Mixed three-point functions of conserved currents in three-dimensional superconformal field theory
FFebruary, 2021
Mixed three-point functions of conserved currents inthree-dimensional superconformal field theory
Evgeny I. Buchbinder and Benjamin J. Stone
School of Physics M013, The University of Western Australia35 Stirling Highway, Crawley W.A. 6009, Australia
Email: [email protected],[email protected]
Abstract
We consider mixed three-point correlation functions of the supercurrent andflavour current in three-dimensional N = 1 and N = 2 superconformal field theories.Our method is based on the decomposition of the relevant tensors into irreduciblecomponents to guarantee that all possible tensor structures are systematically takeninto account. We show that only parity even structures appear in the correlationfunctions. In addition to the previous results obtained in arXiv:1503.04961, it fol-lows that supersymmetry forbids parity odd structures in three-point functions in-volving the supercurrent and flavour current multiplets. a r X i v : . [ h e p - t h ] F e b ontents N = 1 superconformalfield theory 11 N = 1 superconformal field theory 16 (cid:104) LJ L (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . 164.2 The correlation function (cid:104)
J J L (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . 19 N = 2 superconformal field theory 26 (cid:104) LJ L (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . 286.3 The correlation function (cid:104)
J J L (cid:105) . . . . . . . . . . . . . . . . . . . . . . . . 31
A 3D conventions and notation 37References 38 Introduction
It is a well-known property of conformal field theories that the functional form of two-and three-point functions of conserved currents such as the energy-momentum tensor andvector current are fixed up to finitely many parameters. In [1, 2] a systematic formalismwas developed to construct two- and three-point functions of primary operators in diversedimensions. The method was based on properly imposing the relevant symmetries arisingfrom scale transformations and permutations of points as well as the conservation lawsfor the conserved currents, (see also refs. [3–10] for earlier work). More recently it wasshown in [11] that a peculiar feature of three-dimensional (and perhaps in general, odd-dimensional) conformal field theories is the appearance of parity violating contributionsin three-point functions of conserved currents. These structures were overlooked in theoriginal study by Osborn and Petkou [1] (also [2]), and have since been shown to arisein Chern–Simons theories interacting with parity violating matter. Parity violating (orparity odd) structures were also studied in [12–20]. Recently they were also studied inmomentum space [21]. In contrast with the non-supersymmetric case studied in [1, 2], supersymmetry imposesadditional restrictions on the structure of three-point functions of conserved currents. Insupersymmetric field theories the energy-momentum tensor is replaced with the super-current multiplet [31], which contains the energy-momentum tensor, the supersymmetrycurrent and additional components such as the R -symmetry current. Similarly, a con-served vector current becomes a component of the flavour current supermultiplet. Thegeneral formalism to construct the two- and three-point functions of primary operatorsin three-dimensional superconformal field theories was developed in [32–35]. Within thisformalism it was shown in [33] that the three-point function of the supercurrent (and,hence, of the energy-momentum tensor) in three-dimensional N = 1 superconformal the-ory is comprised of only one tensor structure. It was also shown that the three-pointfunction of the non-abelian flavour current (and, hence, the three-point function of con-served vector currents) also contains only one tensor structure. In both cases the tensorstructures are parity even.The aim of this paper is to apply the approach of [33] to the case of mixed correla-tors involving the supercurrent and flavour current multiplets. Our method is based ona systematic decomposition of the relevant tensors into irreducible components, which Parity even correlation functions in momentum space were discussed in [22–30]. A similar formalism in four dimensions was developed in [36–38] and in six dimensions in [39]. N = 1 superconformalsymmetry up to an overall coefficient. We also show that the three-point function in-volving two supercurrents and one flavour current vanishes. In section 5 we present asystematic discussion regarding the absence of parity violating structures in our results.In section 6 we generalise our method to superconformal theories with N = 2 supersym-metry. We show that both mixed correlators are fixed up to an overall coefficient. Inappendix A we summarise our three-dimensional notation and conventions.The non-vanishing of the three-point function of two supercurrents and one flavourcurrent is quite a surprise given that a similar three-point function vanishes in the N = 1case. Naively it appears to be a contradiction, as any theory with N = 2 supersymmetryis also a theory with N = 1 supersymmetry. Hence the number of independent tensorstructures cannot grow as one increases the number of supersymmetries. Nevertheless,we explain that our results in the N = 1 and N = 2 cases are fully consistent.In this paper we concentrate on mixed correlators in theories with N = 1 and N =2 superconformal symmetry. Mixed correlators in conformal field theories with higherextended supersymmetry will be studied elsewhere.3 Superconformal building blocks
The formalism to construct correlation functions of primary operators for conformalfield theories in general dimensions was first elucidated in [1] using an efficient grouptheoretic formalism. In four dimensions the method was then extended to the case of N = 1 supersymmetry in [36, 37, 41], and was later generalised to higher N in [38]. Herewe review the pertinent details of the three-dimensional formalism [32, 33] necessary toconstruct correlation functions of the 3D supercurrent and flavour current multiplets. Let us begin by reviewing infinitesimal superconformal transformations and the trans-formation laws of primary superfields. This section closely follows the notation of [42–44]. Consider 3D N -extended Minkowski superspace M | N , parameterised by coordi-nates z A = ( x a , θ αI ), where a = 0 , , α = 1 , I = 1 , ..., N is the R -symmetry index. The 3D N -extended superconformal group cannotact by smooth transformations on M | N , in general only infinitesimal superconformaltransformations are well defined. Such a transformation δz A = ξz A ⇐⇒ δx a = ξ a ( z ) + i( γ a ) αβ ξ αI ( z ) θ βI , δθ αI = ξ αI ( z ) (2.1)is associated with the real first-order differential operator ξ = ξ A ( z ) ∂ A = ξ a ( z ) ∂ a + ξ αI ( z ) D Iα , (2.2)which satisfies the master equation [ ξ, D Iα ] ∝ D Jβ . From the master equation we find ξ αI = i6 D βI ξ αβ , (2.3)which implies the conformal Killing equation ∂ a ξ b + ∂ b ξ a = 23 η ab ∂ c ξ c . (2.4)The solutions to the master equation are called the conformal Killing supervector fields ofMinkowski superspace [43, 45]. They span a Lie algebra isomorphic to the superconformalalgebra osp ( N | R ). The components of the operator ξ were calculated explicitly in [32],and are found to be ξ αβ = a αβ − λ αγ x γβ − x αγ λ γβ + σx αβ + 4i (cid:15) ( αI θ β ) I + 2iΛ IJ θ αJ θ βI + x αγ x βδ b γδ + i b ( αδ x β ) δ θ − b αβ θ θ − η γI x γ ( α θ β ) I + 2 η ( αI θ β ) I θ , (2.5a)4 αI = (cid:15) αI − λ αβ θ βI + 12 σθ αI + Λ IJ θ αJ + b βγ x βα θ γI + η βJ (2i θ βI θ αJ − δ IJ x βα ) , (2.5b) a αβ = a βα , λ αβ = λ βα , λ αα = 0 , b αβ = b βα , Λ IJ = − Λ JI . (2.6)The bosonic parameters a αβ , λ αβ , σ , b αβ , Λ IJ correspond to infinitesimal translations,Lorentz transformations, scale transformations, special conformal transformations and R -symmetry transformations respectively, while the fermionic parameters (cid:15) αI and η αI cor-respond to Q -supersymmetry and S -supersymmetry transformations. Furthermore, theidentities D I [ α ξ Jβ ] ∝ ε αβ , D I ( α ξ Jβ ) ∝ δ IJ , D ( I [ α ξ J ) β ] ∝ δ IJ ε αβ , (2.7)imply that [ ξ, D Iα ] = − ( D Iα ξ βJ ) D Jβ = λ αβ ( z ) D Iβ + Λ IJ ( z ) D Jα − σ ( z ) D Iα , (2.8) λ αβ ( z ) = − N D I ( α ξ Iβ ) , Λ IJ ( z ) = − D [ Iα ξ J ] α , σ ( z ) = 1 N D Iα ξ αI . (2.9)The local parameters λ αβ ( z ), Λ IJ ( z ), σ ( z ) are interpreted as being associated with com-bined special-conformal/Lorentz, R -symmetry and scale transformations respectively, andappear in the transformation laws for primary tensor superfields. For later use let’s alsointroduce the z -dependent S -supersymmetry parameter η Iα ( z ) = − i2 D Iα σ ( z ) . (2.10)Explicit calculations of the local parameters give [32] λ αβ ( z ) = λ αβ − x γ ( α b β ) γ − i2 b αβ θ + 2i η ( αI θ β ) I , (2.11a)Λ IJ ( z ) = Λ IJ + 4i η α [ I θ J ] α + 2i b αβ θ αI θ βJ , (2.11b) σ ( z ) = σ + b αβ x αβ + 2i θ αI η αI , (2.11c) η αI ( z ) = η αI − b αβ θ βI . (2.11d)Now consider a generic tensor superfield Φ IA ( z ) transforming in a representation T ofthe Lorentz group with respect to the index A , and in the representation D of the R -symmetry group O ( N ) with respect to the index I . Such a superfield is called primarywith dimension q if its superconformal transformation law is δ Φ IA = − ξ Φ IA − qσ ( z )Φ IA + λ αβ ( z )( M αβ ) AB Φ IB + Λ IJ ( z )( R IJ ) I J Φ JA , (2.12)where ξ is the superconformal Killing vector, σ ( z ), λ αβ ( z ), Λ IJ ( z ) are the z -dependentparameters associated with ξ , and the matrices M αβ and R IJ are the Lorentz and O ( N )generators respectively. We assume the representations T and D are irreducible. .2 Two-point functions Given two superspace points z and z , we can define the two-point functions x αβ = ( x − x ) αβ + 2i θ ( α I θ β )2 I − i θ α I θ β I , θ αI = θ αI − θ αI , (2.13)which transform under the superconformal group as follows˜ δ x αβ = (cid:18) δ αγ σ ( z ) − λ αγ ( z ) (cid:19) x γβ + x αγ (cid:18) δ γβ σ ( z ) − λ γβ ( z ) (cid:19) , (2.14a)˜ δθ α I = (cid:18) δ αβ σ ( z ) − λ αβ ( z ) (cid:19) θ β I − x αβ η βI ( z ) + Λ IJ ( z ) θ α J . (2.14b)Here the total variation ˜ δ is defined by its action on an n -point function Φ( z , ..., z n ) as˜ δ Φ( z , ..., z n ) = n (cid:88) i =1 ξ z i Φ( z , ..., z n ) . (2.15)It should be noted that (2.14b) contains an inhomogeneous piece in its transformation law,hence it will not appear as a building block in two- or three-point functions. Due to theuseful property, x αβ = − x βα , the two-point function (2.13) can be split into symmetricand antisymmetric parts as follows x αβ = x αβ + i2 ε αβ θ , θ = θ α I θ αI . (2.16)The symmetric component x αβ = ( x − x ) αβ + 2i θ ( α I θ β )2 I , (2.17)is recognised as the bosonic part of the standard two-point superspace interval. Next letus introduce the two-point objects x = − x αβ x αβ , (2.18a)ˆ x αβ = x αβ (cid:112) x , ˆ x αγ ˆ x γβ = δ αβ . (2.18b)Hence, we find ( x − ) αβ = − x βα x . (2.19)Under superconformal transformations, (2.18a) transforms with local scale parameters,while (2.18b) transforms with local Lorentz parameters˜ δ x = ( σ ( z ) + σ ( z )) x , (2.20a)˜ δ ˆ x αβ = − λ αγ ( z ) ˆ x γβ − ˆ x αγ λ γβ ( z ) . (2.20b)6hus, both objects are essential in the construction of correlation functions of primarysuperfields. We also have the useful differential identities D I (1) γ x αβ = − θ Iβ δ αγ , D I (1) α x αβ = − θ Iβ , (2.21)where D I ( i ) α is the standard covariant spinor derivative (A.16) acting on the superspacepoint z i . Finally, for completeness, the SO( N ) structure of primary superfields in corre-lation functions is addressed by the N × N matrix u IJ = δ IJ + 2i θ Iα ( x − ) αβ θ Jβ , (2.22)which is orthogonal and unimodular, u IK u KJ = δ IJ , det u = 1 . (2.23)The infinitesimal variation of this matrix is˜ δu IJ = Λ IK ( z ) u KJ − u IK Λ KJ ( z ) . (2.24)Hence, (2.22) is expected to appear in the construction of correlation functions of primarysuperfields with SO( N ) indices.The two-point correlation function of a primary superfield Φ IA and its conjugate ¯Φ BJ is fixed by the superconformal symmetry as follows (cid:104) Φ IA ( z ) ¯Φ BJ ( z ) (cid:105) = c T AB ( ˆ x ) D I J ( u )( x ) q , (2.25)where c is a constant coefficient. The denominator of the two-point function is deter-mined by the conformal dimension of Φ IA , which guarantees that the correlation functiontransforms with the appropriate weight under scale transformations. Given three superspace points z i , i = 1 , ,
3, one can define the three-point buildingblocks Z i = ( X i , Θ i ) as follows: X αβ = − ( x − ) αγ x γδ ( x − ) δβ , Θ I α = ( x − ) αβ θ Iβ − ( x − ) αβ θ Iβ , (2.26a) X αβ = − ( x − ) αγ x γδ ( x − ) δβ , Θ I α = ( x − ) αβ θ Iβ − ( x − ) αβ θ Iβ , (2.26b) X αβ = − ( x − ) αγ x γδ ( x − ) δβ , Θ I α = ( x − ) αβ θ Iβ − ( x − ) αβ θ Iβ . (2.26c)7hese objects, along with their corresponding transformation laws, may be obtained fromone-another by cyclic permutation of superspace points. The building blocks transformcovariantly under the action of the superconformal group:˜ δ X αβ = λ αγ ( z ) X γβ + X αγ λ γβ ( z ) − σ ( z ) X αβ , (2.27a)˜ δ Θ I α = (cid:18) λ αβ ( z ) − δ αβ σ ( z ) (cid:19) Θ I β + Λ IJ ( z ) Θ J α . (2.27b)Therefore (2.26a), (2.26b) and (2.26c) will appear as building blocks in three-point cor-relations functions. It should be noted that under scale transformations of superspace, z A = ( x a , θ α ) (cid:55)→ z (cid:48) A = ( λ − x a , λ − θ α ), the three-point building blocks transform as Z = ( X , Θ) (cid:55)→ Z (cid:48) = ( λ X , λ Θ). Next we define X = − X αβ X αβ = x x x , Θ = Θ Iα Θ I α , (2.28)which, due to (2.27a) and (2.27b), have the transformation laws˜ δ X = − σ ( z ) X , ˜ δ Θ = − σ ( z ) Θ . (2.29)We also define the inverse of X , ( X − ) αβ = − X βα X , (2.30)and introduce useful identities involving X i and Θ i at different superspace points, e.g., x αα (cid:48) X α (cid:48) β (cid:48) x β (cid:48) β = − ( X − ) βα , (2.31a)Θ I γ x γδ X δβ = u IJ Θ J β . (2.31b)As a consequence of (2.29), we can identify the three-point superconformal invariantΘ (cid:112) X ⇒ ˜ δ (cid:18) Θ (cid:112) X (cid:19) = 0 . (2.32)Hence, the superconformal symmetry fixes the functional form of three-point correlationfunctions up to this combination. Indeed, using (2.31a) and (2.31b) one can show thatthe superconformal invariant is also invariant under permutation of superspace points, i.eΘ (cid:112) X = Θ (cid:112) X = Θ (cid:112) X . (2.33)8he three-point objects (2.26a), (2.26b) and (2.26c) have many properties similar to thoseof the two-point building blocks. After decomposing X into symmetric and antisymmet-ric parts similar to (2.16) we have X αβ = X αβ − i2 ε αβ Θ , X αβ = X βα , (2.34)where the symmetric spinor X αβ can be equivalently represented by the three-vector X m = − ( γ m ) αβ X αβ . It is now convenient to introduce analogues of the covariantspinor derivative and supercharge operators involving the three-point objects, D I (1) α = ∂∂ Θ α I + i( γ m ) αβ Θ Iβ ∂∂X m , Q I (1) α = i ∂∂ Θ α I + ( γ m ) αβ Θ Iβ ∂∂X m , (2.35)which obey the standard commutation relations (cid:8) D I ( i ) α , D J ( i ) β (cid:9) = (cid:8) Q I ( i ) α , Q J ( i ) β (cid:9) = 2i δ IJ ( γ m ) αβ ∂∂X mi . (2.36)Some useful identities involving (2.35) are D I (1) γ X αβ = − ε γβ Θ I α , Q I (1) γ X αβ = − ε γα Θ I β . (2.37)We must also account for the fact that various primary superfields obey certain differentialequations. Using (2.21) we arrive at the following D I (1) γ X αβ = 2i( x − ) αγ u IJ Θ J β , D I (1) α Θ J β = − ( x − ) βα u IJ , (2.38a) D I (2) γ X αβ = 2i( x − ) βγ u IJ Θ J β , D I (2) α Θ J β = ( x − ) βα u IJ . (2.38b)Now given a function f ( X , Θ ), there are the following differential identities which ariseas a consequence of (2.37), (2.38a) and (2.38b): D I (1) γ f ( X , Θ ) = ( x − ) αγ u IJ D Jα (3) f ( X , Θ ) , (2.39a) D I (2) γ f ( X , Θ ) = i( x − ) αγ u IJ Q Jα (3) f ( X , Θ ) . (2.39b)These will prove to be essential for imposing differential constraints on correlation func-tions, e.g. those arising from conservation equations in the case of correlators involvingthe supercurrent and flavour current multiplets.Finally, for completeness, let us introduce the three-point objects which take care ofthe R -symmetry structure of correlation functions. We define U IJ = u IK u KL u LJ = δ IJ + 2iΘ I α ( X − ) αβ Θ J β , (2.40)9hich transforms as an O( N ) tensor at z ,˜ δU IJ = Λ IK ( z ) U KJ − U IK Λ KJ ( z ) . (2.41)and is orthogonal and unimodular by construction. The others are obtained by cyclicpermutation of superspace points, and are related by the useful identities U IJ = u IK U KL u LJ , U IJ = u IK U KL u LJ . (2.42)As concerns three-point correlation functions; let Φ, Ψ, Π be primary superfieldswith conformal dimensions q , q and q respectively. The three-point function may beconstructed using the general expression (cid:104) Φ I A ( z ) Ψ I A ( z ) Π I A ( z ) (cid:105) = (2.43) T (1) A B ( ˆ x ) T (2) A B ( ˆ x ) D (1) I J ( u ) D (2) I J ( u )( x ) q ( x ) q H J J I B B A ( X , Θ , U ) , where the tensor H I I I A A A is highly constrained by the superconformal symmetry as fol-lows: (i) Under scale transformations of superspace the correlation function transforms as (cid:104) Φ I A ( z (cid:48) ) Ψ I A ( z (cid:48) ) Π I A ( z (cid:48) ) (cid:105) = ( λ ) q + q + q (cid:104) Φ I A ( z ) Ψ I A ( z ) Π I A ( z ) (cid:105) , (2.44)which implies that H obeys the scaling property H I I I A A A ( λ X , λ Θ , U ) = ( λ ) q − q − q H I I I A A A ( X , Θ , U ) , ∀ λ ∈ R \ { } . (2.45)This guarantees that the correlation function transforms correctly under conformaltransformations. (ii) If any of the fields Φ, Ψ, Π obey differential equations, such as conservation lawsin the case of conserved current multiplets, then the tensor H is also constrainedby differential equations. Such constraints may be derived with the aid of identities(2.39a), (2.39b). (iii) If any (or all) of the superfields Φ, Ψ, Π coincide, the correlation function possessessymmetries under permutations of superspace points, e.g. (cid:104) Φ I A ( z ) Φ I A ( z ) Π I A ( z ) (cid:105) = ( − (cid:15) (Φ) (cid:104) Φ I A ( z ) Φ I A ( z ) Π I A ( z ) (cid:105) , (2.46)10here (cid:15) (Φ) is the Grassmann parity of Φ. As a consequence, the tensor H obeysconstraints which will be referred to as “point-switch identities”. To analyse theseconstraints, we note that under permutations of any two superspace points, thethree-point building blocks transform as X αβ ↔ −→ − X βα , Θ I α ↔ −→ − Θ I α , (2.47a) X αβ ↔ −→ − X βα , Θ I α ↔ −→ − Θ I α , (2.47b) X αβ ↔ −→ − X βα , Θ I α ↔ −→ − Θ I α . (2.47c)The constraints above fix the functional form of H (and therefore the correlation func-tion) up to finitely many parameters. Hence the procedure described above reduces theproblem of computing three-point correlation functions to deriving the tensor H subjectto the above constraints. In the next sections, we will apply this formalism to computethree-point correlation functions involving the supercurrent and flavour current multiplets. N =1 superconformal field theory The 3D, N = 1 conformal supercurrent is a primary, dimension 5 / J αβγ , which contains the three-dimensional energy-momentum tensor alongwith the supersymmetry current [45–47]. It obeys the conservation equation D α J αβγ = 0 , (3.1)and has the following superconformal transformation law: δJ αβγ = − ξJ αβγ − σ ( z ) J αβγ + 3 λ ( z ) δ ( α J βγ ) δ . (3.2)The N = 1 supercurrent may be derived from, for example, supergravity prepotentialapproaches [45] or the superfield Noether procedure [48, 49].The general formalism in section 2 allows the two-point function to be determined upto a single real coefficient: (cid:104) J αβγ ( z ) J α (cid:48) β (cid:48) γ (cid:48) ( z ) (cid:105) = i b N =1 x αα (cid:48) x ββ (cid:48) x γ ) γ (cid:48) ( x ) . (3.3)11t is then a simple exercise to show that the two-point function has the right symmetryproperties under permutation of superspace points (cid:104) J αβγ ( z ) J α (cid:48) β (cid:48) γ (cid:48) ( z ) (cid:105) = −(cid:104) J α (cid:48) β (cid:48) γ (cid:48) ( z ) J αβγ ( z ) (cid:105) , (3.4)and also satisfies D α (1) (cid:104) J αβγ ( z ) J α (cid:48) β (cid:48) γ (cid:48) ( z ) (cid:105) = 0 . (3.5)Next let’s consider the 3D N = 1 flavour current, which is represented by a primary,dimension 3 / L α obeying the conservation equation D α L α = 0 . (3.6)It transforms covariantly under the superconformal group as δL α = − ξL α − σ ( z ) L α + λ ( z ) αβ L β . (3.7)We can also consider the case when there are several flavour current multiplets (repre-sented by the flavour index, ¯ a ) corresponding to a simple flavour group. According togeneral formalism in section 2, the two-point function for N = 1 flavour current multipletsis fixed up to a single real coefficient a N =1 (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) (cid:105) = i a N =1 δ ¯ a ¯ b x αβ ( x ) . (3.8)It is easy to see that the two-point function obeys the correct symmetry properties underpermutation of superspace points, (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) (cid:105) = −(cid:104) L ¯ bβ ( z ) L ¯ aα ( z ) (cid:105) . One can also checkthat it satisfies the conservation equation (3.6) D α (1) (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) (cid:105) = 0 . (3.9)Three-point correlation functions of the flavour current and particularly the supercurrentare considerably more complicated, and were derived in [33, 34]. However, correlators of combinations of these fields (mixed correlators) were not studied previously and will beanalysed in section 4. The tensor structure and the conservation law of the N = 1 and N = 2 flavour currents follow fromthe structure of unconstrained prepotentials for N = 1 and N = 2 vector multiplets [50–53]. .2 Correlation functions of conserved current multiplets The possible three-point correlation functions that may be constructed from the con-served N = 1 supercurrent and flavour current multiplets are: (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) , (cid:104) J A ( z ) J B ( z ) J C ( z ) (cid:105) , (3.10) (cid:104) L ¯ aα ( z ) J A ( z ) L ¯ bβ ( z ) (cid:105) , (cid:104) J A ( z ) J B ( z ) L ¯ aα ( z ) (cid:105) , (3.11)where A , B , C each denote a totally symmetric combination of three spinor indices. Thecorrelators (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) and (cid:104) J A ( z ) J B ( z ) J C ( z ) (cid:105) were studied in [33]. Beforewe compute the mixed correlators, let us demonstrate our method on the three-pointfunction (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) , which is comparatively straightforward.The general form of the flavour current three-point function is: (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) = f ¯ a ¯ b ¯ c x αα (cid:48) x ββ (cid:48) ( x ) ( x ) H α (cid:48) β (cid:48) γ ( X , Θ ) , (3.12)The correlation function is required to satisfy the following properties: (i) Scaling constraint: Under scale transformations the correlation function must transform as (cid:104) L ¯ aα ( z (cid:48) ) L ¯ bβ ( z (cid:48) ) L ¯ cγ ( z (cid:48) ) (cid:105) = ( λ ) / (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) , (3.13)which gives rise to the homogeneity constraint on H : H αβγ ( λ X , λ Θ) = ( λ ) − / H αβγ ( X , Θ) . (3.14) (ii) Differential constraints: The conservation equation for the flavour current results in D α (1) (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) = 0 . (3.15)Using identities (2.39a), (2.39b), we obtain a differential constraint on H : D α H αβγ ( X , Θ) = 0 . (3.16)We need not consider the conservation law at z as we can use an algebraic constraintinstead. Here we consider only the contribution proportional to the totally antisymmetric structure constants f ¯ a ¯ b ¯ c . Similarly, one can consider the contribution totally symmetric in flavour indices. However, thiscontribution vanishes [33] so it is omitted here. iii) Point permutation symmetry: The symmetry under permutation of points ( z and z ) results in the followingconstraint on the correlation function: (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) = −(cid:104) L ¯ bβ ( z ) L ¯ aα ( z ) L ¯ cγ ( z ) (cid:105) , (3.17)which constrains the tensor H so that H αβγ ( X , Θ) = H βαγ ( − X T , − Θ) . (3.18)On the other hand the symmetry under permutation of points z and z results in (cid:104) L ¯ aα ( z ) L ¯ bβ ( z ) L ¯ cγ ( z ) (cid:105) = −(cid:104) L ¯ cγ ( z ) L ¯ bβ ( z ) L ¯ aα ( z ) (cid:105) , (3.19)which gives rise to the point-switch identity H αβγ ( X , Θ ) = x γ (cid:48) γ ( x − ) αα (cid:48) x β (cid:48) σ X σβ X x H γ (cid:48) β (cid:48) α (cid:48) ( − X T1 , − Θ ) . (3.20)To solve this problem systematically let’s decompose the tensor H into irreduciblecomponents: H αβγ ( X , Θ) = (cid:88) i c i H i αβγ ( X , Θ) . (3.21)It is also more convenient to work with X m instead of X αβ . We have H αβγ = ε αβ Θ γ A ( X ) , (3.22a) H αβγ = ε αβ ( γ a ) γδ Θ δ B a ( X ) , (3.22b) H αβγ = ( γ a ) αβ Θ γ C a ( X ) , (3.22c) H αβγ = ( γ a ) αβ ( γ b ) γδ Θ δ D ab ( X ) . (3.22d)Here we have used the fact that every matrix anti-symmetric in α, β is proportional to ε αβ , every matrix symmetric in α, β is proportional to a gamma-matrix, and that since H is Grassmann odd it follows that H is linear in Θ due to Θ α Θ β Θ γ = 0. Due to thescaling property (3.14) it follows that the functions A, B, C, D have dimension −
2. Fromeq. (3.18) it also follows that A ( X ) = A ( − X ) , B a ( X ) = B a ( − X ) , (3.23a) C a ( X ) = − C a ( − X ) , D ab ( X ) = − D ab ( − X ) . (3.23b)14t is easy to see that the conservation equation (3.16) splits into the two independentequations ∂ α H αβγ = 0 , (3.24a)( γ t ) ατ Θ τ ∂ t H αβγ = 0 . (3.24b)Imposing (3.24a) results in the algebraic equations A ( X ) = − D aa ( X ) , C a ( X ) = B a ( X ) + (cid:15) amn D mn ( X ) . (3.25)While on the other hand from (3.24b) we obtain ∂ a (cid:8) B a ( X ) + C a ( X ) − (cid:15) amn D mn ( X ) (cid:9) = 0 , (3.26a) ∂ t A ( X ) + (cid:15) tma ∂ m B a ( X ) − (cid:15) tma ∂ m C a ( X ) (3.26b) − ∂ m D mt ( X ) + ∂ t D aa ( X ) − ∂ m D tm ( X ) = 0 . Using eqs. (3.25), (3.26a), (3.26b) we obtain that B a and D ab satisfy ∂ a B a ( X ) = 0 , ∂ a D ab ( X ) = 0 . (3.27)Thus, the problem is reduced to finding transverse tensors B a and D ab of dimension − A and C are then found using eq. (3.25). It is not difficultto show that the solution to this problem is given by A ( X ) = 0 , B a ( X ) = 0 , (3.28a) C a ( X ) = X a X , D ab ( X ) = (cid:15) abc X c X , (3.28b)with c = − c . Hence this correlation function is fixed up to a single real coefficientwhich we denote d N =1 Converting back to spinor notation we find H αβγ ( X , Θ) = i d N =1 X (cid:110) X αβ Θ γ − ε αγ X βδ Θ δ − ε βγ X αδ Θ δ (cid:111) . (3.29)One may also check that this solution satisfies the point-switch identity (3.20). Thisagrees with the result in [33], which was computed in a different way. Our method hasthe advantage that it systematically takes care of all possible irreducible components of H and, hence, is more useful when H is a tensor of high rank. Note that since Θ α Θ β Θ γ = 0 we can replace X with X in (3.29). Mixed correlators in N = 1 superconformal fieldtheory (cid:104) LJ L (cid:105)
Let us first consider the correlation function (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) . Using thegeneral expression (2.43), it has the form (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) = δ ¯ a ¯ b ˆ x αα (cid:48) ˆ x γ γ (cid:48) ˆ x γ γ (cid:48) ˆ x γ ) γ (cid:48) ( x ) / ( x ) / H α (cid:48) β,γ (cid:48) γ (cid:48) γ (cid:48) ( X , Θ ) , (4.1)where H is totally symmetric in three of its indices, H αβ,γ γ γ = H αβ, ( γ γ γ ) . The corre-lation function is also required to satisfy: (i) Scaling constraint: Under scale transformations the correlation function transforms as (cid:104) L ¯ aα ( z (cid:48) ) J γ γ γ ( z (cid:48) ) L ¯ bβ ( z (cid:48) ) (cid:105) = ( λ ) / (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) , (4.2)which implies that we have the following homogeneity constraint on H : H αβ,γ γ γ ( λ X , λ Θ) = ( λ ) − / H αβ,γ γ γ ( X , Θ) . (4.3) (ii) Differential constraints: The differential constraints on the flavour current and supercurrent result in thefollowing constraints on the correlation function: D α (1) (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) = 0 , (4.4a) D γ (2) (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) = 0 . (4.4b)Using identities (2.39a), (2.39b), these result in the following differential constraintson H : D α H αβ,γ γ γ ( X , Θ) = 0 , (4.5a) Q γ H αβ,γ γ γ ( X , Θ) = 0 . (4.5b) (iii) Point permutation symmetry: z and z ) results in the followingconstraint on the correlation function: (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) = −(cid:104) L ¯ bβ ( z ) J γ γ γ ( z ) L ¯ aα ( z ) (cid:105) , (4.6)which results in the point-switch identity H αβ,γ γ γ ( X , Θ ) = − x β (cid:48) β ( x − ) αα (cid:48) x γ (cid:48) δ X δ γ x γ (cid:48) δ X δ γ x γ (cid:48) δ X δ γ X x × H β (cid:48) α (cid:48) ,γ (cid:48) γ (cid:48) γ (cid:48) ( − X T1 , − Θ ) . (4.7)Thus we need to solve for the tensor H subject to the constraints (4.3), (4.5a), (4.5b)and (4.7). To start with we combine two of the three γ indices into a vector index, andimpose a γ -trace constraint to remove the component anti-symmetric in γ , γ , H αβ,γ γ γ = ( γ m ) γ γ H αβ,γ m , ( γ m ) τγ H αβ,γm = 0 . (4.8)Since our correlator is Grassmann odd the function H αβ,γm must be linear in Θ. Justlike the flavour current three-point function, linearity in Θ implies that the differentialconstraints (4.5a) and (4.5b) are respectively equivalent to ∂ α H αβ,γm = 0 , ( γ t ) ατ Θ τ ∂ t H αβ,γm = 0 , (4.9a) ∂ γ H αβ,γm = 0 , ( γ t ) γτ Θ τ ∂ t H αβ,γm = 0 . (4.9b)Now let us decompose H into irreducible components H αβ,γm = (cid:88) i c i H i αβ,γm , (4.10)where H αβ,γm = ε αβ Θ γ A m ( X ) , (4.11a) H αβ,γm = ε αβ ( γ a ) γδ Θ δ B ma ( X ) , (4.11b) H αβ,γm = ( γ a ) αβ Θ γ C ma ( X ) , (4.11c) H αβ,γm = ( γ a ) αβ ( γ b ) γδ Θ δ D mab ( X ) . (4.11d)It follows from eq. (4.3) that the dimension of A, B, C, D is −
3. We now impose thedifferential constraints (4.9a) and (4.9b), along with the gamma-trace constraint (4.8).After imposing (4.9a), (4.9b) the terms O (Θ ) imply A m ( X ) = 0 , C mn ( X ) = 0 , (4.12a) B ma ( X ) = − (cid:15) nra D mnr ( X ) , η na D mna ( X ) = 0 , (4.12b)17hile the terms O (Θ ) give the differential constraints ∂ t B mt ( X ) = 0 , (4.13a) ∂ t D mnt ( X ) = 0 , (4.13b) ∂ t (cid:8) B mt ( X ) + (cid:15) tan D mna ( X ) (cid:9) = 0 , (4.13c) ∂ t (cid:8) D mnt ( X ) + D mtn ( X ) − η tn D maa ( X ) + (cid:15) nta B ma ( X ) (cid:9) = 0 . (4.13d)Imposing the gamma-trace condition (4.8) results in η ma B ma ( X ) = 0 , (cid:15) qma B ma ( X ) = 0 , (4.14a) η ma D mna ( X ) = 0 , (cid:15) qma D mna ( X ) = 0 . (4.14b)One may show that the differential and algebraic constraints above are mutually consistentand reduce to: ∂ t B mt ( X ) = 0 , ∂ t D mnt ( X ) = 0 , (4.15a) η na D mna ( X ) = 0 , η ma D mna ( X ) = 0 , (4.15b) B ma ( X ) = − (cid:15) nra D mnr ( X ) , (4.15c)where B ma is symmetric and traceless, D mna is symmetric in the first and last index. Aftersome calculation one can show that general solutions consistent with the scaling property(4.3) and the above constraints is B ma ( X ) = η ma X − X m X a X , (4.16) D mna ( X ) = (cid:15) ndm X d X a X + (cid:15) nda X d X m X , (4.17)with c = c . Hence, the three-point correlation function is determined up to a single freeparameter which we denote c N =1 . Our solution is then (cid:104) L ¯ aα ( z ) J γ γ γ ( z ) L ¯ bβ ( z ) (cid:105) = δ ¯ a ¯ b x αα (cid:48) x γ γ (cid:48) x γ γ (cid:48) x γ ) γ (cid:48) ( x ) ( x ) H α (cid:48) β,γ (cid:48) γ (cid:48) γ (cid:48) ( X , Θ ) , (4.18)where H αβ,γ γ γ ( X , Θ) = ( γ m ) γ γ H αβ,γ m ( X , Θ) , (4.19) H αβ,γm ( X , Θ) = i c N =1 ( γ a ) γδ Θ δ (cid:110) ε αβ B ma ( X ) + ( γ n ) αβ D mna ( X ) (cid:111) , (4.20)18ith B and D given in eqs. (4.16), (4.17). In spinor notation, this is equivalent to H αβ,γ γ γ ( X , Θ) = i c N =1 (cid:26) ε αβ X (cid:0) ε γ γ Θ γ + ε γ γ Θ γ (cid:1) + 1 X (cid:0) ε γ α X βγ X γ δ Θ δ (4.21)+ ε γ β X αγ X γ δ Θ δ + ε γ α X γ γ X βδ Θ δ + ε γ β X γ γ X αδ Θ δ − ε γ γ X αβ X γ δ Θ δ − X γ γ X αβ Θ γ − ε αβ X γ γ X γ δ Θ δ (cid:1)(cid:27) . Finally, one must check that this solution also satisfies the point-switch identity. Withthe aid of identities (2.31a), (2.31b), it is a relatively straightforward exercise to showthat the point-switch identity (4.7) is indeed satisfied. (cid:104)
J J L (cid:105)
Let us now discuss the remaining mixed correlation function (cid:104) J β β β ( z ) J γ γ γ ( z ) L α ( z ) (cid:105) . (4.22)Here the correlator can exist only if the flavour group contains U (1)-factors, so will assumethat the flavour group is just U (1). At the component level this correlation functioncontains (cid:104) T ab ( x ) T mn ( x ) L c ( x ) (cid:105) , which was shown to vanish in any conformal field theoryafter imposing all differential constraints and symmetries [11]. As we will show, the sameoccurs in the supersymmetric theory. However, we will see that (4.22) vanishes withoutneeding to impose the conservation equation for L α ( z ). The general expression for thiscorrelation function is (cid:104) J β β β ( z ) J γ γ γ ( z ) L α ( z ) (cid:105) = ˆ x β β (cid:48) ˆ x β β (cid:48) ˆ x β ) β (cid:48) ˆ x γ γ (cid:48) ˆ x γ γ (cid:48) ˆ x γ ) γ (cid:48) ( x ) / ( x ) / (4.23) × H β (cid:48) β (cid:48) β (cid:48) γ (cid:48) γ (cid:48) γ (cid:48) α ( X , Θ ) , where H has the symmetry property H β β β γ γ γ α = H ( β β β )( γ γ γ ) α . The correlationfunction is required to satisfy: (i) Scaling constraint: Under scale transformations it transforms as (cid:104) J β β β ( z (cid:48) ) J γ γ γ ( z (cid:48) ) L α ( z (cid:48) ) (cid:105) = ( λ ) / (cid:104) J β β β ( z ) J γ γ γ ( z ) L α ( z ) (cid:105) , (4.24)which results in the constraint H β β β γ γ γ α ( λ X , λ Θ) = ( λ ) − / H β β β γ γ γ α ( X , Θ) . (4.25)19 ii) Differential constraint: The conservation law on the supercurrent implies D β (1) (cid:104) J β β β ( z ) J γ γ γ ( z ) L α ( z ) (cid:105) = 0 , (4.26)which results in a differential constraint on H : D β H β β β γ γ γ α ( X , Θ) = 0 . (4.27) (iii) Point permutation symmetry: The symmetry under permutation of points z and z implies the following constrainton the correlation function: (cid:104) J β β β ( z ) J γ γ γ ( z ) L α ( z ) (cid:105) = −(cid:104) J γ γ γ ( z ) J β β β ( z ) L α ( z ) (cid:105) , (4.28)which results in the identity H β β β γ γ γ α ( X , Θ) = −H γ γ γ β β β α ( − X T , − Θ) . (4.29)Thus, we need to solve for the tensor H subject to the constraints (4.25), (4.27) and(4.29). Note that we also must impose one more differential constraint D α (3) (cid:104) J β β β ( z ) J γ γ γ ( z ) L α ( z ) (cid:105) = 0 , (4.30)which is quite non-trivial in this formalism. Fortunately, constraints (4.25), (4.27) and(4.29) are sufficient to show that correlator (4.22) vanishes, hence we will not need toconsider (4.30).To start, we combine two of the three β , γ indices into a vector index, and impose γ -trace constraints to remove antisymmetric components H β β β γ γ γ α ( X , Θ) = ( γ a ) β β ( γ b ) γ γ H β a,γ b,α ( X , Θ) , (4.31)( γ a ) τβ H βa,γb,α ( X , Θ) = 0 , ( γ b ) τγ H βa,γb,α ( X , Θ) = 0 . (4.32)Now let us split H into symmetric and antisymmetric parts in the first and second pairof indices H βa,γb,α = H ( βa,γb ) ,α + H [ βa,γb ] ,α . (4.33)20ue to the symmetry properties, (4.29) implies that H ( σa,γb ) ,α is an even function of X , while H [ βa,γb ] ,α is odd. Therefore they do not mix in the conservation law (4.27) and maybe considered independently. In irreducible components, H ( βa,γb ) ,α has the decomposition H ( βa,γb ) ,α = (cid:88) i H i ( βa,γb ) ,α , (4.34)where H βa,γb ) ,α = ε βγ Θ α A [ ab ] ( X ) , (4.35a) H βa,γb ) ,α = ε βγ ( γ m ) αδ Θ δ B m [ ab ] ( X ) , (4.35b) H βa,γb ) ,α = ( γ m ) βγ Θ α C m ( ab ) ( X ) , (4.35c) H βa,γb ) ,α = ( γ m ) βγ ( γ n ) αδ Θ δ D mn ( ab ) ( X ) . (4.35d)Here we have made explicit the algebraic symmetry properties of A, B, C and D , whichby virtue of (4.29) are all even functions of X . Now due to linearity in Θ, the differentialconstraint (4.27) is equivalent to the pair of equations ∂ β H βa,γb,α = 0 , ( γ t ) βτ Θ τ ∂ t H βa,γb,α = 0 . (4.36)After imposing (4.36), the terms O (Θ ) imply A m [ ab ] ( X ) = 0 , B m [ ab ] ( X ) = 0 , (4.37a) C m ( ab ) ( X ) + (cid:15) mrs D rs ( ab ) ( X ) = 0 , (4.37b) η mn D mn ( ab ) ( X ) = 0 , η ma D mn ( ab ) ( X ) = 0 , (4.37c)so H βa,γb ) ,α = H βa,γb ) ,α = 0. The terms O (Θ ) then result in the differential constraints ∂ m (cid:8) − C m ( ab ) ( X ) + (cid:15) mrs D rs ( ab ) ( X ) (cid:9) = 0 , (4.38a) (cid:15) ctm ∂ t C m ( ab ) ( X ) − ∂ m D mc ( ab ) ( X ) − ∂ m D cm ( ab ) ( X ) = 0 . (4.38b)Imposing the gamma-trace condition (4.32) results in η ma C m ( ab ) ( X ) = 0 , (cid:15) cma C m ( ab ) ( X ) = 0 , (4.39a) η ma D mn ( ab ) ( X ) = 0 , (cid:15) cma D mn ( ab ) ( X ) = 0 . (4.39b) As in the previous case, our correlator is Grassmann odd which means we can replace X with X . C is a totally symmetric, traceless,transverse and even function of X . Let’s try to construct such a tensor by analysing itsirreducible components. To determine which irreducible components are permitted, let ustrade each vector index for a pair of spinor indices. Since C is completely symmetric andtraceless, it is equivalent to C ( α .... α ) . In addition since C is even in X αβ only irreduciblestructures (that is, totally symmetric tensors) of rank 4 and 0 in X αβ can contribute to thesolution. Going back to vector indices, let us denote these components of C as C mn ) ( X )and C ( X ).Since it is not possible to construct a rank three tensor C ( mnk ) out of C mn ) ( X ) and C ( X ), the tensor C mnk vanishes. Hence, H βa,γb ) ,α = 0.Given this information, the remaining set of equations imply that D is now a totallysymmetric, traceless and transverse tensor that is even in X . Following a similar argu-ment, the symmetries imply that it has irreducible components D mnab ) ( X ), D mn ) ( X )and D ( X ). We are now equipped with enough information to construct an explicit so-lution for D . Using the symmetries and the scaling property (4.25) we have the mostgeneral ansatz D ( mnab ) ( X ) = d X (cid:2) η ma η nb + η mb η na + η mn η ab (cid:3) + d X (cid:2) η mn X a X b + η ma X n X b + η mb X a X n + η na X m X b + η nb X m X a + η ab X m X n (cid:3) + d X X m X n X a X b . (4.40)Requiring that D be traceless and transverse fixes all the d i to 0. Hence, D = 0, and H ( βa,γb ) ,α vanishes.In a similar way we consider H [ βa,γb ] ,α for which we have the following decomposition H [ βa,γb ] ,α = (cid:88) i H i [ βa,γb ] ,α , (4.41)where H βa,γb ] ,α = ε βγ Θ α A ( ab ) ( X ) , (4.42a) H βa,γb ] ,α = ε βγ ( γ m ) αδ Θ δ B m ( ab ) ( X ) , (4.42b) H βa,γb ] ,α = ( γ m ) βγ Θ α C m [ ab ] ( X ) , (4.42c) H βa,γb ] ,α = ( γ m ) βγ ( γ n ) αδ Θ δ D mn [ ab ] ( X ) . (4.42d)22n this case, A, B, C, D are now odd functions in X . Imposing the conservation equationsand vanishing of the γ -trace we obtain the following set of constraints: A ( ab ) ( X ) = 0 , B m ( ab ) ( X ) = 0 (4.43a) D mm [ ab ] ( X ) = 0 , C m [ ab ] ( X ) + (cid:15) mrs D rs [ ab ] ( X ) = 0 , (4.43b) C m [ mb ] ( X ) = 0 , D mn [ mb ] ( X ) = 0 , (4.43c) (cid:15) cma C m [ ab ] ( X ) = 0 , (4.43d) (cid:15) cma D mn [ ab ] ( X ) = 0 . (4.43e)We see that the functions A and B vanish. To show that C m [ ab ] vanishes we considereq. (4.43d) and use the fact that in three dimensions an antisymmetric tensor is equivalentto a vector C m [ ab ] ( X ) = (cid:15) abq ˜ C mq ( X ) . (4.44)Hence from (4.43d) it follows that˜ C ab ( X ) − η ab ˜ C dd ( X ) = 0 . (4.45)Contracting with η ab we find that ˜ C dd = 0, hence ˜ C ab = 0. It also implies that C m [ ab ] =0. Ina similar way using eq. (4.43e) one can show that D mn [ ab ] = 0. This means that H [ βa,γb ] ,α =0. Hence the three-point function of two supercurrents and one flavour current (4.22)vanishes. In [11] it was shown that correlation functions of conserved current in three-dimensionalconformal field theories can have parity violating structures. Specifically, it was definedas follows. Given a conserved current J α α ...α s − α s ( x ) = ( γ m ) α α . . . ( γ m s ) α s − α s J m ...m s ( x ) (5.1)we can construct J s ( x, λ ) = J α α ...α s − α s ( x ) λ α . . . λ α s , (5.2)where λ α are auxiliary commuting spinors. The action of parity is then x → − x , λ → i λ . In theories with a parity symmetry, J µ ...µ s ( x ) acquires a sign ( − s under parity23nd J s ( x, λ ) is invariant. However, as was shown in [11] correlation functions admitcontributions which are odd under parity. In particular, it was shown that a parity oddcontribution to the mixed correlator of the energy-momentum tensor T mn and two flavourcurrents L ¯ ak can arise. Translating their result into our notation it can be written asfollows (cid:104) T mn ( x ) L ¯ ak ( x ) L ¯ bp ( x ) (cid:105) odd = δ ¯ a ¯ b x x x I mn,m (cid:48) n (cid:48) ( x ) I kk (cid:48) ( x ) t m (cid:48) n (cid:48) k (cid:48) p ( X ) , (5.3)where t mnkp ( X ) = (cid:15) npq X q X m X k X + (cid:15) nkq X q X m X p X . (5.4)Here X i are three-point building blocks introduced by Osborn and Petkou in [1], whilethe object I mn ( x ) is the inversion tensor, and I mn,m (cid:48) n (cid:48) ( x ) is an inversion tensor whichextracts the symmetric traceless component. They are defined as follows I mn ( x ) = η mn − x m x n x , (5.5) I mn,m (cid:48) n (cid:48) ( x ) = 12 (cid:8) I mm (cid:48) ( x ) I nn (cid:48) ( x ) + I mn (cid:48) ( x ) I nm (cid:48) ( x ) (cid:9) − η mm (cid:48) η nn (cid:48) . (5.6)An important and specific feature of all parity violating terms is appearance of the (cid:15) -tensor.In N = 1 supersymmetric theories the supercurrent J αβγ and the flavour currentmultiplet L ¯ aα contain the following conserved currents T αβγδ = D ( δ J αβγ ) | , T αβγδ = ( γ m ) ( αβ ( γ n ) γδ ) T mn , ∂ m T mn = 0 , η mn T mn = 0 , (5.7a) Q αβγ = J αβγ | , Q αβγ = ( γ m ) αβ Q mγ , ∂ m Q mα = 0 , ( γ m ) αβ Q mα = 0 , (5.7b) V ¯ aαβ = D ( α L ¯ aβ ) | , V ¯ aαβ = ( γ m ) αβ V ¯ am , ∂ m V ¯ am = 0 , (5.7c)where T mn is the energy-momentum tensor, Q mγ is the supersymmetry current and V ¯ am isa vector current. Hence, the mixed correlators studied in the previous section give rise tothe following correlators in terms of components (cid:104) T mn ( x ) T pq ( x ) V k ( x ) (cid:105) , (cid:104) Q mα ( x ) Q nβ ( x ) V k ( x ) (cid:105) , (cid:104) V ¯ ak ( x ) T mn ( x ) V ¯ bp ( x ) (cid:105) . (5.8)The first two correlators vanish because the entire superspace correlator (4.22) vanishes.The last one is, in general, non-zero and fixed up to one overall coefficient. It can becomputed using eqs. (4.18), (4.21) using the superspace reduction procedure (cid:104) V ¯ ak ( x ) T mn ( x ) V ¯ bp ( x ) (cid:105) = (5.9)116 ( γ k ) α α ( γ m ) ( β β ( γ n ) β β ) ( γ p ) γ γ D (1) α D (2) β D (3) γ (cid:104) L ¯ aα ( z ) J β β β ( z ) L ¯ bγ ( z ) (cid:105) (cid:12)(cid:12) . θ α to zero. We will notperform the reduction explicitly, instead we will indirectly determine whether (5.9) is evenor odd under parity. For this it is sufficient to study whether or not the (cid:15) -tensor appearsupon reduction. Since (cid:15) mnp = 12 tr( γ m γ n γ p ) , (5.10)it is enough to count the number of gamma-matrices: If the number of γ -matrices ap-pearing in the superspace reduction is even the (cid:15) -tensor cannot arise and the contributionis parity even, if the number of γ -matrices is odd the contribution is parity odd. Let usperform the counting. Since in (5.9) we act with just three covariant derivatives beforesetting all θ i = 0 (where i = 1 , , θ i will contribute. Let us concentrate on the terms linear on θ i . Sincethe function H in (4.21) is already linear in θ i we can set θ i = 0 in x ij and X . Thismakes x ij and X symmetric and proportional to a gamma-matrix. Now we have fourgamma-matrices in (5.9), four gamma-matrices coming from x ij in eq. (4.18), zero or twogamma-matrices coming from H in (4.21) and also one more gamma-matrix contained inΘ , see eq. (2.26c). Overall we have odd number of gamma-matrices at this point. How-ever, superspace covariant derivatives also contain gamma-matrices, see eq. (A.16). Sincewe are considering terms linear in θ i and setting θ i = 0 upon differentiating it is easy torealise that in the three derivatives D (1) α D (2) β D (3) γ we must take one derivative withrespect to x i and two derivatives with respect to θ i . This gives one more gamma-matrixmaking the total number even. Terms cubic in θ can be considered in a similar way. Theyalso yield an even number of gamma-matrices. Hence, the entire contribution (5.9) isparity even.In a similar way we can count the number of gamma-matrices in the superspace re-duction of (3.12), (3.29): (cid:104) V ¯ am ( x ) V ¯ bn ( x ) V ¯ ck ( x ) (cid:105) = (5.11) −
18 ( γ m ) α α ( γ n ) ( β β ( γ k ) γ γ D (1) α D (2) β D (3) γ (cid:104) L ¯ aα ( z ) L ¯ aβ ( z ) L ¯ bγ ( z ) (cid:105) (cid:12)(cid:12) . An analysis similar to the above shows that this contribution is also parity even. Fi-nally, one can also consider the superspace reduction of the three-point function of thesupercurrent (cid:104) T mn ( x ) T k(cid:96) ( x ) T pq ( x ) (cid:105) = 164 ( γ m ) ( α α ( γ n ) α α ) ( γ k ) ( β β ( γ (cid:96) ) β β ) ( γ p ) ( γ γ ( γ q ) γ γ ) D (1) α D (2) β D (3) γ (cid:104) J α α α ( z ) J β β β ( z ) J γ γ γ ( z ) (cid:105) (cid:12)(cid:12) (5.12)25nd (cid:104) T mn ( x ) Q kβ ( x ) Q pγ ( x ) (cid:105) =116 ( γ m ) ( α α ( γ n ) α α ) ( γ k ) β β ( γ p ) γ γ D (1) α (cid:104) J α α α ( z ) J ββ β ( z ) J γγ γ ( z ) (cid:105) (cid:12)(cid:12) . (5.13)The three-point function of the supercurrent was found in [33]. We will not repeat ithere since the expression for it is quite long. However, a similar analysis shows that thecontributions (5.12) and (5.13) are parity even. This means that no parity violating structures can arise in three-point functions of T mn , Q mα and V ¯ am in superconformal field theories. Maldacena and Zhiboedov provedin [40] that if a three-dimensional conformal field theory possesses a higher spin con-served current then it is essentially a free theory. Since a free theory has only parityeven contributions to the three-point functions of conserved currents, the correlators in-volving one or more higher spin conserved currents admit only parity even structures.This leads us to conclude that N = 1 supersymmetry forbids parity violating struc-tures in all three-point functions of conserved currents unless the assumptions of theMaldacena–Zhiboedov theorem are violated. The strongest assumption of the theorem isthat the theory under consideration contains unique conserved current of spin two whichis the energy-momentum tensor. Some properties of theories possessing more than oneconserved current with spin two were discussed in [40]. In supersymmetric theory theenergy-momentum tensor is a component of the supercurrent. One can also consider adifferent supermultiplet containing a conserved spin two current, namely J ( α α α α ) , D α J ( α α α α ) = 0 . (5.14)The lowest component of J ( α α α α ) is a conserved spin two current which is not theenergy-momentum tensor. Note that J ( α α α α ) also contains a conserved higher-spincurrent. It will be interesting to perform a systematic study of three-point functions of J ( α α α α ) to see if they allow any parity violating structures. N = 2 superconformal fieldtheory Now we will generalise our method to mixed three-point functions in superconformalfield theory with N = 2 supersymmetry. A specific feature of three-dimensional N = 2 In general, if a superspace three-point function is fixed up to an overall coefficient it is expected tobe parity even because this contribution is expected to exist in a free theory of a real scalar superfield.
The 3D, N = 2 supercurrent was studied in [53, 56–58]. It is a primary, dimension 2symmetric spin-tensor J αβ , which obeys the conservation equation D Iα J αβ = 0 , (6.1)and has the following superconformal transformation law: δJ αβ = − ξJ αβ − σ ( z ) J αβ + 2 λ ( z ) γ ( α J β ) γ . (6.2)The general formalism in section 2 allows the two-point function to be determined up toa single real coefficient (cid:104) J αβ ( z ) J α (cid:48) β (cid:48) ( z ) (cid:105) = b N =2 x αα (cid:48) x β ) β (cid:48) ( x ) . (6.3)It’s then a simple exercise to show that the two-point function has the right symmetryproperties under permutation of superspace points (cid:104) J αβ ( z ) J α (cid:48) β (cid:48) ( z ) (cid:105) = (cid:104) J α (cid:48) β (cid:48) ( z ) J αβ ( z ) (cid:105) , (6.4)and also satisfies the conservation equation D Iα (1) (cid:104) J αβ ( z ) J α (cid:48) β (cid:48) ( z ) (cid:105) = 0 , z (cid:54) = z . (6.5)Similarly, the 3D N = 2 flavour current is a primary, dimension 1 scalar superfield L ,which obeys the conservation equation (cid:0) D α ( I D J ) α − δ IJ D αK D Kα (cid:1) L = 0 , (6.6)and transforms under the superconformal group as δL = − ξL − σ ( z ) L . (6.7)As in the N = 1 case, we assume the N = 2 superconformal field theory in question hasa set of flavour currents L ¯ a associated with a simple flavour group. Due to the absence of27pinor or R -symmetry indices, the N = 2 flavour current two-point function is fixed upto a single real coefficient a N =2 as follows (cid:104) L ¯ a ( z ) L ¯ b ( z ) (cid:105) = a N =2 δ ¯ a ¯ b x . (6.8)The two-point function obeys the correct symmetry properties under permutation of su-perspace points, (cid:104) L ¯ a ( z ) L ¯ b ( z ) (cid:105) = (cid:104) L ¯ b ( z ) L ¯ a ( z ) (cid:105) , and also satisfies the conservation equa-tion (cid:0) D α ( I (1) D J )(1) α − δ IJ D αK (1) D K (1) α (cid:1) (cid:104) L ¯ a ( z ) L ¯ b ( z ) (cid:105) = 0 , z (cid:54) = z . (6.9)In the next section we will compute the mixed correlation functions associated with the N = 2 supercurrent and flavour current multiplets. There are two possibilities to consider,they are (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) , (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) . (6.10)Note that in second case we are considering a U (1) flavour current. (cid:104) LJ L (cid:105)
First let us consider the (cid:104)
LJ L (cid:105) case first. Using the general ansatz, we have (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) = δ ¯ a ¯ b x αα (cid:48) x ββ (cid:48) x ( x ) H α (cid:48) β (cid:48) ( X , Θ ) , (6.11)where H αβ = H ( αβ ) . The correlation function is also required to satisfy the following: (i) Scaling constraint: Under scale transformations the correlation function transforms as (cid:104) L ¯ a ( z (cid:48) ) J αβ ( z (cid:48) ) L ¯ b ( z (cid:48) ) (cid:105) = ( λ ) (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) , (6.12)from which we find the homogeneity constraint H αβ ( λ X , λ Θ) = ( λ ) − H αβ ( X , Θ) . (6.13) (ii) Differential constraints: The differential constraints on the flavour current and supercurrent result in thefollowing constraints on the correlation function: (cid:0) D σ ( I (1) D J )(1) σ − δ IJ D σK (1) D K (1) σ (cid:1) (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) = 0 , (6.14a) D Iα (2) (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) = 0 . (6.14b)28hese result in the following differential constraints on H :( D σ ( I D J ) σ − δ IJ D σK D Kσ ) H αβ ( X , Θ) = 0 , (6.15a) Q Iα H αβ ( X , Θ) = 0 . (6.15b) (iii) Point permutation symmetry: The symmetry under permutation of points ( z and z ) results in the followingconstraint on the correlation function: (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) = (cid:104) L ¯ b ( z ) J αβ ( z ) L ¯ a ( z ) (cid:105) , (6.16)which results in the point-switch identity H αβ ( X , Θ ) = x σσ (cid:48) X σ (cid:48) α x ρρ (cid:48) X ρ (cid:48) β X x H σρ ( − X T1 , − Θ ) . (6.17)The symmetry properties of H allow us to trade the spinor indices for a vector index H αβ ( X , Θ) = ( γ m ) αβ H m ( X , Θ) . (6.18)The most general expansion for H m ( X , Θ) is then H m ( X , Θ) = A m ( X ) − i2 Θ B m ( X ) + (ΘΘ) n C mn ( X ) + 18 Θ D m ( X ) , (6.19)where we have defined(ΘΘ) m = −
12 ( γ m ) αβ (ΘΘ) αβ , (ΘΘ) αβ = Θ Iα Θ Jβ ε IJ , (6.20)and accounted for the N = 2 identityΘ Θ Iα Θ Jβ ε IJ = 0 . (6.21)The prefactors in front of B and D have been chosen for convenience, and as in the N = 1case it is more convenient to work with X m instead of X αβ . Imposing (6.15b) results inthe differential constraints ∂ m A m ( X ) = 0 , (6.22a) ∂ m B m ( X ) = 0 , (6.22b) (cid:15) mnt ∂ n C mt ( X ) = 0 , (6.22c) B q ( X ) + (cid:15) qmn ∂ n A m ( X ) = 0 , (6.22d) D q ( X ) − (cid:15) qmn ∂ n B m ( X ) = 0 , (6.22e) ∂ m (cid:8) C mt ( X ) + C tm ( X ) − η mt C aa ( X ) (cid:9) = 0 , (6.22f)29nd the algebraic constraints C aa ( X ) = 0 , (6.23a) (cid:15) qmt C mt ( X ) = 0 , (6.23b)which imply that C is symmetric and traceless. Furthermore the scaling condition (6.13)allows us to construct the solutions A m ( X ) = a X m X , (6.24a) B m ( X ) = b X m X , (6.24b) C mn ( X ) = c (cid:18) η mn X − X m X n X (cid:19) , (6.24c) D m ( X ) = d X m X . (6.24d)Together (6.22) imply B m ( X ) = D m ( X ) = 0, while a and c remain as two free parameters.Hence the solution for H becomes H αβ ( X , Θ) = ˜ c N =2 X αβ X + i c N =2 (cid:26) Θ Iα Θ Jβ ε IJ X + 32 X αβ X γδ Θ Iγ Θ Jδ ε IJ X (cid:27) . (6.25)After some lengthy calculation it turns out that only the second structure satisfies theconservation equation (6.15a). Hence there is only one linearly independent structure inthe correlation function that is compatible with the differential constraints. Therefore wefind that the final solution is (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) = δ ¯ a ¯ b x αα (cid:48) x ββ (cid:48) x ( x ) H α (cid:48) β (cid:48) ( X , Θ ) , (6.26)with H αβ ( X , Θ) = i c N =2 (cid:26) Θ Iα Θ Jβ ε IJ X + 32 X αβ X γδ Θ Iγ Θ Jδ ε IJ X (cid:27) . (6.27)In deriving this result, we Taylor expanded the denominator in (6.25) using X = X − Θ , which follows from (2.28), (2.34), and then used the N = 2 identity (6.21). It mayalso be shown that this structure satisfies the point-switch identity (6.17).The supercurrent J αβ leads to the following N = 1 supermultiplets (here the bar-projections denotes setting θ I = to zero and D α = D α,I = ): S αβ = J αβ | , D α S αβ = 0 , (6.28a) J αβγ = i D ( α J βγ ) , D α J αβγ = 0 . (6.28b) From here we will use bold R -symmetry indices to distinguish them from other types of indices.
30n these equations J αβγ is the N = 1 supercurrent and S αβ is the additional N = 1 su-permultiplet containing the second supersymmetry current and the R -symmetry current.Similarly, the N = 2 flavour current leads to S = L ¯ a | , (6.29a) L ¯ aα = i D α L ¯ a , D α L ¯ aα = 0 , (6.29b)where L ¯ aα is the N = 1 flavour current and S is unconstrained. Hence, the N = 2three-point function (cid:104) L ¯ a ( z ) J αβ ( z ) L ¯ b ( z ) (cid:105) contains three-point functions of the followingconserved component currents: the energy-momentum tensor, conserved vector currents,the supersymmetry currents and the R -symmetry current. All these three-point functionscan be found by superspace reduction and are fixed by the N = 2 superconformal symme-try up to one overall coefficient (or vanish). A simple gamma-matrix counting proceduresimilar to the one discussed in the previous section shows that all these correlators areparity even. (cid:104) J J L (cid:105)
For this example, the general ansatz gives (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) = x αα (cid:48) x ββ (cid:48) x γγ (cid:48) x δδ (cid:48) ( x ) ( x ) H α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) ( X , Θ ) , (6.30)where H αβγδ = H ( αβ )( γδ ) . The correlation function is required to satisfy the following: (i) Scaling constraint: Under scale transformations the correlation function transforms as (cid:104) J αβ ( z (cid:48) ) J γδ ( z (cid:48) ) L ( z (cid:48) ) (cid:105) = ( λ ) (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) , (6.31)from which we find the homogeneity constraint H αβγδ ( λ X , λ Θ) = ( λ ) − H αβγδ ( X , Θ) . (6.32) (ii) Differential constraints: The differential constraints on the flavour current and supercurrent result in thefollowing constraints on the correlation function: D Iα (1) (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) = 0 . (6.33a) (cid:0) D σ ( I (3) D J )(3) σ − δ IJ D σK (3) D K (3) σ (cid:1) (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) = 0 . (6.33b)31he first equation results in the following differential constraints on H : D Iα H αβγδ ( X , Θ) = 0 . (6.34)The second constraint (6.33b) is more difficult to handle in this formalism, howeverwe will demonstrate how to deal with it later. (iii) Point permutation symmetry: The symmetry under permutation of points z and z results in the following con-straint on the correlation function: (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) = (cid:104) J γδ ( z ) J αβ ( z ) L ( z ) (cid:105) , (6.35)which results in the point-switch identity H αβγδ ( X , Θ) = H γδαβ ( − X T , − Θ) . (6.36)Now due to the symmetry properties of H , we may trade pairs of symmetric spinor indicesfor vector indices H ( αβ )( γδ ) ( X , Θ) = ( γ m ) αβ ( γ n ) γδ H mn ( X , Θ) . (6.37)Now if we split H mn into symmetric and anti-symmetric parts H mn ( X , Θ) = H ( mn ) ( X , Θ) + H [ mn ] ( X , Θ)= H ( mn ) ( X , Θ) + (cid:15) mnt H t ( X , Θ) , (6.38)then the point-switch identity implies H ( mn ) ( X , Θ) = H ( mn ) ( − X T , − Θ) , H t ( X , Θ) = −H t ( − X T , − Θ) . (6.39)General expansions consistent with the index structure and symmetries are H ( mn ) ( X , Θ) = A ( mn ) ( X ) + Θ B ( mn ) ( X ) + (ΘΘ) s C ( mn ) s ( X ) + Θ D ( mn ) ( X ) , (6.40a) H t ( X , Θ) = A t ( X ) + Θ B t ( X ) + (ΘΘ) s C ts ( X ) + Θ D t ( X ) . (6.40b)All the tensors comprising H ( mn ) are even functions of X , while those in the expansionfor H t are odd functions of X . Furthermore, due to symmetry arguments the tensors H ( mn ) and H t do not mix in the conservation law (6.34), hence they may be considered32ndependently. First let us analyse H ( mn ) ; imposing (6.15a) results in the differentialconstraints ∂ m A ( mn ) ( X ) = 0 , (6.41a) ∂ m B ( mn ) ( X ) = 0 , (6.41b) (cid:15) mrs ∂ r C ( mn ) s ( X ) = 0 , (6.41c)2 B ( qn ) ( X ) + i (cid:15) qmt ∂ t A ( mn ) ( X ) = 0 , (6.41d)4 D ( qn ) ( X ) + i (cid:15) qmt ∂ t B ( mn ) ( X ) = 0 , (6.41e) ∂ m (cid:8) C ( mn ) s ( X ) + C ( sn ) m ( X ) − η ms C ana ( X ) (cid:9) = 0 , (6.41f)and the algebraic constraints C mnm ( X ) = 0 , (6.42a) (cid:15) rms C ( mn ) s ( X ) = 0 . (6.42b)The scaling condition (6.32), along with (6.42) imply that C is totally symmetric, tracelessand even in X . Following the argument presented in section 4.2 we find that no such tensorexists, hence C = 0. Furthermore, evenness in X allows us to identify solutions for theremaining tensors A ( mn ) ( X ) = a η mn X + a X m X n X , (6.43a) B ( mn ) ( X ) = b η mn X + b X m X n X , (6.43b) D ( mn ) ( X ) = d η mn X + d X m X n X . (6.43c)Imposing (6.41a) and (6.41b) results in a = − a , b = − b , however for this choice ofcoefficients (6.41d) implies B = 0, while the tensor A survives. It is then easy to see that(6.41e) implies D = 0. Therefore the only solution is A ( mn ) ( X ) = a (cid:18) η mn X − X m X n X (cid:19) . (6.44)Now let us direct our attention to H t ; imposing (6.34) results in the set of equations (cid:15) mnt ∂ m A t ( X ) = 0 , (6.45a) (cid:15) mnt ∂ m B t ( X ) = 0 , (6.45b) ∂ m (cid:8) C mn ( X ) − η mn C ss ( X ) (cid:9) = 0 , (6.45c)2 (cid:15) qts B s ( X ) − i ∂ t A q ( X ) + i η qt ∂ s A s ( X ) = 0 , (6.45d)4 (cid:15) qts D s ( X ) − i ∂ t B q ( X ) + i η qt ∂ s B s ( X ) = 0 , (6.45e)33nd the algebraic constraints (cid:15) nma C ma ( X ) = 0 , (6.46a) C mn ( X ) − η mn C ss ( X ) = 0 . (6.46b)The algebraic constraints (6.46) imply that C = 0. Now since A , B and D are odd in X we can construct the solutions A t ( X ) = a X t X , (6.47a) B t ( X ) = b X t X , (6.47b) D t ( X ) = d X t X . (6.47c)However it is not too difficult to show that imposing (6.45d), (6.45e) requires that A , B and D must all vanish. Hence H t ( X , Θ) = 0.So far we have found a single solution consistent with the supercurrent conservationequation and the point-switch identity, H mn ( X , Θ) = a (cid:18) η mn X − X m X n X (cid:19) , (6.48) H αβγδ ( X , Θ) = ( γ m ) αβ ( γ n ) γδ H mn ( X , Θ)= d N =2 (cid:18) ε αγ ε βδ + ε αδ ε βγ X + 3 X αβ X γδ X (cid:19) . (6.49)Therefore the correlation function is (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) = x αα (cid:48) x ββ (cid:48) x γγ (cid:48) x δδ (cid:48) ( x ) ( x ) H α (cid:48) β (cid:48) γ (cid:48) δ (cid:48) ( X , Θ ) , (6.50)where, after writing our solution in terms of the variable X , H αβγδ ( X , Θ) = d N =2 (cid:26) ε αγ ε βδ X + ε αδ ε βγ X + 38 ε αγ ε βδ Θ X + 38 ε αδ ε βγ Θ X + 3 X αβ X γδ X + 3i2 ε αβ X γδ Θ X + 3i2 ε γδ X αβ Θ X − ε αβ ε γδ Θ X + 158 X αβ X γδ Θ X (cid:27) . (6.51)However it remains to check whether this solution satisfies the flavour current conser-vation equation. As mentioned earlier it is difficult to check conservation laws on the34hird superspace point in this formalism as there are no identities that allow differentialoperators acting on the z dependence to pass through the prefactor of (2.43). To dealwith this we will re-write our solution in terms of the three-point building block X usingidentities (2.31a), (2.33). This ultimately has the effect (cid:104) J αβ ( z ) J γδ ( z ) L ( z ) (cid:105) −→ (cid:104) L ( z ) J γδ ( z ) J αβ ( z ) (cid:105) . (6.52)Written in terms of the variable X , the correlation function is found to be (cid:104) L ( z ) J γδ ( z ) J αβ ( z ) (cid:105) = x γγ (cid:48) x δδ (cid:48) x ( x ) H γ (cid:48) δ (cid:48) αβ ( X , Θ ) , (6.53)where H γδαβ ( X , Θ) = d N =2 (cid:26) X γα X δβ X + X γβ X δα X + 38 X γα X δβ Θ X + 38 X γβ X δα Θ X − X αβ X γδ X − ε αβ X γδ Θ X − ε γδ X αβ Θ X + 34 ε αβ ε γδ Θ X − X αβ X γδ Θ X (cid:27) . (6.54)We are now able to check the conservation equation (6.33b), which after using identitiesequivalent to (2.39a) becomes the constraint (cid:0) D σ ( I D J ) σ − δ IJ D σK D Kσ (cid:1) H γδαβ ( X , Θ) = 0 . (6.55)After a very lengthy calculation one can show that the solution above satisfies this con-servation equation, hence this correlation function is non-trivial and is determined up toa single parameter.This is a peculiar result, as it was shown in section 4.2 that the correlation function (cid:104) J J L (cid:105) vanishes for N = 1. At first glance this appears to be a contradiction sinceany theory with N = 2 supersymmetry is also N = 1 supersymmetric. However, aswas discussed in the previous subsection, the N = 2 current supermultiplets J αβ and L contain not only the N = 1 supercurrent and flavour currents, but also the unconstrainedscalar superfield S and the supermultiplet of currents S αβ . Hence, non-vanishing of the N = 2 three-point function (6.49), (6.50) implies non-vanishing of some of three-pointfunctions involving these additional N = 1 currents. For example, from eqs. (6.49), (6.50)it follows that the following N = 1 correlator is, in general, non-zero: (cid:104) S α α ( z ) J β β β ( z ) L γ ( z ) (cid:105) = − D (2)( β D (3) γ (cid:104) J α α ( z ) J β β ) ( z ) L ( z ) (cid:105)| , (6.56)35here the bar-projection means setting θ i to zero. In components this correlator contains(among others) (cid:104) R m ( x ) T pq ( x ) V s ( x ) (cid:105) , where R m is the U (1) R -symmetry current whichexists in theories with N = 2 supersymmetry. In theories with N = 1 supersymmetrysuch a correlator does not exist because there is no R -symmetry current. On the otherhand, the N = 2 → N = 1 superspace reduction (cid:104) J α α ( z ) J β β ( z ) L ( z ) (cid:105) −→ (cid:104) J α α α ( z ) J β β β ( z ) L γ ( z ) (cid:105) , (6.57)must give zero to be consistent with the result of the previous subsection. Let us checkthat this is indeed the case. To perform the reduction we compute − i D (1)( α D (2)( β D (3) γ (cid:104) J α α ) ( z ) J β β ) ( z ) L ( z ) (cid:105)| . (6.58)That is we must act with three covariant derivatives with respect to θ i and then set all θ i to zero. From the explicit form of the correlator (cid:104) J α α ( z ) J β β ( z ) L ( z ) (cid:105) in eqs. (6.49),(6.50) it follows that it depends on θ i θ j . Since it is Grassmann even it contains onlyeven powers of θ i . Therefore, acting on (cid:104) J α α ( z ) J β β ( z ) L ( z ) (cid:105) with three derivativesas in (6.58) will give a result either linear or higher order in θ i , so it vanishes when weset θ i = 0. This shows that despite being non-zero our result (6.49), (6.50) is consistentwith vanishing of the similar correlator in the N = 1 case. Acknowledgements
The authors would like to thank Jessica Hutomo, Sergei Kuzenko and Michael Pondsfor valuable discussions. The work of E.I.B. is supported in part by the Australian Re-search Council, project No. DP200101944. The work of B.S. is supported by the
Bruceand Betty Green Postgraduate Research Scholarship under the Australian GovernmentResearch Training Program. Note that all component three-point functions contained in (6.49), (6.50) are parity even.
3D conventions and notation
For the Minkowski metric we use the “mostly plus” convention: η mn = diag( − , , , R ) invariant anti-symmetric ε -tensor ε αβ = (cid:32) −
11 0 (cid:33) , ε αβ = (cid:32) − (cid:33) , ε αγ ε γβ = δ αβ , (A.1) φ α = ε αβ φ β , φ α = ε αβ φ β . (A.2)The γ -matrices are chosen to be real, and are expressed in terms of the Pauli matrices σ as follows: ( γ ) αβ = − i σ = (cid:32) −
11 0 (cid:33) , ( γ ) αβ = σ = (cid:32) − (cid:33) , (A.3a)( γ ) αβ = − σ = (cid:32) − − (cid:33) , (A.3b)( γ m ) αβ = ε βδ ( γ m ) αδ , ( γ m ) αβ = ε αδ ( γ m ) δβ . (A.4)The γ -matrices are traceless and symmetric( γ m ) αα = 0 , ( γ m ) αβ = ( γ m ) βα , (A.5)and also satisfy the Clifford algebra γ m γ n + γ n γ m = 2 η mn . (A.6)Products of γ -matrices are then( γ m ) αρ ( γ n ) ρβ = η mn δ αβ + (cid:15) mnp ( γ p ) αβ , (A.7a)( γ m ) αρ ( γ n ) ρσ ( γ p ) σβ = η mn ( γ p ) αβ − η mp ( γ n ) αβ + η np ( γ m ) αβ + (cid:15) mnp δ αβ , (A.7b)where we have introduced the 3D Levi-Civita tensor (cid:15) , with (cid:15) = − (cid:15) = 1. It satisfiesthe following identities: (cid:15) mnp (cid:15) m (cid:48) n (cid:48) p (cid:48) = − η mm (cid:48) ( η nn (cid:48) η pp (cid:48) − η np (cid:48) η pn (cid:48) ) − ( n (cid:48) ↔ m (cid:48) ) − ( m (cid:48) ↔ p (cid:48) ) , (A.8a) (cid:15) mnp (cid:15) mn (cid:48) p (cid:48) = − η nn (cid:48) η pp (cid:48) + η np (cid:48) η pn (cid:48) , (A.8b) (cid:15) mnp (cid:15) mnp (cid:48) = − η pp (cid:48) , (A.8c) (cid:15) mnp (cid:15) mnp = − . (A.8d)37e also have the orthogonality and completeness relations for the γ -matrices( γ m ) αβ ( γ m ) ρσ = − δ αρ δ βσ − δ ασ δ βρ , ( γ m ) αβ ( γ n ) αβ = − η mn . (A.9)Finally, the γ -matrices are used to swap from vector to spinor indices. For example,given some three-vector x m , it may equivalently be expressed in terms of a symmetricsecond-rank spinor x αβ as follows: x αβ = ( γ m ) αβ x m , x m = −
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