Correlation functions of spinor current multiplets in N = 1 superconformal theory

We consider N = 1 superconformal ﬁeld theories in four dimensions possessing an additional conserved spinor current multiplet S α and study three-point functions involving such an operator. A conserved spinor current multiplet naturally exists in superconformal theories with N = 2 supersymmetry and contains the current of the second supersymmetry. However, we do not assume N = 2 supersymmetry. We show that the three-point function of two spinor current multiplets and the N = 1 supercurrent depends on three independent tensor structures and, in general, is not contained in the three-point function of the N = 2 supercurrent. It then follows, based on symmetry considerations only, that the existence of one more Grassmann odd current multiplet in N = 1 superconformal ﬁeld theory does not necessarily imply N = 2 superconformal symmetry.


Introduction
The study of the general structure of correlation functions of primary fields in conformal field theory has a long history.Early theoretical ideas and results can be traced back to the 1970s [1][2][3][4][5][6][7][8][9] (see also a review [10] and references therein).A systematic approach to determining two-and three-point functions of conserved currents, such as the energymomentum tensor and vector current, was undertaken by Osborn and Petkou in [11] and by Erdmenger and Osborn in [12].Such correlation functions are severely constrained by conformal symmetry and the conservation properties and are fixed up to finitely many coefficients.In certain cases these coefficients are related to Weyl or chiral anomalies of a conformal field theory coupled to an external curved background or an external gauge field.
The aim of this paper is to study four-dimensional N = 1 superconformal field theory possessing a conserved spinor current multiplet.Such a current naturally exists in N = 2 superconformal field theory and is a part of the N = 2 supercurrent J [43,44].In general, J contains several conserved N = 1 currents.To describe them let us split the N = 2 covariant derivatives (D i α , D ) and define the N = 1 bar-projection U| = U(x, θ α i , θj α)| θ 2 = θ2 =0 .Then J is composed out of the three independent N = 1 currents [18]: Here J α α is the N = 1 supercurrent containing the energy-momentum tensor, the current of the first supersymmetry and the N = 1 R-symmetry current [25].The spinor current multiplet J α contains the current of the second supersymmetry and two of the three SU(2) R-symmetry currents.Finally, J contains the current corresponding to the combined N = 2 U(1) R-transformations and SU(2) z-rotations which leave θ 1 and θ1 invariant.
In this paper, however, we do not assume N = 2 supersymmetry.We study just N = 1 superconformal field theory which also has a Grassmann odd spinor current multiplet S α and ask the question whether the existence of such a current implies N = 2 superconformal symmetry.For this purpose we study three-point functions involving S α and find their most general form consistent with N = 1 superconformal symmetry.If existence of S α implies N = 2 superconformal symmetry, such three-point functions should be all obtained from the three-point function of the N = 2 supercurrent J by superspace reduction.
A surprising result is that the answer to the above question is negative.More precisely, we find that the three point function S α(z 1 )S β (z 2 )J γ γ (z 3 ) (1.2) is, in general, inconsistent with N = 2 superconformal symmetry because it has more independent tensor structures.That is, despite the fact that S α in components contains a conserved spin-3 2 current, its existence does not imply N = 2 superconformal symmetry.We would like to stress that our results are based on the symmetry considerations only and we do not know how to realise them in local field theory.We were not able to find explicit field theoretic examples of N = 1 superconformal theories possessing a spinor current multiplet, yet without being N = 2 supersymmetric.On the other hand, we demonstrate that the three-point function S α(z 1 )S β (z 2 )L(z 3 ) , (1.3) where L is a flavour current superfield in an N = 1 superconformal theory, is fixed up to two real parameters.This result is in agreement with N = 2 superconformal symmetry, in the sense that the three-point function J α(z 1 )J β (z 2 )J(z 3 ) also possesses two linearly independent structures.
One natural question which arises in this context is whether our results are intrinsic for N = 1 superconformal symmetry or also hold in non-supersymmetric case.That is, let us consider a non-supersymmetric conformal theory which also possesses a conserved Grassmann odd spin- 3  2 current Q mα .Based on symmetries only, does the existence of Q mα imply N = 1 supersymmetry?If the answer is positive the three-point function where T pq is the energy-momentum tensor, must be contained in the three-point function of the N = 1 supercurrent.This question is left for future research. 1One more natural question is whether our results are intrinsic to four-dimensional (super)conformal symmetry or hold in say three and six dimensional (super)conformal theories.This question is also left for future research.
The paper is organised as follows.In section 2 we review the general construction of three-point functions in superconformal theories.In our review we follow [13][14][15]18] where additional details can be found.In section 3 we discuss three-point functions of spinor current multiplets and the supercurrent in N = 1 superconformal theories.Our main result is that the three point function (1.2) is fixed up to three independent parameters.In section 4 we consider three-point functions of spinor current and the flavour current multiplets.In section 5 we discuss remaining three-point correlators involving just the spinor current multiplet.In section 6 we study whether our results in sections 3 and 4 are consistent with N = 2 superconformal symmetry.For this we perform superspace reduction from N = 2 to N = 1.We show that our result for the three point function (1.2), in general, is not consistent with N = 2 superconformal symmetry.Finally, in section 7 we discuss the reduction of the superfield correlator (1.2) to correlators of component currents.More precisely, we concentrate on the three-point functions of two U(1) vector currents contained in S α with the axial current and the energy-momentum tensor contained in J γ γ .We show that our results are in complete agreement with the general structure of these three-point functions derived in [11,12].

Superconformal building blocks
In this section we provide a brief review of two and three-point superconformal structures in 4D N -extended superspace, which were introduced in [13,14] in the N = 1 case, and later generalised to arbitrary N in [15] (see also [18] for a review).These superconformal structures are important for constructing correlation functions of conserved current multiplets in later sections.This section closely follows [18].

Infinitesimal superconformal transformations
where a = 0, 1, are generated by conformal Killing supervector fields [18,45] Eq. (2.3) implies that the spinor parameters can be expressed in terms of the vector ones The vector parameters satisfy which result in the conformal Killing equation The general solution to eq. (2.5) was given in [45] for N = 1 and in [15] for N > 1.
The superalgebra of N -extended conformal Killing supervector fields is isomorphic to su(2, 2|N ).We have the relation Here the superfield parameters are expressed in terms of (2.8) The z-dependent parameters ωαβ (z), σ(z), Λj i (z) correspond to the 'local' Lorentz, scale and SU(N ) R-symmetry transformations, respectively.One may also check the following properties: (2.9) For N = 2, the latter property leads to the analyticity condition The above formalism cannot be directly applied to the case N = 4.In this paper, we are interested in the N = 1 and N = 2 cases.

Two-point structures
Let z 1 and z 2 be two different points in superspace.In constructing correlation functions of primary superfields, all building blocks are composed of the following two-point structures ) where x a ± = x a ± iθ i σ a θi .The former can also be expressed in spinor notation as The notation 'x1 2 ' means that x1 2 is antichiral with respect to z 1 and chiral with respect to z 2 .That is, where D i (1)α and D(1) αi are the superspace covariant spinor derivatives acting on the point z 1 and similarly, D i (2)α and D(2) αi act on the point z 2 , We will also introduce the inverse of x1 2 : (2.15) Note that (x1 2 ) αα (x 2 1) α β = x1 2 2 δ α β .We will also often make use of matrix-like conventions of [14,18] when the spinor indices are not explicitly written.We denote x = (x α α) , x = (x αα ) .
Another important structure is the conformally covariant N × N matrix which satisfies (2.21) The unimodular unitary matrix has the transformation rule As follows from (2.19b) and (2.23), the two-point structures (x 2 1) α α and ûi j (z 12 ) transform covariantly with respect to the superconformal group, i.e., as tensors with Lorentz and SU(N ) indices at both superspace points.
In the N = 2 case, using the SU(2)-invariant tensors ε ij = −ε ji and ε ij = −ε ji , with ε 12 = ε 21 = 1, we can raise and lower isoindices We can then write the condition of unimodularity of the matrix û as or, equivalently To finish this subsection let us also note several useful differential identities: Here and throughout the paper we assume that the space points are not coincident,

Three-point structures
Associated with three superspace points z 1 , z 2 and z 3 are the following superconformally covariant bosonic and fermionic variables Z 1 , Z 2 and Z 3 , where [13,14,18] for details): ) ) Clearly, the structures in (2.28b) and (2.28c) are obtained through cyclic permutations of superspace points.Thus, it suffices to study the properties of (2.28a).We also define and similar relations hold for X2 , X3 .The structures (2.28a) transform as tensors at z 1 ) By cyclic permutation of labels, we may see that Z i transform as tensors at the point z i , i = 1, 2, 3.
Using the matrices u(z rs ), r, s = 1, 2, 3, it is possible to construct unitary matrices transforming covariantly at z 3 only.Their properties are (2.32) In the N = 1 case, there are several properties of Z's which might be useful later.
Note that X is not an independent variable since it can be expressed in terms of X, Θ, Θ.
The variables Z with different labels are related to each other via the identities Making use of (2.30) and (2.34), it follows that and this combination is a superconformal invariant.

Correlation functions of primary superfields
Consider a tensor superfield O A I (z) transforming in a representation T of the Lorentz group with respect to the index A, and in the representation D of the R-symmetry group SU(N ) with respect to the index I. 2 Such a superfield is called primary if its infinitesimal superconformal transformation law reads In the above, ξ is the superconformal Killing vector, while ωαβ (z), σ(z), Λk l (z) are the z-dependent parameters associated with ξ, see eq. (2.8).The superscript 'A' collectively denotes the undotted and dotted spinor indices on which the Lorentz generators M αβ and M α β act.The matrices R i j are the SU(N ) generators.The weights q and q determine the dimension (q + q) and U(1) R-symmetry charge (q − q) of the superfield.Various primary superfields, including conserved current multiplets, obey certain differential constraints imposed by their conservation equations.These constraints need to be taken into account when computing correlation functions.It proves beneficial to make use of these conformally covariant operators [14,18]: (2.37) 2 We assume the representations T and D are irreducible.
They satisfy anti-commutation relations Given an arbitrary function t(X 3 , Θ 3 , Θ3 ), the following differential identities hold The above identities can be derived using these relations: In the N = 1 case, we arrive at the following (2.41d) In accordance with the general prescription of [13][14][15]18], the two-point correlation function of a primary superfield O A I with its conjugate ŌJ B takes the form with C O being a normalisation constant.Furthermore, the three-point correlation function of primary superfields Φ A 1 I 1 , Ψ A 2 I 2 and Π A 3 I 3 has the general expression [13][14][15]18]: The functional form of the tensor H B 1 B 2 A 3 J 1 J 2 I 3 is highly constrained by the superconformal symmetry as follows: (i) It possesses the homogeneity property which guarantees that the correlation function has the correct transformation law under the superconformal group.Note that eq. ( 2.43) by construction has the correct transformation properties at the points z 1 and z 2 .The above homogeneity property implies that it transforms correctly also at the point z 3 .
(ii) If any of the superfields Φ, Ψ and Π satisfy differential equations (e.g.conservation laws of conserved current multiplets), then the tensor 3 is constrained by certain differential equations as well.The latter may be derived using (2.39).
(iii) If any (or all) of the superfields Φ, Ψ and Π coincide, the tensor H J 1 J 2 I 3 B 1 B 2 A 3 obeys certain constraints, as a consequence of the symmetry under permutations of superspace points, e.g.
where ǫ(Φ) is the Grassmann parity of Φ A I .We further note that under permutations of any two superspace points, the three-point building blocks transform as The above conditions fix the functional form of H J 1 J 2 I 3 B 1 B 2 A 3 (and therefore the three-point correlation function) up to a few arbitrary constants.
The main objective of this paper is to determine the constraints on the general structure of three-point correlation functions involving a conserved spinor current multiplet S α (z) (and its conjugate S α(z)), imposed by N = 1 superconformal symmetry.The spinor current superfield S α obeys the following constraints (2.47b) These conditions fix its superconformal transformation law to be and hence, S α is a primary superfield with weights (q, q) = (3 2 , 1) and dimension 5  2 .It was first demonstrated in [18] (and also discussed in the introduction) that the spinor current multiplet S α naturally arises from the reduction of the N = 2 conformal supercurrent [43] to N = 1 superspace. 3However, in this paper we will not assume N = 2 superconformal symmetry.We will consider N = 1 superconformal theory which also has a conserved spinor current multiplet S α .Our aim is to find how three-point correlation functions involving S α are determined by N = 1 superconformal symmetry.A surprising result is that these correlation functions, in general, do not imply N = 2 superconformal symmetry.

Correlation functions of spinor current multiplets with the supercurrent
Let us recall that the N = 1 supercurrent is described by a primary real superfield J α α = Jα α, subject to the conservation law Its superconformal transformation is The reality constraint and conservation law (3.1)imply that J α α has weights (q, q) = ( 3 2 , 3 2 ) and dimension 3.

Correlator
According to the general prescription (2.43), we shall look for the three-point function of the form The tensor H α β, γ γ has the following homogeneity property The correlator must obey the conservation equations (2.47) and (3.1).Specifically, we impose: Upon the use of identities (2.41), along with the constraints on the superspace points z 1 and z 2 are translated to Imposing differential constraints (3.5c) is more complicated.We will take care of (3.5c) at the end.
Our aim is to solve for H α β,γ γ subject to the constraints (3.4) and (3.7).First, noting that H α β,γ γ is Grassmann even, the most general ansatz takes the form It is easy to see that the conditions (3.7b) and (3.7d) lead to and hence, H α β,γ γ (X, Θ, Θ) gets simplified to In order to work out the structures of F α β,γ γ (X) and G α β,γ γ,δ δ (X), it is instructive to make use of vector notation, where we write It is not difficult to construct the most general expressions for these tensors, which are consistent with eq.(3.4).We get ) where c n and d n are some complex coefficients.Our next task is to rewrite (3.7a) and (3.7c) in vector notation to obtain constraints on the coefficients c n and d n .Making use of the expression (3.10), one can show that eq.(3.7a) is fulfilled under the conditions These differential constraints are equivalent to On the other hand, eq.(3.7c) implies that which read which lead to Upon imposing (3.15b), we find that At this stage, we end up with three free coefficients, d 2 , d 3 , d 4 for our function H mn : where we recalled that P α α = −4Θ α Θ α.We can also express our result in spinor notation: It remains to impose conservation equations on the supercurrent J γ γ , eq. (3.5c).Checking conservation laws on z 3 is non-trivial since there are no identities that would allow differential operators acting on the z 3 dependence to pass through the prefactor of (3.3).To simplify our computation, let us first express our correlator (3.3) as where we have defined4 (3.24) On the other hand, the same correlator can be written as follows for some function Using the identity we can also present eq.(3.27) in the form To compute H λ, σσ α , we will use the following identities [14]: Here we have defined After some calculations using the above identities we find that H ρ,δ δ α (X 1 , Θ 1 , Θ1 ) takes the form (3.31) Next, relabelling the superspace points z 2 → z 1 , z 3 → z 2 , z 1 → z 3 , along with eqs.(2.34), allows us to rewrite (3.25) in the equivalent form Now we can apply the conservation condition of J γ γ using eqs.(2.41).We get After some calculations, one may verify that (3.31) satisfies (3.33) for an arbitrary choice of d 2 , d 3 , d 4 .
Finally, we note that our three-point function S α(z 1 )S β (z 2 )J γ γ (z 3 ) satisfies the reality condition S α(z 1 )S β (z 2 )J γ γ (z where * means complex conjugation.It implies the following condition on the function where H αβ γ γ (X 3 , Θ 3 , Θ3 ) is the complex conjugate of the expression (3.22).This condition gives the reality constraint on the coefficients: Therefore, our final expression for the three-point function S α(z 1 )S β (z 2 )J γ γ (z 3 ) is determined up to three independent, real coefficients: (3.37b) In vector notation, this is equivalent to Later we will see that eq.3.2 Correlator S α (z 1 )J β β(z 2 )S γ (z 3 ) In the remaining part of the section we will consider two other correlators involving the supercurrent and spinor currents.They both have non-compensating R-symmetry charge and are expected to vanish.We will check that it is indeed the case.
According to (2.43), we write the general ansatz As a result of imposing the conservation equations (2.47) and (3.1), the tensor H α, δδ γ must satisfy the following constraints It obeys the homogeneity property We also need to make sure that the correlator has the right symmetry property under the replacement of superspace points z 1 ↔ z 3 , since the superfield S α is Grassmann odd.
As in the previous subsection, it is useful to switch to vector notation, Since H mn (X, Θ, Θ) is Grassmann even, the most general form we can write is The requirements Q δ H α, δ δ, γ = 0 and D αH α, δ δ, γ = 0 lead to the algebraic constraints and hence, (3.44b) implies that F mnp (X) is completely symmetric and traceless.On the other hand, imposing Q δ H α, δ δ, γ = 0 and D 2 H α, δ δ, γ = 0 gives us several differential constraints: where Let us solve for F mnp (X).Being completely symmetric in m, n, p, the general solution for F mnp (X) compatible with the homogeneity property (3.40) reads Imposing (3.44b) and (3.45a) leads to Next, we turn to A mn (X), for which we have the general expression It is easy to check that the conditions ∂ m A mn = 0 and ∂ [p A m]n = 0 already give Therefore, we conclude that According to eq. (2.43), the general expression of this three-point function is as follows As a result of imposing the conservation equations (2.47) and (3.1), the tensor H α, δδ γ γ must satisfy the following constraints It also obeys the homogeneity property The correlator also satisfies the symmetry under z 2 ↔ z 3 : Let us now write Since H mn, α(X , Θ, Θ) is Grassmann odd, the most general form we can have is Imposing Q δ H α, δ δ, γ γ = 0 and D αH α, δ δ, γ γ = 0 leads to the requirements ) It can be shown that D 2 H α, δ δ, γ γ = 0 does not give further constraints.
Let us try to construct the explicit solution for A kmn (X).Since it has dimension −3, we can have a general expression Eq. (3.57c) relates the coefficients in the following way Demanding that ∂ k A kmn = 0 leads to a 2 = − 1 2 a 1 , while ∂ [r A k]mn = 0 gives a 1 = 0.As a result, A kmn (X) = 0 . (3.61) Next, we solve for C mn [kp] (X), for which we have the general expression Therefore, we conclude that 4 Correlation functions of spinor current with the flavour current multiplets The N = 1 flavour current multiplet is a primary real superfield L ā = Lā , where ā is the index of a flavour symmetry group, subject to the conservation equation Its superconformal transformation law is It then follows that it has weights (q, q) = (1, 1) and dimension 2.
In this section, we consider three-point functions of spinor current multiplets S α , S α with a U(1) flavour current L. Generalisations to the non-Abelian case are also straightforward.

Correlator
As in the previous section, we start with the general form of the three-point function where H βα has the following homogeneity property By taking the complex conjugate of (4.3), we have the reality condition As a consequence of the spinor current conservation law (2.47)(and its conjugate), the tensor H α β must satisfy the differential constraints The conservation equations for the flavour current, will be imposed at the end, by making use of a similar trick as in subsection 3.1.
Since H α β is Grassmann even we have the following expansion From the conditions D2 H α β = 0, Q 2 H α β = 0 we quickly obtain that which leaves us with Converting to vector notation we see that there are only few possible options for F m and G mn : where c 1 , d 1 , d 2 are arbitrary free coefficients.Substituting eqs.(4.10), (4.11), (4.12) into eq.(4.6) we find that d 2 = −4d 1 .Using the reality property (4.5) we find that c 1 is imaginary and d 1 is real.Hence, we have the following solution where we redefined c 1 → ic 1 so that c 1 and d 1 are now real, free coefficients.
We still must check the flavour current conservation equations (4.7).First, we will express our correlator (4.3) as By rearranging the operators in the three-point function, we may write Comparing eqs.(4.14) and (4.15) we find the relation between H and H: where the identity (3.28) has also been used.Performing a similar calculation as in the previous section we find Upon relabelling superspace points Now the conservation conditions (4.7) become By explicit calculations one can show that H in eq.(4.17) satisfies (4.19) for arbitrary c 1 and d 1 .
Thus, in N = 1 superconformal theory, the three-point function involving spinor current multiplets and a flavour current superfield has two linearly independent functional structures with real coefficients c 1 and d 1 : where In section 6, we will study how this correlator is related to N = 2 superconformal symmetry.

Correlator
This correlator carries a non-vanishing R-symmetry charge and we will show that it vanishes.Here we demonstrate that by imposing the conservation laws for L(z 1 ) and L(z 2 ), it is sufficient to see that L(z 1 )L(z 2 )S α (z 3 ) = 0.As usual, we start with the general form where the tensor H α (X 3 , Θ 3 , Θ3 ) obeys and so its dimension is −3/2.The conservation equations for J(z 1 ) and J(z 2 ) result in Remembering that H α is Grassmann odd, its general expression is given by The second order constraints D2 Using the property (4.23), we can then write From here, it is not hard to verify that the requirements D 2 H α = 0 and Q2 H α = 0 imply a 1 = 0 and c 1 = 0, respectively.Thus, we conclude that

Correlators of the spinor current multiplets
For completeness, we will also consider three-point functions of three spinor current multiplets.Such correlators carry an R-symmetry charge and are expected to vanish.In this section, we will check that it is indeed true.

Correlator
We begin with the general ansatz As a result of imposing (2.47), the tensor H α, γ β must satisfy the following constraints It also obeys the homogeneity property Note that we must impose one more condition: under permutation of superspace points z 1 and z 3 , the correlation function must satisfy However, we will see that the constraints (5.2) and ( 5.3) are sufficient to show that the correlator vanishes.
Let us trade a pair of spinor indices with the vector one, i.e.
where H m, α(X , Θ, Θ) being Grassmann odd has the following expansion One can readily verify that the conditions Q 2 H α, γ β = 0 and D αH α, γ β = 0 give Thus, it suffices for us to determine the general ansatz for A m,n and C m, ( δ α) .In order to be compatible with (5.3), we should write (5.8b)However, when the remaining constraints Q β H α, γ β = 0 and D 2 H α, γ β = 0 are taken into account, we find that a 1 = a 2 = c 1 = c 2 = 0 and, thus, S α (z 1 )S β (z 2 )S γ (z 3 ) = 0 . (5.9) 5.2 Correlator S α (z 1 ) S β(z 2 )S γ (z 3 ) We again begin with the general ansatz As a result of imposing (2.47), the tensor H β, γ α must satisfy the following constraints .11)It also obeys the homogeneity property (5.12) Let us trade a pair of spinor indices with the vector one, i.e.
where H m, β (X, Θ, Θ) being Grassmann odd has the following expansion One can readily verify that the conditions D αH β, γ α = 0 and Q β H β, γ α = 0 give B m, δ, β = 0 , D mn = 0 , ε βδ A m, δ, β = 0 , (5.15a) Thus, it suffices for us to determine the general ansatz for A m, (δβ) and C (mn) only.In order to be compatible with (5.12) and (5.15b), we should write (5.16b)However, it turns out that the second order constraints D 2 H β, γ α = 0 and Q2 H β, γ α = 0 are enough to show that a 1 = a 2 = c 1 = 0. Thus, S α (z 1 ) S β (z 2 )S γ (z 3 ) = 0 . (5.17) 6 N = 2 → N = 1 superspace reduction In this section we will discuss whether the three-point functions studied in the previous sections are consistent with N = 2 superconformal symmetry.For this we will study if they are contained in the three-point function of the N = 2 supercurrent.
6.1 The three-point function of the N = 2 supercurrent Let us recall that in N = 2 superconformal field theory, the supercurrent [43,44] is described by a primary real scalar superfield J (z) of dimension 2, subject to the conservation condition The superconformal transformation law of J , which is uniquely determined from the reality condition J = J and the conservation equation, reads As was discussed in the introduction, the N = 2 supercurrent J is composed of three independent N = 1 multiplets [18].For convenience, we will write them again 3) It follows from eqs. (6.1) and (6.3) that the N = 1 components of J satisfy the following conservation equations D2 J = 0 , D 2 J = 0 (6.4a) From eqs. (6.2) and (6.3) one can also deduce the transformation properties δJ = −ξ J − 2 (σ + σ) J (6.5a) and hence J α α is obtained from J by applying the operator ∆ α α and then switching off the Grassmann variables θ 2 , θ2 .As a first step, let us consider ∆ α αJ (z 1 )J (z 2 )J (z 3 ) .
In [18] it was found that which is a function of Z 3 .Thus, the three-point function (3.3) can be derived by evaluating the bar-projection and making use of the identities (2.39).
After quite a long calculation, the above procedure yields J α(z 1 )J β (z 2 )J γ γ (z 3 ) = (x 3 1) α α(x 2 3) β where Thus, any N = 1 superconformal field theory possessing a conserved spinor current multiplet S α also has N = 2 superconformal symmetry, if and only if the correlator S α(z 1 )S β (z 2 )J γ γ (z 3 ) is of the form (6.14), (6.15).Comparing the above with (3.37), we see that they agree provided where H mn is eq.( 6.15) written in vector notation, that is In section 7, we will present two consistency checks of our result (3.37).
It follows from eqs. (6.3) that in N = 2 superconformal theory, the three-point function J α(z 1 )J β (z 2 )J(z 3 ) can be extracted using the following superspace reduction: along with the identities (2.39), one obtains with That is, the N = 1 correlator S α(z 1 )S β (z 2 )J(z 3 ) is always consistent with N = 2 superconformal symmetry.
6.4 Correlator J α(z 1 )J β (z 2 )L(z 3 ) The N = 2 flavour current superfield [18] is subject to the reality condition L ij = L ij , and the conservation (analyticity) equation These constraints fix its transformation law to be Hence, its superconformal transformation rule is similar to J (see eq. (6.5a)) It was demonstrated in [18] that the three-point function involving two N = 2 supercurrent insertions and a flavour N = 2 current superfield vanishes, As a result, 7 Reduction to component currents In N = 1 superconformal field theory, the spinor current S α and the supercurrent J α α multiplets contain, among others, the following conserved currents: Here the bar-projection corresponds to setting the Grassmann coordinates to zero.The vector current V α α is a complex field, 3) The components R α α and T αβ α β are the U(1) R-symmetry current and the energy-momentum tensor, respectively.This means that the correlator S α(z 1 )S β (z 2 )J γ γ (z 3 ) (7.4) in particular contains the following component correlators The three-point functions of this kind were studied in [11,12]. 5It was shown that the threepoint function of two U(1) vector currents and an axial current has only one independent structure, and the three-point function of two vector currents and the energy-momentum tensor is fixed up to two independent structures.On the other hand, the correlator (7.4) was shown in subsection 3.1 to depend on three linearly independent structures.Hence, it is not trivial to see if our results for the correlator (7.4) are consistent with the general structure of the correlators (7.5) derived in [11,12].In this section we will perform a component reduction of the superfield correlator (7.4) to extract the component correlators (7.5), thus providing consistency checks of our results.
Let us now rewrite the tensor h α α ρρ μµ, λλ (X) in vector notation, where we define The tensor h abcd thus introduced is symmetric and traceless in (a, b).We obtain Our final result in the form (7.20) agrees perfectly with the general structure of the threepoint function involving two vector currents and the energy-momentum tensor constructed by Osborn and Petkou [11].More precisely, we find the following relations: where c and e are the two linearly independent, real parameters appearing in the function tabcd (X) in eq.(3.13) in [11].
.17b) Thus, eqs.(3.15b) and (3.17b) are identical.It follows from eqs. (3.15a) and (3.17a) that Θ1 ).The second line in eq.(3.25) has the same structure as the general expression (2.43) except now we are treating z 1 as the "third point".Since eqs.(3.23) and (3.25) represent the same correlator the functions H and H are related.Knowing H we can find H by comparing eqs.(3.23) and (3.25).For this let us introduce the operators inverse to (3.24):

( 3 .
37) cannot be obtained from three-point function of the N = 2 supercurrent unless the coefficients d 2 , d 3 , d 4 satisfy an additional constraint d 2 + d 3 + d 4 = 0.However, N = 1 superconformal symmetry alone does not impose this constraint.

d 3
= e , (d 2 + 3d 4 ) = −c , (7.21) , • • • N .The Grassmann variables θ α i and θi α are related to each other by complex conjugation: 2 x3 1 2 x2 3 2 x3 2 2 D σl ]H(Z 3 ) as functions of Z 1 , by virtue of (2.34).Upon relabelling z 1 ↔ z 3 (and noting the relation (2.46c)), we then obtain Therefore, we obtain the following surprising result: Existence of a conserved spinor current in N = 1 superconformal field theory, in general, does not imply N = 2 superconformal symmetry.Only if the coefficients d 2 , d 3 , d 4 in (3.37) satisfy the additional constraint (6.17) the three-point function(3.37)iscontained in the three-point function of the N = 2 supercurrent and can be obtained by the superspace reduction.Otherwise, eq.(3.37) contains an extra independent tensor structure incompatible with N = 2 superconformal symmetry.More precisely, the tensor H mn in (3.37) may be expressed in the following way