Bootstrapping $\mathcal{N}=3$ superconformal theories

We initiate the bootstrap program for $\mathcal{N}=3$ superconformal field theories (SCFTs) in four dimensions. The problem is considered from two fronts: the protected subsector described by a $2d$ chiral algebra, and crossing symmetry for half-BPS operators whose superconformal primaries parametrize the Coulomb branch of $\mathcal{N}=3$ theories. With the goal of describing a protected subsector of a family of $\mathcal{N}=3$ SCFTs, we propose a new $2d$ chiral algebra with super Virasoro symmetry that depends on an arbitrary parameter, identified with the central charge of the theory. Turning to the crossing equations, we work out the superconformal block expansion and apply standard numerical bootstrap techniques in order to constrain the CFT data. We obtain bounds valid for any theory but also, thanks to input from the chiral algebra results, we are able to exclude solutions with $\mathcal{N}=4$ supersymmetry, allowing us to zoom in on a specific $\mathcal{N}=3$ SCFT.


Introduction
The study of superconformal symmetry has given invaluable insights into quantum field theory, and in particular into the nature of strong-coupling dynamics. The presence of supersymmetry gives us additional analytical tools and allows for computations that are otherwise hard to perform. A cursory look at the superconformal literature in four dimensions shows a vast number of works on N = 2 and N = 4 superconformal field theories (SCFTs), with the intermediate case of N = 3 almost absent. The main reason for this is that, due to CPT invariance, the Lagrangian formulation of any N = 3 theory becomes automatically N = 4. By now, however, there is a significant amount of evidence that superconformal theories are not restricted to just Lagrangian examples, and this has inspired recent papers that revisit the status of N = 3 SCFTs.
Assuming these theories exist, the authors of [1] studied several of their properties. They found in particular that the a and c anomaly coefficients are always the same, that pure N = 3 theories (i.e., theories whose symmetry does not enhance to N = 4) have no marginal deformations and are therefore always isolated, and also, in stark contrast with the most familiar N = 2 theories, that pure N = 3 SCFTs cannot have a flavor symmetry that is not an R-symmetry. Moreover, since the only possible free multiplet of an N = 3 SCFT is a vector multiplet, the low energy theory at a generic point on the moduli space must involve vector multiplets, and the types of short multiplets whose expectations values can parametrize such branches were analyzed in [1]. When an N = 3 vector multiplet is decomposed in N = 2, it contains both an N = 2 vector and hyper multiplet, which implies that the theories possess both N = 2 Higgs and Coulomb branches that are rotated by N = 3.
Shortly after [1], the authors of [2] presented the first evidence for N = 3 theories by studying N D3-branes in the presence of an S-fold plane, which is a generalization of the standard orientifold construction that also includes the S-duality group. The classification of different variants of N = 3 preserving S-folds was done in [3], leading to additional N = 3 SCFTs. In [4] yet another generalization was considered, in which in addition to including the S-duality group in the orientifold construction, one also considers T-duality. This background is known as a U-fold, and the study of M5-branes on this background leads to N = 3 theories associated with the exceptional (2, 0) theories.
The systematic study of rank one N = 2 SCFTs (i.e., with a one complex dimensional Coulomb branch) through their Coulomb branch geometries [5][6][7][8] has recovered the known N = 3 SCFTs, but also led to new ones [7,9]. Some of these theories are obtained by starting from N = 4 SYM with gauge group U(1) or SU (2) and gauging discrete symmetries, while others correspond to genuine N = 3 SCFTs which are not obtained by discrete gauging. Note that, as emphasized in [3,9], gauging by a discrete symmetry does not change the local dynamics of the theory on R 4 , only the spectrum of local and non-local operators. In particular, the central charges and correlation functions remain the same.
Of the class of theories constructed in [3], labeled by the number N of D3-branes and by integers k, associated to the S-fold, some have enhanced N = 4 supersymmetry, or arise as discretely gauged versions of N = 4. The non-trivial N = 3 SCFT with the smallest central charge corresponds to the theory labeled by N = 1 and = k = 3 in [3], with central charge given by 15 12 . This corresponds to a rank one theory with Coulomb branch parameter of scaling dimension three. Since the Coulomb branch operators of N = 3 theories must have integer dimensions [1], and since theories with a Coulomb branch generator of dimension one or two enhance to N = 4, it follows that dimension three is the smallest a genuine N = 3 theory with a Coulomb branch can have, and that this theory could indeed correspond to the "minimal" N = 3 SCFT. By increasing the number of D3-branes, higher rank versions of this minimal theory can be obtained . More generally, the rank N theories with k = , are not obtained from others by discrete gauging, and have an N dimensional Coulomb branch.
Since pure N = 3 SCFTs have no relevant or marginal deformations, they are hard to study by standard field theoretical approaches. Apart from the aforementioned papers, recent progress in understanding N = 3 theories includes [10][11][12][13]. The classification of all N = 3 SCFTs is not complete yet, and one can wonder if there are theories not arising from the S-fold (and generalizations thereof) constructions. On the other hand, one would like to obtain more information on the spectrum of the currently known theories. In this paper we take the superconformal bootstrap approach to address these questions, and tackle N = 3 SCFTs by studying the operators that parametrize the Coulomb branch. These operators sit in half-BPS multiplets of the N = 3 superconformal algebra, and when decomposed in N = 2 language contain both Higgs and Coulomb branch operators. We will mostly focus on the simplest case of Coulomb branch operators of dimension three.
The bootstrap approach does not rely on any Lagrangian or perturbative description of the theory. It depends only on the existence of an associative local operator algebra and on the symmetries of the theory in question, and is therefore very well suited to the study of N = 3 SCFTs. Since the original work of [14] there have been many papers that study SCFTs through the lens of the numerical bootstrap [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. A basic requirement in any superconformal bootstrap analysis is the computation of the superconformal blocks relevant for the theory in question, although correlation functions of half-BPS operators in various dimensions have been studied [30][31][32], the case of N = 3 has not yet been considered, and calculating the necessary blocks will be one of the goals of this paper. For literature on superconformal blocks see [30][31][32][33][34][35][36][37][38][39].
Also relevant for our work is the information encoded in the 2d chiral algebras associated to 4d SCFTs [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. The original analysis of [40] implies that any four-dimensional N 2 SCFT contains a closed subsector of local operators isomorphic to a 2d chiral algebra. For N = 3 theories, part of the extra supercharges, with respect to a pure N = 2 theory, make it to the chiral algebra and therefore its symmetry enhances to N = 2 super Virasoro [10]. The authors of [10] constructed a family of chiral algebras conjectured to describe the rank one N = 3 theories, generalizing these chiral algebras in order to accommodate the higher-rank cases will be another subject of this work.
The paper is organized as follows. Section 2 studies the two-dimensional chiral algebras associated with N = 3 SCFTs, determining the N = 3 superconformal multiplets they capture, and some of their general properties. We then construct a candidate subalgebra of the chiral algebras for higher rank = k = 3 theories. In section 3 we use harmonic superspace techniques in order to obtain the superconformal blocks that will allow us to derive the crossing equations for half-BPS operators of section 4. We focus mostly on a dimension three operator, but also present the dimension two case as a warm-up. Section 5 presents the results of the numerical bootstrap, both for generic N = 3 SCFTs and also attempting to zoom in to the simplest known N = 3 theory by inputting data from the chiral algebra analysis of section 2. We conclude with an overview of the paper and directions for future research in section 6.

N = 3 chiral algebras
Every 4d N 2 SCFT contains a protected sector that is isomorphic to a 2d chiral algebra, obtained by passing to the cohomology of a nilpotent supercharge [40]. Because N = 3 is a special case of N = 2, one can also study chiral algebras associated to N = 3 SCFTs. This program was started for rank one theories in [10], and here we explore possible modifications such that one can describe higherrank cases as well. We will put particular emphasis on theories containing a Coulomb branch operator with scaling dimension three, since these are the correlators we will study numerically in section 5. We propose a set of generators that, under certain assumptions, describes a closed subalgebra of theories with a dimension three Coulomb branch operator, and write down an associative chiral algebra for them. Associativity fixes all OPE coefficients in terms of a single parameter: the central charge of the theory.
In order to do this we will need extensive use of the representation theory of the N = 3 superconformal algebra; this was studied in [1,[55][56][57][58][59] and is briefly reviewed in appendix A. We will also leverage previous knowledge of chiral algebras for N = 2 SCFTs, and so it will be useful to view N = 3 theories from an N = 2 perspective. Therefore, we will pick an N = 2 subalgebra of N = 3, with the SU(3) R × U(1) r R-symmetry of the latter decomposing in SU(2) R N =2 × U(1) r N =2 × U(1) f . The first two factors make up the R-symmetry of the N = 2 superconformal algebra and the last corresponds to a global symmetry. Therefore, from the N = 2 point of view, all N = 3 theories necessarily have a U(1) f flavor symmetry arising from the extra R-symmetry currents. The additional supercharges and the U(1) f flavor symmetry imply that the Virasoro symmetry expected in chiral algebras of N = 2 theories algebras will be enhanced to a super Virasoro symmetry in the N = 3 case [10].
Let us start reviewing the essentials of the chiral algebra construction (we refer the reader to [40] for more details). The elements of the protected sector are given by the cohomology of a nilpotent supercharge Q that is a linear combination of a Poincaré and a conformal supercharge, In order to be in the cohomology operators have to lie on a fixed plane R 2 ⊂ R 4 . The global conformal algebra on the plane sl(2) ×sl(2) is a subalgebra of the four-dimensional conformal algebra. While the generators of the sl(2) commute with (2.1), those ofsl (2) do not, and an operator in the cohomology at the origin will not remain in the cohomology if translated by the latter. However, it is possible to introduce twisted translations obtained by the diagonal subalgebra of thesl(2) and a complexification, sl(2) R , of the R-symmetry algebra su(2) R , such that the supercharge satisfies From these relations one can prove that Q-closed operators restricted to the plane have meromorphic correlators. We call the operators that belong to the cohomology of Q "Schur" operators. The Schur operators in N = 2 language are local conformal primary fields which obey the conditions The cohomology classes of the twisted translations of any such operator O corresponds to a 2d local meromorphic operator The two important Schur operators that we expect to have in any N = 2 theory with a flavor symmetry are 1 corresponding to the 2d stress tensor T (z).
•B 1 : The highest-weight component J 11 of the moment map operator (∆ = 2, j = = 0, R N =2 = 1 and r N =2 = 0) that is mapped to the affine current J(z) of the flavor group.
These two Schur operators give rise to a Virasoro and an affine symmetry in the chiral algebra respectively, with the two-dimensional central charges obtained in terms of their four-dimensional counter-parts by Note that, since we insist on having unitarity in the four-dimensional theory, the 2d chiral algebra will be necessarily non-unitary. The chiral algebra description of a protected subsector is extremely powerful. By performing the twist of [40] on a four-dimensional correlation function of Schur operators, we are left with a meromorphic 2d correlator that is completely determined by knowledge of its poles and residues. The poles can be understood by taking various OPE limits, thus fixing the correlator in terms of a finite number of parameters corresponding to OPE coefficients. In the cases we will study in this paper (see for example subsection 4.2.1), the meromorphic piece can be fixed using crossing symmetry in terms of a single parameter, which can be identified with the central charge of the theory. Let us emphasize that this can be done without knowledge of which particular chiral algebra is relevant for the SCFT at hand.

Generalities of N = 3 chiral algebras
Let us now study the N = 3 case in more detail. Any local N = 3 SCFT will necessarily contain a stress tensor multiplet, which in table 8 corresponds toB [1,1] . After an N = 2 decomposition of this multiplet (shown in (A.1)) one finds four terms, each contributing with a single representative to the chiral algebra. These four multiplets are related by the action of the extra supercharges enhancing N = 2 to N = 3, and four of these (Q 3 + andQ 3+ and their conjugates) commute with Q [10]. Therefore, acting on Schur operators with these supercharges produces new Schur operators, and the representatives of the four multiplets will be related by these two supercharges. The multiplets and their representatives are: • A multiplet containing the U(1) f flavor currents (B 1 ), whose moment map M IJ gives rise to a two-dimensional current J(z) = [M (z,z)] Q of an U(1) f affine Kac-Moody (AKM) algebra, • Two "extra" supercurrents, responsible for the enhancement to N = 3, contribute as operators of holomorphic dimension 3 2 . These are obtained from the moment map by the action of the supercharges • The stress-tensor multiplet (Ĉ 0,(0,0) ) which gives rise to the stress tensor of the chiral algebra The supercharges Q 3 + andQ 3+ have charges ±1 under the U(1) f flavor symmetry, where we follow the U(1) f charge normalizations of [10]. Therefore the operators G(z) andG(z) have a J charge of +1 and −1 respectively. This multiplet content is exactly the one we would expect from the considerations in the beginning of this section, with the extra supercharges, that commute with Q, producing a global d = 2, N = 2 superconformal symmetry. 3 Moreover, the operator content we just described corresponds precisely to the content of an N = 2 stress tensor superfield which we denote by J , enhancing the Virasoro algebra to an N = 2 super Virasoro algebra [10]. and D 1 2 (0,0) respectively in the notation of [60]. 3 The holomorphic sl(2) that commutes with the supercharge Q, more precisely the Q-cohomology of the superconformal algebra, is enhanced to a sl(2|1).

N = 3 superconformal multiplets containing Schur operators
Our next task is to understand which multiplets of the N = 3 superconformal algebra contribute to the chiral algebra, aside from the already discussed case of the stress-tensor multiplet. Instead of searching for superconformal multiplets that contain conformal primaries satisfying (2.3), we will take advantage of the fact that this was already done in [40] for N = 2 multiplets, and simply search for N = 3 multiplets that contain N = 2 Schur multiplets. To accomplish this, we decompose N = 3 multiplets in N = 2 ones by performing the decomposition of the corresponding characters. In appendix A we present a few examples of such decompositions. Going systematically through the multiplets, 4 we find the following list of N = 3 Schur multiplets: [58], there are no relevant Lorentz invariant supersymmetric deformations of N = 3 theories, while the only such deformations that are exactly marginal are contained in the multipletB [2,0] (and conjugatê B [0,2] ). However, these multiplets also contain additional supersymmetry currents, as can be seen from their N = 2 decomposition, that allow for the enhancement of N = 3 to N = 4, and thus pure N = 3 theories are not expected to have exactly marginal operators. Let us also recall that the multipletŝ C [0,0],(j,) contain conserved currents of spin larger than two, and therefore are expected to be absent in interacting theories [61,62].

Quasi-primaries and Virasoro primaries
Each of the N = 2 multiplets listed above will contribute to the chiral algebra with exactly one global conformal primary (also called quasi-primary), with holomorphic dimension as given in table 1 of [40] and with U(1) f charge f , under the J(z) current, as can be read off from the u f fugacity in the above decompositions. These multiplets generically will not be Virasoro primaries. Only the so-called Hall-Littlewood (HL) operators 5 (B R , D R,(j,0) and D R,(0,) ) are actually guaranteed to be Virasoro primaries. The remaining multiplets will appear in the chiral algebras sometimes as Virasoro primaries, sometimes only as quasi-primaries.

Super Virasoro primaries
Similarly, each N = 3 multiplet gives rise in the chiral algebra to a global supermultiplet consisting of a global superprimary and its three global superdescendants obtained by the action of Q 3 + and Q 3+ . 6 Generically however, these multiplets will not be super Virasoro primaries, even if the global superprimary corresponds to a Virasoro primary. Recall that a super Virasoro primary must, in addition to being a Virasoro primary, have at most a pole of order one in its OPE with both G(z) andG(z), and have at most a singular term of order one in the OPE with J(z). 7 This last condition corresponds to being an AKM primary.
Let us consider the operators which have as a global superprimary a Virasoro primary. For the case ofB [R1,R2] multiplets, we see that its two (or one in case R 1 R 2 = 0) level 1 2 descendants are HL operators, and thus Virasoro primaries. The two-dimensional superconformal algebra then implies that the global superprimary is not only a Virasoro primary, but that it is also annihilated by all the modes G n + 1 2 ,G n + 1 2 . However, this is not enough to make it a super Virasoro primary, as it is not guaranteed that these operators are AKM primaries. An obvious example is the stress tensor multipletB [1,1] , where the AKM current is clearly not an AKM primary. Similar considerations apply to D [R1,R2], multiplets, with the subtlety that even though one of its level 1 2 descendants is not a HL operator, it is still a Virasoro primary [10].
In certain cases it is possible to show that the operators in question are actually super Virasoro primaries, and concrete examples will be given below. For example, if one considers aB [R1,R2] generator that is not the stress tensor multiplet, then the OPE selection rules for the N = 2B R1+R2 2 multiplet [63] imply it is also an AKM primary [42]. 5 Following [40] we refer to operators which are N = 1 chiral and satisfy the Schur condition as Hall-Littlewood operators. 6 Recall that the global superprimary is annihilated only by the G and J n 0 .

Chiral and anti-chiral operators
Finally we note that the multiplets in (2.8) and (2.9) give rise, in two dimensions, to anti-chiral and chiral operators: they are killed byQ 3+ and Q 3 + respectively. These two-dimensional superfields have holomorphic dimension satisfying h = R2 2 = − f 2 and h = R1 2 = f 2 respectively.

[3, 0] chiral algebras
We are now in a position to describe the general features of the chiral algebras associated to the known N = 3 theories. We will describe the chiral algebra in terms of its generators, by which we mean operators that cannot be expressed as normal-ordered products and/or (super)derivatives of other operators. In what follows we assume the chiral algebra to be finitely generated. Although there is yet no complete characterization of what should be the generators of the chiral algebra of a given four-dimensional theory, it was shown in [40] that all generators of the HL chiral ring are generators of the chiral algebra. Moreover, the stress tensor is always guaranteed to be present and, with the exception of cases where a null relation identifies it with a composite operator, it must always be a generator. However this is not necessarily the complete list, and indeed examples with more generators than just the above have been given in [40,42]. 8 The chiral algebras associated to 4d SCFTs do not always correspond to known examples in the literature, and in such situations one must construct a new associative two-dimensional chiral algebra. This problem can be bootstrapped by writing down the most general OPEs for the expected set of generators and then imposing associativity by solving the Jacobi identities. Chiral algebras are very rigid structures and in the cases so far considered [10,42], the Jacobi identities are powerful enough to completely fix all OPE coefficients, including the central charges.

Rank one chiral algebras
In [10], the authors assumed that the only generators of the chiral algebras corresponding to the rank one N = 3 SCFTs described in section 1 (with k = , N = 1) were the stress tensor and the generators of its Higgs branch:B (2.14) Recall the first multiplet gives rise, in two dimensions, to the stress tensor multiplet, and the last two to anti-chiral and chiral operators respectively. With these assumptions they were able to write an associative chiral algebra for the cases = 3, 4 only for a single central charge for the first case and a finite set of values for the second. This set was further restricted to the correct value expected for the known N = 3 theories by imposing the expected Higgs branch chiral ring relation which appears as a null state in the chiral algebra. Note that above, by abuse of notation, we denoted the Higgs branch chiral ring operator by the superconformal multiplet it belongs to. Associativity then fixes all other OPE coefficients of the chiral algebra. The authors of [10] were also able to construct an associative chiral algebra for = 5 and = 6 satisfying the Higgs branch relation if the central charge is given by (2.15). However, as they point out, = 5 does not correspond to an allowed value for an N = 3 SCFTs, as five is not an allowed scaling dimension for the Coulomb branch of a rank one theory, following from Kodaria's classification of elliptic surfaces (see, e.g., [5,10]). The case = 6 is in principle allowed, however no such N = 3 theory was obtained in the S-fold constructions of [3]. 9 Higher rank theories We now attempt to generalize the chiral algebras of [10] to the higher-rank case (with k = , N > 1).
In particular, we focus on the theories whose lowest dimensional generator corresponds to aB [3,0] and its conjugate, since these are the ones relevant for the following sections. To compute OPEs and Jacobi identities we will make extensive use of the Mathematica package [67]. Following its conventions, we use the two-dimensional N = 2 holomorphic superspace with bosonic coordinate z and fermionic coordinates θ andθ, and define the superderivatives as We will denote the two-dimensional generators arising from the half-BPS Higgs branch generatorŝ B [0,3] (B [3,0] ) by W (W). 10 Furthermore, we denote the two-dimensional superfield arising from the stress tensor (B [1,1] ) by J . The OPE of J with itself is fixed by superconformal symmetry, where we defined The OPEs of J with W andW, given in (B.1), are fixed by demanding that these two operators be super Virasoro primaries. As discussed in the previous subsection, W andW could fail to be super Virasoro primaries only if their global superprimary (arising from an N = 2B 3/2 ) failed to be an AKM primary. However, since we are assuming theB 3/2 multiplet to be a generator, and since the AKM current comes from aB 1 N = 2 multiplet, it is clear from the selection rules of N = 2B R operators [63] that these must be AKM primaries. The self OPEs of the chiral (anti-chiral) W (W) superfields are regular, which is consistent with the N = 3 OPE selection rules shown in 3.22. For WW the most general OPE in terms of all of the existing generators is [10] where the sum runs over all uncharged operators, including composites and (super)derivatives. The authors of [10] showed that, considering just these three fields as generators, one finds an associative chiral algebra only if c 2d = −15, which indeed corresponds to the correct value for the simplest known non-trivial N = 3 SCFT (k = = 3 and N = 1 in the notation of [3]). However, there are higher rank versions of this theory (k = = 3 and N > 1), that contain these half-BPS operators plus higher-dimensional ones. The list of half-BPS operators is [3] B [0,R] ,B [R,0] , with R = 3, 6, . . . , 3N , (2.21) giving rise in two dimensions to additional chiral and antichiral operators with charges f = ±6, . . .±6N , and holomorphic dimension h = |f |/2. One can quickly see that the extra generators never appear in the OPEs of W,W, J , as the only OPE not fixed by symmetry is the WW, and U(1) f charge conservation forbids any of theB [R,0] with R 6 to appear. If the generators of the chiral algebras of higher rank theories corresponded only to the half-BPS operators plus the stress tensor, then we would reach a contradiction: W,W, J would form a closed subalgebra of the full chiral algebra, but the central charge would be frozen at c 2d = −15, which is not the correct value for rank greater than one.
To resolve this contradiction we must allow for more generators in the higher-rank case, and at least one of these must be exchanged in the WW OPE. The only freedom in this OPE is to add an uncharged dimension two generator. From the OPE selection rules shown in 3.21 one can see that this operator must correspond to aB [2,2] . There is another possibility, namely aĈ [0,0],0 multiplet, but in four dimensions it contains conserved currents of spin greater than two, which should be absent [61,62] in interacting theories such as the ones we are interested in. The minimal resolution is to add the generator corresponding toB [2,2] . We then assume that the generators of the chiral algebra associated with the = k = 3 theories with N > 1 are • A generator corresponding toB [2,2] which we denote by U.
As before we denote by W andW the generators arising fromB [0,3] and its conjugate. 11 Even though examples are known where the number of generators not arising from generators of the HL ring grows with the number of HL generators [42], the addition of a single operator U is the minimal modification that unfreezes the value of the central charge.
We can now proceed to write down the most general OPEs, it is easy to check that in the ones involving J , W ,W , and U , (2.22) the operators in (2.21) with R 6 cannot be exchanged. Therefore, if our assumption above is correct, the generators in (2.22) form a closed subalgebra.
In what follows we write down the most general ansatz for the OPEs of these operators which, as explained above, are all super Virasoro primaries with the exception of J . The regularity of the self OPEs of W andW follows simply from OPE selection rules, while the OPE between W andW is given by (2.20), allowing for the exchange of U as well. The OPEs involving U are quite long and therefore we collect them in appendix B. Imposing Jacobi identities we were able to fix all the OPE coefficients in terms of a single coefficient: the central charge. In our construction we did not need to impose null states for closure of the algebra.

Fixing OPE coefficients
In the next sections we will study numerically the complete four-point function of twoB [3,0] and twô B [0,3] operators, thanks to the chiral algebra we can compute the OPE coefficients of all operators appearing in the right hand side of the WW OPE. However, we still need to identify the four-dimensional superconformal multiplet that each two-dimensional operator corresponds to. Let us start by examining the low dimensional operators appearing in this OPE: we can write all possible operators with a given dimension that can be made out of the generators by normal ordered products and (super) derivatives. Furthermore, they must be uncharged, since the product WW is. All in all we find the following list: From these operators we are only interested in the combinations that are global superprimary fields, as the contributions of descendants will be fixed from them. 12 Note also that, if we are interested in the four point function of WWWW , we only see, for the exchange of an operator of a given dimension, a sum of the contributions of all global primaries, and we cannot distinguish between individual fields. At dimension h = 1 there is only one operator -the superprimary of the stress-tensor multiplet -and its OPE coefficient squared can be computed to be (after normalizing the identity operator to appear with coefficient one in the four-point function decomposition, and normalizing the J two-point function to match the normalization for the blocks (g 2d N =2 , see (3.26)) that we use in the following sections) This does not depend on the particular chiral algebra at hand, as the OPE coefficient with which the the current J is exchanged, is totally fixed in terms of their charge f and the central charge. As we will show in 4.2.1, the two-dimensional correlation function of the two W and twoW, is fixed in terms of one parameter which we take to be the OPE coefficient of J , and thus related to c 2d . This implies that, for the exchange of operators of dimension larger than one, any sum of OPE coefficients corresponds to a universal function of c 2d . At dimension h = 2 we find two global superprimaries, one corresponding to U itself, and the other containing J J . From the four-dimensional OPE selection rules, shown in (3.21), it follows that both superprimaries must correspond toB [2,2] supermultiplets in four dimensions, as the only other option isĈ [0,0],0 which should be absent in interacting theories. This means that, even from the point of view of the four-dimensional correlation function, these two operators are indistinguishable. Thus, all we can fix is the sum of two OPE coefficients squared: where we used the same normalizations as before, and fixed an orthonormal basis for the operators. This number is again independent of the particular details of the chiral algebra: it only requires the existence of W,W and J . At dimension h = 3, we find four global superprimaries made out of the fields listed above, three of which are Virasoro primaries. In this case, however, these three different operators must belong to two different types of four-dimensional multiplets (once again we are excluding the multiplet containing higher-spin currents). Namely, they must correspond toB [3,3] andĈ [1,1],0 , and distinguishing them from the point of view of the chiral algebra is hard. The two-dimensional operators arising fromB [3,3] are guaranteed to be Virasoro primaries, while those ofĈ [1,1],0 could be or not. Assuming that all Virasoro primaries come exclusively fromB [3,3] we can compute the OPE coefficient with which this multiplet is exchanged by summing the squared OPE coefficients of all Virasoro primaries We can take the large c 4d = − c 2d 12 limit, where the solution should correspond to generalized free field theory. In this case we can find from the four-point function given in appendix C.2 that the OPE coefficient above should go to 1, and indeed this is the case. We could also have assumed that different subsets of the three Virasoro primaries correspond toB [3,3] . Not counting the possibility used in (2.25), there is one possibility which does not have the correct behavior as c 4d → ∞, and two that have: We can now also compute for each of the above cases the OPE coefficient of theĈ [1,1],0 multiplet, and we find that only (2.25) and (2.27) are compatible with 4d unitarity (the precise relation between 2d and 4d OPE coefficients is given by (4.20)).
If we now go to higher dimension, the list of operators keeps on growing, and their four-dimensional interpretation is always ambiguous. A dimension h global superprimary can either be aĈ [2,2],h−4 or â C [1,1],h−3 four-dimensional multiplet, and in this case there does not seem to be an easy way to resolve the ambiguity. 13

Rank one case
Let us now comment on what happens for the case of the rank one theory, where c 2d = −15 and the extra generator U is absent. In this case we find a single (non-null) Virasoro primary at dimension three. 14 This implies that either there is noB [3,3] multiplet and that the OPE coefficient is zero, or, which seems like a more natural option, that the Virasoro primary corresponds to this multiplet, with OPE coefficient Table 1. We used the definitions h := (13 + Vc) −1 andh := (13 + cV ) −1 . The transformations corresponding to G (+,+) , G (−,−) are generated by the ones above. For convenience we give the explicit form of special super- The above corresponds to setting c 2d = −15 in both (2.25) and (2.26), as expected since for this value the extra generator is not needed and decouples. The possibility that there is noB [3,3] multiplet in the rank one theory and thus that the OPE coefficient is zero corresponds to the c 2d = −15 case of (2.27). If this last possibility were true, then we would have that the operator WW ∼ J 3 is not in the Higgs branch, since Higgs branch operators correspond toB R multiplets in N = 2 language. Hence, there would be a relation setting J 3 = 0 in the Higgs branch, which does not seem plausible. In any case, we will allow for (2.27) for generic values of the central charge. It might be possible to select among the two options ((2.25) and (2.27)) by making use of the considerations in [68] about recovering the Higgs branch out of the chiral algebra, but we leave this for future work.

Superblocks
In this section we will use harmonic superspace techniques in order to study correlation functions of half-BPS operators. We will follow closely [39,69], where a similar approach was used to study correlation functions in several superconformal setups.

Superspace
We introduce the superspace M as a coset M SL(4|3) G ≤0 . Here, the factor G ≤0 corresponds to lower triangular block matrices with respect to the decomposition given in (3.1) below. We take E(p) ∈ G >0 as coset representative explicitly given by In the above, α ∈ {1, 2},α ∈ {1, 2} are the familiar Lorentz indices and the coordinates {λ α , πα, θ α ,θα} are fermionic, while the y, v,v are bosonic R-symmetry coordinates. The action of SL(4|3) on this superspace follows from the coset construction and is summarized in table 1. Notice that SL(4|3) acts invariantly within the superspaces M + , We also define X 21 := −X1 2 .

Superfields for superconformal multiplets
The supermultipletsB [R1,R2] correspond to "scalar" superfields on M . Among them, as discussed in the previous section, the ones with R 1 R 2 = 0 are special in the sense that they satisfy certain chirality conditions. We call chiral (anti-chiral) a superfield that depends only on the coordinates {X − , V } ({X + , V }). 15 Within this terminology, the operatorsB [0,R] are chiral while theB [R,0] are antichiral. More general supermultiplets can be described as superfields on M with SL(2|1) × SL(2|1) indices which extend the familiar Lorentz indices. We will not need to develop the dictionary between N = 3 superconformal representations and SL(2|1) × SL(2|1) × GL(1) × GL(1) induced representations in this work and thus leave it for the future.

Examples of two-and three-point functions
We denote superfields and supermultiplets in the same way. Let us list some relevant examples of twoand three-point functions ofB-operators of increasing complexity: where we have defined In (3.5) superspace analyticity implies that S ≤ R and that the correlation function vanishes otherwise. Similarly, in (3.7), C is a superconformal invariant and superspace analyticity implies that P R (C) is a polynomial of degree R in C. Since the three operators are identical, one further imposes Bose symmetry which translates to P R (x) = x R P R (x −1 ). Equation (3.7) specialized to the case R = 1 corresponds to the three-point function of the stress-tensor supermultipletB [1,1] and the argument above implies that P 1 (x) = const × (1 + x). This provides a quick proof of the fact that for N = 3 superconformal theories one has the relation a = c as first derived in [1]. Let us consider the three-point functions relevant for the non-chiral OPEB [R,0] ×B [0,R] . A simple superspace analysis reveals that the three-point function of a chiral and an anti-chiral operator with a generic operator takes the form The quantity ρ O is determined uniquely up to a multiplicative constant by the requirement that (3.9) is superconformally covariant. It is not hard to verify that one can set the coordinates V 1 , V 2 , V 3 , V 3 to zero by an SL(4|3) transformation which is not part of the N = 2 superconformal group SL(4|2) (with the embedding specified in the remark 1 above). This means that (3.9) is zero if its N = 2 reduction (i.e., the result obtained after setting V i = V i = 0) is zero, as confirmed by the selection rules result (3.21) that we derive later in section 3.3.
Turning to the three-point functions relevant for the chiral OPEB [R,0] ×B [R,0] , it is not hard to convince oneself that they take the form and ρ O is fixed by requiring superconformal covariance of (3.10). It is important to remark that, as opposed to (3.9), in this case one cannot set the coordinates V 1 , V 2 , V 3 , V 3 to zero using superconformal transformations. However, they can be set to the values The combinations X and V carry non trivial superconformal weights only with respect to the third point corresponding to the operator O.

Superconformal Ward identities
We will now derive, along the same lines as [31,39,69] are invariant and will be denoted by x 1 , x 2 , y. This can be seen from the fact that all fermionic coordinates in this four-point function can be set to zero by a superconformal transformation. It follows that where is strongly restricted by the requirement of superspace analyticity. Firstly, after setting all fermionic variables to zero polynomiality in the R-symmetry variables implies that G R (x 1 , x 2 , y) is a polynomial of degree R in y −1 . Secondly, one has to make sure that the fermionic coordinates can be turned on without introducing extra singularities in the R-symmetry variables. By looking at the expansion of the eigenvalues of (3.13) in fermions, one concludes that the absence of spurious singularities is equivalent to the conditions . This is a consequence of the protected subsector discussed in section 2, where setting follows from the twisted translations (2.2), as originally discussed in [40]. The chiral algebra further tells us that f R (x 1 ) is a meromorphic function of x 1 , corresponding to a two-dimensional correlation function of the twisted-translated Schur operators, with eachB [0,R] (B [R,0] ) multiplet giving rise to a two-dimensional N = 2 chiral (anti-chiral) operator, as discussed in 2.1.1.
The general solution of the Ward identities can be parametrized as In particular, it is zero for the case R = 1 corresponding to a free theory. For the following analysis it is useful to introduce the variables z,z, w as This change of variable is an involution in the sense that z = x1 x1−1 and so on. They are related to the more familiar cross-ratios as Notice that the WI (3.16) take the same form in the new variables and that moreover for any function f (x).

Selection rules
In this subsection we analyze the possible multiplets allowed by superconformal symmetry in the nonchiral and chiral OPEs. This is a crucial ingredient for the crossing equations and are usually called the OPE selection rules.

Non-chiral channel
The OPE in the non-chiral channelB [ Notice that these relations are remarkably similar to theB R/2 ×B R/2 OPE in the N = 2 case, see [63]. The three upper bounds on the finite summations R, R − 1, R − 2 could be derived by imposing that the three-point function B [R,0]B[0,R] O is free of superspace singularities. Equivalently, it can be derived by requiring that the associated superconformal block takes the form (3.17). We followed the latter strategy as it seemed more economical.

Chiral channel
The chiral channel selection rules are obtained by requiring that a given multiplet can only contribute if it contains an operator annihilated by all the supercharges that annihilate the highest weight of B [R,0] , and if said operator transforms in one of the representations appearing in the tensor product of the R-symmetry representations [R, 0] × [R, 0], and with the appropriate spin to appear in the OPE of the external scalars. We have performed this calculation for R = 2, 3 and based on it we propose that the expression for general R iŝ We have checked the above in several cases for R > 3 and superspace arguments suggest it is indeed the correct selection rule. Note that in (3.22) the B-type multiplets have r = 4R, the C-type multiplets r = 4R − 1, and the A-type multiplets r = 4R − 2. Moreover, if we are considering identicalB [R,0] , then Bose symmetry further constraints the spin of the operators appearing on the right-hand-side according to their SU(3) R representation.

Superconformal blocks
We will now derive the superconformal blocks relevant for the expansion of the four-point function (3.14). The superconformal Ward identities alone turn out not to be sufficient to uniquely determine all the superblocks. We resolve the leftover ambiguity by requiring that they are linear combinations of SL(4|1) × SU(2) (N = 1) superblocks. There are two types of blocks corresponding to the two channels: non-chiral (3.21) and chiral (3.22). The two kinds of blocks are closely connected to N = 2 superconformal blocks relevant for the four-point function ofB-type operators and are collected in tables 2 and 3. When the kinematics is restricted to (z,z, w) = (z, w, w), only superconformal blocks corresponding to the exchange of Schur operators, defined in section 2.1.1, are non-vanishing. Moreover, they reduce to 2d (global) superblocks for the N = 2 superconformal algebra sl(2|1) osp(2|2) in the appropriate channel.

Superconformal blocks for the non-chiral channel.
On general grounds, the N = 3 superconformal blocks contributing to the four-point function (3.14) in the non-chiral channel can be written as an expansion in terms of conformal times SU(3) R-symmetry blocks: The explicit form of the conformal blocks g ∆, is given in Appendix C. The SU(3) R-symmetry blocks take the form The normalization in (3.24) is chosen so that h [a,a] (w) = w −a +. . . for w → 0. The set S χ is determined by considering the decomposition of the N = 3 representation being exchanged into representations of the bosonic subalgebra (this can be done using superconformal characters). The normalization can be fixed by taking for instance c α (χ) = 1 for the label α corresponding to the minimum value of ∆ α in the supermultiplet. Consider the superblocks corresponding to the non-chiral OPE channel of (3.21). Concerning the superblocks for theB [a,a] exchange, it turns out that they are uniquely fixed by imposing the superconformal WI on (3.23). The superblocks corresponding to the exchange of aĈ [a,a], on the other hand are not uniquely fixed by the this procedure. The remaining ambiguity can be resolved by requiring that they reduce to osp(2|2) (this is the global part of the chiral half of the d = 2, N = 2 super Virasoro algebra) superblocks when restricted to (z,z, w) = (z, w, w). Recall that this restriction reduces the correlator to that of 2d N = 2 chiral and anti-chiral operators, and thus the exchange of an operator in the non-chiral channel is captured by the osp(2|2) superblocks of [34]. Specifically, this amounts to requiring where the osp(2|2) superblock is [34] and f (z) corresponds to the parametrization (3.17). 16 The superblocks for the exchange of long operators A ∆ [a,a],r=0, are not uniquely determined by the two conditions given above. The leftover ambiguity can be resolved by studying the Casimir equations. However, we will take a shortcut and use the knowledge of N = 1 superblocks. The relevant superblocks, which were derived in [15,34], are given by It follows from the remark 1, that the N = 3 superblocks can be expanded in N = 1 times SU (2) "flavor symmetry" blocks as (3.28) On the right hand side, the sum runs over the terms (∆, ) , (∆ + 1, ± 1) , (∆ + 2, ± 2) , (∆ + 2, ) , (∆ + 3, ± 1) , (∆ + 4, ) . (3.29) Imposing that the form (3.23), subject to the WI, can be expanded as in (3.28), fixes the leftover am- 16 To each superblock Gχ corresponds a function fχ and a function Hχ by using the parametrization (3.17). the stress-tensor supermultipletsB [1,1] , the function HB [1,1] is zero.
biguity in the N = 3 superblocks and the coefficient functionsd ∆, (w) up to an overall normalization. The solution can then be rewritten in the compact form The simplicity of this expression will be justified in remark 2 below. This concludes the derivation of superconformal blocks relevant for the non-chiral channel, the results are summarized in table 2. Before turning to the discussion of the superblocks relevant for the chiral channel, we perform a consistency check on the blocks just derived. As can be seen in table 2, short blocks can be obtained from the long ones (3.30) at the unitarity bounds by using

Superconformal blocks for the chiral channel.
We denote the superconformal blocks contributing to this channel as G χ (z,z, w), where χ labels the representations being exchanged from the list (3.22). As in the case of the non-chiral channel, we start with an expansion of the superblocks in conformal times SU(3) blocks and impose the superconformal Ward identities, (3.16). Specifically we take (3.32) It appears, perhaps not too surprisingly, that the SU(3) R-symmetry blocks h [2m,n] (w) in this channel coincide with SU(2) blocks, where here and in the following we take m = 2(R − n). They take the form 17 where the normalization is chosen so that h [2m,n] (w) ∼ w m for w ∼ 0, and we omit the label n since it is related to m. The set S χ is determined by looking at the content of the representation χ using characters. Using this information, all the coefficientsc α (χ) are then fixed by the requirement that (3.32) satisfies the superconformal WI (3.16).
With a little inspection on the solutions, one recognizes that the superblocks in this channel are the N = 2 superconformal blocks that contribute to the four-point function ofB N =2 supermultiplets [30][31][32]. The identification is given by where p maps the N = 3 representations being exchanged in the chiral channel, see (3.22), to an N = 2 representations as follows The equality (3.34) is not accidental, we will comment on its origin in the remark below. The resulting superblocks in the parametrization (3.17) are given in table 3. Once again note that the meromorphic function f (z) has a decomposition in 2d blocks, in this case sl(2) blocks These are the blocks relevant for the decomposition of the chiral algebra correlators in the chiral channel, since each 2d N = 2 multiplet contributes with a single sl(2) primary to the OPE of two 2d N = 2 chiral operators.
The unitarity bound relevant for the chiral channel is ,a],0 . Only the underlined term contributes to the superblocks G, as can be seen in table 3.

Remark 2.
In [37], the authors derived superconformal blocks for scalar four-point functions on a super Grassmannian space Gr(m|n, 2m|2n). It is an interesting problem to generalize the analysis of [37] to the more general case of Gr(m|n, M |N ). The example we just studied corresponds to Gr(2|1, 4|3). The example of chiral superfields (in the traditional sense) for N -extended supersymmetry corresponds to the super Grassmannian Gr(2|0, 4|N ) and the corresponding superblocks were given in [34]. The simplicity of the results (3.30) and (3.34) and the one presented in [34] suggests a simple unified picture.

Crossing equations
Equipped with the superconformal blocks relevant for the four-point function of half-BPS operators we are finally ready to write the crossing equations. Most of this section treats the case of arbitrary external dimension R, and in the final section we focus on the cases of a dimension two and three operator. Crossing symmetry can be written in terms of the functions f R (x) and H R (x 1 , x 2 , y) used to parametrize the solution of the Ward identities (3.17). As expected, the chiral algebra correlator f R (x) satisfies a crossing equation on its own, that we solve analytically. This amounts to input about the exchange of Schur operators, that we feed into the full set of crossing equations for H R (x 1 , x 2 , y). These give rise to a system of (three) six equations of the two bosonic cross-ratios, for (R = 2) R = 3, which are the subject of the numerical analysis of section 5.

First equation
Consider the four-point function (3.14), where we take pairwise identical operators. Imposing that it is invariant upon the exchange of points 1 ↔ 3 implies the crossing equation This is due to the fact that the matrix Z, given in (3.13), transforms to its inverse, up to a similarity transform, if points 1 and 3 are exchanged. In terms of the solution of the WI (3.17), equation (4.1) implies that the single variable function f R (x) satisfies a crossing equation on its own: Note that the above is a specialization of (4.1) to (x 1 , x 2 , y) = (x, y, y), and thus corresponds to the crossing equation for the two-dimensional correlator of a dimension R 2 operator. The function f R (x) is easily argued to be a polynomial of degree R in x as we shall show in section 4.2.1. Imposing (4.2), together with the normalization f R (0) = 1, implies this function is fixed in terms of R 2 (respectively R−1 2 ) independent parameters for R even (respectively odd).
The remaining constraints from crossing symmetry (4.1) translate into the following equation where we have made use of (4.2), and defined which is a polynomial in its arguments. Recall that H R (x 1 , x 2 , y) is a polynomial of degree R − 2 in y −1 and thus, with the exception of R = 2, (4.3) encodes a system of crossing equations.

Second equation
In the channel where one takes the OPE of the two chiral operators it is convenient to relabel the points in (3.14) to obtain where we have defined and (z,z,w) := (1 − z 1 , 1 − z 2 , 1 − y 1 ). If the superspace coordinates are V = V = 0, the cross-ratios above are related to the ones entering (3.14) as (z,z,w) = (z,z, w) and ( z 1 , z 2 , y) . The first equality in (4.5) is to be understood as defining the function G R . 18 The second one is a rewriting of (3.14), relating the chiral channel to the non-chiral channel. The function G R satisfies the same superconformal Ward identities as G R . We thus parametrize it as in (3.17), with the functions f R and H R replaced by f R and H R , and the variables x 1 , x 2 , y replaced byz,z,w. An immediate consequence of (4.5) is the relation Note thatz,z,w are the usual cross-ratios for the correlator (4.5), and so we rename them as (z,z,w) → (z,z, w). The relation (4.7) implies a relation for the single variable function f R : which again follows from the fact that the single variable functions are capturing a two-dimensional correlator. For theH function we get from (4.7) that The strange prefactor is the natural supersymmetric completion of (x 2 12 where A R (x, y) was defined in (4.4) and we remind that in (3.18) we set (x 1 , x 2 , y) = ( z z−1 ,z z−1 , w w−1 ). As in the first crossing equation (4.3), the dependence on w disappears from (4.9) for R = 2.

Third equation
Since we consider the case of identicalB [R,0] operators, Bose symmetry under the exchange 1 ↔ 2 in (4.5) requires Plugging in the solution of the WI and using (3.20), the above equation translates into with the first equation again following from Bose symmetry in the chiral algebra.

From the chiral algebra to numerics
In the following subsections we turn the crossing equations (4.3), (4.9), (4.11) into a system ready for the numerical analysis, by fixing all the chiral algebra data. To do so we proceed as follows: • The first step, undertaken in subsection 4.2.1, is to analytically solve the chiral algebra crossing equations for f R (x) andf R (x).
• Decomposing these functions in the blocks of tables 2 and 3 allows us to fix an infinite number of Schur operator OPE coefficients. We recall these operators are the ones in the OPEs (3.21) and (3.22) that contribute to the chiral algebra, see (2.6) and the following.
• From the block decomposition of H R , H R , also given in tables 2 and 3, we see that they receive contributions from some of the multiplets contributing to f R (x) andf R (x).
• Therefore we split the expansion into a sum over the exchange of Schur operatorŝ In general, knowledge of the function f R (x) alone is not sufficient to determine H R,short unambiguously, in contrast with the chiral channel function H R,short , which is fixed in terms off R (x). This is because different N = 3 supermultiplets give the same contribution, in the sense of 2d blocks, to the functions f R (x) andf R (x). As we will see in section 4.2, assuming the absence of supermultiplets containing conserved currents of spin greater than two, the function H R,short and the component of H R,short in the R-symmetry singlet channel can be extracted unambiguously from the knowledge of f R (x).

Summary of the result
Following the procedure that we just outlined one arrives at the following system of crossing equations: (4.13) We now have to make several remarks in order to explain our notation. a) We have defined the functions The explicit form of the functions H χ , H χ is given in tables 2 and 3 for each representation χ. Note that the above functions still have a dependence on the R-symmetry cross-ratio, and thus each equation in (4.13) will give several equations, once this dependence is expanded out.
b) The functions F (0,±) short receive contributions from two sources. The first one comes from the right hand side of (4.3), (4.9) and contains the function f R explicitly. The second one corresponds to the contribution of Schur operators to the left hand side of (4.3), (4.9). Specifically, we have d) The second and third equations in (4.13) are obtained respectively from the antisymmetrization and symmetrization of the superconformal block expansion of (4.9) with respect to the exchange (z,z, w) ↔ (1−z, 1−z, 1−w). An important remark, relevant for the numerical implementation, is that the arguments of four dimensional superconformal and R-symmetry blocks entering (4.15), namely ( z z−1 ,z z−1 , w w−1 ) and their inverses, can be traded for (z,z, w) and (1 − z, 1 −z, 1 − w) using the braiding properties of the conformal blocks (C.3). This fact justifies the use of the suffix b to denote "braided" in (4.15). e) Finally, as customary, the identification |λ χ | 2 = O|χ(O)=χ |λ O | 2 is understood. By χ(O) we mean the representation χ in which the operator O transforms.

Fixing the chiral algebra contributions
We have defined above the functions H R,short and H R,short as the contribution from the exchange of Schur operators to the H R and H R functions, entering (4.3) and (4.9). We will now discuss to which extent these contributions can be extracted from the knowledge of f R (x), or more generally, from the knowledge of the chiral algebra.

Determination of the function f R (x)
The cohomological reduction of the correlator (3.14), which in superspace corresponds to a specialization of the superspace coordinates in (3.2) to X = diag(z, y, y), V = (θ, 0, 0) T and V = (θ, 0, 0), gives the holomorphic correlator where z ± = z ± 1 2 θθ and z 12 = z 1,− − z 2,+ + θ 1θ2 . For the following discussion we set the fermionic coordinates θ =θ = 0. We can view the correlator above as a meromorphic function of z 1 , whose poles correspond to singular terms in the OPE of W(1) with the remaining operators. The chiral OPE is non-singular, so there is no pole when z 1 ∼ z 3 (corresponding to x ∼ 1). The singularity for z 1 ∼ z 2 (corresponding to x ∼ 0), on the other hand, is already taken care of by the prefactor in the right hand side of (4.18). Finally, for z 1 ∼ z 4 (corresponding to x ∼ ∞) we have f R (x) ∼ x R . There is no other singularity and so f R (x) is a polynomial of degree R in x, that we normalize as f R (0) = 1, subject to the crossing relation (4.2). It is thus fixed in terms of R 2 constants. If follows from the exchange of the two-dimensional stress tensor that the small x expansion of the correlator takes the form f R (x) = 1 + R 2 4c 4d x + . . . , where c 4d ≡ c is the central charge of the four-dimensional theory, thus fixing one of the R 2 constants. 20 For R = 1 the crossing relation (4.2) implies that f 1 (x) = 1 + x, forcing the central charge to take the value c 4d = 1 4 , which corresponds to N = 4 SYM with gauge group U(1).

Non-chiral channel
Consider the expansion of the function f R (x) in holomorphic N = 2 (global osp(2|2)) blocks as Using the result given in table 2, and the selection rules (3.21), it is clear that in general one cannot reconstruct the four-dimensional OPE coefficients corresponding to Schur operators (4.12) from the knowledge of the expansion (4.19). This is best illustrated by looking at the following examples (4.20) and so on. Above we used the compact notation λB a ≡ λ and λĈ a, ≡ λ though this is not captured by the notation. The general pattern is quite simple and one finds b (R) Note that compared to the results that can be obtained from table 2 and the selection rules (3.21), we omitted by hand the supermultipletsĈ [0,0], for external fields with R ≥ 2, because they are the supermultiplets that contain higher spin conserved currents. They are included only in the free field case R = 1. For R ≥ 2, while allowed by the selection rules (3.21), we want to demand that they are absent in order to focus on interacting theories. We remark further that the OPE coefficient corresponding to the exchange of the stress-tensor supermultipletB [1,1] can be extracted unambiguously. It follows from the above considerations that also |λB 2 | 2 can be extracted without ambiguity. However, in general, the four-dimensional OPE coefficients cannot be extracted uniquely from the expansion (4.19). As discussed in section 2.3 and section 5.3.2 using the knowledge of the chiral algebra and some extra assumptions one can find, in the case R = 3, only two allowed values for λ (3) B [3,3] and λ In (4.23) the first summation starts from h = 2, since HB [1,1] = 0. In writing this equation we allowed for higher-spin currents to have a non-vanishing OPE coefficient, such that it becomes clear how they would contribute to the crossing equations. Looking at table 2 we see that if higher-spin currents are present they contribute exactly the same way as the R-symmetry singlet long multiplet at the unitarity bound ∆ = + 2. After setting them zero |λ  (z,z) is easily worked out, but will not be relevant here. We finally remark that the summation of the first term in (4.24) can be done explicitly for any R. See appendix D.2 for details.
Example: For R = 2, we find where from (4.20) we take, |λB [1,1] (4.26) Note that if higher-spin currents are present the above identification of OPE coefficients with b h cannot be made for h > 1.

Chiral channel
Now we expand the function f R , related to f R by (4.8), in N = 2 holomorphic blocks, which in this channel coincide with ordinary sl(2) blocks, see (3.36). Specifically where we note that the sum starts from h = R, which is due to the fact that the relevant OPE is non-singular. Moreover, the index h has the same parity as R as follows from the braiding relations of individual blocks (C.5) together with (4.11). By looking at the selection rules given in (3.22), and after a quick look at table 3, one concludes that where ≥ 0. Note that in this channel we can reconstruct the four-dimensional OPE coefficients of Schur operator completely in terms of the OPE coefficients of the cohomologically reduced problem. We can thus uniquely determine the contribution of these operators to H R : R+2n+2 HĈ . (4.29) The summation of this expression is straightforward and similar to the one done in appendix D.2. The final result is given by where we have defined the kinematic factor with k β (z) and h SU(2) a (w) given in (C.2) and (3.33) respectively. We have now obtained explicit expressions for H R,short and H R,short , and can compute the functions F (0,±) short entering the crossing equations (4.13). See appendix D.1 for more details.
The case R = 2 The bootstrap equations (4.13) in this case are independent of the R-symmetry variables w. Using the R = 2 specializations of the tables 2 and 3, namely table 4 and 5, we obtain ∆> +2 (4.33) Here λ ∆, andλ ∆, denote the OPE coefficients of the longs multiplets with dimension ∆ and spin appearing in the non-chiral and chiral channels respectively, and F short , see appendix D.1 .

The case R = 3
For R = 3 we unpack the superconformal blocks given in tables tables 2 and 3 in tables 6 and 7.
To write down the crossing equations (4.13) in components we need to fix a basis in the space of . When considering identicalB [2,0]  . R-symmetry polynomials. There is a natural choice which follows by noticing that  The equations for B and C in the chiral channel follow from the last two entries in (4.34) at the unitarity bound, as can be seen from (4.35) More explicitly, we extract the coefficients of (1 − w) −1 (1 ± y −1 ) of the first line of (4.13) and the coefficients of  a),a], respectively. As a consistency check, we verified that the bootstrap equations above are satisfied with positive coefficients for the cases of free U(1) N = 4 SYM (considered a special N = 3 theory) and for the generalized free theory discussed in appendix C.2. [1,1] (w) H χ (z,z, w) B [6,0] g 2d 3 (z) g 5,1 (z,z) + g 6,0 (z,z)h

Numerical results
We are finally ready to apply the numerical bootstrap machinery to our crossing equations. Our goal is to chart out the allowed parameter space of N = 3 theories, but also to "zoom in" to particular solutions of the crossing equations that correspond to individual N = 3 SCFTs. After a short review of numerical methods we start by considering the multiplet containing a Coulomb branch operator of dimension two, which we recall also contains extra supercharges. This is a warm-up example that will allow us to check the consistency of our setup. In the remainder of the section we then focus on a Coulomb branch operator of dimension three for various values of the central charge. In general it is hard to exclude solutions that have enhanced N = 4 symmetry, and also to impose that the Coulomb branch operator is a generator. 21 In order to avoid N = 4 solutions, at the end of this section we input knowledge of the specific chiral algebra [10] that is conjectured to correspond to the simplest known N = 3 SCFT .

Numerical methods
The crossing equations written in (4.13) are too complicated to study exactly, beyond focusing on special limits, or protected subsectors, as done in section 2. Therefore we proceed to analyze these equations using the numerical techniques pioneered in [14] (see e.g. [70,71] for reviews).
Very schematically, we have a system of crossing equations (three (4.33) and six (4.34) for thê B [2,0] andB [3,0] respectively) of the form where V fixed (z,z) collects the part of F short that is completely fixed from the chiral algebra, with the remainder of F short moved to the left-hand side. We use the SDPB solver of [72], and rule out assumptions on the spectrum {O i } of local operators and their OPE coefficients |λ Oi | 2 (CFT data), by considering linear functionals acting on the crossing equations. In the crossing equation (4.33) and (4.34) we will be taking derivatives ∂ m z ∂ n z of F ± , F b ± , F ± and from their symmetry properties under z → 1 − z,z → 1 −z we see that only even (odd) derivatives of 22 The numerical bounds will be obtained for different values of the cutoff Λ, which effectively means we are considering a truncation of the Taylor series expansion of the crossing equations around z =z = 1 2 . We rule out assumptions on the CFT data by proving that they are inconsistent with the truncated system of crossing equations at order Λ . Therefore, for each cutoff we find valid bounds, that will improve as we send Λ → ∞. We refer the reader to the by now extensive literature on these numerical techniques, e.g. [16,72], for all the other technical details and approximations needed for the numerical bootstrap.

The case R = 2
As a warm-up, let us consider external operatorsB [2,0] ,B [0,2] , which contain the extra supercharges allowing for an enhancement to N = 4. For this case we will only bound the minimal allowed central charge c. We recall that the OPE selection rules in this case are given bŷ with each multiplet contributing with a superblock as given in tables 4 and 5, with a positive OPE coefficient squared, and the crossing equations are given in (4.33). To obtain central charge bounds, we allow for all operators consistent with unitarity that have not been fixed by the chiral algebra.
In the chiral channel this amounts to allowing all long operators consistent with unitarity, together with the short multiplets which sit at the long unitarity bound (which are not Schur operators). In the non-chiral channel the OPE coefficient ofB [1,1] is fixed unambiguously from the chiral algebra in terms of the central charge. For the remaining Schur operators the chiral algebra is not constraining enough and we are left with some ambiguities. As shown in equation (4.24) we can fix universally the OPE coefficients ofĈ [1,1], andB [2,2] in terms of those of theĈ [0,0], multiplets. These last multiplets contain higher-spin currents and should be absent thereby resolving the ambiguity. Nevertheless, as is also clear from (4.24) and table 4, the contribution of theĈ [0,0], multiplets is identical to that of long multiplets at the unitarity bound, and thus, by allowing for long multiplets to have a dimension arbitrarily close to the unitarity bound, we allow for these currents to appear with arbitrary coefficient. Therefore, we do not truly exclude free theories in the bootstrap, and we should expect to recover the solution corresponding to U(1) N = 4 SYM theory. 22 As usual, the equations are antisymmetric in z ↔z and so we only need derivatives with m < n. The numerical lower c bound is shown in figure 1 as a function of Λ −1 , where Λ is the cutoff on the number of derivatives taken of the crossing equation, as defined in (5.2). The solid yellow and blue lines correspond to various linear fits to subsets of points, and attempt to give a rough estimate of the Λ = ∞ bound. It seems plausible that the bound is converging to c = 3 12 which corresponds to the central charge of U(1) N = 4 SYM. Recall that for this value of the central charge the coefficient b is negative, which means that it cannot be interpreted as arising only from â B [2,2] multiplet, and that the conserved current multipletĈ [0,0],0 has to be present. But this is exactly what our crossing equations are allowing for, as when we solve for the OPE coefficient ofB [2,2] in terms of b 2 and let the OPE coefficient ofĈ [0,0],0 be arbitrary we find it contributes just as a long at the unitarity bound. Naturally, if one wanted to obtain dimension bounds on the long operators for c = 3 12 we would have to allow for the multipletĈ [0,0],0 to be present by adding their explicit contribution, but if no gap is imposed, then allowing for long multiplets of arbitrary dimension automatically allows for these currents.

The case R = 3
We now turn our attention to the correlation function ofB [3,0] ,B [0,3] multiplets, whose crossing equations are given in equation (4.35). We recall that in the chiral channel the OPE coefficients of all of the Schur multipletsĈ [4,0],( +1 2 , 2 ) andB [6,0] were fixed universally from the chiral algebra correlation function. Therefore, the undetermined CFT data in this channel amounts to , where the last multiplets contribute the same way as the longs at the unitarity bound as seen in (3.37) and table 7.
The Schur multiplets in the last line end up contributing to the crossing equations in the same way as the long multiplets in the line above at the unitarity bound (see (4.24) and table 6), following from the long decomposition at the unitarity bound (3.31). This implies that, unless we impose a gap in the spectrum of the corresponding long multiplets, we can never truly fix the OPE coefficients of these Schur operators. As usual, the multipletsĈ [0,0], should be set to zero for interacting theories. However this is not enough to resolve all the ambiguities, and we must resort to numerics in order to study the OPE coefficient of the remaining operators. In the last part of this section we will see how these ambiguities turn out to be useful to exclude N = 4 solutions to the crossing equations by inputting the OPE coefficient ofB [3,3] computed from the chiral algebra of an N = 3 SCFT.

Central charge bounds
Let us start by placing a lower bound on c, allowing again for the presence of all operators consistent with unitarity. We recall once again that long multiplets A ∆ [0,0], of arbitrary dimension allow for conserved currents of spin larger than two, and thus not excluding free theories from the analysis. Naturally then, the U(1) N = 4 SYM theory is also a solution to the crossing equations we study. Therefore, the strongest bound one could possibly hope to find corresponds to the central charge of U(1) N = 4 SYM. This value is smaller than the smallest central charge of all known, nontrivial, N = 3 theories, which is c = 15 12 . In figure 2 we show the minimal allowed central charge as a function of Λ −1 , the inverse of the number of derivatives. Extrapolation for infinitely many derivatives this time does not seem to converge to the value of the U(1) N = 4, which is c = 1 4 = 0.25. 24 Since the value of the minimal allowed central charge is smaller than that of the free N = 4 theory one might suspect the solution to this set of crossing equations that saturates the central charge bound does not correspond to a physical SCFT, and could imagine a mixed correlator system, e.g., adding the stress tensor multiplet, would improve on this.

Bounding OPE coefficients
Apart from the central charge, there are other OPE coefficients of physical interest, which were not fixed analytically and can be bounded numerically. Let us emphasize that the N = 3 stress-tensor multipletB [1,1] cannot recombine to form a long multiplet, unlike the N = 2 stress-tensor multiplet. This has the important consequence that, when we add the stress tensor multiplet with a particular coefficient, we are truly fixing the central charge to a particular value. In comparison, in N = 2 theories this was only accomplished when one imposed a gap in a particular channel, preventing those long multiplets to hit the unitarity bound and mimic the stress tensor. Therefore, we will bound the OPE coefficients as a function of the central charge for the range 1 4 c ∞. The lower end of the interval corresponds to the central charge of U(1) N = 4 SYM, although interacting theories should have higher central charges. In particular there is an analytic lower bound for interacting N 2 SCFTs of c 11 30 ≈ 0.37 [46]. Furthermore it can be shown, by considering the N = 3 stress tensor four-point function in the chiral algebra, that any interacting N 3 SCFT must obey c 13 24 ≈ 0.54 [74]. These two bounds will be depicted as vertical dashed lines in all the numerical results. In the limit c → ∞ the stress tensor decouples and we expect that the numerical bounds converge to the values of generalized free field theory (see appendix C.2).
The Schur operatorB [3,3] A particularly interesting operator to consider is theB [3,3] appearing in the non-chiral channel. Despite being captured by the two-dimensional chiral algebra, is not possible to fix its OPE coefficient universally from the chiral algebra four-point function, due to the ambiguities described in 4.2. Making assumptions about what particular chiral algebra corresponds to a given 4d theory, one can try to resolve this ambiguity, as done in section 2.3, which gave two seemingly consistent possibilities. However, we will first take an agnostic viewpoint, and ask what numerical constraints crossing symmetry and unitarity place on the squared OPE coefficient of this operator (|λB [3,3] | 2 ). These are shown in figure 3 as a function of the inverse of the central charge.
Since this operator is protected, we can compare the value of the bound to the well known N = 4 solutions. We extracted the OPE coefficient of this multiplet from the four-point function of half-BPS operators in the [0, 3, 0] representation of SU(4) R given in [37], after projecting the N = 4 multiplets to the particular N = 3 multiplet we are considering. It turns out to have a constant value of one, irrespective of the central charge of the theory. We depicted this as a red line in figure 3 and, to give an idea of where the physical N = 4 theories sit, we also added red dots in the positions corresponding to the central charge of N = 4 SYM with gauge group SU(n) (c = n 2 −1 4 ) for n ∈ {3, 4, . . .}. The value expected from the block decomposition of both the U(1) N = 4 SYM (c = 1 4 ) and the generalized free field theory (c = ∞, given in appendix C.2) is also one, and is marked by red dots as well. A rough extrapolation of our results for infinite central charge and for c = 1 4 suggests the numerical bounds could converge to the values expected for these theories. 24 Similar results were also observed in the case of chiral correlators in N = 2 theories [19,73]. Finally, we compare the numerical bounds with the results that can be extracted from a particular chiral algebra. Let us first consider the chiral algebra of [10] that is conjectured to correspond to the simplest known N = 3 SCFT with c = 15 12 (in their notation this corresponds to = 3, where of course this has no relation to the spin). As discussed in section 2.3 we can construct candidate operators, in the chiral algebra, to correspond to aB [3,3] . In this case there is only one candidate, and if one assumes it to be in fact aB [3,3] we find λB [3,3] 2 = 22 85 , (5.5) which is shown as a green dot in figure 3. 25 Note that this value lies well inside the numerical bounds, and in particular it is also smaller than the continuation to arbitrary c of the value corresponding to SU(N ) N = 4 SYM. Since the N = 4 SYM correlation function of [37] which we decomposed in blocks is a solution of the crossing equations for any value of c, the best numerical bound one can hope to obtain is λB [3,3] 2 ≤ 1. In fact, our numerical results appear consistent with the upper bound converging to one for c = 15 12 . Therefore, to be able to reach the known nontrivial N = 3 SCFT with c = 15 12 we must go inside these bounds, and fix the OPE coefficient ofB [3,3] to a value that is incompatible with the N = 4 solution to the crossing equation; we will do this at the end of this section.
We now turn to the chiral algebra constructed in 2.2, with the goal of understanding the higher rank versions of the aforementioned theory. Recall that we assumed the chiral algebra of the higher rank theories to be generated solely by the Higgs branch generators, the stress tensor, and an additional dimension three operator. Under this assumption, we were able to construct a closed subalgebra of all of these chiral algebras, which is associative for generic values of c. In that setting we can attempt to compute |λB [3,3] | 2 , and there were two options consistent with the large central charge behavior of the generalized free field theory and unitarity, given in (2.25) and (2.27), which are plotted as green curves in figure 3. Equation (2.27) is the one that does not go through the expected value for c = 15 12 , but that we kept for arbitrary values of c. If our assumptions are correct, then we see that the value of |λB [3,3] | 2 lies well inside the numerical bounds, and is weaker than that of N = 4 SYM. This is not necessarily a downside, as one of our goals must be to determine ways to exclude the N = 4 solutions to our crossing equations, and this provides such a way. By imposing the value of the OPE coefficient corresponding to (2.25) or (2.27) we are sure to exclude N = 4 from our analysis. We will come back to this point at the end of this section.
Note that both (2.25) and (2.27) diverge at c = 13 24 , which corresponds to the analytic central charge bound obtained in [74], following from the fact that the norm of one of the candidateB [3,3] operators is going to zero. We note that the chiral algebra in 2.2 was constructed with a generic central charge in mind and care was not given to possible null states arising at specific values of c. It is not clear that the solution we have is consistent for c = 13 24 , as null states are expected to decouple. It is also worth noting the interesting interplay between analytical and numerical results. The analytical OPE coefficient is only consistent with the (current) numerical bounds for 1/c 1.33 − 1.36 depending on which curve one takes. This provides a lower bound c 0.74−0.75 on the central charge of any N = 3 SCFT with a dimension three Coulomb branch operator (B [3,0] ) of which the chiral algebra presented in 2.2 is a closed subalgebra, improving over the analytical bound c 13 24 ≈ 0.54 of [74]. On the other hand, this bound is lower than the one obtained using the sum-rule of [75,76] for a rank one theory with a generator of dimension three, namely c 15 12 . Although there are known cases where this sum rule does not hold (see [3,9]), they correspond to theories obtained by gauging discrete symmetries, so this bound could be valid for theories which are not of this type. 26 The multipletsB [2,2] andC Next we turn our attention to the short multiplets in the chiral OPE that sit at the unitarity bound of the long multiplets, and are not captured by the chiral algebra. As representatives, we show the upper bounds on the OPE coefficients squared of the multipletsB [2,2] andC We show in figure 4 the value of these OPE coefficients in the case of the generalized free field theory, and of the U(1) N = 4 SYM as the two red dots at c −1 = 0 and c −1 = 4 respectively. The convergence of our numerical results is rather slow and one cannot conclude if they will converge for these central charges to the known solutions, although they are not incompatible with this possibility. The green lines in the plots mark the central charge of the "minimal" N = 3 SCFT (c = 15 12 ) with the green dot providing a valid upper bound for the OPE coefficients of this theory.
Finally, to better understand what is failing in the crossing symmetry equations if one tries to go below the minimal numerically allowed central charge (c min in figure 2), it is instructive to look at the OPE coefficient bounds near those central charges. One finds (not shown), that while the bounds on the squared OPE coefficients of bothB [3,3] andB [2,2] have a very sharp drop near c min , the upper bound on the squaredC   Figure 4. Upper bound on the OPE coefficient squared ofB [2,2] (|λB [2,2] | 2 , depicted on the left) and of C c min , which is inconsistent with unitarity. This suggests it is this last multiplet that is responsible for the lower bound on the central charge, and that the solution at c min would have the other two short operators present. Note that bothB [2,2] andC have zero OPE coefficient for the U(1) N = 4 SYM theory.

Dimension bounds
Next we turn to the dimensions of the lowest lying scalar long operators in the various channels. In doing so we must worry about the short multiplets whose OPE coefficients we bootstrapped in the previous subsection, as they all sit at the unitarity bound of the different long multiplets we study (see the tables 6 and 7). By allowing for long multiplets with arbitrary dimension, these short multiplets can appear with any coefficient. Even if we were to explicitly add by hand the short multiplets with a given OPE coefficient, the long multiplet at the bound would mimic those shorts, and in practice we would only be imposing the OPE coefficient of the short multiplets to be greater or equal to a given value. However, once we impose a gap in the spectrum of the long operator, then we can truly fix the OPE coefficient of the corresponding short multiplet.
In the non-chiral channel, we focus on the dimension of the first scalar long of each type  Table 7 contains also long multiplets A ∆>5+ [0,1],10, but for those the spin must be odd by Bose symmetry.

Non-chiral channel
The upper bounds on the dimensions ∆ [0,0] and ∆ [1,1] , of the first long multiplets A ∆>2 [0,0],0 and A ∆>4 [1,1],0 respectively, as functions of the inverse central charge are depicted in figure 5. Once again red dots mark the dimension of the lowest dimensional operator in the generalized free field theory and the U(1) N = 4 SYM solutions. In both cases the green vertical line ending on a dot marks the central charge of the simplest known nontrivial N = 3 SCFT, and provides an upper bound for the dimension of these two operators in this theory. We will improve on the latter bound at the end of this section.   [46] and N = 3 SCFTs [74].
At the unitarity bound, the long multiplet of type A ∆>2 [0,0],0 mimics a higher spin conserved current multiplet (Ĉ [0,0] =0 ), expected to be absent in an interacting theory, and therefore when obtaining the bound on the left side of 5 we do not allow for such a multiplet to be present. This explains why the upper bound is presumably converging to the unitarity bound ∆ = 2 for c −1 = 4, since such currents should be present in the U(1) N = 4 solution, as indicated by the red dot. For larger central charges the upper bound is far away from unitarity, and thus theories saturating the upper bound do not contain theĈ [0,0] =0 multiplet, although they could have the higher spin versions of this multiplet which also contain higher-spin conserved currents.
On the other hand, the multiplet that sits at the unitarity bound of A ∆>4 [1,1],0 is theB [3,3] discussed in the previous subsection, and in obtaining the bounds for ∆ [1,1] we allowed the short multiplet to be present with arbitrary OPE coefficient. We can obtain a stronger bound for specific N = 3 SCFTs by fixing the short OPE coefficient according to section 2.3, as we shall do later in figure 7 for the case of c = 15 12 ⇒ c −1 = 0.8.

Chiral channel
Turing to the scalar long operator appearing in the chiral OPE, we obtain an upper bound for the first A ∆>5 [2,0],10,0 multiplet. In imposing a gap in this channel we must decide on whether the short multiplet B [2,2] is present or not. Recall that, unlike the shorts at the unitarity bound of long operators appearing in the non-chiral channel, this short is not captured by the chiral algebra and thus we have no reason to expect it to be present or absent. Therefore, we show a bound on the dimension ∆ [2,0] of this long multiplet both allowing for (left plot in 6) and disallowing for (right plot in 6) the presence ofB [2,2] . Once again the red dots depict the value of these dimensions expected for the U(1) N = 4 SYM and generalized free field theories. We observe that for c → ∞, the right hand side of figure 6 comes close  [2,0],10,0 , allowing for (left) and disallowing for (right) the short multipletB [2,2] . The strongest bound corresponds to 24 derivatives, and they are reduced in steps of two. The red dots mark the dimension of the first long operators for generalized free field theory and U(1) N = 4 SYM, in the right plot the red dot of generalized free field theory is at the unitarity bound, meaning that the short multiplet is present in this solution. The green line marks the central charge c = 15 12 of the simplest known N = 3 SCFT, with the green dot providing an upper bound for this theory. The two dashed lines correspond to the minimum central charges for an interacting N = 2 [46] and N = 3 SCFTs [74].
to the unitarity bound ∆ = 5. In fact, a simple extrapolation seems to suggest that for Λ → ∞ the bound will converge to around 5. This is consistent with the fact that this multiplet is present in the generalized free theory solution (see appendix C.2), i.e. the bounds force the long multiplet to "become short" for c → ∞. (Said multiplet is absent in the U(1) N = 4 SYM solution.) For values of c around the value relevant for the "minimal" N = 3 SCFT, marked as green lines in the plots, there seems to be a solution of the crossing equations with this multiplet absent.

Carving out solutions inside the bounds
As the final point of this section we come back to the issue of distinguishing N = 4 solutions to the crossing equations from pure N = 3 ones. One possibility is to extract the spectrum of the extremal solution [77] saturating each of the above bounds and check if it is consistent or inconsistent with N = 4 supersymmetry. However, we would like to do better, and to be able to exclude the N = 4 solution altogether. Our explorations in the first part of this section provide such a way, namely by fixing the OPE coefficient ofB [3,3] to the value expected to correspond to the N = 3 theories of interest (see section 2.3). This value is smaller than the one of SU(N ) N = 4 SYM and in its derivation in the chiral algebra we did not allow for the currents enhancing the supersymmetry to N = 4. As usual, because a long multiplet at the unitarity bound (A ∆=4 [1,1],0 ) mimics the contribution of this short multiplet to the crossing equations, we cannot really fix its OPE coefficient unless we impose a gap. This is what is done when bounding the lowest dimensional A ∆>4 [1,1],0 , and so we repeat the analysis leading to the right side of figure 5, but now fixing the OPE coefficient ofB [3,3] . The result is shown in figure 7, where we plot the upper bound on the dimension as a function of the OPE coefficient for  fixed c = 15 12 . The red line marks the value of the OPE coefficient for the N = 4 solution with this particular value of c. While the green line marks the value of the OPE coefficient expected for the N = 3 SCFT we are interested in (5.5), and provides an upper bound for the dimension ∆ [1,1] in this theory, which improves significantly on the one obtained from figure 4. This shows that, at least in figure 4, the theory saturating the bound does not correspond to the N = 3 SCFT we were after, and thus, to zoom in to this specific theory we must carve further inside the bounds as done here. This however does not guarantee the theory now sits at the bound. 28 Similarly, we can repeat this analysis for the central charges of the higher rank theories and we find that, at fixed |λB [3,3] | 2 , if the central charge is (increased) decreased the bound seems to get (stronger) weaker (not shown). Due to the dependence of (2.25) and (2.27) on c the upper bound on ∆ [1,1] does not change significantly.

Conclusions
In this paper we have initiated the N = 3 superconformal bootstrap program with two goals in mind. First, to constrain the space of four-dimensional N = 3 SCFTs, and second, to focus on specific examples of N = 3 theories with the hope of obtaining information about their spectrum. In order to zoom in on the known N = 3 SCFTs we relied on a combination of numerical bootstrap results and 28 We seem to observe a small bump for |λB [3,3] | 2 ≈ 0.33, and preliminary functional analysis suggest this is correlated to the fact that to the left of the bump a conserved currentĈ [0,0] =0 is allowed, and to the right disallowed. This does not necessarily imply that the conserved currents are present for the c = 15 12 extremal solution, but could mean that to get closer to an interacting N = 3 SCFT we should simultaneously impose a gap in the A ∆>2 [0,0],0 long channel.
analytical results from two-dimensional chiral algebras, with particular emphasis on the "minimal" N = 3 SCFT, and its higher-rank versions. We approached these theories from the point of view of the Coulomb branch, focusing mostly on a half-BPS operator of dimension three, which is the only Coulomb branch generator of the "minimal" N = 3 SCFT, and which is also present in its higher-rank versions.
A basic requirement for any bootstrap study is the conformal block expansion of the four-point function. In section 3 we showed that for N = 3 half-BPS operators there are no nilpotent invariants, allowing us to concentrate on superconformal primaries without any loss of information. Demanding the absence of singularities when turning on the fermionic coordinates places strong restrictions on the form of the four-point function, giving rise to the Ward identities. In the case at hand, these identities were not enough to completely fix the superblock (unlike the cases of N = 2 and N = 4 half-BPS superblocks [31,32]). For superblocks associated to short multiplets, we used information coming from the 2d chiral algebra, while for long blocks we leveraged knowledge of N = 1 blocks. In the end, we packaged our solution in an elegant way in terms of a single N = 1 conformal block with shifted arguments.
The existence of a protected subsector captured by the 2d chiral algebra allowed us to solve the crossing equations exactly within the subsector. Which in turn fixed the OPE coefficients of certain short operators universally, i.e., without needing to specify a particular four-dimensional theory. However, some operators appear indistinguishable at the level of the chiral algebra, leading to ambiguities in fixing the corresponding OPE coefficients. Some of these ambiguities can be resolved by knowledge of the specific chiral algebra associated to the N = 3 theory in question, but this is not always the case.
An important question is the defining characteristics of the chiral algebra associated to N = 3 SCFTs. To that end, we determined which N = 3 superconformal multiplets are captured by the 2d chiral algebra, and some of their general properties, which could allow distinguishing between the aforementioned operators. Taking advantage of the chiral algebra conjectured to correspond to the "minimal" SCFT [10], we were able to compute the OPE coefficient |λB [3,3] | 2 . Moreover, we proposed, under certain assumptions, a closed subsector for the higher-rank versions of this theory, and used it to compute |λB [3,3] | 2 in this case. To go beyond the protected subsector, or even to distinguish between operators appearing identically in the chiral algebra, one needs numerical bootstrap techniques. These provided constraints on the spectrum of unprotected long operators, and on the OPE coefficients of various short operators. For the particular OPE coefficient |λB [3,3] | 2 that we were able to fix from the chiral algebra, we compared the general numerical results valid for any N = 3 SCFT, with the ones of the specific theories captured by the chiral algebra that we constructed. This comparison provided a numerical lower bound on the central charge for theories captured by our chiral algebra.
A natural limitation of any N = 3 bootstrap program, as it was also for the N = 2 bootstrap, is that theories with more supersymmetry will generically be solutions of the crossing equations we consider. In order to restrict to pure N = 3 SCFTs, one would have to exclude the presence of superconformal multiplets containing the currents allowing for this enhancement. However, the multiplets that are physically relevant for the study of these theories (for example the ones considered in [19,46,73] in the N = 2 case) usually do not allow for the multiplets containing the extra supercurrents to be exchanged in their OPEs, and therefore we cannot set them to zero. To overcome this limitation we input into the numerical bootstrap information arising from the chiral algebras of pure N = 3 SCFTs, namely, the OPE coefficient |λB [3,3] | 2 . This allowed us explore inside the numerical bounds, and zoom in on the N = 3 solutions with this particular value of the OPE coefficient. By fixing the central charge to that of the "minimal" N = 3 theory, and fixing the OPE coefficient accordingly, it is plausible that this theory sits at the bound of figure 7, although currently there is no evidence this has to be the case, and we would have to provide more information (such as adding stress tensors as external operators). Nevertheless, the ambiguity in fixing OPE coefficients turned out to be crucial in excluding the N = 4 solution to the crossing equations. For the higher rank versions, one would have to also consider the four-point functions of the additional Coulomb branch operator, which is a natural next step in the N = 3 bootstrap, along with the study of stress tensor four-point functions. Name Name in [58] Superconformal primary Conditions   Table 8. We list here the unitary representations of N = 3 with the name that we give them in the present work accompanied by the one that they have in [58], which was based on the type of shortening condition that they obey. The third column shows the charges of the superconformal primary in the representation, while the fourth one lists the conditions that the charges have to obey. The A2, respectivelyĀ2 shortening cases are obtained by putting j = 0, respectively = 0. This changes the null states drastically, but not our labels.
decomposes in SU(2) R N =2 × U(1) r N =2 × U(1) f , where the first two factors are the R-symmetry of the N = 2 superconformal algebra, and the last corresponds, from the N = 2 point of view, to a global symmetry. Therefore when viewed as N = 2 theories, all N = 3 theories have a U(1) f flavor symmetry, and we will keep this flavor grading when decomposing N = 3 representations in N = 2.
We follow the conventions of [10] for the definition of the flavor charges. We note that we follow the naming conventions of Dolan and Osborn [60] for the representations of N = 2, which are summarized for instance in Appendix A of [19]. While the interpretation of most of these multiplets might be obscure, the following have a natural physical interpretation 30 •Ĉ 0,(0,0) is the stress tensor multiplet of an N = 2 SCFT, containing in addition to the stress tensor, the SU(2) R N =2 and U(1) r N =2 currents, • B R are closely related to the Higgs branch of the theory, in particular theB 1 multiplet contains conserved currents of spin one, associated to flavor currents of the theory, • E r,(0,0) are N = 2 chiral operators, and are related to the Coulomb branch of the theory, • D 1 2 ,(0,0) (and conjugate) which are additional supercurrent multiplets, •Ĉ 0,(j>0,>0) contain conserved currents of spin greater than two, which signal free theories [61,62].
In addition, the multiplets dubbed "Schur" operators in [40], that is, the ones captured by the twodimensional chiral algebra reviewed in section 2, also play an important role. These areB R , D R(0,) , D R(j,0) andĈ R(j,) , giving rise to two-dimensional sl(2) primaries of scaling dimension R, R + + 1, R + j + 1 and R + j + + 2 respectively. The N = 3 multiplets that contain such operators are 30 For a more detailed description see, e.g., section 2 of [19] listed in equations (2.6)-(2.13), together with their decomposition in N = 2, but where we omitted all N = 2 multiplets not containing Schur operators. Below we present a few examples of the complete N = 2 decomposition. These decompositions are obtained by computing the characters of the N = 3 multiplets of table 8, following the method described in appendix C of [78], and re-writing it in terms of characters of N = 2 representations, which can be obtained from the tables of [60].
The stress-tensor multiplet decomposes in the expected way, containing only Schur multipletŝ and their conjugates. Note that the above decomposition contains "exotic" N = 2 E r,(j,0) operators with spin j > 0, which do not seem to occur in known N = 2 SCFTs (see [79] for a discussion). Similarly in [1] the question of which N = 3 operators could contain operators whose vevs parametrized the Coulomb branch was addressed. The authors of [1] argue that the only type of such multiplets arê B [R1,0] and conjugates, since the B [0,0],r,0 multiplet would not be consistent with the three different N = 2 subalgebras N = 3 contains. We finish this appendix with the example of the decomposition of a generic long N = 3 multiplet. Considering a multiplet whose highest weight transforms in the symmetric traceless representation for simplicity, A ∆ [R1,R2],r, , there appears to be a simple prescription for the decomposition into N = 2 multiplets, which we have checked in a variety of cases. Namely, we first decompose the SU(3) R ×U(1) r representation ([R 1 , R 2 ], r) of the superconformal primary of the N = 3 multiplet in representations of SU(2) R N =2 × U(1) r N =2 × U(1) f . Let {(R , r , F )} be the list of representations appearing in that decomposition. To each such representation we associate an N = 2 multiplet A ∆ R ,r , , graded by the corresponding U(1) f charge (u f ) F . Finally, in the decomposition of the N = 3 multiplet, each of these N = 2 multiplets will be accompanied by the following list of long multiplets: 2 ) u f + A with h [1,1] (w) given in (3.24). The block expansion of the two-dimensional correlator f gft R (x) (i.e., in the non-chiral channel) is explicitly given by (C.9) Note (and compare with the discussion around (4.20)) that in the generalized free theory example we have b (R) h<R = 0. In particular, there is no stress tensor being exchanged. Similarly, we can decompose the two-dimensional correlator in the chiral channel to find (C.10)

D Short contributions to crossing
Here collect some bulky equations used in the crossing equations (4.13) and summarize the computation used in section 4.2 for the function H R,short .

D.1 Explicit expressions for F
(0,±) short Here we collect the expressions for F (0,±) short that we need in the crossing equations (4.13). Using the definition (4.4) for the function A R , we write