Superconformal field theory in three dimensions: Correlation functions of conserved currents

For N-extended superconformal field theories in three spacetime dimensions (3D), with N=1,2,3, we compute the two- and three-point correlation functions of the supercurrent and the flavour current multiplets. We demonstrate that supersymmetry imposes additional restrictions on the correlators of conserved currents as compared with the non-supersymmetric case studied by Osborn and Petkou in hep-th/9307010. It is shown that the three-point function of the supercurrent is determined by a single functional form consistent with the conservation equation and all the symmetry properties. Similarly, the three-point function of the flavour current multiplets is also determined by a single functional form in the N=1 and N=3 cases. The specific feature of the N=2 case is that two independent structures are allowed for the three-point function of flavour current multiplets, but only one of them contributes to the three-point function of the conserved currents contained in these multiplets. Since the supergravity and super-Yang-Mills Ward identities are expected to relate the coefficients of the two- and three-point functions under consideration, the results obtained for 3D superconformal field theory are analogous to those in 2D conformal field theory. In addition, we present a new supertwistor construction for compactified Minkowski superspace. It is suitable for developing superconformal field theory on 3D spacetimes other than Minkowski space, such as S^1 x S^2 and its universal covering space R x S^2.


Abstract
For N -extended superconformal field theories in three spacetime dimensions (3D), with 1 ≤ N ≤ 3, we compute the two-and three-point correlation functions of the supercurrent and the flavour current multiplets. We demonstrate that supersymmetry imposes additional restrictions on the correlators of conserved currents as compared with the non-supersymmetric case studied by Osborn and Petkou in hep-th/9307010. It is shown that the three-point function of the supercurrent is determined by a single functional form consistent with the conservation equation and all the symmetry properties. Similarly, the three-point function of the flavour current multiplets is also determined by a single functional form in the N = 1 and N = 3 cases. The specific feature of the N = 2 case is that two independent structures are allowed for the three-point function of flavour current multiplets, but only one of them contributes to the three-point function of the conserved currents contained in these multiplets. Since the supergravity and super-Yang-Mills Ward identities are expected to relate the coefficients of the two-and three-point functions under consideration, the results obtained for 3D superconformal field theory are analogous to those in 2D conformal field theory.
In addition, we present a new supertwistor construction for compactified Minkowski superspace. It is suitable for developing superconformal field theory on 3D spacetimes other than Minkowski space, such as S 1 × S 2 and its universal covering space R × S 2 .

Introduction
One of the well-known implications of conformal invariance in d ≥ 2 dimensions [1,2,3] is that the functional form of the two-and three-point correlation functions of primary fields is fixed up to a finite number of parameters. 1 In general, however, it is a nontrivial technical exercise to determine explicitly the three-point functions of constrained tensor operators, examples of which are the energy momentum-tensor or conserved vector currents. The point is that such operators obey certain differential constraints and some 1 In the Euclidean case, these results follow from the following well-known mathematical observation.
For any three distinct points p 1 , p 2 and p 3 on the d-sphere S d = R d {∞}, there exists a conformal transformation g ∈ SO 0 (d + 1, 1) that maps these points to 0, (1, 0, . . . , 0) and ∞, respectively. Here S d is understood to be the conformal compactification of Euclidean space R d , i.e. the set of all null straight lines through the origin in R d+1,1 . The above observation can be rephrased as the statement that there is no conformal invariant of three points. work is required in order to classify those functional contributions to the given threepoint function, which are consistent with all the constraints. Building on the theoretical ideas and results that may be traced back as early as the 1970s (see, e.g., [4,5,6,7,8] and references therein), Osborn and Petkou [9] presented the group-theoretic formalism to construct the three-point functions for primary fields of arbitrary spin in d dimensions. They analysed in detail the restrictions on the correlation functions imposed by the conservation equations for the energy-momentum tensor and conserved currents, and the outcomes of their study include the following conclusions: The three-point function of the energy-momentum tensor contains three linearly independent functional forms for d > 3, while for d = 3 there are two and for d = 2 only one. 2 The three-point function of vector currents contains two linearly independent completely antisymmetric functional forms, which is in accord with the result obtained in 1971 by Schreier [2]. In the d = 4 case, an additional completely symmetric structure is allowed, which reflects the presence of anomalies [13]. In the same d = 4 case, the three parameters describing the three-point function of the energy-momentum tensor were demonstrated [9,13] to be related to two coefficients in the trace anomaly of a conformal field theory in curved space.
When conformal symmetry is combined with supersymmetry, the story of the two-and three-point functions of conserved currents described in [9,13] has to be supplemented with a sequel, for there appear conceptually new fermionic conserved currents. In the realm of supersymmetric field theories, the energy-momentum tensor is replaced with the supercurrent [14]. The latter is a supermultiplet containing the energy-momentum tensor and the supersymmetry current, along with some additional components such as the Rsymmetry current. Thus the supercurrent contains fundamental information about the symmetries of a given supersymmetric field theory.
The supercurrent is the source of supergravity [15,16,17], in the same way as the energy-momentum tensor is the source of gravity. For every superconformal field theory, the supercurrent is an irreducible multiplet that may be coupled to the Weyl multiplet of conformal supergravity. For non-superconformal theories, the supercurrent is reducible and contains the so-called trace multiplet which includes the trace of the energymomentum tensor. Different supersymmetric theories may possess different trace multiplets that correspond to different off-shell formulations for supergravity.
For completeness, it is pertinent to recall the structure of (non-)conformal supercur-rents in four spacetime dimensions (4D). The N = 1 conformal supercurrent [14] is a real vector superfield J αβ constrained byDβJ αβ = 0 or, equivalently, D α J αβ = 0. The simplest non-conformal supercurrent was given by Ferrara and Zumino [14]; the corresponding conservation equation isDβJ αβ = D α T , where the trace multiplet T is chiral (see [18] for a review). The Ferrara-Zumino supercurrent proves to be well defined on a dense set in the space of N = 1 supersymmetric field theories. For recent discussions of the most general 4D N = 1 non-conformal supercurrents, see [19,20,21]. The N = 2 conformal supercurrent [22,23] is a real scalar superfield J constrained byD iαDαj J = 0 or, equivalently, D αi D j α J = 0, see [24] for more details. Numerous N = 2 supersymmetric theories are characterised by the following conservation equation [24,25]: Here the trace multiplets T ij = T ji = T ij and Y are linear and reduced chiral superfields 3 respectively; g ij = g ji = g ij is a constant iso-triplet that might be thought of as an expectation value of the tensor multiplet, one of the two supergravity compensators, see [25] for the details.
Superconformal symmetry imposes additional restrictions on the structure of threepoint functions of conserved currents as compared with the non-supersymmetric case studied in [9,13]. In 4D N = 1 superconformal theories, the three-point function of the supercurrent is the sum of two linearly independent functional structures [26] as compared with the three functional forms in the non-supersymmetric case [9]. The corresponding coefficients a and c were shown [26] to be related to those constituting the super-Weyl anomaly in curved superspace studied theoretically in [27] and computed explicitly in [28] (see [18] for a review). 4 The same conclusion holds for the three-point function of the supercurrent in 4D N = 2 superconformal field theories [24], while N = 4 superconformal symmetry is known to demand a = c. As concerns the three-point function of the flavour current multiplets, there exist two independent structures in the 4D N = 1 case [26] (as compared with three in the non-supersymmetric case [9,13]), while N = 2 superconformal symmetry allows only one [24].
The present paper is the first in a series devoted to the correlation functions of conserved currents in N -extended superconformal field theories in three spacetime dimensions. We start with a brief discussion of 3D conformal supercurrents and flavour current multiplets. The 3D N -extended conformal supercurrents have been described in [32] us- 3 The linear superfield is constrained by D (i α T jk) =D (i α T jk) = 0, while the reduced chiral superfield obeys the chirality constraintD iα Y = 0 and the reality condition D αi D j α Y =D iαDαjȲ . 4 In his analysis [26], Osborn used the realisation of superconformal transformations in 4D N = 1 Minkowski superspace described in [18]. Earlier works on superconformal transformations in superspace include [29,30,31].
ing the conformal superspace formulation for N -extended conformal supergravity given in [33]. For every N = 1, 2 . . . , the supercurrent is a primary real superfield of certain tensor type and dimension, which obeys some conservation equation formulated in terms of covariant derivatives. Denoting by D I α the spinor covariant derivatives of N -extended Minkowski superspace R 3|2N , the conformal supercurrents 5 for N ≤ 4 are specified by the following properties:

SUSY Type Supercurrent Dimension
Conservation Equation For N > 4, the conformal supercurrent is a completely antisymmetric dimension-1 superfield, J IJKL = J [IJKL] , subject to the conservation equation The above results follow from the analyses carried out in [33,32]. Given an N -extended superconformal field theory, it may be coupled to the Weyl multiplet of N -extended conformal supergravity. In curved superspace, the supercurrent J (with its indices suppressed) of the matter model with action S matter is J ∝ δS matter δH , (1.3) with H being an unconstrained prepotential for conformal supergravity. The latter has the following index structure: H αβγ for N = 1 [36], H αβ for N = 2 [37,38], H α for N = 3 and H for N = 4 [33,32]. 6 The 3D N -extended flavour current multiplets constitute another family of primary real superfields obeying certain conservation equations. For N ≤ 3, they have the following structure:
In the N = 4 case, there are two inequivalent flavour current multiplets, L IJ + and L IJ − . Each of them is described by a primary antisymmetric dimension-1 superfield, L IJ = −L JI , which obeys the conservation equation (1.5) What differs between the two flavour current multiplets, L IJ + and L IJ − , is that they are subject to different self-duality constraints (1.6) The above results naturally follow from the known structure of unconstrained prepotentials for the N -extended vector multiplets given in the following publications: [40,36] for N = 1, [40,41,37] for N = 2, [42] for N = 3 and [43,44] for N = 4.
The general group-theoretic formalism to construct the two-and three-point functions of primary superfields in 3D N -extended Minkowski superspace was developed by Park [45], as a natural extension of earlier 4D [26,46] and 6D [47] constructions. Instead, we will re-derive the two-and three-point superconformal building blocks, originally given in [45], by making use of the 3D N -extended supertwistor construction of [48]. Such a reformulation makes it possible to apply the formalism for computing correlation functions on more general (conformally flat) superspaces than the standard Minkowski superspace used in [45].
In this paper we study the correlation functions of conserved current multiplets in superconformal field theories with N ≤ 3, while the case N > 3 will be considered elsewhere. The main outcomes of our study are as follows: For N ≤ 3, the three-point function of the supercurrent is determined by a single functional form consistent with the conservation equation and all the symmetry properties. The same conclusion holds for the three-point function of flavour current multiplets in the N = 1 and N = 3 cases. As concerns the N = 2 case, two independent structures are allowed for the three-point function of flavour current multiplets, but only one of them contributes to the threepoint function of the conserved currents contained in these multiplets. Thus the 3D superconformal story is analogous to that in 2D conformal field theory.
In 3D N = 2 superconformal field theories, of special importance are contact terms of the supercurrent and conserved current multiplets [49]. Such contributions to correlation functions are associated with certain Chern-Simons terms for background fields. In this paper, we will concentrate on studying the correlation functions at distinct points where the contact terms do not contribute.
Before we turn to the technical aspects of this paper, it is worth discussing one more conceptual issue: the symmetry structure of extended supersymmetric field theories from the point of view of "less extended" supersymmetry. Every N -extended superconformal field theory is a special theory with (N − 1)-extended superconformal symmetry. It is worth elucidating the structure of (N − 1)-extended supermultiplets contained in the N -extended supercurrent or flavour current multiplet. To uncover this, we split the Grassmann coordinates θ α I of N -extended Minkowski superspace M 3|2N onto two subsets: (i) the coordinated θ α I , withÎ = 1, . . . , N − 1, corresponding to (N − 1)-extended Minkowski superspace M 3|2(N −1) ; and (ii) two additional coordinates θ α N . The corresponding splitting of the spinor derivatives D I α is DÎ α and D N α . Given a superfield V on M 3|2N , its bar-projection onto Consider the N = 2 case. The spinor covariant derivatives DÎ α and D N α introduced above, now become D α and D 2 α respectively. The supercurrent J αβ leads to the following N = 1 supermultiplets: Here J αβγ is the N = 1 supercurrent, while the additional superfield S αβ contains the U(1) R-symmetry current (the θ-independent component of S αβ ) and the second supersymmetry current (the top component of S αβ ).
The N = 3 supercurrent J α leads to the following N = 2 supermultiplets: Here J αβ is the N = 2 supercurrent, while R α contains the third supersymmetry current and two R-symmetry currents corresponding to SO(3)/SO(2).
Next, the N = 4 supercurrent J contains the following N = 3 supermultiplets: Here J α is the N = 3 supercurrent, while S contains the fourth supersymmetry current and three R-symmetry currents corresponding to SO(4)/SO(3). Upon reduction to N = 2 superspace, the scalar S generates two primary N = 2 superfields: (i) the scalar S| θ 2 =0 , which is an N = 2 flavour current multiplet; and (ii) the spinor D 3 α S| θ 2 =0 , which is of the type (1.8a).
Finally, we just mention the N → (N −1) decomposition of flavour current multiplets. The N = 2 multiplet L leads to the following N = 1 real supermultiplets: (1.10a) Here L α is an N = 1 flavour current multiplet, and the real scalar S is unconstrained. The N = 3 multiplet L I leads to an N = 2 flavour current multiplet L and a chiral scalar.
Therefore, if one studies N -extended superconformal field theories in (N −1)-extended superspace, it is not sufficient to analyse the correlation functions of those currents which correspond to the manifestly realised symmetries.
There is a remarkable difference between superconformal field theories and ordinary conformal ones in diverse dimensions. For the action of the conformal group on (compactified) Minkowski space, there is no conformal invariant of three points. The situation is different in superspace. On (compactified) Minkowski superspace, the superconformal group does not act transitively on the set consisting of triples of distinct superspace points. As a result, there exist nilpotent superconformal invariants of three points. Such invariants have been constructed by Park in diverse dimensions [50,45,46,47]. This paper is organised as follows. Following [48], in section 2 we review the supertwistor construction of N -extended compactified Minkowski superspace M 3|2N . Minkowski superspace M 3|2N originates as a dense open subset of M 3|2N . It is shown that M 3|2N is a homogeneous space for the superconformal group OSp(N |4, R), while only the infinitesimal superconformal transformations are well defined on M 3|2N . Section 3 describes a different isomorphic realisation for OSp(N |4, R). Using this realisation, we construct a global supermatrix parametrisation of M 3|2N as well as a smooth metric on M 3|2N , which only scales under the superconformal transformations. The supertwistor formalism is used in section 4 to derive all building blocks in terms of which the two-and three-point functions of primary superfields are constructed. The general structure of the two-and three-point functions of primary superfields is described in section 5 following [45]. The two-and three-point functions for the supercurrent and flavour current multiplets in superconformal field theories with N = 1, N = 2 and N = 3 are computed in sections 6, 7 and 8 respectively. Concluding comments are given in section 9.
We have also included several technical appendices. Appendix A gives a summary of our 3D notation and conventions. Appendix B is devoted to the correlation functions involving (anti)chiral superfields. Appendix C is concerned with the N = 2 → N = 1 superspace reduction of the three-point functions for the N = 2 supercurrent and flavour current multiplets. In Appendix D we reduce to components the three-point function for N = 1 flavour current multiplets.

Supertwistor construction
In this section we describe the supertwistor construction of N -extended compactified Minkowski superspace. Our presentation mostly follows the construction given in [48] and inspired by [51] (see also [52]).

Supertwistors and the superconformal group
In three spacetime dimensions, the N -extended superconformal group 7 is OSp(N |4; R). It naturally acts on the space of real even supertwistors and on the space of real odd supertwistors.
An arbitrary supertwistor is a column vector In the case of even supertwistors, Tα is bosonic and T I is fermionic. In the case of odd supertwistors, Tα is fermionic while T I is bosonic. The even and odd supertwistors are called pure. We introduce the parity function ε(T ) defined as: ε(T ) = 0 if T is even, and ε(T ) = 1 if T is odd. It is also useful to define Then the components T A of a pure supertwistor have the following Grassmann parities A pure supertwistor is said to be real if its components obey the reality condition 3) The space of complex (real) even supertwistors is naturally identified with C 4|N (R 4|N ), while the space of complex (real) odd supertwistors may be identified with C N |4 (R N |4 ).
Introduce a graded antisymmetric supermatrix 4) where ½ N denotes the unit N × N matrix. With the aid of J, we may define a graded symplectic inner product on the space of pure supertwistors by the rule: for arbitrary pure supertwistors T and S, the inner product is where the row vector T sT is defined by and is called the super-transpose of T . The above inner product has the following symmetry property where ε i stands for the Grassmann parity of T i . If T and S are real supertwistors, then applying the complex conjugation gives By definition, the supergroup OSp(N |4; C) consists of those even (4|N ) × (4|N ) supermatrices which preserve the inner product (2.5) under the action Such a transformation maps the space of even (odd) supertwistors onto itself. The condition of invariance of the inner product (2.5) under (2.10) is (2.11) The subgroup OSp(N |4; R) ⊂ OSp(N |4; C) consists of those transformations which preserve the reality condition (2.3), This is equivalent to In conjunction with (2.11), this reality condition is equivalent to is a row vector that transforms under OSp(N |4; R) such that Z A T A is invariant for every supertwistor T , (2.14) A dual supertwistor Z is even (odd) if Z A T A is a c-number for every even (odd) supertwistor T . Given a pure dual supertwistor Z, its super-transpose Z sT will be defined to be the following column vector The superconformal algebra osp(N |4; R) consists of real supermatrices Ω obeying the master equation The general solution of this equation is The bosonic parameters λ α β , σ, a αβ , b αβ and Λ IJ , as well as the fermionic parameters ǫ α I ≡ ǫ I α and η αI ≡ η Iα in (2.17) are real.

Compactified Minkowski superspace
In accordance with [48], the compactified N -extended Minkowski superspace M 3|2N is defined to be the set of all Lagrangian subspaces of R 4|N , the space of real even supertwistors. We recall that a Lagrangian subspace of R 4|N is a maximal isotropic subspace of R 4|N . By definition, such a subspace is spanned by two even supertwistors T µ with the properties that (i) the bodies of T 1 and T 2 are linearly independent; (ii) they obey the null condition (iii) they are defined only modulo the equivalence relation Equivalently, the space M 3|2N can be defined to consist of rank-two supermatrices of the form 20) which are defined modulo the equivalence relation Here F and G are 2 × 2 real bosonic matrices, and Υ is a N × 2 real fermionic matrix. The null condition (2.18) can be rewritten as It may be shown that the superconformal group OSp(N |4; R) acts transitively on the compactified Minkowski superspace. Thus M 3|2N can be identified with the coset space OSp(N |4; R)/G P 0 , where G P 0 denotes the isotropy group at a given two-plane P 0 ∈ M 3|2N .

Minkowski superspace
As discussed in [48], Minkowski superspace M 3|2N is identified with a dense open subset U F of M 3|2N spanned by supermatrices (2.20) under the condition det F = 0 . (2.23) Every null two-plane in M 3|2N may be described by a supermatrix of the form where the real matrixx is constrained bŷ The points of M 3|2N are naturally parametrised by the variables z A = (x a , θ α I ). Given a group element g ∈ OSp(N |4; R), its action P → gP on the two-plane P(z) ∈ M 3|2N can be represented as provided the transformed two-plane, gP(z), belongs to M 3|2N . In general, this property holds only locally, since Ξ(g; z) may become singular for certain group elements g (special conformal transformations) and some spacetime points x.
As pointed out in the previous subsection, M 3|2N can be identified with the homogeneous space OSp(N |4; R)/G P 0 , where G P 0 denotes the isotropy group at a given two-plane P 0 ∈ M 3|2N . Consider a special null two-plane P 0 ∈ M 3|2N which corresponds to the origin of M 3|2N , that is P 0 = P(z = 0). Its isotropy group G P 0 is the subgroup of OSp(N |4; R) generated by supermatrices (2.17) of the form: The isotropy group G P 0 consists of the followings supermatrices where the 2 × 2 matrixb is constrained by As follows from (2.29), G P 0 includes space reflections. Choosing i = 1 or i = 3, let us consider the following element of G P 0 : with σ i being the i-th Pauli matrix, det σ i = −1. Associated with this group element is the transformation on M 3|2N which is a reflection about one of the coordinate axes in two-space.
It is also seen from (2.29) that G P 0 includes arbitrary R-symmetry transformations from the group O(N ) and not necessarily from its connected component of the identity, SO(N ), as discussed by [45].
A complement of the subalgebra (2.28) in osp(N |4; R) generates a subgroup of the superconformal group consisting of all supermatrices of the form: Such a supermatrix describes a spacetime translation (ǫ = 0) and a Q-supersymmetry transformation (a = 0) when acting on M 3|2N .
The N -extended Minkowski superspace can be also realised as a homogeneous space. The standard realisation is where P(3|N ) denotes the N -extended super-Poincaré group and SL(2, R) the spin group in three spacetime dimensions. Every group element g ∈ P(3|N ) can uniquely be represented in the form g = s(a, ǫ) h(M), where Here the supermatrix h(M) describes a Lorentz transformation. The points of M 3|2N can be parametrised by the following coset representative The Q-supersymmetry transformation s(0, ǫ) acts on M 3|2N according to the law s(z) → s(z ′ ) = s(0, ǫ)s(z), and thus These results can be rewritten as where we have introduced the supersymmetry generators From here we immediately read off the spinor covariant derivatives which anti-commute with the supercharges, {D I α , Q J β } = 0, and obey the anti-commutation relations We introduce the 3D extension of the Volkov-Akulov supersymmetric one-form [57,58] It is obviously invariant under the Q-supersymmetry transformation (2.36). Every null two-plane in U G may be described by a supermatrix of the form

Twin Minkowski superspace
where the real 2 × 2 matrixy is constrained by y −y T = 2iρ Tρ ⇐⇒ y αβ = y αβ + iρ Iα ρ Iβ , y αβ = y βα . (2.43) One may think of U G as a twin of U F obtained by replacing the spacetime translations and Q-supersymmetry transformations with the special conformal boosts and Ssupersymmetry transformations, respectively. The two-plane P 0 , which is the origin of U F , is replaced with P ∞ corresponding to y αβ = 0 and ρ Iα = 0, the origin of U G . The two-plane P ∞ is an infinitely separated point from the viewpoint of U F . The isotropy group P ∞ , denoted G P∞ , consists of the following supermatrices    whereâ is defined in (2.32). The following one-form is invariant under the S-supersymmetry transformations.
In the intersection of the two charts introduced, U F U G , the transition functions arě where the matrix J is defined by (2.4).
The two charts U F and U G do not cover the compactified Minkowski superspace. It may be shown [48] that the bosonic body of M 3|2N \ {U F U G } is topologically S 1 . Additional charts are required if we are interested in the global description of M 3|2N . Instead of introducing such additional charts, there is actually a better way out. It turns out that there exists an isomorphic realisation for OSp(N |4; R) that is ideally suited for a global description of M 3|2N . It will be presented in the next section.

Alternative definition of compactified Minkowski superspace
We would like to introduce one more refinement of the formalism that will be rather useful for the discussion in next sections. Following [48], we have defined the compactified N -extended Minkowski superspace to be the space of null two-planes (through the origin) in R 4|N . However, every two-plane in R 4|N is a real two-plane in C 4|N , the space of complex even supertwistors. A two-plane in C 4|N is described by a supermatrix P = (T 1 , T 2 ), where the supertwistors T 1 and T 2 constitute a basis of the two-plane. This supermatrix is defined modulo the equivalence relation The equivalent supermatrices define the same two-plane. A two-plane P in C 4|N is said to be real if it possesses a basis P 0 consisting of real supertwistors, which means P † 0 = P sT 0 . Given an arbitrary basis P of the real two-plane, it holds that P = P 0 Ξ, for some nonsingular matrix Ξ, and hence P † ∼ P sT . We will adopt this new point of view in what follows. It allows us to define the compactified N -extended Minkowski superspace M 3|2N to be the space of all real Lagrangian subspaces of C 4|N , the space of even supertwistors. In this section we present a different isomorphic realisation for the superconformal group, which allows us to construct (i) a global supermatrix parametrisation of M 3|2N ; and (ii) a globally defined smooth metric, ds 2 , on M 3|2N with the property that ds 2 only scales under the superconformal transformations. The crucial feature of this realisation is that it is suitable for developing superconformal field theory on 3D spacetimes more general than Minkowski space, such as S 1 × S 2 and its universal covering space R × S 2 . Our results in this section are analogous to those for the supersphere S 3|4n [53].

Algebraic aspects
The superconformal group possesses an alternative realisation based on the isomorphism Here the supergroup on the right consists of all even (4|N ) × (4|N ) supermatrices g constrained by where we have introduced The condition (3.2a) defines the supergroup U(2, 2|N ). It should be pointed out that for N = 4 the supergroup SU(2, 2|N ) is the N -extended superconformal group in four spacetime dimensions, as defined in [24] (PSU(2, 2|4) in the N = 4 case). In what follows, the supergroup on the right of (3.1) will be denoted OSp(N |4; R) U .
The proof of (3.1) is based on considering the following correspondence for every supertwistor T and dual supertwistor Z. Here the supermatrix U is defined by The symmetric 4 × 4 matrix U is unitary, U † U = U U = ½ 4 , and symplectic, UJU = J.
Another important property is UJU −1 = −iI. These properties have obvious counterparts in terms of U: Of special importance for us will be the identity The above properties of U imply that g defined by (3.4a) obeys the conditions (3.2), and hence g ∈ OSp(N |4; R) U , for every g ∈ OSp(N |4; R), and vice versa.
Associated with the supergroup OSp(N |4; R) U are two invariant inner products defined as follows: for arbitrary pure supertwistors T and S.
An important feature of the supergroup OSp(N |4; R) U is that one can define an involution ⋆ that acts on the space of supertwistors and commutes with the superconformal transformations. Associated with a pure supertwistor T is its star-image, ⋆T , defined by Explicitly the map ⋆ acts as follows: This shows that ⋆(⋆T ) = T , for every supertwistor T .

Compactified Minkowski superspace
Let us see how the compactified Minkowski superspace is described within the supergroup realisation introduced above. The null two-plane P ∈ M 3|2N turns into P = UP.
Since P obeys the null condition P sT JP = 0 and is real, P † = P sT , the two-plane P enjoys the two null conditions The reality condition P † = P sT turns into It may be seen that this reality condition preserves its form only under the real equivalence transformations However, making use of the identities (3.6) and (3.7), it may be rewritten in a different but equivalent form ⋆ P ∼ P , (3.14) which does not change its form under arbitrary complex equivalent transformations see subsection 2.5. Thus, the compactified N -extended Minkowski superspace M 3|2N is equivalently defined as the set of all two-planes in the space of even supertwistors C 4|N which obey (i) the null conditions (3.11); and (ii) the reality condition (3.14).
We can represent where F and G are bosonic 2 ×2 matrices, and the remaining N ×2 matrix Λ is fermionic. Then the null condition (3.11a) tells us that In conjunction with the fact that the supermatrix P has rank two, this condition implies that det F = 0 and det G = 0, see [51] for the proof. As a result, making use of the equivalence relation (3.15) allows us to bring every two-plane P ∈ M 3|2N to the form Now, the null conditions (3.11a) and (3.11b) turn into Moreover, the reality condition (3.14) gives The relations ( The general solution of these constraints is for real parameters ϕ and a, b, c parametrising, respectively, S 1 and S 2 . We also have

Superconformal metric
As shown in the previous subsection, every null two-plane P ∈ M 3|2N is uniquely represented in the form (3.18) for some matrices h and ζ constrained according to (3.19) and (3.20). This means that, given a group element g ∈ OSp(N |4; R) U , it acts on P as for some nonsingular matrix ϕ(g, h, ζ). Explicitly, if we represent g in the block form then h ′ and ζ ′ are seen to be fractional linear functions of h and ζ, Cartan's one-form takes its values in the superalgebra osp(N |4; R) U and possesses the superconformal transformation law where the shorthand notation ϕ = ϕ(g, h, ζ) has been used. We can introduce a superinterval which is a globally defined tensor field over M 3|2N . It follows from (3.26) that ds 2 only scales under the superconformal transformations, In the Minkowski chart, it may be seen that the variables h and ζ are expressed in terms of the superspace coordinates as A direct calculation of E gives the following expression: whereê is the supersymmetric one-form (2.40).

Pseudo inversion
Consider a particular superconformal transformation where σ 2 is the second Pauli matrix. It acts on M 3|2N as follows In the real realisation of the superconformal group, the supermatrix (3.31) turns into Its action on the two-plane P(z) defined by (2.24) is This leads tox Let g = ½ + κΩ be an infinitesimal superconformal transformation, where κ is an infinitesimal parameter and Ω is an arbitrary element of the superconformal algebra osp(N |4; R) given by (2.17). In Minkowski superspace, its action is given by eq. (2.27). It is an instructive exercise to check that the transformation F (½ + κΩ)F generates the following infinitesimal transformation: As compared with g = ½ + κΩ, the transformation F (½ + κΩ)F swaps the spacetime translations and special conformal boosts, as well as the Q-supersymmetry and S-supersymmetry transformations. It also changes the sign of the scale parameter σ.
The transformation F has properties analogous to those of the superinversion (i.e., a supersymmetric extension of the conformal inversion), see e.g. [18] for the 4D case. However, the restriction of F to compactified Minkowski space is a transformation that belongs to the connected component of the identity of the conformal group, and thus it differs from the 3D conformal inversionx which belongs to the other component of the conformal group. This is why it is appropriate to call F "pseudo inversion." The transformation (3.35) was called "superinversion" in [45]. Our consideration shows that this terminology is somewhat misleading. An extension of conformal inversion (3.37) is unclear to us.

Fibre bundles over compactified Minkowski superspace
Fibre bundles over M 3|2N , such as compactified harmonic/projective superspaces in three spacetime dimensions [48], can be obtained by generalising the construction of subsection 3.2 to include odd supertwistors. 8 Odd supertwistors are destined to parametrise fibres over M 3|2N . In the unitary realisation of the superconformal group, given an odd supertwistor Σ, it is defined by the following two conditions: • it is orthogonal to the even supertwistors T µ which form a basis of the null two-plane P ∈ M 3|2N , with respect to the inner products (3.8a) and (3.8b), • it is defined modulo the equivalence relation for arbitrary a-numbers a µ (i.e. odd elements of the Grassmann algebra).
If the null two-plane P is chosen in the form (3.18), then imposing the first null condition (3.38) and making use of the equivalence relation (3.39), the odd supertwistor may be brought to the form with v = (v I ) being a bosonic N -vector. Here we have used the reality conditions (3.20).
It is important to point out that the second null condition (3.38) also leads to the same explicit expression (3.40) for Σ. Thus the space of odd supertwistors at P may be identified with C N .
As simple examples of fibre bundles over M 3|2N , we can consider odd supertwistor Grassmannians G(m, N ), where m may take values from 1 to N . By definition, the points of G(m, N ) are described by m odd supertwistors Σ i , with i = 1, . . . , m, such that (i) the bodies of Σ i are linearly independent; and (ii) the supertwistors Σ i are defined modulo the equivalence relation In this paper, we are mostly interested in the m = N case, for which Σ i may be chosen in the form: In the remainder of the paper, we use the real realisation of the superconformal group described in section 2.1. In this realisation, only one of the two null conditions (3.38) remains, We recall that for every point z in Minkowski superspace, the supermatrix P = (T 1 , T 2 ) can be chosen as Therefore, instead of the representation (3.40), now every odd supertwistor Σ from the fibre at P(z) can be brought to the form: Finally, the expression (3.42) turns into 4 Two-point and three-point building blocks Here we derive those two-and three-point functions of superspace coordinates which are building blocks for the correlation functions of primary superfields. An alternative derivation was given by Park [45].
As is seen from (3.44) and (3.46), the coset representative s(z), defined by (2.35), is built from the supermatrices corresponding to the even two-plane P(z) and the odd N -plane O(z) according to the rule Here P ∞ denotes the null two-plane corresponding to the origin of the chart U G ⊂ M 3|2N , see subsection 2.4. The two-plane P ∞ is an infinitely separated point from the viewpoint of an observer living in Minkowski superspace.

Infinitesimal superconformal transformations
For our subsequent analysis, it is advantageous to recast the superconformal transformation laws of P(z) and O(z) in terms of the coset representative (4.1). Before discussing the transformation of s(z), we first point out that the infinitesimal transformation z A → z A + δz A , with δz A given by (2.27), can be rewritten as where ξ denotes a first-order differential operator The components of this operator are as follows: Following the terminology of [18], the supervector field ξ is called a conformal Killing supervector field. It may equivalently be defined [48] as the most general solution to the Here the coefficient functions in the right-hand side read and may be thought of as the parameters of local Lorentz, R-symmetry and scale transformations, respectively. The explicit calculation of these parameters gives Under the infinitesimal superconformal transformation associated with Ω, eq. (2.17), the even two-plane P(z) and the odd N -plane O(z) vary as ξP(z) = P(z + δz) − P(z) and ξO(z) = O(z + δz) − O(z), respectively. These variations are computed by the rule 9 where we have introduced the z-dependent S-supersymmetry parameter As concerns the coset representative (4.1), it follows from first principles that for some supermatrix ω(z) belonging to the isotropy subalgebra (2.28). Making use of (4.8), the explicit form of ω(z) is

Two-point functions
By construction, it holds that P sT (z)JP(z) = 0 and P sT (z)JO(z) = 0. Consider two different superspace points z 1 and z 2 . Then we can define two-point functions Making use of (2.25),x 12 andθ 12 can be rewritten with explicit spinor indices as follows: According to (4.8), the above two-point functions transform semi covariantly under the superconformal group Here the variation δ is defined by its action on an n-point function Φ(z 1 , . . . , z n ) to be We note that the definitions (4.12) can be recast in terms of the coset representative (4.1) by introducing the following two-point supermatrix [45] S(z 1 , z 2 ) := (s(z 1 )) sT Js(z 2 ) . (4.16) Using the transformation law of s(z), eq. (4.10), we read off the superconformal transformation of S(z 1 , z 2 ) Let us introduce the following objects with ε = (ε αβ ). Using (4.14) it is easy to check that x 12 2 transforms only under local scale transformations whilex 12 varies only with the local Lorentz parameters Thus, they will naturally appear as two-point building blocks in the correlation functions of primary superfields to be studied in the next sections.
Since the two-point function x αβ 12 has the symmetry property it can be divided into the symmetric and antisymmetric parts The symmetric part is nothing but the bosonic component of the standard two-point superspace interval We stress that both x αβ 12 and x αβ 12 are invariant under supersymmetry while only the latter transforms covariantly under the superconformal group according to (4.14).
To introduce one more important building block, we point out that the pseudo inversion F acts on P(z) by the rule (3.34). One may also work out the action of F on O(z). These results allow us to compute the action of F on the coset representative (2.35) where h(F ; z) is a supermatrix from the isotropy group G P 0 . The latter supermatrix is of the type (2.29) with the following block matrix elements: (4.24b) The N × N matrix R is orthogonal, R T R = ½ N , and unimodular, det R = 1.
Let us denote the two-point analog of the matrix (4.24b) as u 12 . It is defined by and has the properties The sign difference in the right-hand sides of (4.24b) and (4.25) follows from the fact that x 12 has the symmetry propertyx which differs by sign from (2.25).
Here the inverse matrixx −1 12 is expressed in terms ofx 12 aŝ With the use of (4.14) one can check that this matrix transforms as or, equivalently, This shows that u IJ 12 is an invariant tensor two-point function of the superconformal group (compare with the transformation law (5.1) describing a primary superfield). Therefore this object will naturally appear in correlation functions of primary superfields with O(N ) indices.

Three-point functions
Given three superspace points z 1 , z 2 and z 3 , we construct the following three-point functionsX The structures in (4.30b) and (4.30c) follow from (4.30a) by applying cyclic permutations of superspace points. This is why it suffices to study the properties of (4.30a).
With the use of (4.14) one can check thatX 1 andΘ 1 transform as a tensors at the superspace point z 1 and scalar at z 2 and z 3 , Thus, they turn out to be essential building blocks for correlation functions of primary superfields.
Let us consider the squares of the structures in (4.30a) The variations of these objects involve only the parameter of local scale transformation As a consequence, the combination [45] is a superconformal invariant and the superconformal symmetry can fix the form of correlations functions only up to this combination.
The two-point function (4.13a) has the following distributive property As a consequence, the three-point functions (4.30a) obey Hence, similar to the two-point function (4.21), the decomposition of the three-point function X αβ 1 into symmetric and antisymmetric parts reads Given the symmetric object X 1αβ we construct a vector by the standard rule, X 1m = − 1 2 γ αβ m X 1αβ , and introduce analogs of the covariant spinor derivative (2.39) and the corresponding supercharge operator (4.38) They obey standard anticommutation relations where we omit the subscript (1) labeling the superspace point.
There are various identities that involve the three-point functions X i αβ and Θ i α at different points, e.g., These relations allow us to prove the following properties of three-point functions (4.30c) Here D I (i)α is the conventional covariant spinor derivative (2.39) which acts on the superspace coordinates at the point z i .
Given a function f (X 3 , Θ 3 ) depending on the objects (4.30c) one can prove the following differential identities Note that on the left of these identities there are standard covariant spinor derivatives (2.39) while on the right there are generalized derivative and supercharge given in (4.38). The above properties (4.42) will be important in the next sections.
Using the relations (4.40) one can also check that the object (4.34) is invariant under permutations of superspace points, (4.43) Finally, we introduce the following three-point functions which have simple transformation properties. One may see that U IJ 1 transforms as an O(N ) tensor at superspace point z 1 By construction, the matrix U 1 is orthogonal, U T 1 U 1 = ½ N , and unimodular, det U 1 = 1.
It can be expressed in terms of the three-point functions (4.30c) Analogous results hold for U 2 and U 3 .
The matrices U 2 and U 3 are related to U 1 as

Correlation functions of primary superfields
Consider a superfield Φ I A (z) that transforms in a representation T of the Lorentz group with respect to its index A and in a representation D of the R-symmetry group O(N ) with respect to the index I. Such a superfield is called primary of dimension q if its superconformal transformation law is Here ξ is the conformal Killing supervector (4.3), and the z-dependent parameters σ(z), λ αβ (z) and Λ IJ (z) associated with ξ are given in (4.7). The matrices M αβ and R IJ are the Lorentz and O(N ) generators, respectively.
In the non-supersymmetric case, the formalism to construct the correlation functions of primary fields in conformal field theories in diverse dimensions was developed in [9] (see also [13]). In four dimensions, this approach was generalised to N = 1 superconformal field theories formulated in superspace in [26] (see also [50]) as well as to higher N [46]. The correlation functions of primary superfields in three and six dimensions were studied in [45] and [47], respectively. Here we briefly review the 3D formalism of [45] as it will be employed further for constructing correlation functions of conserved current multiplets in 3D superconformal field theories.
The two-point correlation function of the primary superfield Φ I A and its conjugateΦ A I is fixed by the superconformal symmetry up to a single coefficient c and has the form Let Φ, Ψ and Π be primary superfields (with indices suppressed) of dimensions q 1 , q 2 and q 3 , respectively. The three-point correlation function for these superfields can be found with the use of the ansatz where H J 1 J 2 I 3 B 1 B 2 A 3 is a tensor constructed in terms of the three-point functions (4.30) and (4.44). The functional form of this tensor is highly constrained by the following conditions: (i) It should obey the scaling property in order for the correlation function to have the correct transformation law under the superconformal group.
(ii) When some of the superfields Φ, Ψ and Π obey differential equations such as the conservation conditions of conserved current multiplets, the tensor H J 1 J 2 I 3 B 1 B 2 A 3 is constrained by certain differential equations as well. In deriving such equations the identities (4.42) may be useful.
(iii) When two of the superfields Φ, Ψ and Π (or all of them) coincide, the tensor H should obey certain constraints originating from the symmetry under permutations of superspace points, e.g.
where ǫ(Φ) is the Grassmann parity of Φ A I .
These constraints fix the functional form of the tensor H (and, hence, the three-point correlation function) up to a few arbitrary constants.
The procedure described reduces the problem of computing thee-point correlation functions to deriving the single function H subject to the above mentioned constraints. In the next sections we will apply this procedure to compute the two-and three-point correlation functions of the supercurrents and the flavour current multilpets in superconformal field theories with 1 ≤ N ≤ 3.

Correlators in N = 1 superconformal field theory
To start with, we give an example of a classically N = 1 superconformal field theory. It is described by n primary real scalars ϕ of dimension 1/2 with action with λ a coupling constant. This action is invariant under the superconformal transformation The supercurrent of this model [59] is 3) The flavour current multiplet reads A natural generalisation of (6.1) is the most general off-shell 3D N = 1 superconformal sigma model given in [48]. 10 6.1 N = 1 flavour current multiplets In N = 1 supersymmetric field theory, the flavour current multiplet is described by a primary real spinor superfield L α of dimension 3/2 (with its flavour index suppressed) which transforms under superconformal group as and obeys the conservation equation Let us assume that the superconformal field theory under study has several flavour current multiplets Lā α , withā the flavour index. According to the general formula (5.2), the two-point function of such operators is fixed up to one real coefficient a N =1 , assuming that the flavour group is simple. With the relation (4.20) it is easy to see that (6.7) obeys the right symmetry property under the permutation of superspace points, Lā α (z 1 )Lb β (z 2 ) = − Lb β (z 2 )Lā α (z 1 ) . Next, using the explicit expression for x 12αβ given in (4.13a), one may check that (6.7) respects the conservation condition (6.6) Consider now the three-point correlation function Lā α (z 1 )Lb β (z 2 )Lc γ (z 3 ) . Since the superspace coordinates do not carry any flavour group indices, the dependence of the correlation function onā,b andc should factorise in the form of an invariant tensor of the flavour group, which is completely antisymmetric, fābc = f [ābc] , or completely symmetric, dābc = d (ābc) . In principle, the correlator may be a sum of two terms, one of which is proportional to fābc and the other to dābc. In four dimensions, contributions with dābc arise as a consequence of anomalies. In three dimensions, gauge theories are anomaly-free. Therefore, it is natural to expect that the part of Lā α (z 1 )Lb β (z 2 )Lc γ (z 3 ) with dābc should vanish as it was observed in the non-supersymmetric case studied in [9]. Nevertheless, here we start by considering the most general expression for the correlation function including contributions of both types, with fābc and dābc, and then show that the latter vanishes upon imposing all the relevant constrains.
According to the general formula (5.3), we have to look for the three-point correlator in the form Recall that the superfield Lā α is Grassmann odd. Hence, the correlator (6.9) changes its sign when we interchange any pair of superfields in it, e.g. : The most general expressions for these tensors subject to the constraints (6.10) and (6.12) read where c n and d n are some real coefficients and Recall that we use the notation in which X 2 = − 1 2 X αβ X αβ and X k ≡ (X 2 ) k/2 . Note that there is no need to add one more addmisible structure 1 X 3 (X αγ Θ β +X βγ Θ α ) to the list (6.14a), since it is linearly dependent of the others, To fix the values of the coefficients c n and d n in (6.13) we have to take into account the conservation condition (6.6), Making use of (4.42a), this equation imposes the following constraint on the tensors H αβγ (f,d) : Here D α is the generalized covariant spinor derivative defined in (4.38). The equation (6.17) leads to the following constraints on the coefficients c n and d n : To find further constrains on the coefficients, we recall that the correlation function changes its sign if we swap any two superfields in it, e.g., Using the identities (4.40), we find the following corollaries of (6.19): where the right-hand side should be taken with the plus sign for H (f ) and with minus for H (d) . The constraints (6.20) are satisfied under the conditions Thus, from (6.18) and (6.21) we see that all d-coefficients vanish, d n = 0, and therefore H αβγ (d) = 0. Furthermore, only one independent coefficient remains among c n , which we denote by b N =1 = c 2 = −c 3 . Our final expression for the correlator (6.9) is The superfield operator Lā α (z) contains an ordinary conserved flavour current Lā m (x) as its linear in θ component, where | indicates that we have to set θ = 0. From (6.22) we can extract the three-point function Lā αα ′ (x 1 )Lb ββ ′ (x 2 )Lc γγ ′ (x 3 ) by the rule: It is instructive to compare the flavour current correlation function (6.22) with the corresponding non-supersymmetric expression found in [9]. After a straightforward but lengthy calculation (see Appendix D for the technical details) we find Here we have defined According to (D.17), the coefficients b 1 and b 2 are given in terms of b N =1 as follows This is the same result as the one obtained in [9] except for the fact that the two coefficients b 1 and b 2 , which were completely independent in the non-supersymmetric case, are now equal to each other due to supersymmetry.
As pointed out in [10], in 3D conformal field theories an additional parity violating 11 structure can arise in the three-point correlator of flavour currents, However, this structure does not appear upon the N = 1 → N = 0 reduction of our result (6.22) and, hence, it is not consistent with supersymmetry. The same conclusion holds in all cases considered below in this paper. Specifically, the correlators of both flavour current multiplets and supercurrents contain only parity even contributions. 12
(6.29) 11 The parity transformation in question is x m → −x m . The correlator of three flavour currents acquires a minus sign under this transformation. 12 One way to check whether a given contribution is even or odd under parity is to reduce it to the N = 0 case to see whether or not it contains an ε mnp tensor. This is easy to see from the general structure of the supersymmetric result without performing the reduction in detail. We will not discuss details of the reduction of our results to N = 0 in other sections of this paper.
This conservation equation is invariant under the superconformal transformation of J αβγ , which is According to the general discussion in section 5, the two-point function of the supercurrent is given by It is easy to show that the two-point function (6.31) has the right symmetry property under the change of superspace points, Similarly, we can write the most general form for the three-point function that is consistent with the superconformal symmetry. Let us denote by A = (α, α ′ , α ′′ ) a symmetric combination of the three spinor indices, J A ≡ J αα ′ α ′′ . Then If we exchange the first and the second superspace points z 1 ↔ z 2 it follows that X 3 → −X T 3 , Θ 3 → −Θ 3 . Since the supercurrent J αβγ is Grassmann odd the correlation function (6.33) has to change the sign under z 1 ↔ z 2 , A ↔ B. It implies that the function H ABC (X, Θ) has to satisfy The three-point function (6.33) now has the right symmetry property under z 1 ↔ z 2 , but it does not necessarily has the right symmetry under z 1 ↔ z 3 and z 2 ↔ z 3 . Additionally, the function H ABC (X, Θ) is constrained by the conservation law (6.29). Upon the use of identity (4.42a) the latter is translated to (6.37) 13 Here and below we sometimes use a comma to separate various groups of indices. Now our aim is to find the most general solution for H. The standard approach used in 4D superconformal field theories with N = 1 [26] and N = 2 [24] is based on writing the most general ansatz in terms of X and Θ consistent with the symmetries and the scaling property (6.35) and constrain it by the conservation law (6.37). However, because of a large number of tensorial indices it appears to be inefficient as such an ansatz would require to analyse quite a considerable number of possible terms. Hence, we will take a slightly indirect route.
Here we use the fact that every symmetric in (α, β) matrix is proportional to a gammamatrix. We also indicated that the matrices A and B are antisymmetric in (m, n) and C and D are symmetric. The tensors A, B, C, D depend on X m and are symmetric under X m → −X m . Now we will impose the conditions (6.39) as well as the conservation law (6.42a), (6.42b). To begin with, we will impose ∂ α H (αm,βn),γk = 0. Then it is easy to see that That is we find two symmetric traceless tensors C (mnp),k and D (mnp),(kr) . Substituting now (6.45) into (6.44) we find that C (mnp),k and D (mnp),(kr) are related to each other as follows Quite remarkably, eqs. (6.46) allow us to fully solve for D (mnp),(kr) in terms of C (mnp),k . In order to do this we will decompose C (mnp),k and D (mnp),(kr) into irreducible components. To understand which irreducible components are relevant it is convenient to trade each vector index for a pair of spinor ones. Since C (mnp),k and D (mnp),(kr) are symmetric and traceless they become equivalent to symmetric tensors C (α 1 ...α 6 ),(β 1 β 2 ) and D (α 1 ...α 6 ),(β 1 ...β 4 ) . Hence, C contains irreducible components (that is, totally symmetric tensors) of rank 8, 6, 4 and 2, whereas D contains irreducible components of rank 10, 8, 6, 4 and 2 (note that neither C nor D contains the rank 0 representation since the number of α and β indices is different). Now let us recall that all irreducible components must be even functions of X αβ . This means that irreducible tensors of rank 10, 6 and 2 must vanish since they contain an odd number of X αβ . Therefore, in both C and D only irreducible components of rank 8 and rank 4 can contribute. Going back to the vector indices, let us denote the irreducible components of C as C . By construction, all these tensors are symmetric and traceless. It is not hard to construct explicit decompositions of C (mnp),k and D (mnp),(kr) into the irreducible components. The decomposition of C (mnp),k reads (6.47) Here we have taken into account that C (mnp),k is symmetric in (m, n, p). Recalling now that it is also traceless (see eq. (6.45)), η mn C (mnp),k = 0, gives Similarly, we have the following decomposition of D (mnp),(kr) : D (mnp),(kr) = ε mks η ss ′ T (np),r,s ′ + ε nks η ss ′ T (mp),r,s ′ + ε pks η ss ′ T (mn),r,s ′ + ε mrs η ss ′ T (np),k,s ′ + ε nrs η ss ′ T (mp),k,s ′ + ε prs η ss ′ T (mn),k,s ′ , (6.49) where T (np),r,s is given in terms of D . However, it is straightforward to show that such terms will cancel when we substitute them in (6.49) and, hence, they can be ignored. Substituting the irreducible decompositions (6.47), (6.48), (6.49), (6.50), (6.51) into (6.36) yields the solution Thus, the tensor D is fully determined in terms of C.
Finally, let us consider the equation (6.42b) which involves the derivative with respect to X. It is possible to show using (6.46) that (6.42b) is equivalent to a pair of simple equations ∂ m C (mnp),k = 0 , ∂ m D (mnp),(kr) = 0 . (6.53) Now we are ready to construct an explicit solution. It is enough to consider C (mnp),k since D (mnp),(kr) is fully expressed in terms of it. Using the symmetry in (m, n, p), the scaling property (6.35) and the fact that it is an even function of X m we have the following most general ansatz Here we adopt the vector notation X 2 = X m X m and a, b, c, d are some coefficients. Imposing η mn C (mnp),k = 0 gives c = −5a, d = 10a − 5b. Imposing ∂ m C (mnp),k = 0 gives b = 3a, d = −10b − 5c. Thus, we obtain that there is only one independent coefficient which we choose to be a and the remaining three coefficients are given by b = 3a , c = d = −5a . Then, the free parameter, which we denote as d N =1 , will show up as an overall coefficient in the final answer for the correlation function presented below.
To summarise, we have shown that the three-point function of supercurrents in N = 1 superconformal theories is fixed up to one overall coefficient d N =1 . The explicit form of the function H αm,βn,γk is given by The tensors C (mnp),k and D (mnp),(kr) are given by (6.54), (6.56) and (6.49), (6.57), respectively.
Obviously, the correlation function (6.33) changes its sign under permutation of the superspace points z 1 and z 3 with the simultaneous swap of indices A and C As a consequence, the tensor H should obey the following equation However, it appears to be very difficult to check that the tensor (6.58) obeys this equation because of its complicated structure. Alternatively, in Appendix C we demonstrate that the expression (6.58) can be derived as a result of reduction of the N = 2 supercurrent correlation function which will be computed in subsection 7.2. This will proved that (6.58) obeys the required property (6.60).

Correlators in N = 2 superconformal field theory
We start with an example of a classically N = 2 superconformal field theory. It is described by n primary chiral scalars Φ (viewed as a column vector) of dimension 1/2, D α Φ = 0, and their conjugate antichiral superfields Φ † with action 15 Here λ is a dimensionless coupling constant. The supercurrent of this model is [34,35] The action is obviously O(n) invariant. The corresponding flavour current multiplet is with Σ a being the generator of the flavour O(n) group. It is not difficult to check that on-shell the currents (7.2) and (7.3) obey the N = 2 conservation equations in (1.1) and (1.4), respectively. In the free case, λ = 0, the action is U(n) invariant; the corresponding flavour current multiplet is given by (7.3), in which Σ a now stands for the generator of the U(n) group. The free model is trivially superconformal at the quantum level.
A natural generalisation of (7.1) is the most general off-shell 3D N = 2 superconformal sigma model given in [48]. 16

N = 2 flavour current multiplets
The N = 2 flavour current is described by a primary scalar L of dimension 1, which means its superconformal transformation is δL = −ξL − σ(z)L . (7.4) This transformation law is uniquely fixed by requiring the conservation equation to be superconformal.
As in the N = 1 case, we assume that the N = 2 superconformal field theory under study has a set of flavour current multiplets Lā associated with a simple flavour group. Since the superfields Lā carry neither spinor nor R-symmetry indices, their two-point correlation function is simply where a N =2 is a free coefficient. It is straightforward to check that this correlator is symmetric, Lā(z 1 )Lb(z 2 ) = Lb(z 2 )Lā(z 1 ) , and respects the conservation equation (7.5), Our next goal is to work out the most general expression for the three-point function Lā(z 1 )Lb(z 2 )Lc(z 3 ) compatible with all the physical requirements. According to (5.3), we have to make the ansatz and obey the conservation condition The latter constraint is obtained from (7.5) with the use of (4.42a).
The correlation function (7.8) is invariant under exchange of the superspace points z 1 and z 2 and the flavour indicesā andb. As a consequence, the functions H (f ) and H (d) are constrained by The general solutions of the equations (7.9), (7.10) and (7.11) prove to be Here b N =2 andb N =2 are two real coefficients. One can also check that the functions H (f ) and H (d) obey the equations which are corollaries of the following symmetry property Finally we point out that the functions (7.12) can be rewritten in terms of the covariant object X αβ with the use of (4.37) In verifying eq. (7.15a), the N = 2 identity ε IJ Θ I α Θ J β Θ 2 = 0 may be useful. We point out that the expression in parentheses in (7.15b) involves the square of the superconformal invariant (4.34).
It should be stressed that the appearance of the d-term in the flavour current correlation function (7.8) is a novel feature which distinguishes the N = 2 superconformal field theories from the N = 1 ones considered in section 6.1 and from non-supersymmetric ones studied in [9]. In contrast to the four-dimensional theories, in three dimensions this part of the correlation function cannot be considered as an anomaly induced contribution. To understand the role of this part of the correlation function it would be interesting to consider some examples of N = 2 theories in which this contribution is nontrivial. 17 We leave this issue for further studies.

N = 2 supercurrent
The N = 2 supercurrent is described by a primary symmetric second-rank spinor J αβ = J (αβ) of dimension 2, hence its superconformal transformation is This transformation law is uniquely fixed by the condition that the supercurrent conservation equation D α I J αβ = 0 (7.17) is superconformal.
According to the general prescription (5.2), the two-point function for the supercurrent is given by 18) where c N =2 is a real coefficient. It is not difficult to see that this correlator is symmetric, , and respects the conservation equation (7.17), The most general expression for the three-point function for the supercurrent is where, by construction, the function H ρρ ′ ,σσ ′ γγ ′ (X, Θ) obeys the symmetry property Since both the supercurrent J αβ and H αα ′ ,ββ ′ ,γγ ′ are Grassmann even, the three-point function (7.21) has to be symmetric under the exchange z 1 ↔ z 2 , α, α ′ ↔ β, β ′ . Hence, H αα ′ ,ββ ′ ,γγ ′ (X, Θ) satisfies the following symmetry property In addition, H αα ′ ,ββ ′ ,γγ ′ is characterised by the scaling property With the use of eq. (4.42a), the supercurrent conservation condition (7.17) is translated to the following equation for H αα ′ ,ββ ′ ,γγ ′ : Just like in the problem of the three-point correlator for the N = 1 supercurrent considered in section 6.2, it is convenient to trade each pair of spinor indices for vector ones, where H mnp (X, Θ) satisfies the same scaling property as (7.23) as well as H mnp (−X T , −Θ) = H nmp (X, Θ) (7.26) and (γ m ) αβ D I α H mnp (X, Θ) = 0 . (7.27) Unlike the N = 1 case, now it is not hard to list all possible structures consistent with the symmetry (7.26) and the scaling property (7.23). This makes the analysis considerably simpler than in the previous section. Just like in the N = 1 case, it is more convenient to view H as function of X m rather than X αβ . Then the building blocks which can appear in H are Note that there is the following N = 2 identity Taking into account the symmetry property (7.26) and (7.23) we get the following general expression for H: where A i , B i , C i are some coefficients and the tensors H i,mnp , H i,mnp , H i,mnp are explicitly given by , H 4,mnp = ε mnr X p X r X q (ΘΘ) q X 6 ; (7.32) It is easy to realize that H i,mnp , H i,mnp and H i,mnp do not mix in the equation (7.27) and, hence, they must satisfy the conservation law independently. Let us now substitute (7.31), (7.32) and (7.33) in (7.27). This equation will lead to two types of terms: terms linear in Θ and terms proportional to Θ 3 . Clearly, these terms must vanish separately. Let us first consider the terms linear in Θ. Using the identities it is straightforward to show that Thus, H i,mnp and H i,mnp can be ignored and we can concentrate only on H i,mnp in eq. (7.31). Substituting (7.31) in (7.27) and considering only the terms linear in Θ gives the following constraints on the coefficients A i : Similarly, concentrating on the terms cubic in Θ, after straightforward but lengthy calculations we obtain the following system: To derive the system of equations (7.37), it is important to make use of the following N = 2 identity which can easily be obtained by differentiating (7.29).
The systems (7.36) and (7.37) turn out to be consistent and can be solved in terms of one independent coefficient which we choose to be A 1 ≡ A: Thus, the three-point function of the supercurrent is fixed up to a single coefficient A.
Since the three-point function has only one overall coefficient, our result should possess the right symmetry properties under the exchange z 1 ↔ z 3 , z 2 ↔ z 3 . However, since the final result is rather simple and contains only a few terms listed in (7.31), the symmetry under, say, the z 1 ↔ z 3 exchange is not hard to verify. The invariance of the three-point function implies the following equation 18 on the tensor H Using the formulae (4.40) we can relate X 3 with X 1 and Θ 3 with Θ 1 It is now straightforward to substitute eq. (7.31) into (7.41) and verify that it is indeed fulfilled if the coefficients A i satisfy (7.39). More precisely, eq. (7.41) constrains the coefficients A i as follows The system of equations (7.43) is weaker than the system (7.36), (7.37) and is contained there. That is why the conservation law alone fully constrains the coefficients.
To conclude this section, we rewrite explicitly the final result for the tensor H in terms of the objects (4.30): Here we have denoted the overall coefficient by d N =2 .

Correlators in N = 3 superconformal field theory
The off-shell N = 3 superconformal sigma model in three dimensions proposed in [48] is a nontrivial example of classically N = 3 superconformal theories. Its formulation is based on the projective superspace techniques [64,65] (see [66] for a review). An alternative approach to describe off-shell N = 3 hypermultiplets in three dimensions [67] is provided by the harmonic superspace formalism [68,39], see, e.g., [69] for the formulation of the ABJM models [70] in N = 3 harmonic superspace. In the present paper, we will not discuss the harmonic and the projective superspace formulations for the off-shell hypermultiplet as it goes beyond our goals. Here we will simply provide examples of N = 3 supercurrent and flavour current multiplets, and for this it suffices to consider a free on-shell massless N = 3 hypermultiplet. 19 It is described by a primary superfield q i , and its conjugateq i , subject to the equation of motion [67] which is the 3D N = 3 analogue of the famous 4D N = 2 hypermultiplet constraints due to Sohnius [71]. Here D ij α is obtained from D I α by replacing its isovector index with a pair of isospinor ones by the general rule [48] with σ being the Pauli matrices. The hypermultiplet q i transforms in the defining representation of SU(2), which is the double cover of the R-symmetry group SO(3). The SU (2) indices are raised and lowered with the antisymmetric tensors ε ij and ε ij , ε 12 = −ε 12 = 1.
Let us consider a system of n free on-shell hypermultiplets. It is described by a column n-vector q i constrained by (8.1) and its conjugate q † i . The supercurrent J α and a flavour current multiplet Lā ij are given by where Σā is a flavour group generator. With the use of (8.1) it is possible to check that the operators (8.3) obey the N = 3 conservation equations given in (1.1) and (1.4). In the SU(2) notation, these equations read We now turn to studying the correlation functions of the supercurrent and flavour current multiplets in quantum N = 3 superconformal models.

N = 3 flavour current multiplets
The N = 3 flavour current multiplet is described by a primary real isovector L I of dimension 1, which transforms under the superconformal group as and obeys the conservation equation Similar to the N = 1 and N = 2 cases, we assume that the N = 3 superconformal field theory under study has a set of flavour current multiplets L Iā associated with a simple flavour group. According to (5.2), the two-point correlator for these multiplets is with some coefficient a N =3 . It may be checked that this two-function is symmetric, L Iā (z 1 )L Jb (z 2 ) = L Jb (z 2 )L Iā (z 1 ) , and respects the conservation law (8.6), For the three-point function L Iā (z 1 )L Jb (z 2 )L Kc (z 3 ) , we follow (5.3) and make the ansatz where fābc and dābc are antisymmetric and symmetric invariant tensors of the flavour group. The functions H IJK (f,d) should obey the following scaling property The three-point function under consideration has to possess the symmetry property which obeys both equations (8.10) and (8.12). However this term is linearly dependent of the others, (8.19) and therefore the list (8.16) is complete.
The tensor H IJK (f ) in (8.13) is determined by two sectors with functions H IJK (f )n and H IJK (f )n . As will be seen further, these pieces should independently obey the constraints imposed by the conservation condition and symmetry of the correlation function under the permutation of superspace points.
With the aid of the identity (4.42a), the supercurrent conservation law (8.6) leads to the following constraint on H IJK (f,d) Computing the derivatives of the tensors (8.15) and (8.16) and substituting them in the equation (8.20) we find the following constraints on the coefficients c n and d n : In deriving these equations we have used the following N = 3 identities: which are differential consequences of the more general N = 3 identity As is seen from (8.21a), the part of the correlation function with the symmetric tensor dābc vanishes since H IJK (d) = 0. In the rest of this section we concentrate on the derivation of the part of the flavour current correlator with the antisymmetric tensor fābc.
The three-point correlation function has to possess the symmetry property It imposes the following constraint on the tensor H IJK as a consequence of (4.47). This equation gives additional relations among coefficients c i and d i , which are: Comparing these equations with (8.21b) and (8.21c) we see that all coefficients d i vanish while all c i can be expressed in terms of Taking into account these relations, we can rewrite the resulting expression for the tensor H IJK (f ) (8.13) in terms of the matrix (4.44). Our final result for the three-point function is where Here we have used the following relation between the matrix U IJ given by (4.46) and the composites in (8.17) The three-point function (8.28) looks more complicated than its 4D N = 2 counterpart [24]. The reason for that is the isovector notation used for the R-symmetry indices. Switching to the isospinor notation, following the prescription (8.2), should simplify the structure of the correlation function.

N = 3 supercurrent
The N = 3 supercurrent is described by a primary real spinor J α of dimension 3/2, which is characterised by the superconformal transformation law and obeys the conservation equation where c i are some coefficients and One can also check that this expression obeys the equation which is a consequence of the symmetry property

Concluding remarks
In this paper, we have demonstrated that for three-dimensional N -extended superconformal field theories with 1 ≤ N ≤ 3 each of the two-point and three-point functions for the supercurrent is fixed by the symmetries and by the conservation law up to a single overall coefficient. In particular, our results imply that each of the two-and three-point functions for the stress-energy tensor in 3D superconformal theories are fixed up to one coefficient. It is natural to expect that the coefficients in the two-and three-point functions for the supercurrent are related to each other through a Ward identity just like in 4D (super)conformal theories [9,26]. Although the required Ward identities may be derived using the known prepotential formulations for 3D N = 1 supergravity [36] and 3D N = 2 supergravity [37,38], we postpone the study of such a relation for future work. The fact that such correlation functions are constrained up to an overall coefficient makes 3D superconformal theories similar to the well-studied case of 2D conformal field theory.
We have also proved that the three-point function of the flavour current multiplets is determined by a single functional form in the N = 1 and N = 3 cases. The specific feature of the N = 2 case is that two independent structures are allowed for the threepoint function of the flavour current multiplets, but only one of them contributes to the three-point function of the conserved currents contained in these multiplets. This is explicitly demonstrated in Appendix C.
As was shown in [10,11], 3D non-supersymmetric conformal theories can have certain odd parity contributions to three-point functions of the stress-energy tensors and flavour currents. They do not exist for an arbitrary number of space-time dimensions but are a specific 3D (and, perhaps, in general, an odd-dimensional) feature. Such terms do not arise in free conformal field theories but can appear in interacting Chern-Simons theories coupled to parity violating matter. Some general constructions of the parity violating terms in N = 1 superconformal theories were later discussed in [72]. There it was shown that in some examples correlators of conserved currents can also contain parity odd contributions. In our approach we did not distinguish whether various allowed structures are even or odd under parity. For N = 1, 2, 3 we always assumed the most general ansatz. Hence, our analysis demonstrates that odd parity contributions do not appear in the supersymmetric cases for both the flavour current and supercurrent correlators. 20 This is explicitly proved in Appendix D for the N = 1 flavour current multiplets.
It is an interesting problem to generalise the present results to the superconformal theories with N ≥ 4 supersymmetry. This is also postponed for future work.
We hope that the techniques developed in our paper will be useful in the context of generalised higher spin superconformal theories formulated on hyper-superspaces, see [73] and references therein.
Recently, the so-called superembedding formalism in four dimensions [74,75,76], which was originally introduced by Siegel [77] and fully elaborated in [52] under the name bi-supertwistor formalism, has been applied to compute correlation functions of multiplets containing conserved currents in 4D N = 1 superconformal theories [76,78,79]. 21 The 3D N -extended bi-supertwistor formalism was presented in [52]. It would be of interest to apply this approach for an alternative computation of the correlation functions of the supercurrent and flavour current multiplets derived in our paper. The results given in section 3 might be useful for that.
Given a three-vector x m , it can be equivalently described by a symmetric second-rank spinor x αβ defined as The same convention is adopted for the spacetime derivatives, Note also that the square of a vector in terms of spinor indices reads

correlation functions in chiral basis
The case N = 2 is special since the R-symmetry group SO(2) is isomorphic to U(1), and one can define a chiral subspace of the full superspace on which the superconformal group OSp(2|4; R) acts by holomorphic transformations. This appendix is devoted to a brief discussion of the correlation functions involving (anti)chiral superfields.

B.1 (Anti)chiral two-point functions
Instead of the real Grassmann coordinates θ Iα = (θ 1α , θ 2α ), we introduce new complex variables, which have definite U(1) charges with respect to the R-symmetry group. The corresponding spinor covariant derivatives obey the anti-commutation relations which guarantee the existence of a chiral subspace of the full superspace. The crucial features of the chiral subspace are that (i) it is invariant under the N = 2 super-Poincaré group; and (ii) its bosonic y a and fermionic θ α coordinates are annihilated by the operators D γ . Its bosonic coordinate is The superconformal transformation law of the real superspace coordinates, eq. (2.27), implies that the superconformal group acts by holomorphic transformations on the chiral subspace. The superconformal variations of the chiral coordinates y αβ and θ α are δy αβ = a αβ + 4iǭ (α θ β) − λ α γ y γβ − y αγ λ γ β + σy αβ + y αγ y βδ b γδ +iy αγ θ γη β + iy αγ θ βη γ + 2iy βγ θ αη γ + iy αβ θ γη γ , (B.5a) The parameter Λ of U(1) R-symmetry is related to the SO(2) parameters Λ IJ as Λ IJ = ε IJ Λ, where ε IJ is the antisymmetric tensor.
Let us consider the following Q-supersymmetric two-point function which is chiral in its first superspace argument and antichiral in the other, It is related to the two-point function (4.13a) as y αβ 12 = x αβ 12 + 2iθ α 12 θ β 12 . (B.8) For its square we have Using this formula, we get the conjugation rule for y 12 2 : One also finds the following useful identity for the product of y 12 2 andȳ 12 2 y 12 2ȳ 12 Now, consider the orthogonal matrix u IJ 12 given by (4.25) and transforming by the rule (4.29). In this transformation, the matrix Λ IJ (z) is antisymmetric, and thus has one independent component which we denote by Λ(z), Given the matrix u IJ 12 , we construct a complex scalar two-point function v 12 which transforms as δv 12 = i(Λ(z 1 ) − Λ(z 2 ))v 12 . (B.14) Using the explicit expressions for the two-point functions (4.13a) and (4.25) it can be shown that (B.9) and (B.13) are related to each other as Aapplying (4.19) and (B.14), we find the transformation of (B.9) to be δy 12 2 = (σ(z 1 ) + σ(z 2 ))y 12 2 , where the chiral superfield σ includes parameters of local scale and U(1) transformations [48] σ(z) = σ(z) + iΛ(z) ,D α σ(z) = 0 . In this section, we consider some simple correlation functions which involve chiral and antichiral primary superfields. First, we will consider a two-point correlator with chiral and antichiral superfields, and then we will derive the general expression for the three-point correlation function with chiral, antichiral and linear superfields.
Let Φ be a chiral superfield of dimension q with no spinor indices. Its superconformal transformation reads where σ is given by (B.17). This transformation preserves chirality since σ is chiral. Using the two-point function (B.9) it is straightforward to construct the two-point correlator of the chiral superfield Φ and its conjugateΦ, where c is an arbitrary coefficient. Owing to (3.38) this expression automatically possesses correct chirality properties with respect to both arguments and has the right transformation rule because of (B.16).
As an example of a three-point function, we consider the correlator of a linear superfield Gā, a chiral superfield Φā and an antichiral oneΦā. Here the indexā can be considered as a flavour group index. In this case these superfields can be identified with N = 2 superfield components of the N = 3 flavour current studied in sect. 8.1. Assuming that all these three superfields have dimension one, we look for the correlation function with the use of the standard ansatz where the functions H (f,d) should have the following homogeneity property Using the identity (4.42a), the linearity of the superfield Gā, D 2 Gā =D 2 Gā = 0, turns into the following equations for the functions H (f,d) This appendix is devoted to the N = 2 → N = 1 superspace reduction of the threepoint functions for the N = 2 supercurrent and flavour current multiplets.

C.1 The supercurrent correlation function
As discussed in section 1, every N = 2 superconformal field theory is a special N = 1 superconformal field theory. The N = 1 supercurrent J αβγ for this theory is related to its N = 2 supercurrent J αβ by the first equation in (1.7b). As a consequence, the N = 1 supercurrent correlation function (6.33) appears as a result of the N = 2 → N = 1 reduction from (7.20) Recall that here the symbol | means that we have to set θ 2 α to zero after computing the derivatives. In this section we denote the values of the SO(2) indices I = 1, 2 with boldface font to distinguish them from indices of other types.
According to (6.33), the N = 1 supercurrent correlation function is expressed in terms of the tensor H αα ′ α ′′ ,ββ ′ β ′′ ,γγ ′ γ ′′ (or H αm,βn,γk if we trade the pairs of the spinor indices into the vector ones by the rule (6.38)) which was found in the form (6.58) with the tensors C and D given by (6.54), (6.56) and (6.49), (6.57), respectively. In its turn, the N = 2 supercurrent correlation function is represented by the tensor H αα ′ ,ββ ′ ,γγ ′ given explicitly by (7.44). In this section we will show that the former can be derived from the latter by means of the equation (C.1).
To start with, we point out that the expression (7.44) for the tensor H αα ′ ,ββ ′ ,γγ ′ with spinor indices converted into Lorentz ones can be rewritten as where the tensor C mnp,k has the form In the formula (C.2) the dependence on the Grassmann variables is only through the factor γ µν p Θ 1 µ Θ 2 ν while the rest is described by the tensor (C.3) which is a function of X.
It is interesting to note that (C.3) coincides with the similar tensor in eqs. (6.54), (6.56) which was encountered in sect. 6.2. As was already shown there, this tensor is symmetric and traceless over the first three indices C mnp,k = C (mnp),k , η mn C mnp,k = 0 , (C.4) and obeys the differential equation where ∂ m = ∂ ∂X m . We also showed in sect. 6.2 that the equations (C.4) and (C.5) define the form of the tensor (C.3) uniquely, up to an overall coefficient. Now we substitute the expression (7.20) for the N = 2 supercurrent correlation functions into (C.1) and represent it in the following form where in these two terms A and B the derivatives are distributed as follows It is easy to see that the terms with the covariant spinor derivatives distributed in other ways vanish since the expressions like D 2 (1)α x 13ββ ′ = −2iθ 2 13β ′ ε βα die in the |-projection. We will analyse the A and B sectors separately.
We begin by considering the A term. Using the explicit expression for x, eq. (4.13a), we find Next, with the use of (4.42) we get Now we substitute (C.9) and (C.10) into (C.7) and apply simple identities like to represent the A sector of the supercurrent correlation function in the form Here the symmetrisation involves only the underlined indices.
We stress that the covariant spinor derivatives D 2 α in the expression (C.13) act only on the Grassmann variable Θ 2 ν and do not hit the X-dependent tensor C mnp,k since the action of the covariant spinor derivative on any combination of X m is proportional to Θ 2 which dies in the |-projection. Hence, after converting pairs of spinor indices into vector ones and using identities with three-dimensional gamma-matrices (A.3,A.4), the expression (C.13) can be rewritten in the form H αm,βn,γk where Θ δ ≡ Θ 1 δ . The first term here coincides (up to the factor −6) with the corresponding term in (6.58). To match the other terms we need to consider also contributions to H αm,βn,γk from the B part given by (C.8).
In the B sector of the correlation function we need to compute three covariant spinor derivatives of the tensor (C.2), (C.15) In this expression, u 21 23 and u 21 13 are components of the matrix (4.25) which appear in (C.15) owing to the identities (4.42). The factor (x −1 23 ) σ β (x −1 13 ) ρ α in the right-hand side of (C.15) is the right one which is required to form the expression (6.33). Now we have to analyse the remaining piece of this expression.
Using the explicit expressions (4.38) for Q 2 σ and D 2 ρ the first term in the right-hand side of (C.15) can be rewritten as 16) To get the last line we used the fact that in the |-projection only those terms survive in which the derivative D 2 (3)γ acts on Θ 2µ and produces the factor [(x −1 13 ) µ γ − (x −1 23 ) µ γ ]. However, the latter structure is non-covariant in the sense that it cannot be expressed solely in terms of X αβ and Θ I α . Indeed, using the identity (4.35) and the definition of X αβ (4.30), we represent the factor [(x −1 13 ) µ γ − (x −1 23 ) µ γ ] in (C. 16) as The last two terms here are non-covariant. Therefore, they must cancel against similar terms coming from the last two terms in (C.15) To prove the cancellation of non-covariant terms we have to use the fact the tensor H is linear in Θ 1 and can be represented in the form H = Θ 1 κ h κ , for some h κ . Here we suppress all indices of the tensor H as they do not play role in this consideration. Then, using the explicit expression for Θ 1 α (4.30), we observe that the non-covariant terms have the same structure and cancel against each other where D (mn),k,p,r = 1 5 ε pqr η qq ′ C mnq ′ ,k + ε nqk η qq ′ C mq ′ r,t + ε nqr η qq ′ C mq ′ p,k + ε mqp η qq ′ C q ′ nr,k +ε mqr η qq ′ C q ′ np,k + ε lsr X l ∂ s C mnp,k + ε qpl η qq ′ X l ∂ r C mnq ′ ,k −ε qpl η qq ′ X r ∂ l C mnq ′ ,k − ε lqr η qq ′ X q ∂ p C mnq ′ ,k . (C.26) Our final task is to match the tensor (C.26) with the last two terms in (6.58). A straightforward comparison is rather complicated and we will give an indirect proof. For this we will show that (C.26) satisfies all the same equations as D (mnp),k,r from sect. 6.2.
To continue, we contract (C.26) with η pr and ε prq to obtain η pr D (mnp),k,r = 0 , ε rpq D (mnp),k,r + η qq ′ C mnq ′ ,k = 0 , (C. 30) which, using (C.29), imply (6.46). Finally, using eqs. As a result, we found that D (mnp),(kr) satisfies exactly the same equations as in sect. 6.2. On the other hand, we have shown in sect. 6.2 that these equations allow us to fully solve for D in terms of C and such a solution is unique. Since the tensor C in (C.2) coincides with the one from sect. 6.2 we conclude that (C.26) is the same as D (mnp),k,r found in sect. 6.2. This completes our proof.
As a byproduct of the above computations, we find that the coefficients in the threepoint functions of N = 1 and N = 2 supercurrents derived in the sections 6.2 and 7.2 are related to each other as d N =2 = −5d N =1 . (C.32)

C.2 The flavour current correlation function
The N = 2 → N = 1 superspace reduction of the flavour current correlation functions given by (7.8) and (6.9) goes the same way as in the previous section. Therefore here we mention only the essential details of this derivation.
(C. 36) Finally, applying the identity (6.15), the tensor (C.43) can be brought to the form Comparing the last expression with (6.22) we conclude that the coefficients b N =1 and b N =2 are related to each other as b N =1 = 2b N =2 . (C.45)

D Component reduction
The correlation functions of the energy-momentum tensor and flavour currents originate as components in the θ-expansion of the correlation functions for the supercurrent and flavour current multiplets, respectively. In this section, we consider a particular example in which we demonstrate how to derive the correlation function of the flavour current from the corresponding superfield correlator obtained in section 6.1.
We start with the N = 1 flavour current correlator in the form (6.9). Substituting the latter into (6.24), we represent the correlation function as a sum of two pieces In the A sector, we compute the derivatives of the objects x 13 and x 23 using their explicit form (4.13a), With the use of (4.42), the derivatives of the tensor H in (D.2) can be written as Substituting (D.4) and (D.5) into (D.2) we represent the A part in the form The symmetrisation here involves only the underlined indices.