Three-point functions of a superspin-2 current multiplet in 3D, N=1 superconformal theory

We consider $\mathcal{N=1}$ superconformal field theories in three-dimensions possessing a conserved current multiplet $\mathcal{F}_{ (\alpha_{1} \alpha_{2} \alpha_{3} \alpha_{4}) }$ which we refer to as the superspin-2 current multiplet. At the component level it contains a conserved spin-2 current different from the energy-momentum tensor and a conserved fermionic higher spin current of spin 5/2. Using a superspace formulation, we calculate correlation functions involving $\mathcal{F}$, focusing particularly on the three-point function $ \langle \mathcal{F} \mathcal{F} \mathcal{F} \rangle $. After imposing the constraints arising from conservation equations and invariance under permutation of superspace points, we find that the parity-even and parity-odd sectors of this three-point function are each fixed up to a single coefficient. The presence of the parity-odd contribution is rather non-trivial, as there is an apparent tension between supersymmetry and the existence of parity-odd structures.


Introduction
A peculiar feature of three-dimensional conformal field theories is the presence of parity violating, or parity-odd, structures in the three-point functions of conserved currents such as the energy-momentum tensor and vector currents [1]. These structures were not considered in the systematic studies of [2,3], which utilised a group-theoretic approach to solve for the correlation functions of conserved currents in a generic number of space-time dimensions. 1 Parity-odd structures are not present in free theories but have been shown to arise in Chern-Simons theories interacting with parity violating matter. In various approaches and contexts they were studied in [14][15][16][17][18][19][20][21][22][23][24][25][26].
In general, besides the energy-momentum tensor and vector currents, conformal field theories also possess currents of higher spin. In [17] Maldacena and Zhiboedov proved under certain assumptions (see below) that all correlation functions of higher-spin currents in three-dimensional conformal field theory are equal to that of a free theory. In particular, it implies that they do not have parity-odd contributions. This theorem was later generalised to higher-dimensional cases in [27][28][29]. These results can be viewed as the analogue of the Coleman-Mandula theorem [30] for conformal field theories.
In this paper we will be interested in N = 1 superconformal field theories in three dimensions. The general formalism to construct the two-and three-point functions of conserved currents in three-dimensional superconformal field theories was developed in [31][32][33][34] (a similar formalism in four dimensions was developed in [35][36][37] and in six dimensions in [38]). In supersymmeric theories, conserved currents are contained within supermultiplets. The energy-momentum tensor lies in the supercurrent multiplet [39], which in three dimensions also contains a fermionic supersymmetry current. On the other hand, a vector current becomes a component of the flavour current multiplet. As was pointed out in [32,40] there is an apparent tension between supersymmetry and the existence of parity violating structures in the three-point functions of conserved currents. In particular, three-point functions containing the supercurrent and flavour current multiplets admit only parity-even contributions. Combining this with the Maldacena-Zhiboedov theorem, it follows that supersymmetric conformal field theories do not admit parity-odd contributions to the three-point functions of conserved currents for any spin unless the assumptions of the theorem are violated.
The strongest assumption of the Maldacena-Zhiboedov theorem is that the conformal field theory under consideration possesses a unique conserved spin-2 current -the energymomentum tensor. However, in the same article [17] Maldacena and Zhiboedov showed that the existence of a conserved fermionic higher-spin current implies that there is more than one conserved current of spin 2. In supersymmetric theories conserved currents belong to supermultiplets which contain both bosonic and fermionic currents. This implies that a supersymmetric conformal field theory possessing a bosonic higher spin current also possesses a fermionic higher spin current (and vice-versa), thus it is conceivable that there exists another conserved current of spin 2. This, in turn, implies that in three-dimensional superconformal field theories the assumptions of [17] might be violated and the properties of correlation functions of higher spin currents might be more subtle.
In this paper, we will assume that the N = 1 superconformal field theory under consideration possesses a spin-2 conserved current different from the energy-momentum tensor. It naturally sits in the supermultiplet F α 1 α 2 α 3 α 4 = F (α 1 α 2 α 3 α 4 ) = F α(4) , (1.1) and satisfies the conservation equation here particular realisations of superconformal theories possessing a conserved superspin-2 multiplet, our interest here is to explore how the N = 1 superconformal symmetry constraints the three-point correlation functions involving F α (4) .
Our main result is that the three point function, is fixed by the N = 1 superconformal symmetry up to one parity-even and one parity-odd structure. Our analysis is technically quite involved; the analytic superfield consideration turns out to be quite intractable and we were required to complete both superfield and component analysis with the aid of the xAct package [41] for Mathematica, which contains an advanced suite of tools designed for tensor analysis. The three-point function (1.4) contains two independent component correlators (all others can found in terms of these two by virtue of the conservation law (1.2)) These two correlators were analysed analytically, however, to provide a complete check that all the necessary conditions are satisfied we had to also perform some numerical analysis. We also discuss some basic mixed three-point functions involving F α (4) . 2 In particular we analyse the three-point function O(z 1 ) F α(4) (z 2 ) O(z 3 ) , where O(z) is a scalar superfield of dimension ∆. We found that it is fixed up to a single parity-even tensor structure. We also compute the three-point function where Lā α (z) is the non-abelian flavour current multiplet. We found that this three-point function is also fixed up to a single parity-even tensor structure, which is in disagreement with the result previously reported in [21], which used a different approach (see Subsection 5.2 for details). In our approach the analysis of this correlation function is relatively straightforward as it can be studied analytically, so we are confident in our result.
The paper is organised as follows. In Section 2 we introduce the superconformal building blocks that are essential to the construction of two-and three-point correlation functions of primary operators. In Section 3 we analyse the structure of the supermultiplet F; in particular we define the component fields in the multiplet and determine the constraints on them resulting from the superfield conservation equations. Section 4 is devoted to studying the three-point function FFF . First we impose the constraints resulting from the conservation of F and invariance under permutation of superspace points z 1 and z 2 ; we show that these constraints are sufficient to fix the parity-even and parity-odd sectors each up to a single coefficient. Next we check invariance under permutation of superspace points z 1 and z 3 which requires a combination of both analytic and numerical methods. As a result, we show that the three-point function is fixed by the N = 1 superconformal symmetry up to two independent tensor structures, one is parityeven while the other is parity-odd. Section 5 is devoted to the study of mixed correlation functions involving the superfield F. We compute the three-point function of F with two scalar superfield insertions, and the three-point function of F with two non-abelian flavour current multiplets. In Section 6 we provide a brief summary of the work and some future directions. The appendices A, B and C are devoted to our conventions, technical details and some consistency checks.

Superconformal building blocks
In this section we will review the pertinent details of the group theoretic formalism used to compute correlation functions of primary superfields. For a more detailed review of our conventions the reader may consult [32,40].
Now consider a generic tensor superfield Φ A (z) transforming in a representation T of the Lorentz group with respect to the index A. 3 Such a superfield is called primary with dimension q if its superconformal transformation law is where ξ is the superconformal Killing vector, and the matrix M αβ is the Lorentz generator. The z-dependent parameters σ(z), λ αβ (z) associated with ξ are defined as follows

Two-point and three-point building blocks
Two-point building blocks: Given two superspace points z 1 and z 2 , we can define the two-point functions Note that x αβ 21 = −x βα 12 . It is convenient to split the two-point function (2.7) into symmetric and antisymmetric parts as follows where y αβ 12 is the symmetric part of x αβ 12 , It can also be represented by the three-vector y m 12 = − 1 2 (γ m ) αβ y αβ 12 . Next we introduce the two-point objects Hence, we find (2.11) These objects are essential in the construction of correlation functions of primary superfields. We also have the useful differential identities where D (i)α is the standard covariant spinor derivative (A.16) acting on the superspace point z i .

Three-point building blocks:
Given three superspace points z i , i = 1, 2, 3, one can define the following three-point building blocks and, similarly, (X 2 , Θ 2 ), (X 3 , Θ 3 ) which can be found from (2.13) by cyclic permutation. Next we define We also define the normalised building block,X 1 , and the inverse of X 1 , There are also useful identities involving X i and Θ i at different superspace points, e.g., The three-point objects (2.13a) and (2.13b) have many properties similar to those of the two-point building blocks. Now if we decompose X 1 into symmetric and antisymmetric parts similar to (2.8) we have where the symmetric spinor X 1 αβ can be equivalently represented by the three-vector X 1 m = − 1 2 (γ m ) αβ X 1 αβ . Now let us introduce analogues of the covariant spinor derivative and supercharge operators involving the three-point objects, which obey the standard anti-commutation relations Some useful identities involving (2.18) are We must also account for the fact that various primary superfields obey certain differential equations. Using (2.12) we arrive at the following Now given a function f (X 3 , Θ 3 ), there are the following differential identities which arise as a consequence of (2.20), (2.21a) and (2.21b): These identities are essential for imposing differential constraints on correlation functions.

Building blocks in components:
For future reference we will also review the non-supersymmetric conformal blocks detailed in [2]. These objects will appear in component reduction of superspace correlation functions. The two-point and three-point structures are defined as follows These objects may be obtained by bar-projection of the superspace variables defined in Section 2 as follows Here (i, j, k) is a cyclic permutation of (1, 2, 3). That is, In addition, we introduce the inversion tensor, I a 1 a 2 , and its representation acting on rank-2 symmetric traceless tensors, I a 1 a 2 ,m 1 m 2 where we have introduced the projection operator

Correlation functions of primary superfields
The two-point correlation function of a primary superfield Φ A and its conjugateΦ B is fixed by the superconformal symmetry as follows where c is a constant coefficient. The denominator of the two-point function is determined by the conformal dimension q of Φ A , which guarantees that the correlation function transforms with the appropriate weight under scale transformations.
Concerning the three-point functions; let Φ, Ψ, Π be primary superfields with conformal dimensions q 1 , q 2 and q 3 respectively. The three-point function may be constructed using the general expression where the tensor H A 1 A 2 A 3 is highly constrained by the superconformal symmetry as follows: the three-point building blocks transform as Z = (X, Θ) → Z = (λ 2 X, λΘ). As a consequence, the correlation function transforms as which implies that H obeys the scaling property This guarantees that the correlation function transforms correctly under conformal transformations.
(ii) If any of the fields Φ, Ψ, Π obey differential equations, such as conservation laws in the case of conserved current multiplets, then the tensor H is also constrained by differential equations. Such constraints may be derived with the aid of identities (2.22a), (2.22b).
(iii) If any (or all) of the superfields Φ, Ψ, Π coincide, the correlation function possesses symmetries under permutations of superspace points, e.g.
where (Φ) is the Grassmann parity of Φ. As a consequence, the tensor H obeys constraints which will be referred to as "point-switch identities".
The constraints above fix the functional form of H (and therefore the correlation function) up to finitely many parameters. Hence the procedure described above reduces the problem of computing three-point correlation functions to deriving the tensor H subject to the above constraints.
3 Component structure of a superspin-2 current multiplet In this paper we will be interested in three-point functions of a superspin-2 current multiplet described by the totally symmetric superfield F α(4) := F α 1 α 2 α 3 α 4 (z), satisfying the conservation equation In three dimensions this superfield admits the follwing Taylor expansion It can be convenient to express some of these fields in vector notation as follows where J a 1 a 2 is symmetric and traceless; a similar treatment follows for the other fields in the multiplet. Imposing the conservation equation is then tantamount to the following constraints on the component fields Hence we see this multiplet contains only two independent component currents: a conserved spin-2 field J a 1 a 2 satisfying the same properties as the energy momentum tensor, and a conserved spin-5/2 field Q a 1 a 2 ,α which is conserved and gamma-traceless (the latter guarantees that Q is totally symmetric in spinor notation). Let us stress that J a 1 a 2 is different from the energy-momentum tensor T a 1 a 2 , the latter is a component of the supercurrent multiplet J α (3) . The independent components of F α(4) may be extracted by bar-projection: In addition, under infinitesimal superconformal transformations, the superfield F transforms as where q is the scaling dimension of F. The conservation equation (3.1) then uniquely fixes the dimension of the field as follows: if we compute δD α 1 F α 1 α 2 α 3 α 4 (z), and use the definitions (2.5), (2.6) we obtain Hence we see that we require q = 3 for covariant conservation of F.

Correlation function FF F
In this section we will derive an explicit solution for the three-point function FFF . In Subsection 4.1 we impose the constraints which arise due to the superfield conservation equations and invariance under permutation of superspace points z 1 and z 2 . This is already to sufficient to fix the three-point function up to one parity-even and one parityodd structure. In Subsection 4.2 we computationally analyse the constraints arising from invariance of the three-point function under permutation of superspace points z 1 and z 3 ; this is done by considering the independent component correlators contained within FF F : JJJ and QJQ . This is followed by a numerical analysis of the point-switch identity for consistency. Most of the tensor expressions are too large to manipulated efficiently by hand, so we make use of Mathematica to do most of the lengthy calculations.

Superfield analysis
The ansatz for the correlation function FFF is where the tensor H is independently totally symmetric in the α i , β i and γ i , and is required to satisfy covariant constraints which arise due to conservation equations and invariance under permutations of superspace points. The constraints are summarised below:

(i) Homogeneity constraint
Covariance of the correlation function under scale transformations of superspace results in the following constraint on H which implies that H is a homogeneous tensor field of degree −3. This constraint ensures conformal covariance of the three-point function.
(ii) Differential constraints The conservation equation (3.1) implies that the correlation function must satisfy the following constraint Application of the identities (2.22a) results in the following differential constraint on H Invariance under permutation of the superspace points z 1 and z 2 results in the following constraint on the correlation function which results in the condition There is an additional point-switch identity obtained from imposing invariance under permutation of the points z 1 and z 3 , however it is considerably more complicated so we will discuss it in detail later.
To make subsequent calculations more tractable, it is often convenient to express H in terms of its vector equivalent by factoring out gamma matrices as follows This equality holds provided that H (in vector notation) is symmetric and traceless in the pairs a i , b i and c i respectively. This is seen by requiring that the components antisymmetric in α 2 , α 3 (and other combinations involving β i and γ i ) vanish. Further, since H is Grassmann even it admits the Taylor expansion At this step it is more convenient to view F and G as functions of the three-vector X m rather than of X αβ . The point-switch identity (4.6) then implies the following constraints on F and G On the other hand, the differential constraint (4.4) results in 4 In the next subsections we will computationally solve for the tensor H subject to the constraints listed above using the xAct package.

Parity-even sector
In the parity-even sector, we begin by constructing a solution for F that is an even function of X, hence (4.9a) implies (4.11) A general expansion for F consistent with the symmetry property (4.11) may be obtained by introducing the symmetric and traceless basis tensors found in [2]. Explicit expressions for the elements of the tensor basis {Υ i }, i = 1, ..., 8, are as follows (4.12e) These tensors each possess a variety of symmetry properties, in particular they are symmetric and traceless in pairs of indices. Using this basis we can construct the following set of rank 6 tensors The t i a 1 a 2 b 1 b 2 c 1 c 2 each possess the symmetry property (4.11), hence, the ansatz for the tensor F is a linear combination of these tensor structures: where we have used the homogeneity constraint (4.2). It now remains to impose the differential constraint (4.4), which results in the following relations Hence, we see that the differential constraint immediately fixes the parity-even sector down to three independent coefficients. It is at this step where the linear dependence of the first five tensor structures can be noticed, as the k 1 dependence can be removed by shifting the variables as follows: Alternatively it may be shown that the following linear dependence relation holds It is now clear that the k 1 term is redundant, hence, it can be completely removed from our analysis. This reduces our system of equations to Therefore, the parity-even sector of the three-point function is fixed at this stage up to two independent coefficients, k 2 and k 4 , and the explicit solution for F is The tensor G is then determined in terms of F using (4.10). However, we have not yet imposed the condition (4.9b). Since G is an odd function of X by virtue of (4.10), the constraint (4.9b) implies After some calculations one can show that this results in an additional relation between the coefficients k 2 and k 4 : Thus, the conservation equations and the proper transformation under the z 1 ↔ z 2 exchange fix the parity-even sector up to a single overall coefficient. Note that so far we have not imposed the z 1 ↔ z 3 point-switch identity. It will be imposed later.

Parity-odd sector
Let us now construct the parity-odd sector of the correlation function, where we begin by assuming that the tensorF is an odd function of X. Due to (4.9a), this implies that F must satisfyF (4.21) Now let us construct an explicit solution for the tensor F ; it must be an odd function of X, and each term must contain at most one instance of the Levi-Civita tensor (as products of the latter may be expressed in terms of the metric). We may decomposeF as follows: where each P i must have the symmetry property P i m,a 1 a 2 b 1 b 2 (X) = P i m,(a 1 a 2 )(b 1 b 2 ) (X). Requiring that the expansion (4.22) is consistent with the properties of pairwise index symmetry and (4.21) implies that the P i must be identical. Hence we need to find a general expansion for a tensor P m,a 1 a 2 b 1 b 2 which is homogeneous degree 0 and is composed solely ofX and the metric tensor. Using Mathematica we can generate an ansatz consistent with the symmetry properties: Only 9 of these structures contribute when substituted into (4.22), in particular the terms with c 9 , c 4 and c 10 may be neglected. Imposing tracelessness on each pair of indices is tantamount to the following constraints on the coefficients It remains to impose the differential constraint forF in (4.10), from which we find the additional relations Hence, the solution forF is fixed up to a single coefficient, b = c 6 . 5 The solution forF becomesF where the explicit solution for P is The tensorG is found using eq. (4.10). However, we still need to impose the symmetry property (4.9b). SinceG is an even function of X, (4.9b) implies After some calculations we find that eq. (4.28) is satisfied automatically and does not result in any restrictions on b. Thus, the conservation equations and the proper transformation under the z 1 ↔ z 2 exchange fix the parity-odd sector up to a single overall coefficient.

Point-switch identity
The last constraint to be imposed on the correlation function FFF is invariance under the permutation of points z 1 and z 3 , i.e we must have This results in the following constraint on H H α(4)β(4)γ(4) (X 3 , It is clear that direct calculation of (4.30) is inefficient due to i) the large number of tensor structures in the solution for H, and ii) the linear dependence between the structures. Therefore, we will need to consider some alternative approaches, which will be explored in the next subsections.
The superfield correlator FFF contains only two independent component correlators These may be obtained by bar-projection of the three-point function FFF as follows 6 All correlators involving the components S α(3) and B α(4) in eq. (3.2) either vanish or are expressed in terms of (4.31) by virtue of (3.4b). From eq. (4.29) it follows that the component correlators (4.31) satisfy the following point-switch identities These relations will be studied analytically (though with extensive use of Mathematica) in subsections 4.2.1 and 4.2.2. However, proving eqs. (4.33) is not sufficient to prove eq. (4.29). The reason is that we cannot use eqs. (3.4b) because we have not yet proven that the conservation law on the third point is satisfied. In fact, it will follow once we prove eq. (4.29). Hence, to prove eq. (4.29) at the component level we must consider all component correlators obtained from (4.29) by the action of the superspace covariant derivatives followed by bar-projection. This is, clearly, impractical. Therefore, our approach will be to study eq. (4.29) at higher orders in θ i numerically, which we do in 4.2.3. For this we will keep θ i arbitrary but use various numeric values for the space-time points x 1 , x 2 , x 3 . Then the components of F α(4) (z 1 ) F β(4) (z 2 ) F γ(4) (z 3 ) will be polynomials in θ i with numeric coefficients. Since these polynomials are quite complicated we are confident in our results despite the proof not being fully analytic.

Component correlator JJJ
The computation of the component correlator JJJ is relatively straightforward, explicitly we have Since bar-projections of any objects involving Θ vanish, combined with the result we obtain If we convert this result into vector notation by combining pairs of spinor indices, and apply the identity we obtain the result If we now use (4.14), then this component correlator can be put in the covariant canonical form It then follows that the constrained tensors t a 1 a 2 b 1 b 2 c 1 c 2 (X 12 ) appearing in JJJ satisfy all the same properties as those present in the energy-momentum tensor three-point function T T T , so we can simply use the known results.
It was shown in [2] that the parity-even contribution to the three-point function T T T in general dimensions is fixed up to three independent coefficients. However, in threedimensional theories there is linear dependence between the tensor structures due to the identity (4.16). This reduces the number of independent structures down to two. The solution for t a 1 a 2 b 1 b 2 c 1 c 2 (X) found in [2] is the same as given in our eqs. (4.14), (4.18). It was also shown in [2] that this solution satisfies the z 1 ↔ z 3 point-switch identity. Since the solution for the correlator JJJ is identical to that of T T T it follows that the three-point function JJJ in (4.35) with F α(4)β(4)γ(4) defined in (4.18) is compatible with the point-switch identity (4.33a) for arbitrary k 2 and k 4 . In this case there is a further relation between k 2 and k 4 in eq. (4.20) however it does not affect the z 1 ↔ z 3 point-switch identity.
The parity odd sector of the energy-momentum tensor three-point function was obtained in [1]. It was shown that it is fixed up to one independent structure given in eqs. (4.26), (4.27). 7 Hence, eqs. (4.26), (4.27) are also compatible with the point-switch identity (4.33a). In the remaining subsections we will consider the relation (4.29) at higher orders in θ i .

Component correlator QJQ
The correlator QJQ can be computed as follows After evaluating the derivatives, one finds that the calculation is broken up into two relevant parts: the A contribution is due to the derivatives hitting the prefactor, while the B contribution arises due to the derivatives hitting H, Other combinations in which either derivative acts on the prefactor and H result in terms that are at least linear in θ 1 , θ 3 or Θ 3 , which vanish upon bar projection, so they may be neglected. The A contribution can be written in the form with T A defined as T A α,α(4) β(4) γ,γ(4) (X) = −10i ε γ(α F α 1 α 2 α 3 α 4 )β(4)γ(4) (X) . Similarly if we evaluate the B contribution, we find it can be written in the form with T B given by the expression Hence, we see that the correlation function QJQ may be written in the following covariant canonical form with T α,α(4)β(4)γ,γ(4) (X) = T A α,α(4)β(4)γ,γ(4) (X) + T B α,α(4)β(4)γ,γ(4) (X) . Additional details regarding this calculation are contained in appendix B. It is worth commenting that it is not immediately obvious that T is totally symmetric in the α indices; indeed it may shown by direct calculations that this symmetry is manifest by virtue of (4.3) (and by extension (4.10)). 8 To make subsequent calculations more tractable, we convert this entire expression into vector notation. The component three-point function may then be written in the following form It is convenient to decompose the tensor T into the symmetric and antisymmetric parts T m,a 1 a 2 b 1 b 2 c 1 c 2 ,αγ (X) = ε αγ A m,a 1 a 2 b 1 b 2 c 1 c 2 (X) + (γ n ) αγ S n m,a 1 a 2 b 1 b 2 c 1 c 2 (X) . We find the following expressions for the tensors A and S: The differential operator D and the constant "projection" tensors Π, Ξ naturally arise when expressing QJQ in the covariant form (4.48). They have the following definitions The point switch identity on QJQ , eq. (4.33b) can be written in terms of the following two equations involving only vector indices: To recall, here the tensors A and S are given by eqs. (4.50), (4.51), where the tensor F is given by eq. (4.18) in the parity-even case and in eqs. (4.26), (4.27) in the parity-odd case, and the tensor G in both cases is obtained from F using eq. (4.10).
Now the task is to substitute F and G into eqs. (4.50), (4.51) in the parity-even and in the parity-odd cases separately to determine if there are additional, different from eq. (4.20) constrains on the coefficients k 2 , k 4 and b in eqs. (4.18) and (4.26). Since A and S have rather complicated definitions, it is futile to attempt to impose them by hand, however computation of these identities is possible in Mathematica using the xAct package [41]. The package allows for symbolic manipulation of tensors using index notation, and contains a suite of "canonicalisation" functions which can essentially manipulate tensor structures down to their simplest form. In this way the computations are completely symbolic and are exactly the same as if they were done "by hand". Once a given tensor is canonicalised, we can then convert the expression into an array using in-built functions.
Parity-even sector: Evaluating (4.55a) using definitions (4.50) and the solution (4.18) results in ≈ 400 terms after canonicalisation. On the other hand, (4.55b) results in ≈ 800. The tensor structures in each identity should cancel amongst each other for some relation between the coefficients k 2 and k 4 . However if we naively just collect all of the tensor structures one would find that k 2 = k 4 = 0, as there is a hidden linear dependence between the terms. A way around this is to go into a coordinate basis and check every component of the LHS of (4.55a) and (4.55b). If we carry out this computation, the identities are satisfied for the choice k 2 = −2k 4 . Hence, we do not get any new relations in the parity-even sector and it is still fixed up to an overall coefficient.
Parity-odd sector: We now carry out an identical analysis for the parity-odd solution (4.26), which turns out to be more computationally intensive. In this case there are ≈ 800 tensor structures after canonicalisation of the LHS of (4.55a), while there are ≈ 1600 for (4.55b). If one goes into a coordinate basis the identities are satisfied for an arbitrary choice of the coefficient b. Hence, the parity-odd sector is also fixed up to a single tensor structure.

Numerical analysis
To supplement the results above, we will carry out a numerical analysis of the point switch identity by substituting in various configurations of points. To do this, first we convert the ansatz (4.1) into vector notation. This can be done by introducing the following N = 1 object To recall, x 12 is given in eq. (2.8), the vector y 12 is given in (2.9) andŷ m 12 = y m 12 /y 12 . I ab (x 12 ) may be thought of as the supersymmetric generalisation of (2.26b). It obeys some useful properties such as (4.57) Using this new object, the ansatz (4.1) can be written in the form (4.58) Now to check the point switch identity, we will introduce null vectors λ 1 , λ 2 , λ 3 , and contract them with the ansatz to obtain: Essentially our approach is to pick a configuration of points x 1 , x 2 , x 3 , and null vectors λ 1 , λ 2 , λ 3 , then expand out (4.59) to all orders and combinations of the fermionic superspace coordinates θ 1 , θ 2 , θ 3 . This simplifies the point switch identity to a polynomial expression in the fermionic coordinates. We then check whether the point switch identity is satisfied for both the parity-even and parity-odd solutions that we found in Subsection 4.1. To carry out these computations we must make use of the following expansions for the fermionic two-point (three-point) functions, which follow from the definitions (2.8), (2.13b): where we have used the notation Expansions for the other building blocks are obtained by cyclic permutations of superspace points. Hence we see that the resulting polynomial from (4.59) will be a function of θ 2 i , θ i · θ j , (θ ij · γ · θ jk ) a , and combinations/products of these objects. 9 All the θ expansions and numerical calculations are done computationally. We performed a numeric analysis for various configurations of points and null vectors and always found the same result. Below we present one example when the polynomials are relatively simple.
Let us pick the following points and null vectors We now substitute the above values into (4.60). For the parity-even solution (denoted by 9 Not all these objects are linearly independent since θ i are Grassmann odd but one can choose a convenient basis. subscript E) we obtain Clearly, this expression vanishes at each order of θ for the choice k 2 = −2k 4 which is the same condition as found previously in eq. (4.20). Hence, the numerical evaluations agree with our previous calculations and does not give any new relations.
Next we perform the same calculation for the parity odd solution (denoted by O). Explicit evaluation of F(z 1 ) F(z 2 ) F(z 3 ) O yields the following polynomial When we similarly compute F(z 3 ) F(z 2 ) F(z 1 ) O we find the same result. Hence, we observe cancellation at every order and therefore the odd solution also satisfies the point switch identity for an arbitrary coefficient b.
Note that the polynomials in both parity-even and parity-odd cases are quite nontrivial even for a simple choice of the points and null vectors. We performed a similar numeric analysis for various other choices and obtained the same result as above. However, in all other cases the polynomials are quite large so we will not present them here. The complexity of the polynomials makes any accidental cancellations highly unlikely. Hence, we are confident that the point-switch identity is satisfied for k 2 = −2k 4 and arbitrary b.

Summary of results
Since our analysis is rather technical and involves analytic and numeric computations of the superfield and component expressions, we will collect all the pieces together and summarise our results. We found that the correlation function FFF contains two independent tensor structures after imposing all of the constraints; one of them is parity-even, while the other is parity-odd. In particular, we found that the parity-odd contribution is present unlike in all cases of three-point functions involving the supercurrent and flavour current multiplets [32,40].
The correlation function found above has the following structure where H can also be written as follows The tensor H in vector notation then may be split into parity-even and parity-odd sectors where each solution admits the following expansion with G,G determined in terms of F ,F by the equations (4.10), After imposing all the constraints, we find that the solution for the tensor F in the even sector is where we have relabeled k 4 → a and t i (X) are given by eqs. (4.12), (4.13).
On the other hand, we find the solution in the odd sector to bẽ with P defined as in (4.27).

Mixed correlators
In this section we will discuss some basic examples of three-point functions of F with other fields such as a scalar superfield O of dimension ∆, and the non-abelian flavour current superfield Lā α of dimension 3/2. The calculations are straightforward compared to the FF F case, so we will not require computational methods here. These three-point functions were also previously studied in [21], however, our method is different and more explicit.

Correlation function OFO
Let us now compute the correlation function OFO , which admits the general ansatz which implies that H is a homogeneous tensor field of degree −3.
(ii) Differential constraints The conservation equation (3.1) implies that the correlation function must satisfy the following constraint Application of the identities (2.22a) results in the following differential constraint on H Q σ H σα(3) (X, Θ) = 0 . Invariance under permutation of the superspace points z 1 and z 3 results in the following constraint on the correlation function which results in the following constraint on H Now we must construct an explicit solution; analogous to the FFF case, we combine symmetric pairs of spinor indices into vector ones as follows: where it is required that H in vector notation is both symmetric and traceless. It has the expansion H a 1 a 2 (X, Θ) = F a 1 a 2 (X) + Θ 2 G a 1 a 2 (X) .
The component fields F and G are both required to be symmetric and traceless. If we now impose (5.3), we obtain the constraints Therefore we need only solve for the field F . A general expansion consistent with the tensor symmetries and homogeneity is Note that no parity violating structures are permitted as there is simply not enough indices on the tensor F to allow for such contributions. Substituting this solution into (5.9) shows that it is satisfied for any value of c, while G = 0. The final solution for the tensor H in spinor notation is Indeed, substitution of this solution into (5.6) demonstrates that it is compatible with the point-switch identity. Hence this correlation function is determined up to a single parity-even tensor structure. A similar result was obtained in [21].

Correlation function FLL
In this subsection we will compute the correlation function FLL , where L is the non-abelian flavour current superfield of dimension 3/2 which obeys the conservation equation The correlation function admits the general ansatz The constraints on this three-point function are summarised below:

(ii) Differential constraints
The conservation equations (3.1), (5.12) imply the following constraints Application of the identities (2.22a) then gives Invariance under permutation of the superspace points z 2 and z 3 is equivalent to the condition which results in the following constraint on H As before we combine symmetric pairs of spinor indices into vector ones as follows: The above decomposition holds provided that H a 1 a 2 , βγ is symmetric and traceless in a 1 , a 2 . 10 We then expand this in irreducible components as follows: Here the A i and S i are both symmetric and traceless in a 1 , a 2 . Imposing the differential relation (5.16a) results in the following constraints on the tensors A i and S i while (5.16b) gives the additional relations Hence, we may treat A 1 and S 1 as independent. The only solution for the tensor A 1 compatible with the symmetries is while for S we have the general ansatz Imposing tracelessness in a 1 , a 2 on this expansion results in the constraint The solution then becomes It remains to impose the differential constraints. In particular, the equations (5.22a), while A 2 vanishes. After making the replacement k 2 →c, the solutions for the tensors These solutions are consistent with the remaining constraints (5.23a), (5.23b) for the choicec = −3c. It can also be shown by direct substitution that this solution is consistent with the point-switch identity (5.18). Hence this correlator is determined up to a single tensor structure.
Let us comment on the absence of parity-odd contributions. 11 They could only potentially come from the following terms contained in S 1 which are odd under X m → −X m . However, this expression cannot be at the same time traceless and transverse for any choice of the coefficient k 1 , k 3 , k 4 , which can be easily checked.
This result is contrary to the computation carried out using the polarisation spinor formalism in [21], where it was shown that a parity violating contribution can exist. A direct comparison with the results obtained in [21] is difficult as our approach and notation are quite different. Our formalism, however, has the benefit that it is analytic and rather explicit. 12 As a consistency check, in Appendix C we reformulate this problem and use the LLF ansatz. The evaluation procedure is slightly different but the same conclusion is obtained.

Conclusion
In this paper we analysed various correlation functions involving a conserved superspin-2 current multiplet F α (4) . The case of FFF is particularly challenging due to the proliferation of tensor structures in the solution; indeed we found that it could only be studied efficiently using computational methods. We obtained that the three-point function FFF contains one parity-even and one parity-odd structure.
The appearance of a single parity-even structure can be understood intuitively and is somewhat expected. Indeed, the superfield F α(4) contains a conserved spin-2 current J α(4) as the lowest component which, though being different from the energy-momentum tensor, satisfies the same conservation equation. Its three-point function has two parity-even structures which can be attributed to contributions from a free boson and a free fermion. Since supersymmetry relates bosons and fermions, it is reasonable to expect that these structures become related, giving rise to a single independent contribution. On the other hand, the existence of the parity-odd structure in FFF is rather non-trivial because, as was pointed out in the introduction, there is an apparent tension between parity-odd structures and supersymmetry: all three-point functions involving the energy-momentum tensor and vector currents admit parity-odd structures in the non-supersymmetric case [1] but not in the supersymmetric one [32,40].
Let us now clarify a possibly confusing point. The three-point function of the energymomentum tensor T does not allow parity-odd structures in the supersymmetric case, whereas the three-point function of the similar spin-2 current J does. This might look paradoxical because T and J have the same symmetry properties and satisfy the same conservation equation. However, it is important to remember that T and J belong to different supermultiplets and, hence, transform differently under supersymmetry. Therefore, restrictions on their correlation functions due to supersymmetry are different.
A natural extension of our results is to study the three-point functions of higher-spin current multiplets of (arbitrary) higher (super)spin. For non-supersymmetric conformal field theories, the three-point functions of bosonic higher spin currents were found in [44][45][46]. In four-dimensional supersymmetric conformal field theories correlation functions of higher-spin spinor currents were recently studied in [47] (see also [48]). Deriving explicit solutions becomes increasingly difficult for fields with higher-spins. It is possible that other approaches, for example, based on supersymmetric generalisations of the embedding formalism [49][50][51][52] or of the spinor-helicity formalism [25,53,54], can be more efficient. It would be interesting to explore them as well.
Another natural question is to find explicit realisations of superconformal field theories possessing a conserved superspin-2 current multiplet. Since this multiplet also contains a higher-spin current one should expect that these theories possess infinitely many conserved higher-(super)spin currents [55]. ε-tensor The γ-matrices are chosen to be real, and are expressed in terms of the Pauli matrices σ as follows: The γ-matrices are traceless and symmetric and also satisfy the Clifford algebra γ m γ n + γ n γ m = 2η mn . (A.6) Products of γ-matrices are then where we have introduced the 3D Levi-Civita tensor , with 012 = − 012 = 1. It satisfies the following identities: We also have the orthogonality and completeness relations for the γ-matrices Finally, the γ-matrices are used to swap from vector to spinor indices. For example, given some three-vector x m , it can be expressed equivalently in terms of a symmetric second-rank spinor x αβ as follows: The same conventions are also adopted for the spacetime partial derivatives ∂ m (A.14) We also define the supersymmetry generators Q I 15) and the covariant spinor derivatives which anti-commute with the supersymmetry generators, {Q α , D β } = 0, and obey the standard anti-commutation relations

C.1 Correlator LLO
In this sub-appendix we derive the general form of the correlation function LLO . We also demonstrate that our solution is consistent with the results of [21] in terms of the number of independent tensor structures. The ansatz for LLO is (C.1) The constraints on this three-point function are summarised below:
(ii) Differential constraints The conservation equations (3.1) imply the following constraints Hence we may treat A 1 and S 1 as independent. Explicit solutions for the tensors A 1 and S 1 are These solutions are trivially compatible with (C.9a), (C.9b). Using (C.10a), (C.10b) we obtain expressions for A 2 and S 2 A 2 (X) = ib 2 (2 − τ ) 1 X τ +1 , S 2 a (X) = ia 2 τ X a X τ +2 . (C.12) Following [21] we set ∆ = 1 (τ = 2) and obtain the set of solutions A 1 (X) = a X 2 , A 2 (X) = 0 , (C.13a) The solution for H in spinor notation is then which clearly contains both parity even and parity odd contributions. Our notation is quite different so it is difficult to make a direct comparison, however we agree on the number of independent tensor structures.

C.2 Correlator FLL -alternative ansatz
In this subsection we investigate an alternative formulation of the correlation function FLL , it serves as a consistency check of our result in Subsection 5.2. The starting point is the alternative ansatz LLO : Lā β (z 1 )Lb γ (z 2 )F α(4) (z 3 ) = δābx The constraints on this three-point function are summarised below:
(ii) Differential constraints The conservation equations (3.1) imply the following constraints Imposing (5.15b) is rather non-trivial, it will be handled later in this section.