times (see the discussion in section 2.4.2). The 4-point function in the t-channel is therefore obtained by exchangingȳ 1 ↔ȳ 3 in (3.24) and multiplying the result by a factor of (−1), which gives In deriving (3.25) we used the identity Equating (3.24) to (3.25) we obtain (after a slight rearrangement) our first 1d crossing This equation has the unique solution

The Stress-Tensor Multiplet
Moving forward to a non-trivial example we will now consider the OPE of the twisted version Including all of the contributions mentioned above, the OPE of O 2 can be written as where the numerical factors were chosen such that the OPE coefficients match the conventions of [41]. We emphasize again that up to these coefficients, the form of (3.30) is trivially fixed by demanding invariance under the global su(2) 1 symmetry.
Evaluating the O 2 4-point function in the s-channel gives ( The t-channel expression is obtained by takingȳ 1 ↔ȳ 3 under whichw → 1 −w. Equating the two channels results in the crossing equation The solution of (3.32) is given by (1.2), which we reproduce here for the convenience of the reader: In Table 3 we list the values of these OPE coefficients in the theory of a free N = 8 multiplet and in mean-field theory (MFT) (corresponding, for instance, to the large N limit of the  19 We consider these facts to be non-trivial checks on our formalism. 19 Correlators with 6 or more insertions of O 2 depend on more OPE coefficients on top of the ones appearing in (3.33). theory is an example for which λ (B,2) = 0 and we will next consider an interacting theory of this sort.

The Twisted Sector of
We will now consider the U (2) 2 × U (1) −2 ABJ theory and show that the OPE coefficients in its twisted sector can be computed explicitly. This theory is believed to arise in the IR of N = 8 supersymmetric Yang-Mills theory with gauge group O(3), and as such is expected to be a strongly coupled SCFT. However, it also shares some similarities with the free U (1) 2 ×U (1) −2 ABJM theory. Indeed, the moduli space and the spectrum of chiral operators in both theories are identical [53]. In particular, the spectrum of operators contributing to the Q-cohomology is the same in both theories, though we stress that the correlators are generally different.
In Appendix D we show that the contribution to the cohomology in both theories arises from a single (B, +) multiplet transforming in the [00k0] irrep for any even k. 20 In other words, there is one twisted operator O k for every even k. With this spectrum, the most general twisted OPE that we can write down up to Q-exact terms is given by . (3.37) Moreover, it follows that the OPE coefficients of this system can be determined completely since λ 2,2,2 is calculable by using supersymmetric localization. In particular, λ 2 2,2,2 = 1 2 λ 2 Stress = 128 c T and recall that c T is the coefficient of the 2-point function of the canonicallynormalized stress-tensor. In [41], by using supersymmetric localization it was found that in the U (2) 2 × U (1) −2 ABJ theory c T = 64/3 ⇒ λ 2 2,2,2 = 6. We conclude that the coefficients λ m,n,p in (3.34), or equivalently, the 3-point functions of 1 2 -BPS operators in the U (2) 2 × U (1) −2 ABJ theory are calculable. Some specific values of these OPE coefficients are listed in Table 4.

4-point Correlation Functions and Superconformal Ward Identity
In this section we will show that in the particular case of 4-point functions of (B, +) type operators O k ( x, Y ) in N = 8 SCFTs, the results obtained by using the topological twisting procedure can be reproduced by using the superconformal Ward identity derived in [62].
This will provide a check on some of the computations of the previous sections that involve such 4-point functions. Note, however, that the topological twisting method applies more generally to any N ≥ 4 SCFT and to any n-point function of twisted operators. In general, in the notation we used above: λ 2,n,n = n 2 λ 2,2,2 . - where the variables z,z and w,w are related, respectively, to the sp(4) and so(8) R cross-ratios defined by The function G k (z,z; w,w) in (3.38) is symmetric under z ↔z and under w ↔w. Moreover, it is a general degree k polynomial in 1 U and V U , as follows from the fact that the 4-point function must be polynomial in all the Y i variables. The full osp(8|4) superconformal algebra imposes additional constraints on G k (z,z; w,w), which are encapsulated in the superconformal Ward identity. This Ward identity was computed in [62] and takes the form Let us now discuss how to obtain the 4-point function in the topologically twisted sector directly in terms of the variables z,z, w, andw. To do that we restrict the external operators in (3.38) to a line by taking x i = (0, x i , 0) with 0 = x 1 < x 2 < x 3 = 1 and x 4 = ∞. In particular, this implies that z 1d =z 1d = x 2 . In addition, using the projection of the polarizations Y i , which was given in (3.12) and (3.14), we find that Note that z = x 12 x 34 x 13 x 24 is the single SL(2, R) cross-ratio, and for the ordering x 1 < x 2 < x 3 < x 4 we have that 0 < z < 1. In addition, recall that ȳ 1 ,ȳ 2 ȳ 3 ,ȳ 4 ȳ 1 ,ȳ 3 ȳ 2 ,ȳ 4 is the single SU (2) crossratio, which was denoted byw in (3.23) for reasons that now become obvious. We conclude that in terms of the variables z,z, w andw the 1d topological twisting is equivalent to setting z =z = w, and identifyingw with the SU (2) cross-ratio (3.23).
Since from our general arguments the full 4-point function in (3.38) must be constant after the 1d twisting, and the pre-factor of G k (z,z; w,w) in (3.38) projects to a constant (up to ordering signs), we conclude that where the a j are some numbers and the same relation must hold for G n (z, z; w, z) as follows from the w ↔w symmetry of G k . In fact, one can prove (3.44) directly from the superconformal Ward identity (3.41) by a simple application of the chain rule. 22 Indeed, z∂ z G k (z, z; z,w) = (z∂ z +z∂z + w∂ w )G n (z,z; w,w) z→z w→z (3.41) where in the next to last equality we used symmetry of G k under z ↔z.
To make contact with the 1d OPE methods of Section 3.3 we must find the contribution of each superconformal multiplet to the function G k (z, z; z,w) = G k (w). For that purpose consider the s-channel expansion of the 4-point function (3.38): while contributions arising from the B-type multiplets are non-vanishing. This confirms 22 The analogous statement in the context of N = 4 theory in four dimensions is more familiar (see e.g. [64] and references therein). In that case the Ward identity for the 4-point functions of 1 2 -BPS operators transforming in the [0k0] ∈ SU (4) R , is (3.45) The holomorphic functions f k (z,w) were interpreted in [3] as correlation function in 2d chiral CFT.
the general cohomological arguments of section (3.1) that only those multiples survive the topological twisting.
Let us now compute the 1d projection of a given superconformal block. Superconformal primary operators of type B have zero spin and those that transform in the [0(a − b)(2b)0] irrep have dimension ∆ = a. It follows that the full contribution to G k (z,z; w,w) from such an operator is λ 2 Y ab (w,w)g a,0 (z,z) (see (3.47)). Our normalization convention for conformal blocks is defined, as in [41], to be In addition, from the SO(8) Casimir equation satisfied by the Y ab (see e.g., [64]), one can show that where P n (x) are the Legendre polynomials and the overall constant was fixed to match the conventions of [41].
We conclude that the contribution from any B-type multiplet to G k is given by where the higher order contributions in the expansion around z = 0 must all cancel, as the projected superconformal block is independent of z. One can verify that the contributions from each multiplet to the 4-point functions of O 1 in (3.24) and those of O 2 in (3.31), which were obtained by using the 1d OPE directly, match precisely those same contributions obtained using the prescription in (3.50).

Numerics
In this section, we present improved numerical bootstrap bounds for generic N = 8 SCFTs, extending the work of [41]. We obtain both upper and lower bounds on OPE coefficients of protected multiplets appearing in the OPE of O Stress with itself, and find that the allowed regions are small bounded areas. The characteristics of such bounds, including the appearance of the kinks observed in [41], can be understood by combining the analysis of the Q-cohomology discussed in previous sections with the general considerations regarding product SCFTs that will be described in this section. Throughout this section, we denote each multiplet by the multiplet types listed in Table 5, in particular we call (B, +) [0020] 'Stress', which denotes the stress-tensor multiplet. In

Formulation of Numerical Conformal Bootstrap
Let us briefly review the formulation of the numerical conformal bootstrap for 3d CFTs with maximal supersymmetry. For further details, we refer the reader to [41].
where M ranges over all the superconformal multiplets that appear in the OPE of O 35c with itself as listed in Table 5, d M are functions of superconformal blocks, and λ 2 M are squares of OPE coefficients that must be positive by unitarity. As in [41], we will normalize the OPE coefficient of the identity multiplet to λ Id = 1, and we will parameterize our theories by the value of λ Stress . The latter OPE coefficient is simply related to the coefficient c T that represents the normalization of the two-point function of the canonically-normalized stress tensor (1.3). 23 In particular, we have c T = 256/λ 2 Stress in conventions where c T = 1 for a 23 In the case of an SCFT with more than one stress-tensor multiplet, which are all assumed to be of (B, +) [0020] type, c T corresponds to the total (diagonal) canonically normalized stress tensor. theory of a free real scalar field or a free Majorana fermion in three dimensions. See Table 6 for the few lowest values of c T for known N = 8 SCFTs.   [41] for a derivation as well as analytical formulas for these coefficients.
In [41], the numerical bootstrap was used to find upper bounds on scaling dimensions of long osp(8|4) multiplets as well as upper bounds on OPE coefficients of short multiplets.
Here, we extend the results on upper bounds to semi-short multiplets and we also provide lower bounds on the OPE coefficients of both short and semi-short multiplets. To find upper/lower bounds on a given OPE coefficient of a multiplet M * that appears in the provided that the scaling dimensions of each long multiplet satisfies ∆ ≥ ∆ * j . Here we choose the spectrum to only satisfy unitarity bounds ∆ * j = j + 1, which provides no restrictions on the set of N = 8 SCFTs. To obtain the most stringent upper/lower bound on λ 2 M * , one should then minimize/maximize the RHS of (4.4) under the constraints (4.3). Note that a lower bound can only be found this way for OPE coefficients of protected multiplets, as shown in [20]. For long multiplets, the condition α( d M * ) = −1 is inconsistent with the requirement α( d M ) ≥ 0, because it is possible to have a continuum of long multiplets arbitrarily close to M * .
The numerical implementation of the minimization/maximization problem described above requires two truncations: one in the number of derivatives used to construct α and one in the range of multiplets M that we consider. We have found that considering multiplets M with spins j ≤ 20 and derivatives parameter Λ = 19 as defined in [41] leads to numerically convergent results. The truncated minimization/maximization problem can now be rephrased as a semidefinite programing problem using the method developed in [20]. This problem can be solved efficiently by freely available software such as sdpa gmp [65].

Bounds for Short and Semi-short Operators
In Figure 1 we show upper and lower bounds for λ 2 (B,+) and λ 2 (B,2) in N = 8 SCFTs, and in Figure 2 we show upper and lower bounds on OPE coefficients in the semi-short (A, 2) and (A, +) multiplet series for the three lowest spins 1, 3, 5 and 0, 2, 4, respectively. We plot these bounds in terms of λ 2 Stress /16 instead of c T (as was done in [41]), because the allowed region becomes bounded by straight lines. Recall that for an SCFT with only one stresstensor multiplet, λ 2 Stress /16 can be identified with 16/c T ; this quantity ranges from 0, which corresponds to the mean-field theory obtained from large N limit of ABJ(M) theories with c T → ∞, to 1, which corresponds to the free U (1) k ×U (1) −k ABJM theory with c T = 16 that was shown in [41] to be the minimal possible c T for any consistent 3d SCFT-see Table 6.
For SCFTs with more than one stress tensor, one can also identify λ 2 Stress /16 with 16/c T , where c T is the coefficient appearing in the two-point function of the canonically-normalized diagonal stress tensor, but, as we will see in the next subsection, more options are allowed.  Table 7.
There are a few features of these plots that are worth emphasizing: • The bounds are consistent with and nearly saturated by the free and mean-field theory limits. In these limits, the OPE coefficients of the (B, +) and (B, 2) multiplets are given in Table 3.
The mean-field theory values can be derived analytically using large N factorization. In Similarly, the OPE coefficients for the first few A-type multiplets can also be computed analytically by expanding the four point function of O 35c into superconformal blocks.
We give the first few values in Table 7.  • The numerical bounds for λ 2 (B,+) and λ 2 (B,2) can be mapped onto each other under the exact relation (1.2) that is implied by crossing symmetry in Q-cohomology. This mapping suggests that the relation (1.2) is already encoded in the numerical bootstrap constraints, and indeed, we checked that the numerical bounds do not improve by imposing it explicitly before running the numerics. The apparent visual discrepancy in the size of the allowed region between the two plots in Figure 1 comes from the factor of 5 difference between λ 2 (B,+) and λ 2 (B,2) in (1.2).
• The lower bounds for λ 2 (B,+) as well as for the OPE coefficients of the A-series are strictly positive for all N = 8 SCFT. Therefore, at least one multiplet of each such kind must exist in any N = 8 SCFT-the absence, for instance, of (A, 2) multiplets of spin j = 3 would make the theory inconsistent.
• The lower bounds in Figures 1 and 2 are saturated (within numerical uncertainties) in the mean field theory limit c T → ∞, while the upper bounds are less tight. In the free theory limit c T = 16, it is the upper bounds that are saturated (within numerical uncertainties), while the lower bounds are less tight for the A-series OPE coefficients.
In the case of the (B, +) and (B, 2) multiplets, the lower bounds are also saturated in the free theory limit c T = 16, simply because there the relation (1.2) combined with λ 2 (B,2) ≥ 0 forces the lower bounds to coincide with the precise values of the OPE coefficients.
• The lower bound for λ 2 (B,2) vanishes everywhere above λ 2 Stress /16 ≈ 0.701 (or, equivalently, below c T ≈ 22.8). Consequently, the lower bound for λ 2 (B,2) shows a kink at c T ≈ 22.8, and upon using (1.2) this kink also produces a kink in the lower bound for The previous analysis suggests that the kink is caused by the disappearance of (B, 2) multiplets, and therefore λ 2 (B,2) = 0. The only known N = 8 SCFT aside from the free theory that lies in the range where λ 2 (B,2) is allowed to vanish, namely 16 ≤ c T ≤ 22.8, is U (2) 2 × U (1) −2 ABJ theory, which has c T ≈ 21.33-see Table 6. In the Appendix we calculate the superconformal index of this theory and show explicitly that it indeed does not contain any (B, 2) multiplets that is in [0200] irrep. While this theory has c T slightly smaller than the observed locations of the kinks, the lower bounds are observed to be less accurate near the free theory as noted above, so the location of the kink may be caused by the existence of the U (2) 2 × U (1) −2 ABJ theory that lacks (B, 2) multiplets.

Analytic Expectation from Product SCFTs
As all known constructions of N = 8 SCFTs provide discrete series of theories, one may expect that only discrete points in Figures 1 and 2  Let us denote these scalars by O 1 ( x, Y ) and O 2 ( x, Y ) for the two SCFTs, respectively, where x is a space-time coordinate and Y is an so(8) polarization. Moreover, let us normalize these operators such that (4.6) In the product SCFT we can consider the operator (4.8) Apart from this normalization condition, the linear combination in (4.7) is arbitrary.
We can easily calculate the four-point function of this operator given (4.6) and the four- The term in the parenthesis is the four point function of a 35 c operator in mean field theory. In In particular, if we are looking at the coefficient of the stress tensor block itself, we have

16
, λ 2 2 that must lie within the region allowed by our bounds. Instead, the curve must lie within the allowed region. This curve is an arc of a parabola.
An example is in order. Let us consider the (B, 2) multiplet, and the following three  Since the filled region in Figure 3 is realized in the product SCFT between known theories, it must lie within the region that is not excluded by the numerical bounds presented in Figure 1. It is not hard to see that it does. What is remarkable though is that the numerical bounds are almost saturated by a large part of the region in Figure 3, suggesting that it is likely that the allowed region in Figure 1 is what it is because of the existence of the three SCFTs denoted by (A), (B), and (C) above, as well as their product SCFT.
Note that the region between the x-axis and the outer curve that extends between points (B) and (C) in Figure 3 is an allowed region according to the numerical exclusion plot in Figure 1. However, none of the known N = 8 SCFTs lie within this region, suggesting that perhaps all N = 8 SCFTs sit within the filled region in Figure 3. In future work, it would be interesting to verify whether or not this statement is correct.

Summary and Discussion
In this paper, we have studied a certain truncation [3] of the operator algebra of threedimensional N = 4 SCFTs obtained by restricting the spectrum of operators to those that are nontrivial in the cohomology of a certain supercharge Q. The local operators that represent non-trivial cohomology classes are certain 1 2 -BPS operators that are restricted to lie on a line, and whose correlation functions define a topological quantum mechanics. More specifically, these 1 2 -BPS operators are superconformal primaries that are charged only under one of the su(2) factors in the so(4) R ∼ = su(2) L ⊕su(2) R R-symmetry. These are precisely the operators that contribute to the Higgs (or Coulomb) limits of the superconformal index [66].
What is special about the truncation we study is that the correlation functions in the 1d theory are very easy to compute and are in general non-vanishing. In particular, the crossing symmetry constraints imposed on these correlation functions can be solved analytically and may lead to non-trivial constraints on the full 3d N = 4 theory.
We worked out explicitly some of these constraints in the particular case of N = 8 SCFTs.
These N = 8 SCFTs can be viewed as N = 4 SCFTs with so(4) flavor symmetry. One of our main results is the relation (1.2) between the three OPE coefficients λ Stress , λ (B,+) , and λ (B,2) that appear in the OPE of the N = 8 stress-tensor multiplet with itself. Since every local N = 8 SCFT has a stress-tensor multiplet, the relation (1.2) is universally applicable to all local N = 8 SCFTs! As explained in Section 3.2, this relation is only a particular case of more general relations that also apply to all local N = 8 SCFTs and that can be easily derived using the same technique.
In particular cases, additional information about a given theory combined with the exact relations we derived may be used to determine exactly the OPE coefficients. For instance, in U (1) k ×U (1) −k ABJM theory at level k = 1, 2 and in U (2) 2 ×U (1) −2 ABJ theory, we can show using the superconformal index that many multiplets must be absent-see Appendix D. In this case, we can determine many OPE coefficients exactly. While the case of U (1) k ×U (1) −k ABJM theory is rather trivial (for k = 1 the theory is free, while for k = 2 we have a free theory coupled to a Z 2 gauge field), the case of U (2) 2 × U (1) −2 ABJ theory seems to be nontrivial. This latter theory is the IR limit of O(3) N = 8 super-Yang-Mills theory in three dimensions. In this theory, for instance, we find that the coefficient λ (B,+) mentioned above equals 8/ √ 5, while in the free theory it equals 4 (in our normalization). We believe that for those theories all other OPE coefficients of operators participating in the cohomology can also be computed and we list many more values in Table 4. As far as we know, our results for the U (2) 2 × U (1) −2 ABJ theory constitute the first three-point functions to be evaluated in an interacting N = 8 SCFT beyond the large N limit. It would be very interesting to see if there are other non-trivial theories where the conformal bootstrap is as predictive as in this case. We hope to return to this question in the future.
Our analysis leads us to conjecture that in an N = 8 SCFT with a unique stress tensor, there are many super-multiplets that must actually be absent, even though they would be allowed by osp(8|4) representation theory. The argument is that the so(4) ∼ = su(2) 1 ⊕ su(2) 2 flavor symmetry of the 1d topological theory is generated by operators coming from N = 8 stress-tensor multiplets. But each such stress-tensor multiplet gives the generators of only one of the two su(2) flavor symmetry factors in our topological theory. Therefore, if the 3d theory has a unique stress tensor that corresponds to, say, the su(2) 1 factor, then the 1d topological theory will likely be invariant under su(2) 2 . Consequently, the topological theory must not contain any operators charged under su(2) 2 . The absence of such operators in 1d implies the absence of many BPS multiplets in 3d. For more details on which 3d BPS multiplets are conjectured to be absent, see the discussion in Appendix D.
In Section 4, we complement our analytical results with numerical studies. In particular, we improve the numerical results of [41] by providing both upper and lower bounds on various OPE coefficients of BPS multiplets appearing in the OPE of the superconformal primary of the stress-tensor multiplet, O 35c , with itself. We find that these bounds are pretty much determined by three theories: the U (1) k × U (1) −k ABJM theory, the U (2) 2 × U (1) −2 ABJ theory, and the mean-field theory obtained in the large N limit of ABJM and ABJ theories, as well as their product SCFT. Interestingly, these results also provide an intuitive explanation for the kink observed in [41] to occur at c T ≈ 22.8: this kink is related to a potential disappearance of the (B, 2) multiplet appearing in the O 35c × O 35c OPE. Indeed, we checked in Appendix D that this multiplet is absent from the U (2) 2 × U (1) −2 ABJ theory, which has c T = 64 3 ≈ 21.33, a value very close to where the kink occurs. As of now, the exact relations between OPE coefficients that we derived in this paper from the conformal bootstrap seem to be complementary to the information one can obtain using other techniques, such as supersymmetric localization. It would be interesting to understand whether or not one can derive them in other ways that do not involve crossing symmetry.
We leave this question open for future work.
It would be very interesting to understand the implications of our result to the M-theory duals of the N = 8 ABJ(M) theories. In particular, in the large N limit these gravity duals are explicitly given by classical eleven-dimensional supergravity. The exact relations between OPE coefficients must then translate into constraints that must be obeyed by higherderivative corrections as well as by quantum corrections to the leading two-derivative elevendimensional classical supergravity theory.
Another open question relates to the nature of the 1d topological theory that represents the basis for the exact relations we derived. One can hope to classify all such 1d topological theories and relate them to properties of superconformal field theories in three dimensions.
In particular, it might be possible that such a study will shed some light on the problem of classifying all N = 8 SCFTs.
Another future direction represents an extension of our results to theories with N < 8 supersymmetry. In this paper, we carried out explicitly both an analytical and numerical study that was limited to N = 8 SCFTs. One might expect a richer structure in the space of SCFTs with smaller amounts of supersymmetry. From the point of view of the AdS/CFT correspondence, the N = 6 case would be particularly interesting, as such SCFTs are still rather constrained, but there exists a large number of them that have supergravity duals.

Acknowledgments
We thank Ofer Aharony, Victor Mikhaylov, and Leonardo Rastelli for useful discussions. The The results of this work rely heavily on properties of the osp(N |4) symmetry algebra of 3d CFTs with N supersymmetries. In this appendix we give a brief review of the representation theory of osp(N |4) (for more details the reader is refered to e.g., [67][68][69]).
Representations of osp(N |4) are specified by the scaling dimension ∆, Lorentz spin j, and so(N ) R-symmetry irrep [a 1 · · · a N 2 ] of the superconformal primary, as well as by various shortening conditions. The other operators in the multiplet can be constructed from the so(2, 1) ⊕ so(N ) R highest-weight state |h.w. = ∆, j; [a 1 · · · a N 2 ] of the superconformal primary by acting on it with the Poincaré supercharges Q w m . The Q w m transform in the [10 · · · 0] fundamental representation of so(N ) R , labeled by the weight vector w, and the spin- 1 2 Lorentz representation labeled by m = ± 1 2 . States of the form Q w 1 m 1 · · · Q w k m k |h.w are called "descendants at level k", and their norm is determined in terms of the norm of |h.w.  Table 8. In words, for j > 0 the type A shortening condition implies that a state of spin j − 1/2 at level one becomes null, while for j = 0 the first state that becomes null is a spin zero descendant at level 2. In the j = 0 case we also have the type B multiplets which admit stronger shortening conditions than type A. The type B shortening conditions are specified by requiring that a spin-1 2 state at level one and also a spin zero state at level two both become null.
The two classes of shortening conditions A and B are each further subdivided into more groups, since each of the corresponding conditions in table 8 can be applied several times for different so(N ) R weights w ∈ [10 · · · 0] of the supercharges Q w α . The allowed weights w are restricted by unitarity and depend on the particular so(N ) R irrep of the superconformal primary. The full list of multiplet types for N = 8 SCFTs are listed in Table 2, and for 24 The particular linear combination Q w was chosen such that this state is annihilated by J + .
|h.w. = 0 The last two lines contain the commutation relations of the so(N ) and sp(4) algebras, respectively. In (B.1), δ represents the Kronecker delta symbol, and ω is the sp(4) symplectic form.
Since sp(4) ∼ = so (3,2), the generators M AB can be easily written in terms of a more standard presentation of the generators of so (3,2), which in turn can be written in terms of the generators of angular momentum, translation, special conformal transformations, and dilatation in three dimensions. The latter rewriting is more immediate, so we will start with it. LetM IJ be the generators of so (3,2) satisfying where the indices I, J, . . . run from −1 to 3 and η IJ is the standard flat metric on R 2,3 with signature (−, −, +, +, +). TheM IJ are anti-symmetric. Writing we obtain the usual presentation of the conformal algebra: where η µν = diag(−1, 1, 1) and µ, ν = 0, 1, 2.
Then writing the matrix M AB as from the last line of (B.1) one obtains the following rewriting of the conformal algebra 26 (B.17) In this notation, the conjugation properties of the generators (B.5) are The extension of the conformal algebra to the osp(N |4) superconformal algebra is given by where R rs are the anti-symmetric generators of the so(N ) R-symmetry. In addition to (B.18), we also have The relation between the odd generators Q α and S β appearing here and the supercharges 26 Parentheses around indices means symmetrization by averaging over permutations.

B.2 osp(4|4)
In the following we are going to focus on N = 4. We project the so (4)  where in the last line we used the definitions (σ nm ) b a ≡ 1 4 (σ nσm − σ mσn ) b a and (σ nm )˙a˙b ≡ 1 4 (σ n σ m −σ m σ n )˙a˙b. We turn vectors into bi-spinors using v aȧ ≡ σ r aȧ v r . The so(4) rotation generators R rs can be decomposed into dual and anti-self-dual rotations using The N = 4 superconformal algebra in this notation is given by 27 and also In this notation, the conjugation properties (B.25) become are superconformal primaries (SCPs) of (B, +) multiplets.
First, let us show that such operators cannot belong to A-type multiplets. A-type mul- 27 We only list the commutators which involve R-symmetry indices as the others remain as before.
tiplets satisfy the unitarity bound ∆ ≥ j L + j R + s + 1 , for SCPs of A-type multiplets .
We can in fact show that ∆ > j L + j R , for all CPs of A-type multiplets .

C.2 N = 8
We now examine how an N = 4 superconformal primary that satisfies (C.1) can appear as part of an N = 8 supermultiplet. We have already shown that such an operator must have Since such an operator is a superconformal primary of a B-type multiplet in N = 4, it must also be in a B-type multiplet in N = 8.
If (w 1 , w 2 , w 3 , w 4 ) is an so(8) weight, then we can take the su(2) L ⊕ su(2) R quantum numbers to be An operator satisfying (C.7) must therefore have The states of the superconformal primary of any B-type multiplet satisfy ∆ ≥ w 1 , for any SCP of a B-type multiplet ,

D Cohomology Spectrum from Superconformal Index
In this appendix we describe a limit of the superconformal index that is only sensitive to non-trivial states in the cohomology of Q. For any N = 2 SCFT one can define the superconformal index [68,71] as where ∆ is the conformal dimension and m s is the su(2) Lorentz representation weight.
Here, the quantities F i and R are the charges under the flavor symmetries indexed by i and the R symmetry, respectively, while the Q a are topological charges that exist whenever the fundamental group π 1 of the gauge group is non-trivial. It can be shown that the index does The theory has three commuting Abelian flavor symmetries. Two of these Abelian flavor symmetries are easy to describe: under them the fields (A 1 , A 2 , B 1 , B 2 ) have charges normalized as in (D.3) for later convenience. The third Abelian symmetry is more subtle.
Since both U (N ) and U (Ñ ) have non-trivial π 1 , there exist two topological symmetries whose currents are respectively, where F 1,2 are the vector multiplet field strengths. (In this normalization, the corresponding charges Q 1 and Q 2 satisfy Q a ∈ k 2 Z.) One can also define a U (1) b symmetry under which the fields (A 1 , A 2 , B 1 , B 2 ) have charges In computing the superconformal index, we can introduce fugacities for all the symmetries discussed above and compute where z 1 , z an SO(4) F ∼ = SU (2) 1 ×SU (2) 2 flavor symmetry. The four chiral multiplets in N = 2 language are assembled, in our conventions, into a hypermultiplet whose scalars (A 1 , B † 2 ) transform as a doublet of SU (2) L and a twisted hypermultiplet whose scalars (A 2 , B † 1 ) transform as a doublet of SU (2) R . Let (m L , m R , m 1 , m 2 ) be the magnetic quantum numbers for SU (2) L × SU (2) R ×SU (2) 1 ×SU (2) 2 and (j L , j R , j 1 , j 2 ) the corresponding spins. We have the following identification of charges 28 With all the normalization factors taken into account, the superconformal index (D.8) is where z = xz 1 ,z = x/z 1 , p = √ z 2 z b , and q = z 2 /z b , and we used the fact that only states with ∆ = R + m s contribute to the index.
The limit in which only the states that are non-trivial in Q-cohomology contribute to the index isz → 0 with z, p, and q held fixed: 29 I(z, p, q) = lim z→0 I(z,z, p, q) = tr z ∆ q 2m 1 p 2m 2 , (D.14) where the trace is over the states with ∆ = j L and j R = s = m R = m s = 0. Indeed, in this limit only states with m s + m R = 0 give a non-vanishing contribution in (D.13). Since only states with ∆ = m s + m L + m R contributed to I in the first place, we have that in the limitz → 0 only states with ∆ = m L contribute. These are precisely the states that are non-trivial in the Q-cohomology described in Section 2.3.
At each order in z, the q and p dependence should organize itself into a sum of characters of SU (2) 1 and SU (2)  theory that has c T = 64 3 ≈ 21.33, and the free U (1) k × U (1) −k ABJM theories with k = 1, 2, which have c T = 16. 30 We already know that from Section 4.2 the (B, 2) [0200] multiplet is absent in the free theories, and in this appendix we will show that the same is true also for the U (2) 2 × U (1) −2 ABJ theory.
Let us start by computingĨ for the U (1) k × U (1) −k ABJM theory with k = 1, 2. In this case we have only two integration variables λ andλ, one for each gauge group, and the GNO monopoles are labeled by a single number n =ñ. We obtaiñ I(z, p, q) = We see thatĨ only depends on the combination z 2 /y = q and is independent of p = √ z 2 y. .
(D. 28) In deriving this expression, we used .

(D.29)
This identity is a consequence of (D.21) applied term by term to the series expansion of Let us consider the BLG theories. The superconformal index of BLG theories can be computed using the expression for the U (2) k × U (2) −k ABJM index with some small modifications (see e.g., [75]). Since the gauge group is now SU (2) × SU (2) we must impose that the Cartan elements in each SU (2) sum to zero (i.e. λ 1 + λ 2 =λ 1 +λ 2 = 0). In addition, since the SU (2) has trivial π 1 there is no notion of a topological charge, and we must impose that the sum of GNO charges for each SU (2) factor vanishes. The baryon number symmetry (D.5) is now a global symmetry of the theory.