Bounds on $\mathcal{N}=1$ Superconformal Theories with Global Symmetries

Recently, the conformal-bootstrap has been successfully used to obtain generic bounds on the spectrum and OPE coefficients of unitary conformal field theories. In practice, these bounds are obtained by assuming the existence of a scalar operator in the theory and analyzing the crossing-symmetry constraints of its 4-point function. In $\mathcal{N}=1$ superconformal theories with a global symmetry there is always a scalar primary operator, which is the top of the current-multiplet. In this paper we analyze the crossing-symmetry constraints of the 4-point function of this operator for $\mathcal{N}=1$ theories with $SU(N)$ global symmetry. We analyze the current-current OPE, and derive the superconformal blocks, generalizing the work of Fortin, Intrilligator and Stergiou to the non-Abelian case and finding new superconformal blocks which appear in the Abelian case. We then use these results to obtain bounds on the coefficient of the current 2-point function.


Introduction
Recently, there has been much interest in generating numerical constraints on conformal field theories using the conformal bootstrap [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In these works the existence of a scalar primary operator φ of dimension ∆ is assumed. Using the conformal bootstrap on the 4-point function of φ, it is then possible to generate numerically bounds on operator dimensions and OPE coefficients of operators in the φ × φ OPE as a function of ∆. The crucial ingredient which allows us to generate these bounds is the knowledge of all the scalar conformal blocks [17,18], which encode the dependence of the 4-point function of φ on each conformal family in the φ × φ OPE 1 .
It would interesting to apply these methods without introducing any assumption on the operator spectrum. For instance, one would like to analyze the four-point function of the stress-tensor, which exists for any CFT. More generally, assuming the CFT has some global symmetry one would like to understand the constraints of conformal invariance arising from application of the conformal bootstrap to the 4-point function of the global symmetry current. Unfortunately, to this date there are no closed form expressions for the conformal blocks of non-scalar operators (see however [20,21]), so these interesting directions cannot be pursued in a straightforward way yet.
However, in supersymmetric theories the situation is better since in some cases, symmetry currents reside in multiplets whose superconformal-primary (sprimary) is a scalar field. For instance in N = 4 supersymmetric Yang-Mills the energy-momentum tensor resides in a multiplet whose sprimary is a scalar in the 20 representation of the SU (4) R R-symmetry group. The bootstrap constraints for this case were recently analyzed in [12].
Similarly, in any four-dimensional N = 1 superconformal theory, a global symmetry current j a µ resides in a real multiplet J a (z) = J a (x, θ,θ) = J a (x) + iθj a (x) − iθj a (x) − θσ µθ j a µ (x) + · · · , (1.1) which satisfies D 2 J a =D 2 J a = 0, and the omitted terms in the equation above are determined by this constraint. The sprimary J a is a dimension two real scalar field in the adjoint representation of the symmetry group.
In this paper we will use the conformal bootstrap to constrain four dimensional N = 1 superconformal theories with an SU (N ) global symmetry 2 . In particular, we will place lower bounds on the current "central charge" τ defined as the coefficient of the current 2-point function 3 j a µ (x)j b ν (0) = 3τ δ ab x 2 δ µν − 2x µ x ν 4π 4 x 8 . (1. 2) The decomposition of the 4-point function of J a into conformal blocks is constrained by supersymmetry. In particular, the OPE coefficients in J a × J b of different primary operators in a super-multiplet are not independent and the corresponding conformal blocks are re-packaged into the so-called superconformal blocks. These constraints were already analyzed in detail in [22] for the U (1) case, and will generalize those results to the non-Abelian case. In addition, we find new operators which generally appear in the OPE which were not found in [22].
The form of the bounds we find is τ > f (N ). Qualitatively, the existence of a lower bound means there is a minimal amount of "charged stuff" which must exist in any such theory. A free chiral superfield has, in our normalization, τ = 1. We do not know of any theory with τ < 1 and it would be very interesting to understand whether those exist, or alternatively to prove that τ ≥ 1 in general.
The paper is organized as follows. In section 2 we briefly review the conformal bootstrap and set up our conventions. In addition, we determine the sum-rules which result from applying crossing-symmetry to the 4-point function of a scalar primary in the adjoint representation of SU (N ). In section 3 we discuss the constraints imposed by N = 1 superconformal invariance on the J a × J b OPE and superconformal blocks. In section 4 we present the lower bounds we obtained on τ and a short discussion.

Conformal Bootstrap
In this section we spell out our normalization conventions and briefly summarize the conformal bootstrap constraint for a general CFT. The reader is referred to [1] for a more extensive treatment.
Consider a general CFT in four Euclidean dimensions, and in particular the subset of operators consisting of spin-primary operators O ( ) , which are symmetrictraceless rank-tensors (i.e. in the ( /2, /2) representation of the Lorentz group SO(4)). The index I labels the primary operators in the CFT, and we will denote the complex conjugate operator by a barred indexŌJ ≡ (O J ) † .
We set the normalization of such operators by demanding that their 2-point function is of the form where on the RHS the indices (µ 1 , . . . , µ ) and (ν 1 , . . . , ν ) should be symmetrized with the traces removed, and ∆ I denotes the dimension of O ( ) I . The 3-point function of a spinprimary with two scalar primaries φ a , φ b of equal dimension ∆ 0 is where again the Lorentz indices on the RHS should be symmetrized with the traces removed, and a, b are arbitrary labels. The information on the 2-point and 3-point In above equation the identity operator 1 contains the information on the 2-point function and the sum over primaries encodes all the information on 3-point functions. The operator are complex then the OPE coefficients are generally complex and satisfy λ ab,Ī = (λ ab,I ) * . If we choose a real basis of operators, then the OPE coefficients λ ab,I must be real λ ab,Ī = λ ab,I . The 3-point function is non-zero only for O I of integer spin, and (odd) even spins correspond to the (anti-)symmetric combination of φ a and φ b (i.e. λ ab,I is (anti-)symmetric in a, b for (odd) even spins).

Bootstrap for Scalars in the Adjoint of SU (N )
In this section we discuss a specific case of the general bootstrap constraint (2.6) in which φ a is a real scalar primary in the adjoint representation 5 of SU (N ). We will later apply the results of this section to the case in which this scalar is the top of the current multiplet in N = 1 theories. The crossing-symmetry relations in CFTs with global symmetries were considered in full generality in [6], and we apply these results to our case of interest. The operators which appear in the φ a × φ b OPE can be decomposed into any of the 7 irreducible representations in the product of two adjoint representations of SU (N ) (see Appendix A). Each such representation arises from either the symmetric or anti-symmetric product. Operators in the φ a × φ b OPE which are in a (anti-)symmetric representation must be of (odd) even spin from Bose symmetry. 4 If an operator is complex then its complex conjugate should also be included in the sum as an independent primary operator. Let O r I be an operator in representation r which appears in the φ × φ OPE, with I, J, . . . , labeling the elements of the representation. We denote the corresponding OPE coefficient (defined in (2.3)) as λ Or ab ,I and split it to a universal group factor times some coefficient, ab,I , (2.10) ab,I is the relevant Clebsch-Gordan coefficient, and is the same for any operator in the representation r, while the coefficient λ O is the same for each element of the representation. The sum-rule in (2.6) becomes Each term in the second sums in equation (2.12) has the same sign from the (−2) factor, as this only depends on whether r is in the symmetric or anti-symetric product of two adjoints 6 . We use this property to write the above sum-rule as where G r (u, v) is the sum over conformal blocks in a given representation, and (1 r ) ab,cd = ±δ IJ C (r) ab,IC (r) cd,J is just the identity matrix in the representation r projected to adjoint representation indices up to a sign, which can be determined by reflection positivity. Explicit expressions for these identity matrices are given in (A.5).
After plugging (A.5) into the sum-rule (2.13), it can be decomposed into several equations by equating the coefficients of the independent delta-functions in the identity matrices. We do this in the next subsections paying attention to the special cases SU (2) and SU (3).
The resulting sum-rules are conveniently expressed in terms of the functions For U (1) the sum-rule is the usual one for the 4-point function of a real scalar operator: where we separated the contribution of the identity operator for which p 1 = 1 and g 0,0 = F 0,0 = −1.

SU (2)
For SU (2) we have 3 × 3 = 5 s + 3 a + 1 s corresponding to the representations (S,S) s , (Adj) a and the trivial representation. Setting all the terms which correspond to the other representations in (2.13) to zero, plugging in the expressions for the identity matrices (A.5) and equating independent coefficients, we can express the result as three independent sumrules 7 , For SU (3) the (A,Ā) s representation does not exist so we set it to zero in (2.13). The resulting sum-rules are given by Equivalent sum-rules were also worked out in [6] for scalars in the fundamental of SO(3). Our result is consistent with [6], but we work in slightly different convention such that G here , which amounts to a rescaling of all the OPE coefficients in the trivial representation by a factor of 2.

SU (N ) for N > 3
For N > 3 all the 7 representations listed in appendix A can appear in the OPE, and we find

Conformal Bootstrap for Conserved Currents in N = 1 SCFTs
Consider an N = 1 superconformal field theory with global symmetry group G. In this section we will analyze the bootstrap constraints for the 4-point function of J a (x), which is the top of the current multiplet J a (z) defined in (1.1). In particular, we extend the results of [22] for U (1) to the non-Abelian case, and also find additional possible operators in the J a × J b OPE. We use the notations and conventions of [22].

Current-Current OPE in N = 1 SCFTs
The general form of the 3-point function of sprimary operators was found in [23]. For the 3point function of two conserved currents with some other sprimary O in some representation r the result is where the superspace coordinates are z j = (x j , θ j ,θ j ), and we define The quantities X, Θ andΘ are functions of the superspace coordinates given by and i labels the Lorentz representation of O.
The function t i (X, Θ,Θ) has to scale appropriately with respect to dilatations and U (1) R transformations 8 . Moreover, because of current conservation, D 2 J a =D 2 J a = 0, 8 The scaling is t(λλX, λΘ,λΘ) = λ 2aλ2ā t(X, Θ,Θ), where a − 2ā = 2 − q, andā − 2a = 2 −q. The R-charge and dimension of O are related to (q,q) by RO = 2 3 (q −q) and ∆O = q +q. the correlator (3.1) satisfies a differential equation. As shown in [23], this equation can be translated to the following differential equation for t: ab,I is either symmetric or anti-symmetric under a ↔ b, we need to find t i (X, Θ,Θ) which is either symmetric or anti-symmetric under which are manifestly odd and even under z 1 ↔ z 2 , respectively. The above constraints are sufficient to completely determine t(X, Θ,Θ) up to an overall numerical factor. In particular, [22] found 9 two structures corresponding to spin-sprimary operators with zero R-charge, which take the form 10 Under z 1 ↔ z 2 the structures (3.8) and (3.9) transform as ab,I is (anti-)symmetric in a and b, then in (3.1), the structure t µ 1 ···µ + appears for (odd) even and t µ 1 ···µ − for (even) odd . The = 0 case is special since there is no structure for ∆ = 2 which is odd under z 1 ↔ z 2 (see appendix B). Therefore in this case only scalar sprimaries in representations which arise from the symmetric product of two adjoints can contribute to (3.1) with the structure The ∆ = 2 scalar in the adjoint representation corresponds to 11 O I = J a . In that case the 3-point function is completely determined (for the canonically normalized current) by the Ward identities to be [23], where κ is the Tr G 3 't Hooft anomaly and τ is defined through the 2-point function of the canonically normalized current (1.2).
In addition, we find various contributions to (3.1) corresponding to operators which are not in spin-Lorentz representations. Those are collected in 1. Let us discuss some properties of the operators listed in table 1. The 1 2 , 0 structure in the second entry of table 1 (and its 0, 1 2 conjugate) actually arises from a larger family of structures t Θα , which satisfies all the constraints 12 . These structures correspond to operators with dimension ∆ = 5 2 − which violate the unitarity bound for = 1, ∆ ≥ | 3 2 R − j +j| + j +j + 2 = + 5 2 . The = 1 structure however, corresponds to a chiral operator (QαΨ α = 0), in which case the unitarity bound is modified to ∆ = 3 2 R ≥ j + 1. The corresponding operator saturates this bound, so it is in fact a free chiral fermion.
Short representations such as 2 , −1 2 short , can certainly appear, at least for free theories. They can be constructed in the following way, using the current J a as the basic building block (3.14) Symmetrization over Lorentz is to be understood. One can verify that these are superconformal primaries and satisfy the shortening condition Q α 1 O ab α 1 ···α ,α 1 ···α −1 = 0, by using the superconformal algebra and the fact that Q 2 J a (x) =Q 2 J a (x) = 0. When the indices a, b are not contracted to the adjoint these are short multiplets in interacting theories as well, since no short distance singularities can appear between J a and J b .
In the following section we will describe the application of these results to the conformal block decomposition of the 4-point function of J a (x).

Superconformal Blocks
The structures for the 3-point function J a (z 1 )J b (z 2 )O i I (z 3 ) , found in the previous section, relate the J a × J b OPE coefficients of primary super-descendants of O i I , to the coefficient of the superconformal primary. The sum over primary operators in the conformal block decomposition of the current 4-point function, can then be rearranged as a sum over superconformal primary operators, with "superconformal blocks" replacing the usual conformal blocks. The superconformal blocks are linear combinations of the usual conformal blocks, which take into account the relations between OPE coefficients of the primary operators in each super-multiplet.
For the purposes of this paper, we are interested in these relations for the J a × J b OPE. These can be obtained by setting θ 1,2 =θ 1,2 = 0 in the various expressions for (3.1), expanding in θ 3 andθ 3 and disentangling the various primary super-descendants in this expansion.
The superconformal blocks for spin-sprimaries, corresponding to the t + and t − structures in equations (3.8) and (3.9), were computed in [22]. The result is 13  table 1 there is only one primary super-descendant which can contribute to the J a ×J b OPE (i.e. which is in a spin-Lorentz representation). Therefore, there are no special relations between OPE coefficients inside each multiplet in those cases. In particular, the ±1 2 , ∓1 2 sprimaries contain a spinprimary super-descendant of dimension ∆ ≥ + 4, which is obtained by acting on the sprimary with Q α andQα appropriately 14 .
To summarize, we can write the s-channel decomposition as follows: where we separated the sum over representations to sums over symmetric and anti-symmetric representations. The sums in the square brackets are over sprimary operators in the J a ×J b OPE in the indicated Lorentz representation 15 . Note that for the operators in table 1 16 only (odd) even appears for (anti-)symmetric representations. A similar expression holds for the t-channel. The final result for the sum-rules in the N = 1 case is obtained using the appropriate conformal or superconformal blocks in the adjoint scalar sum-rules written in section 2.2. We wrote the coefficient p O in (3.19) with some abuse of notation to avoid clutter. It should be understood that it denotes the coefficient which was defined in (2.13) for the 14 When the unitarity bound is saturated the multiplet decomposes to two short multiplets as in (3.13).
The above spin-primary superdescendant sits in the second factor on the RHS of (3.13) as can be seen e.g., in [24]. 15 In the sum over operators in the Lorentz reprsentations ±1 2 , ∓1 2 , we implicitly include also the short 2 , −1 2 and −1 2 , 2 operators. 16 For spin-sprimaries the summation is over even and odd spins regardless of whether the representation of O is in the symmetric or anti-symmetric product of the current operators. This is because an even (odd) spin sprimary contains odd (even) super-descendant conformal primaries. appropriate operator. In particular, for a spin-sprimary it denotes the coefficient of the sprimary if G + ∆, is used, while if G − ∆, is used, it denotes the coefficient of the spin-+ 1 super-descendant. For the sprimaries in table 1 it denotes the coefficient of the spinsuper-descendant.
A check of the above results can be obtained by decomposing the N = 2 superconformal blocks found in [25]. This was described in some detail [22], though not carried out explicitly. We verified that this decomposition is consistent only if we include the operators in Table 1.

Bounds on Current Central Charges
Having written down the sum-rules, including the SUSY constraints, we are now ready to apply any of the methods developed in [1,4,5,8] to find bounds on OPE coefficients. The basic strategy for obtaining such bounds involves converting the problem into a system of constraints for every possible operator in the spectrum and is reviewed in Appendix C.1.
In SCFTs, the ∆ = 2 adjoint scalar sits at the top of the current supermultiplet. Thus, one can effectively use it to place a bound applicable for every N = 1 theory with SU (N ) global symmetry. Specifically, we have obtained a bound on τ , the coefficient of the current two-point function, which according to the AdS/CFT dictionary translates to a bound on g/R AdS , g being the coupling of the SU (N ) gauge theory in the bulk.

Bounds on OPE coefficients in U (1) SCFTs
The leading terms in the JJ OPE, when J is canonically normalized, take the form We first attempt to obtain a bound for the OPE coefficient λ J . In our normalization (2.1), (2.3), this is nothing but λ J = κ 4τ 3/2 . Performing the procedure described above using the same parameters as in [8], with k = 10, one obtains an upper bound for the OPE coefficient |λ J | < 1.546.

Bounds on OPE Coefficients and on τ in SU (N ) SCFTs
The major difference from the U (1) case arises, from the fact that there are now several different tensor structures appearing in the OPE, three in the case of SU (2), five in the case of SU (3) and seven in the generic case SU (N ) for N > 3. As shown, for example, in [5][6][7][8], one can use a vectorial linear functional in order to obtain a bound when several sum-rules are involved.
In the non-abelian case, the JJ OPE for the canonically normalized current takes the form We want to place a bound on τ . This can be done by isolating the contribution of j a µ in the sum-rules and placing a bound on its OPE coefficient. In our normalization (2.1), Therefore, |λ jµ | = 3 τ in (2.10). The OPE coefficient enters the sum-rule as the coefficient p jµ of the super-conformal block G − 2,0 (3.19), and due to our normalization of the conformal blocks chosen without the (2) − factor we have We can obtain an upper bound on p jµ , which translates into a lower bound on τ . Figure  1 shows the lower bounds on τ obtained for different values of the gauge group size N . Note that the bounds increase with N , as one would expect. E.g., we can think of SU (2) as a subgroup of SU (N ) with N > 2. In that case the generators of SU (N ), which are not part of the SU (2) subgroup, would appear in the singlet representation of the SU (2) current-current OPE. Thus, for consistency, the bound for SU (2) must be weaker than the bound for N > 2. This is indeed the case.
For comparison, the free theory of a single chiral multiplet in the fundamental of SU (N ) has τ chiral = 1 in our normalization [8]. For large values of N we in fact find a bound which excludes the free theory with one chiral multiplet. In Figure 2 we show the bounds for very large values of N as well. It has been more difficult to obtain them numerically. It is somewhat surprising that for large values of N we find f (N ) > 1, inconsistently with the existence of the free theory with one chiral multiplet. This may, of course, be the result of a mistake in our analysis, but here we offer an alternative explanation. Recently, [26] found analytically a bound on τ for N = 2 SCFT. Specifically, for SU (N ) flavor symmetry it was shown that τ ≥ N for N > 3. The analysis of [26] excludes free theories by removing conserved higher spin currents from the analysis 17 (whose existence implies that the theory is free [27]). In particular, this implies that free theories with τ of order N 0 are isolated in the space of N = 2 SCFTs.
It seems possible that a similar result applies also in the N = 1 case, and even though we allow conserved higher spin currents in the numerics, free theories are not found by the numerical constraint solvers as they are isolated points. Some support for this suggestion can be found in the lower bounds on τ in the presence of a chiral sprimary found in [8]. There, it is shown that there is a very sharp drop in the bound, toward τ min = 1 when approaching ∆ Φ = 1, the free theory. It seems likely that this becomes steeper upon increasing N . Furthermore, it has been shown in [29] that there is no exactly marginal deformation of the theory with a free chiral field in the fundamental of SU (N ). While this does not, of course, prove that there is no interacting theory with small τ − 1, there seems to be some evidence in favor of this claim. It would be interesting to investigate this point further.
Finally, if the theory has a gravity dual, then in our normalization we have [28] τ = 8π 2 R AdS g 2 , (4.6) where g is the coupling constant of the non-abelian gauge theory in the bulk, which matches the SU (N ) global symmetry. Thus, one can obtain an upper bound on g 2 /R AdS , meaning that the gauge coupling cannot become arbitrarily large in the bulk theory. This argument has been used in [5] to claim that such a bound exists in a bulk theory in the presence of a charged scalar. Here we see that it exists regardless of the type of excitation, and it is just a consequence of having a holographic dual. It would be very interesting to understand why such a bound exists from the bulk perspective. As an order of magnitude estimation, a lower bound of τ ≥ 1 at N → ∞ translates to g 2 /R AdS ≤ 8π 2 .

Acknowledgments
We are grateful to O. Aharony for many suggestions, enlightening conversations and collaboration at early stages of this project. We would also like to thank D. Simmons-Duffin and B.C. van Rees for discussions and helpful suggestions. We are especially grateful to J.F. Fortin, K. Intriligator and A. Stergiou for useful comments on an early version of this manuscript.

A Product of Two SU (N ) Adjoints
Let us decompose the tensor product of two SU (N ) adjoints into irreducible representations. Generally this decomposition contains 7 irreducible representations:  (2). In addition the adjoint representation in the product of two SU (2) adjoints comes only from the anti-symmetric combination.

A.1 Identity Matrices
Let us determine the identity matrices (1 r ) ab,cd defined around (2.13). In particular we will write those matrices in the fundamental representation basis, This is more convenient since the symmetry properties of the representations r in the tensor product are most easily expressed in the fundamental basis. The identity matrix can be constructed by symmetrizing and removing traces appropriately from the tensor δ p i δ q j δ m k δ n l . This determines the identity matrices up to an overall normalization. The overall sign of the identity matrices is determined by reflection-positivity as described in [6].
Up to an overall positive normalization (which can be absorbed in the OPE coefficients) we find 18 By expanding t in the grassmann variables and using the constraints it is easy to see that t(X, Θ,Θ) = t 0 (X) + t µ (X)Θσ µΘ .

(B.4)
Now, using the anti-symmetry in z 1 ↔ z 2 (under which X → −X) we obtain the equations In addition, we have the scaling constraints which follow from the ones for t(X) Solving the above scaling constraints in terms of polynomials in X µ and plugging in the differential equations (B.5),(B.6) we see that there is no solution unless ∆ = 2.

C.1 Obtaining Numerical Bounds on OPE coefficients
We now review, briefly, how one obtains upper bounds on OPE coefficients. To find a bound for the OPE coefficient of a superconformal primary O 0 , with conformal dimension ∆ 0 and spin 0 , we first isolate it from the sum in (2.16), moving all other operators to the RHS, obtaining Next, we apply a linear functional α : f (z,z) → R to the functions F ∆, (z,z), demanding that the following conditions hold: ≥ 0 for all other operators satisfying the unitarity bounds .
For all linear functionals satisfying these constraints we have For the last inequality we have used the fact that both the OPE coefficients squared p ∆, and the functionals applied to the functions F are positive. Then, minimizing α[1] over all functionals satisfying the constraints in (C.2) can yield an upper bound on |λ O | 2 , if any such functionals can be found. In four dimensions, lacking analytical tools to solve the infinite-dimensional minimization problem, one is forced to perform this minimization while limiting the search space to a finite dimensional subset of all possible functionals. This procedure yields a valid, though not necessarily tight, bound. Previous works have found it useful, due to the special properties of the functions F ∆, (z,z), to use the following test functionals Here, q(z,z) is some function which does not depend on ∆ or and k is a positive integer limiting the size of the search space. More details can be found in Appendix A of [8]. The functions F ∆, are symmetric with respect to z ↔z, and we must also have m + n even. The minimization is then over all possible values of a mn . When there is a global symmetry [7] there are several such sum-rules. The test functionals now take the form Instead of minimizing over a mn , with m and n labeling the number of z andz derivatives in the linear functional, respectively, we now minimize over a r mn . One now writes down a positivity constraint for each representation appearing with each spin. Note that the number of structures is equal to the number of sum-rules.
In principle, any integer spin and any conformal dimension ∆ satisfying the unitarity bound can appear in the spectrum. Recall also that real supermultiplets are limited to having ∆ ≥ 2, rather than the unitarity bound ∆ ≥ 1, due to the unitarity constraint on their current superdescendant (cf. 2.3 of [8]). Solving the problem numerically requires one to reduce the number of constraints to finite size. This has been achieved in [1,4,5] by discretizing the continuous parameter ∆ and setting an upper limit on the spins and scaling dimension for each spin. This essentially reduces the problem to a finite-dimensional linear programming problem, and the minimization can be solved by known algorithms. Notice that such a reduction necessarily omits the constraints for high spins and scaling dimensions, and the resulting bound may be invalid if it violates these constraints. In order to somewhat alleviate this concern, one can check that the constraints are not violated at high spins and conformal dimensions using the known asymptotics of the conformal blocks.
A more recent approach, taken in [8], avoids the need for discretization altogether, and instead makes clever use of Semi-Definite Programming (SDP) to handle all values of ∆ simultaneously. This is done by approximating the (super-)conformal blocks by polynomials of ∆ times a positive factor, than the constraints translate to demanding that these polynomials are positive for ∆ larger than the unitarity bound. In order to apply the method to the superconformal blocks for the conserved currents, we should find a good approximation for them. The details are similar to those found in Appendix A of [8]. Our suprconformal blocks are simpler than those obtained in the case of a chiral four-point function, as they contain at most two conformal blocks. Thus, one easily read off their polynomial approximations from A.18 in [8], albeit with different coefficients. More recent works can also be used to obtain polynomial approximations for (super)conformal blocks in any dimension, following [13].
In this formulation, the bound is valid for arbitrarily high scaling dimensions, but an upper limit must still be set on the spin to render the problem finite. We have chosen to use ≤ 20, as well as including = 100, 101, 1000, 1001, in order to capture high spin behavior. We solve the SDP numerically using SDPA-GMP 7.1.2 [31], using Mathematica 8.0 to set up the SDP input. The rest of the details, including the input parameters for SDPA-GMP, are exactly as described in Appendix B of [8].