Superconformal Symmetry and Correlation Functions

Transcript

1 KIAS hep-th/ Superconformal Symmetry and Correlation Functions arxiv:hep-th/ v3 26 Oct 1999 Jeong-Hyuck Park School of Physics, Korea Institute for Advanced Study Cheongryangri-dong, Dongdaemun-gu Seoul , Korea Abstract Four-dimensional N-extended superconformal symmetry and correlation functions of quasi-primary superfields are studied within the superspace formalism. A superconformal Killing equation is derived and its solutions are classified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. In general, due to the invariance under supertranslations and special superconformal transformations, superconformally invariant n-point functions reduce to one unspecified n 2-point function which must transform homogeneously under the remaining rigid transformations, i.e. dilations, Lorentz transformations and R-symmetry transformations. Based on this result, we are able to identify all the superconformal invariants and obtain the general form of n-point functions for scalar superfields. In particular, as a byproduct, a selection rule for correlation functions is derived, the existence of which in N = 4 super Yang- Mills theory was previously predicted in the context of AdS/CFT correspondence [1]. Superconformally covariant differential operators are also discussed. PACS: Pb; Hf Keywords: Superconformal symmetry; Correlation functions; Selection rule; Superconformally covariant differential operators address: jhp@kias.re.kr

2 1 Introduction & Summary Superconformal field theories have been of renewed attention after the Maldacena conjecture that the string/m theory on AdS d+1 backgrounds is dual to a conformal field theory in a spacetime of dimension, d, which is interpreted as the boundary of AdS d+1 [2 4]. As all the known nontrivial conformal field theories in higher than two dimensions are supersymmetric theories [5 16], it is natural to consider a group which combines supersymmetry and conformal symmetry together, i.e. the superconformal group. In fact, the pioneering work on supersymmetry in four-dimensions [17] introduced the N = 1 superconformal symmetry, though it is broken at quantum level. Contrary to the ordinary conformal symmetry, not all spacetime dimensions allow superconformal symmetry. The standard supersymmetry algebra admits an extension to a superconformal algebra only if d 6 [18]for a review see [19]. In particular in fourdimensions, which is of our interest in this paper, the bosonic part of the superconformal algebra is o2,4 un. 1.1 Hence the four-dimensional superconformal group is identified with a supermatrix group, SU2,2 N [20,21] or its complexification, SL4 N;C [22,23]. Normally N 5 cases are excluded from the renormalization point of view, as theories with more than four supercharges must have spins higher than one such as graviton/gravitino and it is unlikely that supergravity theories are renormalizable. However, such a restriction on the value of N is not relevant to our work. According to the conjecture [2 4], four-dimensional N = 4 super Yang-Mills theory with gauge group SUN is dual to type IIB string theory on AdS 5 S 5 in the limit of small g YM and large but fixed t Hooft coupling, gym 2 N. In this limit, the string theory can be effectively described by tree level type IIB supergravity, while the field theory dual is strongly coupled. As the perturbative approach breaks down in the strongly coupled CFT side, to check the conjectured duality it is desirable to have non-perturbative understanding on super Yang-Mills theory. This motivates us to explore four-dimensional N-extended superconformal symmetry and correlation functions subject to the symmetry as done in the present paper. In our previous work [21], six-dimensional N, 0 superconformal symmetry was analyzed in terms of coordinate transformations on superspace and through dimensional reduction basic features of four-dimensional N-extended superconformal symmetry were obtained. In the present paper, in a similar fashion to [21,24] but in a self-contained manner, we analyze four-dimensional N-extended superconformal symmetry on superspace. Our main results concern the general forms of superconformally invariant n-point functions for 1

3 quasi-primary superfields. In particular, as a byproduct, we obtain a selection rule for correlation functions of the component fields, ψ I x, appearing in the power series expansions of quasi-primary superfields in Grassmann coordinates, θ and θ. The selection rule states that if the the sum of the R-symmetry charge, κ i, is not zero then the correlation function of the component fields vanishes ψ I 1 1 x 1 ψ In n x n = 0 if n κ i i=1 The existence of this kind of selection rule in N = 4 super Yang-Mills theory was previously predicted by Intriligator within the context of AdS/CFT correspondence, as the dual IIB supergravity contains a corresponding U1 symmetry [1]. Therefore our results provide a supporting evidence for the Maldacena conjecture, as the selection rule here is derived by purely considering the symmetry on CFT side without referring to the string side. The contents of the present paper are as follows. In section 2, we first define the four-dimensional N-extended superconformal group in terms of coordinate transformations on superspace as a generalization of the definition of ordinary conformal transformations. We then derive a superconformal Killing equation, which is a necessary and sufficient condition for a supercoordinate transformation to be superconformal. The general solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations, where R-symmetry is given by UN as in eq.1.1 and supertranslations and special superconformal transformations are dual to each other through superinversion map. The four-dimensional N-extended superconformal group is then identified with a supermatrix group, SU2, 2 N, having dimensions 15+N 2 8N as known. However, we point out that for N = 4 case an equivalence relation must be imposed on the supermatrix group and so the four-dimensional N = 4 superconformal group is isomorphic to a quotient group of the supermatrix group. In section 3, we obtain an explicit formula for the finite nonlinear superconformal transformations of the supercoordinates, z, parameterizing superspace and discuss several representations of the superconformal group. We also construct matrix or vector valued functions depending on two or three points in superspace which transform covariantly under superconformal transformations. For two points, z 1 and z 2, we find a matrix, Iz 1,z 2, which transforms covariantly like a product of two tensors at z 1 and z 2. For three points, z 1,z 2,z 3, we find tangent vectors, Z i, which transform homogeneously at z i, i = 1,2,3. These are crucial variables for obtaining two-point, three-point and general n-point correlation functions later. In section 4, we discuss the superconformal invariance of correlation functions for quasiprimary superfields and exhibit general forms of two-point, three-point and n-point func- 2

4 tions. Explicit formulae for two-point functions of superfields in various cases are given. In general, due to the invariance under supertranslations and special superconformal transformations, n-point functions reduce to one unspecified n 2-point function which must transform homogeneously under the rigid transformations - dilations, Lorentz transformations and R-symmetry transformations. We then identify all the superconformal invariants and obtain the general form of n-point functions of scalar superfields. As a byproduct, we derive the selection rule for correlation functions 1.2. In section 5, superconformally covariant differential operators are discussed. The conditions for superfields, which are formed by the action of spinor derivatives on quasi-primary superfields, to remain quasi-primary are obtained. In general, the action of differential operator on quasi-primary fields generates an anomalous term under superconformal transformations. However, with a suitable choice of scale dimension and R-symmetry charge, we show that the anomalous term may be cancelled. We regard this analysis as a necessary step to write superconformally invariant actions on superspace, as the kinetic terms in such theories may consist of superfields formed by the action of spinor derivatives on quasi-primary superfields. In the appendix, the explicit form of superconformal algebra and a method of solving the superconformal Killing equation are exhibited. Recent review on the implications of N = 1 superconformal symmetry for four-dimensional quantum field theories is contained in [25] and some related works on superconformally invariant correlation functions can be found in [26 33]. N = 1 superconformal symmetry on curved superspace is studied in [34 36] and conformally covariant differential operators in non-supersymmetric theories are discussed in [37, 38]. 2 Superconformal Symmetry in Four-dimensions In this section we first define the four-dimensional N-extended superconformal group on superspace and then discuss its superconformal Killing equation along with the solutions. 3

5 2.1 Four-dimensional Superspace The four-dimensional supersymmetry algebra has the standard 1 form with P µ = H, P {Q aα, Q ḃ α} = 2δ b a σ µ α αp µ, [P µ,p ν ] = [P µ,q aα ] = [P µ, Q ȧ α ] = {Q aα,q bβ } = { Q ȧ α, Q b β} = 0, 2.1 where 1 α, α 2, 1 a N and Q aα satisfies Q aα = Q ȧ α. 2.2 P µ,q aα and Q ȧ α generate a supergroup, G T, with parameters, z M = x µ,θ aα, θ α a, which are coordinates on superspace. The general element of G T is written in terms of these coordinates as gz = e ix P+θa Q a+ Q a θ a. 2.3 Corresponding to eq.2.2 we may assume θ aα to satisfy so that where θ aα = θ α a, 2.4 gz = gz 1 = g z. 2.5 The Baker-Campbell-Haussdorff formula with the supersymmetry algebra 2.1 gives gz 1 gz 2 = gz 3, 2.6 x µ 3 = x µ 1 +x µ 2 +iθ a 1 σµ θ2a iθ a 2 σµ θ1a, θ a 3 = θa 1 +θa 2, θ3a = θ 1a + θ 2a. 2.7 Lettingz 1 z 2 wemaygetthesupertranslationinvariantoneforms, e M = e µ,dθ aα,d θ α a, where e µ z = dx µ +idθ a σ µ θa iθ a σ µ d θ a. 2.8 The exterior derivative, d, on superspace is defined as d dz M z M = em D M = e µ µ +dθ aα D aα d θ α a D ȧ α, See Appendix A for our notations and some useful equations. 4

6 where D M = µ,d aα, D ȧ α are covariant derivatives µ = x µ, D aα = θ aα iσµ θa α x µ, satisfying the anti-commutator relations Dȧ α = θ α a +iθ a σ µ α x µ, 2.10 {D aα, D ḃ α} = 2iδ b a σ µ α α µ Under an arbitrary superspace coordinate transformation, z z, e M and D M transform as e M z = e N zr N M z, D M = R 1 M N zd N, 2.12 so that the exterior derivative is left invariant e M zd M = e M z D M, 2.13 where R M N z is a 4+4N 4+4N supermatrix of the form with R M N z = Rµ νz µθ bβ β µ θ b B aαz µ D aα θ bβ D aα θ β b B aµ α z D ȧ α θ bβ D ȧ α θ β b, 2.14 Rµ ν x ν z = x µ +i θ a x µσν θ a iθ a σ ν θ a x, 2.15 µ B µ aαz = D aα x µ +id aα θ b σ µ θ b +iθ b σ µ D aα θ b, 2.16 B aµ α z = D ȧ α x µ +i D ȧ α θ b σ µ θ b +iθ b σ µ Dȧ α θ b = B µ aα z Superconformal Group & Killing Equation The superconformal group is defined here as a group of superspace coordinate transformations, z z, that preserve the infinitesimal supersymmetric interval length, e 2 = η µν e µ e ν g up to a local scale factor, so that e 2 z e 2 z = Ω 2 z;ge 2 z,

7 where Ωz;g is a local scale factor. This requires B aα µ aµ z = B α z = 0 D aα x µ +id aα θ b σ µ θ b +iθ b σ µ D aα θ b = 0, D ȧ α x µ +i D ȧ α θ b σ µ θ b +iθ b σ µ Dȧ α θ b = 0, 2.19 and e µ z = e ν zr µ ν z;g, 2.20 R λ µ z;gr ρ ν z;gη λρ = Ω 2 z;gη µν, detrz;g = Ω 4 z;g Hence R M N in eq.2.14 is of the form 2 R M N z;g = R ν µ z;g µθ bβ µ θ β b 0 D aα θ bβ D aα θ β b 0 D ȧ αθ bβ D ȧ α θ β b Rµ ν z; g is a representation of the superconformal group. Under the successive superconformal transformations, z z g g z giving z g z, we have Infinitesimally z z +δz, eq.2.19 gives where we define Infinitesimally R ν µ Rz;gRz ;g = Rz;g D aα h µ = 2iσ µ λa α, Dȧ α h µ = 2iλ a σ µ α, 2.24 λ a = δθ a, λa = δ θ a, h µ = δx µ +iδθ a σ µ θa iθ a σ µ δ θ a. from eq.2.15 is of the form 2.25 R ν µ δ ν µ + µh ν, 2.26 so that the condition 2.21 reduces to the ordinary conformal Killing equation 2 More explicit form of R M N is obtained later in eq µ h ν + ν h µ η µν

8 Eq.2.27 follows from eq Using the anti-commutator relation for D aα and D ḃ α 2.11 we get from eq.2.24 and hence δa b ν h µ = 1 Daα λ b σ µ σ 2 ν α σ ν σ µ Dḃ α α λa, 2.28 δ b a µ h ν + ν h µ = D aα λ bα D ḃ α λ α aη µν, 2.29 which implies eq Thus eq.2.24 is a necessary and sufficient condition for a supercoordinate transformation to be superconformal. With the notation as in eq.a.15 we write h α α = h µ σ µα α, h αα αα = h µ σ µ, 2.30 and using eq.a.7, eq.2.24 is equivalent to D aα h β β = 0, Da α h β β = 0, 2.31 or while λ aα, λ α a are given by D aα h ββ = 1 2 δ β α D aγ h βγ, Dȧ α h ββ = 1 2 δ β α D ȧ γ h γβ, 2.32 λ aα = i 1 8 D ȧ α h αα, λ α a = i 1 8 D aα h αα Eq.2.31 may therefore be regarded as the fundamental superconformal Killing equation and its solutions give the generators of extended superconformal transformations in fourdimensions. The general solution is 3 hz = x b x + x w 1 2 λ+4ρ aθ a +w λ 4 θ a ρ a x + 4i θ a t a b θb +2Ω θ a θ a +4i ε a θ a θ a ε a +ã, 2.34 where a µ,b µ,λ R, Ω S 1, t sun and for w µν = w νµ we define w = 1w 4 µν σ µ σ ν, w = 1w 4 µνσ µ σ ν For later use it is worth to note ǫ w t ǫ 1 = w, ǫ 1 w t ǫ = w, w = w A method of obtaining the solution 2.34 is demonstrated in Appendix B. 7

9 Eq.2.34 also gives and λ a = ε a λ+iωθa θ a w+t a b θb +θ a b x + i ρ a x + 4θ a ρ b θ b, 2.37 δ x + = x + b x + 4 x + ρ a θ a +λ x + +w x + x + w+4i ε a θ a +ã Note that δ x +,λ a are functions of x +,θ a only, which can be also directly shown from eq In fact, the superconformal group can be obtained alternatively by imposing the super-diffeomorphisms to leave the chiral subspaces of superspace invariant. The chiral structures are given by z M + + = x µ +,θ aα, z M = x µ, θ ȧ α In this approach, one needs to solve a reality condition [39,40] δ x + x +,θ δ x x, θ = 4i λ a x, θθ a +4i θ a λ a x +,θ Extended Superconformal Transformations In summary, the generators of extended superconformal transformations in four-dimensions acting on the four-dimensional superspace, R 4 4N, with coordinates, z M = x µ,θ a, θ a, can be classified as 1. Supertranslations, a, ε, ε δx µ = a µ +iε a σ µ θa iθ a σ µ ε a, δθ a = ε a, δ θ a = ε a This is consistent with eq Dilations, λ δx µ = λx µ, δθ a = 1 2 λθa, δ θ a = 1 2 λ θ a Lorentz transformations, with w, w defined in eq.2.35 δx µ = w µ ν xν, δθ a = θ a w, δ θ a = w θ a R-symmetry transformations, UN, of dimension N 2, t,ω δx µ = 0, δθ a = t a b θb +i 1 2 Ωθa, δ θ a = θ b t b a i1ω θ 2 a, 2.44 where the N N matrix, t, is a SUN generator, i.e. t = t, t a a = 0 and Ω S1. 8

10 5. Special superconformal transformations, b, ρ, ρ δx µ = 2x bx µ x 2 b µ +θ a σ µ x + ρ a + ρ a x σ µ θa +2θ a b θ b θ b σ µ θa, δθ a = θ a b x + i ρ a x + 4θ a ρ b θ b, Superinversion δ θ a = x b θ a +i x ρ a 4 θ b ρ b θa. In four-dimensions we define superinversion, z M is z M = x µ,θ aα, θ a α R 4 4N, by x µ ± = xµ, θ at = i 1 ǫ 1 x θb ζ ba, x 2 x 2 θ a = i 1 ǫ 1 x t x +θ bt 2 ζba, where N N matrices, ζ ab, ζ ab satisfy Eq.2.46 may be rewritten as ζ ab ζbc = δ a c, ζab = ζ ba, ζab = ζ ab θ a = i 1 θ a x x 2, θ a = ζ ba θt b ǫ, θ a = i 1 x + θa, x 2 θa = ǫθ bt ζba It is easy to verify that superinversion is idempotent i 2 s = Using we get under superinversion ẽ = e µ σ µ = d x + 4i θ a dθ a, e = e µ σ µ = dx + +4id θ a θa, 2.50 ẽz = x 1 + ezx 1, ez = x 1 ẽz x 1 +, 2.51 and hence e 2 z = Ω 2 z;i s e 2 z, Ωz;i s = 9 1. x 2 +x

11 Eq.2.51 can be rewritten as e µ z = e ν zr ν µ z;i s, R µ ν z;i s = 1 2 tr x 1 σ ν x 1 + σ µ = 1 2 trx 1 + σ ν x 1 σ µ, 2.53 or using eq.a.6b R µν z;i s = 1 x µ x +x ν 2 +x 2 +xµ x ν + ηµν x + x +iǫ µν λρx λ xρ i s g i s If we consider a transformation, z z, where g is a four-dimensional superconformal transformation, then we get from eqs.2.37, 2.38 δ x + = x + a x + 4 x + ε aθ a λ x + +w x + x + w+4i ρ a θ a + b, δθ a = ρ a 1 2 λ+iωθ a θ a w ζt t ζ a b θ b +θ a a x + i ε a x + 4θ a ε b θ b, 2.55 where ρ a = ζ ba ρ t b ǫ 1, ρ a = ǫ 1 ρ bt ζba, ε a = ǫε bt ζba, ε a = ζ ba ε t b ǫ. Hence, under superinversion, the superconformal transformations are related by K a µ b µ ε a ρ a λ Ω w µ ν t a b b µ a µ ρ a ε a λ Ω w µ ν ζt t ζ a b In particular, special superconformal transformations 2.45 can be obtained by where b, ρ is a supertranslation and u M = b µ, ρ aα, ρ α a. z i s b, ρ i s z z s z;u,

12 2.5 Superconformal Algebra The generator of infinitesimal superconformal transformations, L, is given by If we write the commutator of two generators, L 1,L 2, as then h µ 3, λ aα 3, λ α 3a are given by L = h µ µ +λ aα D aα λ α a D ȧ α [L 2,L 1 ] = L 3 = h µ 3 µ +λ aα 3 D aα λ α 3a D ȧ α, 2.60 h µ 3 = h ν 2 νh µ 1 +2iλ a 1 σµ λ2a 1 2, λ aα 3 = h µ 2 µ λ aα 1 +λ bβ 2 D bβ λ aα 1 1 2, 2.61 λ α 3a = h µ 2 µ λ α 1a λ β 2b D ḃ β λ α 1a 1 2, and h µ 3, λ aα 3, λ α 3a satisfy eq.2.24 verifying the closure of the Lie algebra. Explicitly with eqs.2.34,2.37 we get a µ 3 = w µ 1νa ν 2 +λ 1a µ 2 +2iε a 1 σµ ε 2a 1 2, ε a 3 = εa 2 w 1 + 1λ 2 1ε a 2 i ρa 1ã2 +t a 1bε b 2 +i1ω 2 1ε a 2 1 2, ε 3a = w 1 ε 2a λ 1 ε 2a +iã 2 ρ 1a ε 2b t b 1a i 1 2 Ω 1 ε 2a 1 2, λ 3 = 2a 2 b 1 +2 ρ a 1 ε 2a +ε a 2ρ 1a 1 2, w µν 3 = w µ 1λw λν 2 +2a µ 2b ν 1 aν 2 bµ 1+2ε a 2 σ[µ σ ν] ρ 1a ρ a 1 σ[µ σ ν] ε 2a 1 2, b µ 3 = w µ 1νb ν 2 λ 1 b µ 2 +2i ρ a 1 σ µ ρ 2a 1 2, 2.62 ρ 3a = w 1 ρ 2a 1λ 2 1ρ 2a +ib 2 ε 1a ρ 2b t b 1a i 1Ω 2 1ρ 2a 1 2, ρ a 3 = ρa 2 w 1 1λ 2 1 ρ a 2 iεa 1 b 2 +t a 1b ρ b 2 +i1ω 2 1 ρ a 2 1 2, t a 3b = t 1 t 2 a b +4ε a 1 ρ 2b ρ a 2 ε 1b 4 N εc 1 ρ 2c ρ c 2 ε 1cδ a b 1 2, Ω 3 = 2i 4 N 1εa 2ρ 1a ρ a 1 ε 2a

13 From eq.2.62 we can read off the explicit forms of four-dimensional superconformal algebra as exhibited in Appendix C. Now, we consider N 4 case and N = 4 case separately. For N 4 case, if we define a 4+N 4+N supermatrix, M, as where M = w+ 1 2 λ+i1 2 ψ iã 2 ε b ib w 1 2 λ+i1 2 ψ 2ρ b 2 ρ a 2ε a t a b +i 2 N ψδa b, 2.63 ψ = Ω, then the relation above 2.62 agrees with the matrix commutator 4 N [M 1,M 2 ] = M This can be verified using eqs.a.7, A.11, A.12. In general, for N 4, M can be defined as a 4,N supermatrix subject to and a reality condition BMB 1 = M, B = strm = 0, Supermatrix of the form 2.63 is the general solution of these two equations. The 4 4 matrix appearing in M, w + 1 λ iã 2 ib w 1λ, corresponds to a generator of O2,4 = SU2,2 as demonstrated in Appendix D. Thus, the N 4 superconformal group in four-dimensions may be identified with the supermatrix group generated by supermatrices of the form M 2.63, which is SU2,2 N G s having dimensions 15+N 2 8N. 12

14 When N = 4, similar analysis is also possible with a subtle modification. In this case, M is a 4, 4 supermatrix satisfying the reality condition 2.67 and, instead of eq.2.66, Such a supermatrix, M, is of the general form M = M+i 1 2 Ω δ a b strm = 2iΩ i 1 2 ψ δ a b, 2.70 where M is of the form 2.63 with ψ = 0, and ψ in eq.2.70 is just an arbitrary real bosonic variable. Essentially we add the Ω term to eq.2.63 to get eq Same as N 4 case, the matrix commutator of M 1 and M 2 reproduces eq.2.62 as in eq.2.65, though ψ is arbitrary. As the variable, ψ, is auxiliary, one might be tempted to fix its value, or more generally let it be a function of the parameters of superconformal transformations, a µ, b µ, Ω, and try to determine the function. However, this is not possible. The commutator of M 1 and M 2 includes ψ type term [M 1,M 2 ] = M 3 +i 1 2 ψ 31, ψ 3 = 2iε a 2 ρ 1a ε a 1 ρ 2a + ρ a 2 ε 1a ρ a 1 ε 2a, 2.71 and ψ 3 can not be expressed in terms of the parameters, a µ 3, b µ 3, Ω 3,, appearing in eq Hence it is not possible to put ψ as a function of the superconformal transformation parameters. 4 Therefore four-dimensional N = 4 superconformal algebra is represented by 4, 4 supermatrices, M, satisfying the reality condition 2.67 with an equivalence relation,, imposed M 1 M 2 if M 1 M 2 = iψ1 for some ψ R We note that an extra condition, strm = 0, defines an invariant subalgebra of the whole four-dimensional N = 4 superconformal algebra. This invariant subalgebra forms a simple Lie superalgebra. In the literature the four-dimensional N = 4 superconformal algebra is often identified with this simple Lie superalgebra, the R-symmetry of which is su4 rather than u4 [18], as the Ω term in eq.2.70 is neglected. However, we emphasize here that the whole N = 4 superconformal algebra may contain a u1 factor which has non-trivial 4 An alternative approach may be taken as in [41], where a modified supermatrix commutator is introduced for SLm m. 13

15 commutator relations with other generators as seen in eq.2.62 or eq.c.9. The N = 4 superconformal group in four-dimensions, G s, is now identified with a quotient group of the supermatrix group, as it is isomorphic to the supermatrix group generated by supermatrices of the form M 2.70 with an equivalence relation imposed on the supermatrix group element, G, from eq.2.72 G 1 G 2 if G 1 1 G 2 = e iψ 1 for some ψ R We also note that the four-dimensional N = 4 superconformal group has dimensions and is isomorphic to a semi-direct product of U1 and a simple Lie supergroup. Therefore, by breaking the U1 symmetry, the four-dimensional N = 4 superconformal group can be reduced to the simple Lie subgroup having dimensions Coset Realization of Transformations In this section, we first obtain an explicit formula for the finite nonlinear superconformal transformations of the supercoordinates and discuss several representations of the superconformal group. We then construct matrix vector valued functions depending on two three points in superspace which transform covariantly under superconformal transformations. These are crucial variables for obtaining two-point, three-point and general n-point correlation functions later. 3.1 Superspace as a Coset To obtain an explicit formula for the finite nonlinear superconformal transformations, we first identify the superspace, R 4 4N, as a coset, G s /G 0, where G 0 G s is the subgroup generated by matrices, M 0, of the form 2.63 with a µ = 0, ε a = 0 and depending on parameters b µ, ρ a, ρ a, λ, Ω, w µν, t a b. The group of supertranslations, G T, parameterized by coordinates, z M R 4 4N, has been defined by general elements as in eq.2.3, with the group property given by eqs.2.6,2.7. Now we may represent it by supermatrices 5 G T z = exp 0 i x 2 θ b θ a 0 = 1 i x + 2 θ b θ a δ a b. 3.1 Note that G T z 1 = G T z. In general an element ofg s canbe uniquely decomposed asg T G 0. Thus for any element 5 The subscript, T, denotes supertranslations. 14

16 g Gg G s we may define a superconformal transformation, z z, and an associated element G 0 z;g G 0 by If Gg G T then clearly G 0 z;g = 1. Infinitesimally eq.3.2 becomes Gg 1 G T zg 0 z;g = G T z. 3.2 δg T z = MG T z G T z ˆM 0 z, 3.3 where M is given by eq.2.63 or eq.2.70 and ˆM 0 z, the generator of G 0, has the form ˆM 0 z ŵz+ 2ˆλz+i 1 1 ˆψz = 1 ib ŵz 2ˆλz+i 1 ˆψz 2ˆρ 2 b z. 2 ˆρ a z 0 ˆt a bz+i 2 ˆψz+δ 4 N N Ωδ a b 3.4 The components depending on z are given by ŵz = w 4 θ a ρ a + x b+ 1 2 tr4 θ a ρ a x b1, ŵz = w +4ρa θ a b x tr4ρ aθ a b x + 1 = ŵz, ˆλz = λ+2b x+2θ a ρ a + ρ a θa = 1 4 µh µ z, ˆψz = ψ +2θ a b θ a +2iθ a ρ a ρ a θa, 3.5 ˆt a bz = t a b +4iθ a b θ b +4 ρ a θb θ a ρ b 1 N 4iθc b θ c +4 ρ c θc 4θ c ρ c δ a b, ˆρ a z = ρ a ib θ a = i 1 4 σµ µ λa z, ˆρ a z = ρ a +iθ a b = ˆρ a z. Writing δg T z = LG T z we may verify that L is identical with eq ŵz, ŵz can be also written as ŵz = 4ŵµνz σ 1 µ σ ν, ŵz = 1 4ŵµνzσ µ σ ν, with ŵ µν z = w µν +4x [µ b ν] +θ a σ [µ σ ν] 2ρ a ib θ a 2 ρ a +iθ a b σ [µ σ ν] θa = [µ h ν] z

17 The definitions3.5 can be summarized by and they give D bβ λ aα = 1δ a 2 b δ β αˆλz+iˆωz δ b a ŵ α β z+δβ αˆt a b z, 3.7 [D aα,l] = 1 2 ˆλz+iˆΩzD aα ŵ α β zd aβ +ˆt b azd bα, 3.8 where ˆΩz = 4 N 1ˆψz+δ 4 N Ω. 3.9 For later use we note D aα ŵ µν z = 2σ [µ σ ν] αβˆρ aβ z, D aαˆλz = 2ˆρaα z, D aαˆψz = 2iˆρaα z, 3.10 D aαˆt b cz = 4δ abˆρ cα z+ 4 N δb cˆρ aα z. The above analysis can be simplified by reducing G 0 z;g. To achieve this we let and then M 0 Z 0 = Z 0 H 0, H 0 = Now if we define Z 0 = , 3.11 w 1 2 λ+i1 2 ψ 2ρ b 0 t a b +i 2 N ψ +δ4 N Ωδ a b i x + 2 θ b Zz G T zz 0 = 1 0, θ a δ a b then Zz transforms under infinitesimal superconformal transformations as δzz = LZz = MZz ZzHz,

18 where Hz is given by ŵz 1 ˆM 0 zz 0 = Z 0 Hz, Hz = 2ˆλz+i 1 ˆψz 2ˆρ 2 b z 0 ˆt a bz+i 2 ˆψz+δ. 4 N N Ωδ a b 3.15 From eqs.2.60, 2.65 considering we get [L 2,L 1 ]Zz = L 3 Zz, 3.16 H 3 z = L 2 H 1 z L 1 H 2 z+[h 1 z,h 2 z], 3.17 which gives separate equations for ŵ, ˆλ, ˆψ, ˆρ a and ˆt a b, thus ˆλ 3 = L 2ˆλ1 L 1ˆλ2, and so on. As a conjugate of Zz we define Zz by Zz = Zz B = 1 i x 2 θ b 0 2θ a δ a b This satisfies Zz = Z0G T z 1, 3.19 and corresponding to eq.3.14 we have δ Zz = L Zz = Hz Zz ZzM, 3.20 where ŵz+ 1 Hz = 2ˆλz+i 1 ˆψz ˆρ a z ˆt a bz++i 2 ˆψz+δ N N Ωδ a b 3.2 Finite Transformations Finite superconformal transformations can be obtained by exponentiation of infinitesimal g transformations. To obtain a superconformal transformation, z z, we therefore solve the differential equation d dt zm t = L M z t, z 0 = z, z 1 = z, 3.22 where, with L given in eq.2.59, L M z is defined by L = L M z M

19 From eq.3.14 we get which integrates to where Kz, t satisfies d dt Zz t = MZz t Zz t Hz t, 3.24 Zz t = e tm ZzKz,t, 3.25 d dt Kz,t = Kz,tHz t, Kz,0 = Hence for t = 1 with Kz,1 Kz;g the superconformal transformation, z eq.3.25 becomes g z, from Zz = Gg 1 ZzKz;g, Gg 1 = e M G 0 z;g in eq.3.2 is related to Kz;g from eq.3.27 by In general Kz;g is of the form Kz;g = G 0 z;gz 0 = Z 0 Kz;g Lz+ ;g 2Σ b z;g 0 u a bz;g From 1 ŵz 2ˆλz+i 1 ˆψz = w λ+i1ψ +4ρ 2 aθ a b x +, 3.30 Lz + ;g is defined on chiral superspace, and since Lz + ;g is a 2 2 matrix, we have Infinitesimally this is consistent with eq We decompose Lz + ;g as Lz + ;gǫlz + ;g t = detlz + ;gǫ ˆLz + ;g = Ω + z + ;g 1 2Lz + ;g, 3.32a Ω + z + ;g = detlz + ;g, 3.32b where ˆL SL2,C, the 2 2 matrices with determinant one. Since d uz,t = uz,tˆtz dt t +i 2 ˆψz N t +i 1 2 δ4 N Ω, u = u 1 and hence u UN. If we write uz; g ûz;g =, 3.33 detuz;g 1 N 18

20 then û SUN. From eq.3.27 Z transforms as Zz = Kz;g ZzGg, 3.34 where Kz;g = Kz;g = Lz ;g 0 2 Σz;g u 1 z;g Lz ;g = Lz + ;g, Σz;g = Σz;g., 3.35 In a similar fashion to eq.3.32a we write ˆ Lz ;g = Ω z ;g 1 2 Lz ;g = ˆLz + ;g SL2,C, 3.36a Ω z ;g = det Lz ;g = Ω + z + ;g. 3.36b If we define for superinversion, z is z, 2.48 Gi s 1 = ǫ ǫ ζ ab, Kz;i s = an analogous formula to eq.3.27 can be obtained 1 0 Gi s 1 ZzKz;i s = Similarly we have where i x t 2θ at 2 θ b t δb a i ǫ x+ 1 2i x 1 θ c ζ cb 0 v a c zζcb, 3.37 = Zz t Kz;i s ZzGi s = Zz t, 3.39 i x ǫ Kz;i s = i ζ ac θc x + ζ ac v 1c bz For later use, we also define with eq.3.9,3.22 d dt Υz,t = iˆωz t Υz,t, Υz,0 = 1, Υz;g Υz,1,

21 and Note that Since Ωz;g Ω + z + ;gω z ;g Ωz;g = Ωz;g, Υz;g = Υz;g sdetg = expstrlng, 3.44 when N = 4, Υz;g is related to the superdeterminant of Gg G s Υz;g = e iω = sdetgg If σ ν R ν µ z;g = Lz +;gσ µ Lz ;g, 3.46 then R ν µ z;g is identical to the definition 2.20, since infinitesimally ˆλzσ µ ŵzσ µ +σ µ ŵz = ˆλzσ µ σ ν ŵ ν µ z = σ ν ν h µ z, 3.47 which agrees with eq Furthermore eq.3.46 shows that the definition2.21 of Ωz;g is consistent with eq We may normalize R ν µ z;g as well ˆR ν µ z;g = Ωz;g 1 R ν µ z;g SO1,3 = SL2,C/Z Representations Basedontheresultsintheprevioussubsection, itiseasytoshowthatthematrix,r M N z;g, given in eq.2.22 is of the form R M N z;g = Rµ ν z;g i Σ b z;g σ µ Lz + ;g β i Lz ;g σ µ Σ b z;g β 0 Ωz;gΥz;g 1 β 2 ˆL α z + ;gû 1b a z;g Ωz;g Υz; g 1 2 ˆ L β αz ;gû a bz;g 3.49 Since R M N z;g is a representation of the four dimensional N-extended superconformal group, each of the following also forms a representation of the group, though it is not a faithful representation Ωz;g D, Υz;g U1, ˆRz;g SO1,3, ˆLz + ;g SL2,C, ˆ Lz ;g SL2,C, ûz;g SUN,

22 where D is the one dimensional group of dilations. Under the successive superconformal transformations, g : z g z g z, they satisfy Ωz;gΩz ;g = Ωz;g, and so on We note that when N 4, Ω + z + ;g and Ω z ;g can be written as Ω + z + ;g = Ωz;gΥz;g N N 4, Ω z ;g = Ωz;gΥz;g N N Hence, they also form representations of the N 4 superconformal group. On the other hand, in the case of N = 4, due to the arbitrariness of ψ in eq.2.70 Lz + ;g and Lz ;g do not form representations. They do so only if the equivalence relation2.73 is imposed, but this will give just ˆLz + ;g,ˆ Lz ;g and Ωz;g. 3.4 Functions of Two Points In this subsection, we construct matrix valued functions depending on two points, z 1 and z 2, in superspace which transform covariantly like a product of two tensors at z 1 and z 2 under superconformal transformations. If Fz is defined for z R 4 4N by Fz = Z0G T zz0 = i x+ 2 θ b 2θ a δ a b, 3.53 then Fz satisfies F z = Fz = i x 2 θ b 2θ a δ a b, 3.54 and the superdeterminant of Fz is given by sdetfz = x If we consider iθa x + 1 Fz 1 2i x 1 + θ b 0 1 = i x+ 0 0 v a b z,

23 then this defines v a b z as v From eqs.3.55, 3.56 it is evident that a bz = δ a b +4i 1 θ a x θb x 2 detvz = x x 2 + It is useful to note v a b z = v 1a bz = v a b z = δa b 4i 1 θ a x + θb x 2 + Now with the supersymmetric interval for R 4 4N superspace defined by G T z 2 1 G T z 1 = G T z 12, z M 12 = x µ 12,θ a 12, θ 12a = z M 21, x µ 12 = x µ 1 x µ 2 +iθ1σ a µ θ2a iθ2σ a µ θ1a, θ12 a = θ1 a θ2 a, θ12a = θ 1a θ 2a, 3.60 we may write and where Zz 2 Zz 1 = Fz 12 = i x 21 2 θ 21b 2θ21 a δ a b sdetfz 12 = x 2 12,, 3.61 detvz 21 = x2 21, x x µ 21 = xµ 2 x µ 1+ 2iθ a 1σ µ θ2a = x µ 21 +iθ a 21σ µ θ21a = x 21 µ, x µ 12 = xµ 1 x µ 2+ 2iθ a 2 σµ θ1a = x µ 12 +iθ a 12 σµ θ12a = x 12 µ. From eqs.3.27,3.34 Fz 12 transforms as 3.63 Fz 12 = Kz 2 ;gfz 12 Kz 1 ;g In particular, with eqs.3.29,3.35, this gives transformation rules for x 12 and x 21 x 12 = Lz 1 ;g x 12Lz 2+ ;g, 3.65a x 21 = Lz 2 ;g x 21Lz 1+ ;g, 3.65b 22

24 so that x 2 12 = Ω z 1 ;gω + z 2+ ;gx 2 12, 3.66a x 2 21 = Ω z 2 ;gω + z 1+ ;gx 2 21, 3.66b and in particular x 2 12 x 2 21 = Ωz 1;g 2 Ωz 2 ;g 2 x2 12 x2 21, 3.67a x 2 12 x N 1 = Υz 1;g 2 Υz 2 ;g 2 x 2 12 x N b From eqs.3.46,3.65a trσ µ x 12σ ν x 21 transforms covariantly as trσ µ x 12σ ν x 21 = trσ λ x 12σ ρ x 21R µ λ z 1 ;gr ν ρ z 2 ;g Since v a bz 21 transforms infinitesimally as δvz 21 = ˆtz 2 vz 21 vz 21 ˆtz 1 +i 2 ˆψz N 2 ˆψz 1 vz 21, 3.69 finitely it transforms as vz 21 = u 1 z 2 ;gvz 21 uz 1 ;g From eqs.3.38,3.39 Fz 12 transforms under superinversion as Kz 2 ;i s Fz 12 Kz 1 ;i s = F z 12 t, 3.71 which gives and x 1 2 x 21 x 1 1+ = x 12, x 2 12 = x2 21, 3.72 x 2 2 x 2 1+ ζv 1 z 2 vz 21 vz 1 ζ = vz 21 t Eq.3.72 shows that eq.3.68 holds for superinversion as well trσ µ x 12σ ν x 21 = trσ λ x 12σ ρ x 21R λ µ z 1 ;i s R ρ ν z 2 ;i s

25 3.5 Functions of Three Points In this subsection, for three points, z 1,z 2,z 3 in superspace, we construct tangent vectors, Z i, which transform homogeneously at z i, i = 1,2,3. i With z s 21 z21 i, z s 31 z31, we define Z M 1 = X 1,Θ µ a 1, Θ 1a R 4 4N by G T z 31 1 G T z 21 = G T Z Explicit expressions for Z M 1 can be obtained by calculating Zz 31 Zz 21 i X1+ 2 Θ = FZ 1 = 1b 2Θ a 1 δ a b We get Using one can assure Θ a 1 = i θa 21 x 1 21 θa 31 x 1 31, X 1+ = x x 23x 21, Θ1a = ix 1 13 θ 13a x 1 12 θ 12a x 13 +x 21 +4i θ 13a θa 21 = x 23, 3.78 X 1 = X 1+ 4i Θ 1a Θ a 1 = x x 32x 31 = X 1+, X 1 = 1 2 X 1+ + X 1 = X µ 1 σ µ It is evident from eq.3.75 that under z 2 z 3, Z 1 Z 1. Associated with Fz given in eq.3.53 we define Fz by Fz = ǫ 0 0 ζ Fz t ǫ 0 0 ζ = ix+ 2 θ b 2 θa δ a b With this definition we may write ix1+ 2 Θ 1b FZ 1 = 2 Θa 1 δ a b, 3.81 where Θ 1a = i x 1 31 θ 31a x 1 21 θ 21a, X 1+ = x x 23 x 13, Θa 1 = iθ a 12 x θa 13 x

26 FZ 1 transforms infinitesimally as δ FZ 1 = ŵz1 1 2ˆλz 1 +i 1 2 ˆψz ˆtz 1 +i 2 N ˆψz 1 +δ 4 N Ω FZ 1 FZ 1 and hence for finite transformations FZ 1 = Lz1+ ;g uz 1 ;g 1 ŵz ˆλz 1 +i 1 2 ˆψz ˆtz 1 +i 2 N ˆψz 1 +δ 4 N Ω FZ 1 Lz1 ;g uz 1 ;g Thus Z 1 transforms homogeneously at z 1, as tangent vectors do. Explicitly we have from eq.3.84 X 1+ = Ωz 1 ;g 1ˆLz1+ ;g 1 X 1+ˆ Lz1 ;g 1,, Θ 1a = Ωz 1;g 1 2Υz 1 ;g 1 2 ˆLz1+ ;g 1 Θ1b û b a z 1;g, 3.85 Θ a 1 = Ωz 1 ;g 1 2Υz 1 ;g 1 2û 1a bz 1 ;g Θb 1ˆ Lz1 ;g 1. X 1 also transforms in the same way as X 1+ in eq.3.85 and hence X µ 1 = Ωz 1 ;g 1 X ν 1 ˆR ν µ z 1 ;g, 3.86a From eq.3.55 we get Θ a 1 σµ Θ 1a = Ωz 1 ;g 1 Θ a 1 σν Θ1a ˆRν µ z 1 ;g. 3.86b If we define a function ṽz UN by then a direct calculation leads sdet FZ 1 = sdetfz 1 = X1 2 = x x2 12 x2 31 ṽ a b z = ζvzt ζ a b = δ a b 4i 1 θ a x θb, 3.88 x 2 ṽz 1 = vz 13 vz 32 vz

27 Similarly for R ν µ z;i s given in eq.2.53 we have RZ 1 ;i s = x 2 12 x2 21 x2 31 x2 13 Rz 12;i s Rz 23 ;i s Rz 31 ;i s From eqs.3.68,3.70 ṽz 1,RZ 1 ;i s transform homogeneously at z 1 under superconformal g transformation, z z, ṽz 1 = u 1 z 1 ;gṽz 1 uz 1 ;g, 3.91a RZ 1 ;i s = Ωz 1 ;g 2 R 1 z 1 ;grz 1 ;i s Rz 1 ;g. 3.91b i Under superinversion, z s j z j, j = 1,2,3, Z 1 transforms to Z 1, from eq.3.72, as and hence X 1+ = x 1+X 1 x 1, Θ a 1 = iv 1a bz 1 Θb 1 x 1, 3.92 X µ 1 = Ωz 1 ;i s 2 X ν 1 R ν µ z 1 ;i s, 3.93a Θ a 1 σ µ Θ 1a = Ωz 1 ;i s 2 Θ a 1σ ν Θ1a R ν µ z 1 ;i s. 3.93b Note the minus sign in eq.3.93b. By taking cyclic permutations of z 1,z 2,z 3 in eq.3.77 we may define Z 2,Z 3. We find Z 2,Z 3 are related to Z 1 in a simple form FZ 2 = FZ 3 = i x vz 21 ix ṽz 13 FZ 1 i x vz 12 FZ 1 ix ṽz 31 where Z = X,Θ a, Θ a is defined by superinversion, Z is Z. Explicitly we have,, 3.94 X 2 + σ = x 21X 1+ x 12, Θ 2 a = iv a bz 21 Θb 1 x 12, 3.95a X 3+ = x 1 31 X 1 1+ σx 13, Θa 3 = iṽa bz 13 Θ1 b x b 26

28 From eqs.3.77, 3.79 we get X 1+ X 1 X 2 1+X 2 1 X 2 1+ X 2 1 = x2 12 x2 23 x2 31, 3.96a x2 21 x2 13 x2 32 = trx 21 x 23x 13 x 12x 32 x b x2 12 x2 21 x2 23 x2 32 x2 31 x2 13 These expressions are invariant under cyclic permutations of z 1,z 2,z 3 and hence X 1+ X 1 X 2 1+ X 2 1 X 2 1+ X 2 1 = X2 2+ X 2 2 = X 2+ X 2 X 2 2+ X 2 2 = X2 3+, 3.97a X3 2 = X 3+ X b X 2 3+ X3 2 From eq.3.86a these are invariants for any continuous superconformal transformation and furthermore from eq.3.93a the latter is invariant under superinversion along with X X X1 2 X1+ 2 Note that such invariants, depending on three points, do not exist in ordinary conformal theories and that in the case of N = 1 due to the identity A.19a those two variables are not independent [24, 25]. 4 Superconformal Invariance of Correlation Functions In this section we discuss the superconformal invariance of correlation functions for quasiprimary superfields and exhibit general forms of two-point, three-point and n-point functions. 4.1 Quasi-primary Superfields We first assume 6 that there exist quasi-primary superfields, Ψ I z, which under the superconformal transformation, z z, transform g as Ψ I Ψ I, Ψ I z = Ψ J zdj I z;g In [24] it was explicitly shown that the chiral/anti-chiral superfields and supercurrents in some N = 1 theories are quasi-primary. 27

29 Dz; g obeys the group property so that under the successive superconformal transformations, g g : z z g z, it satisfies and hence also Dz;gDz ;g = Dz;g, 4.2 Dz;g 1 = Dz ;g We choose here Dz;g to be a representation of SL2,C SUN U1 D, which is a subgroup of the stability group at z = 0, and so we decompose the spin index, I, of superfields into SL2,C index, ρ, and SUN index, r, as Ψ I Ψ ρ r. Now DJ I z;g is factorized as D I J z;g = D σ ρ ˆLz + ;gd r s ûz;gωz;g η Υz;g κ, 4.4 where Dρ σˆl, D r s û are representations of SL2,C, SUN respectively, while η and κ are the scale dimension and R-symmetry charge of Ψ ρ r respectively. Infinitesimally δψ ρ rz = L+ηˆλz+iκˆΩzΨ ρ rz Ψ σ rz 1s 2 µν σρŵ µν z Ψ ρ sz 1 2 sa b s rˆt b a z, 4.5 where s µν, s a b are matrix generators of SO1,3, SUN satisfying and hence [s µν,s λρ ] = η µλ s νρ +η µρ s νλ +η νλ s µρ η νρ s µλ, From eqs.3.15, 3.17 using eq.4.6 we have [s a b,sc d ] = 2δa d sc b δc b sa d, 4.6 [ 1 2 sa bt b 1a, 1 2 sc dt d 2c] = 1 2 sa b[t 1,t 2 ] b a. 4.7 δ 3 Ψ ρ r = [δ 2,δ 1 ]Ψ ρ r. 4.8 It is useful to consider the complex conjugate superfield of Ψ ρ r Ψ ρr z transforms as Ψ ρr z = Ψ ρ rz. 4.9 Ψ ρr z = Ωz;g η Υz;g κ Dρ σ ˆ Lz ;gd r s ûz;g 1 Ψ σs z

30 Superconformal invariance for a general n-point function requires Ψ I 1 1 z 1Ψ I 2 2 z 2 Ψ In n z n = Ψ I 1 1 z 1Ψ I 2 2 z 2 Ψ In n z n In superconformal field theories on chiral superspace, the representation of U1 D is given by Ω + z + ;g η, 4.12 so that for N 4, η and κ are related by, from eq.3.52, η + 4 N 1κ = On the other hand when N = 4, as shown in subsection 3.3, Ω + z + ;g does not form a representation of the N = 4 superconformal group, and hence there is no conventional way of defining quasi-primary chiral/anti-chiral superfields in N = 4 superconformal theories. We speculate that this fact makes it difficult to construct four-dimensional N = 4 superconformal theories on chiral superspace. 4.2 Two-point Correlation Functions The solution for the two-point function of the quasi-primary superfields, Ψ ρ r, Ψ ρr, has the general form 7 where we define Ψ ρr z 1 Ψ σ I ρσ ˆ x 12I r sˆvz 12 sz 2 = C Ψ, 4.14 x η 4 N 1κ x η+ 4 1κ N ˆ x 12 = x 12 x SL2,C, 1 x ˆv a N 2 21 bz 12 = δ a x 2 12 b +4iθ 12 x 1 a θ 12 12b SUN, 4.15 and I ρσ ˆ x 12, I r sˆvz 12 are tensors transforming covariantly according to the appropriate representations of SL2, C, SUN which are formed by decomposition of tensor products of ˆ x 12, ˆvz See subsection 4.4 for a proof. 29

31 Undersuperconformaltransformations,I ρσ ˆ x 12andI r s vz 12satisfyfromeqs.3.65a,3.70 Dˆ Lz1 ;giˆ x 12DˆLz 2+ ;g = Iˆ x 12, 4.16a Dûz 1 ;g 1 Iˆvz 12 Dûz 2 ;g = Iˆvz b As examples, we first consider the chiral/anti-chiral scalar and spinorial fields, Sz +, Sz, φ α z +, φ α z in N 4 theories which transform as S z + = Ω + z + ;g η Sz +, 4.17a S z = Ω z ;g η Sz, 4.17b φ α z + = Ω +z + ;g η φ β z + ˆL α β z +;g, φ α z = Ω z ;g ηˆ L α βz ;g φ βz, 4.17c 4.17d so that from eq.4.13 η + 4 N 1κ = 0 and s µν 1 2 σ [µ σ ν] for the spinorial fields. The two-point functions of them are Sz 1 Sz 2+ = C S 1 x 2 12 η, 4.18 φ α z 1 φ α z 2+ = C φ i ˆ x 12 αα x 2 12 η For a real vector field, V µ z, where the representation of SL2,C is given by ˆR ν µ z;g and the R-symmetry charge is zero, κ = 0, we have V µ z 1 V ν z 2 = C V I µν z 12 x 2 12 x η, Iµν z 12 = 1 2 trσµˆ x 12σ νˆ x From eq.a.10 one can show I µν z 12 = 1 2 trσ µˆ x 12σ νˆ x 21 = 1 2 tr σ µˆx 21 σ νˆx 12, 4.21 where ˆx = x x = ˆ x

32 Hence I µν z 12 satisfies I µν z 12 I λν z 12 = δ µ λ Note that Iz 12 Rz 12 ;i s, where Rz;i s is given by eq For gauge fields, ψ a z, ψ a z, which transform as ψ a z = Ωz;g η Υz;g κ ψ b zû b a z;g, 4.24a ψ a z = Ωz;g η Υz;g κ û 1a b z;g ψ b z, 4.24b the two-point function of them is ψ a ˆv a bz 12 z 1 ψ b z 2 = C ψ x η 4 N 1κ x η+ 4 1κ N Note that to have non-vanishing two-point correlation functions, the scale dimensions, η, of the two fields must be equal and the R-symmetry charges must have the same absolute value with opposite signs, κ, κ, as shown in subsection 4.4 later. For a real vector superfield, V µ z, if we define from eq.4.20 we get From with D aα = ζ ab ǫ 1αβ D bβ we get V α α z = σ µα α V µ z, 4.26 V α α z 1 V β βz 2 = 2C V ˆx 21 α βˆx 12 β α x 2 12 x η D aα z 1 x µ 21 = 2iσµ θ12a α, D aα z 1 x µ 12 = 0, 4.28 θ a D aα z 1 V α α z 1 V βz 12 β 2 = 4iC V η 3 β x β α, 4.29 x 2 12 x2 21 η and hence V α α z 1 V β βz 2 is conserved if η = 3 D aα z 1 V α α z 1 V β βz 2 = 0 if η = The anti-commutator relation for D aα, D ḃ α 2.11 implies also x µ 1 V µ z 1 V ν z 2 = 0 if η =

33 4.3 Three-point Correlation Functions The solution for the three-point correlation function of the quasi-primary superfields, Ψ ρ r, has the general form 8 Ψ ρ 1rz 1 Ψ σ 2sz 2 Ψ τ 3tz 3 = H ρ rσ s τ t Z 1I σ σ ˆ x 12I τ τ ˆ x 13I s sˆvz 12 I t tˆvz 13 x η 2 4 N 1κ 2 x η 2+ 4 N 1κ 2 x η 3 4 N 1κ 3 x η 3+ 4 N 1κ 3, 4.32 where Z 1 M = X µ 1,Θ a 1, Θ 1a R 4 4N is given by eq Superconformal invariance 4.11 is now equivalent to H ρ rσsτtzd ρ ρ ˆL = H ρ rσ sτ tz D σ σˆ L Dτ τˆ L, Z M = X ν ˆRν µ ˆL, Θ aˆl, ˆ L Θa, 4.33a H ρ r σs τt ZDr rûd s sûd t tû = H ρ rσsτtz, Z M = X µ, Θ b ζûζ b a, ζû 1 ζ a b Θb, 4.33b H ρ rσsτtz = λ η 2+η 3 η 1 H ρ rσsτtz, Z M = λx µ, λ 1 2Θ a, λ 1 2 Θa, 4.33c H ρ rσsτtz = e iκ 1+κ 2 +κ 3 Ω H ρ rσsτtz, Z M = X µ, e i1 2 Ω Θ a, e i1 2 Ω Θa. 4.33d Note that ˆL SL2,C, û SUN, λ R, Ω S 1 and ˆR ν µ ˆL is given from eq.3.46 by ˆR ν µ ˆL = 1 2 tr σ νˆlσ µˆ L In general there are a finite number of linearly independent solutions of eq.4.33a, and this number may be reduced by imposing extra restrictions on the correlation function. 8 See subsection 4.4 for a proof. 32

34 As an example, we consider the three-point correlation function of a real vector superfield, V µ z, where κ = 0. From eq.4.32 we may write V µ z 1 V ν z 2 V λ z 3 = Hµν λ Z 1 I ν ν z 12 I λ λ z 13 x 2 12 x2 21 x2 31 x η Since eq.4.33a is obtained by considering invariance under continuous superconformal transformations, invariance under superinversion which is a discrete map may give an extra restriction. Besides the superconformal invariance, the three-point function has additional symmetry under permutations of the superfields. Furthermore, for supercurrents we may require the correlation function to satisfy the conservation equations like eqs.4.30,4.31. More explicitly, under superinversion we may require V µ z transforms to V µ z = V ν zˆr ν µ z;i s Ωz;i s η The occurrence of the minus sign in N = 1 Wess-Zumino model and vector superfield i theory was verified in [24]. Invariance under superinversion, z s j z j, j = 1,2,3, implies using eqs.2.52, 3.72, 3.74 H µ ν λ Z 1 ˆR µ µ z 1 ;i s ˆR ν ν z 1 ;i s ˆR λ λ z 1 ;i s = Ω η z 1 ;i s H µνλ Z 1, 4.37 which also implies using eqs.3.92, 3.95a Ωz 12 ;i s η H µ ν λ Z 1 ˆR µ µ z 12 ;i s ˆR ν ν z 12 ;i s ˆR λ λ z 12 ;i s = H µνλ Z 2, 4.38 where Z 2 i is given by superinversion, s Z 2 Z2. From V µ z 1 V ν z 2 V λ z 3 = V µ z 1 V λ z 3 V ν z 2 we have H µνλ Z = H µλν Z, 4.39 andfrom V µ z 1 V ν z 2 V λ z 3 = V ν z 2 V λ z 3 V µ z 1 wehaveusingeqs.3.90,4.38with Z is Z H µνλ Z = ΩZ;i s η H νλ µ ZˆR λ λ Z;i s Imposing these extra conditions it was shown that the three-point correlation functions of supercurrents in N = 1 theories have two linearly independent forms [25]. Similarly, the three-point functions of real scalar superfields in N = 1 theories have also two linearly independent solutions [24]. Invariance under R-symmetry transformations 4.33b,d implies that H µνλ Z is a function of X µ,θ aα Θ α a or equivalently X µ, Θ a σ µ Θa, as demonstrated in subsection 4.5, and hence we may put H µνλ Z = H µνλ X λ,θ a σ λ Θa

35 4.4 n-point Correlation Functions - in general In this subsection we show that the solution for n-point correlation functions of the quasiprimary superfields, Ψ ρ r, has the general form Ψ ρ 1 1 r 1 z 1 Ψ ρn n r n z n = H ρ 1 r1 ρ 2 r 2 ρ n r n Z 11,,Z 1n 2 n k=2 I ρ k ρ k ˆ x 1kI r k r k ˆvz 1k x 2 1k 1 2 η k 4 N 1κ k x 2 k1 1 2 η k+ 4 N 1κ k, 4.42 i where in a similar fashion to eq.3.75 Z 11,,Z 1n 2 are given, with z s k1 zk1, k 2, by G T z n1 1 G T z j1 = G T Z 1j 1, j = 2,3,,n We note that all of them are tangent vectors at z 1. Superconformal invariance 4.11 is equivalent to H ρ 1 r 1 ρ 2 r 2 ρ nr n Z 1,,Z n 2 D ρ 1 ρ 1 ˆL = H ρ 1 r1 ρ 2 r 2 ρ n rn Z 1,,Z n 2 n Z M j = Xν j ˆR ν µ ˆL, Θ a jˆl, ˆ L Θja, k=2 D ρ k ρ k ˆ L, 4.44a H ρ 1 r 1 ρ 2 r 2 ρnr n Z 1,,Z n 2 n k=1 D r k r k û = H ρ 1 r1 ρ 2 r 2 ρ nr n Z 1,,Z Z M j = X µ j, Θb j ζûζ b a, ζû 1 ζ a b Θjb, n 2, 4.44b H ρ 1 r1 ρ 2 r 2 ρ nr n Z 1,,Z n 2 = λ η 1+η 2 + +η n H ρ 1 r1 ρ 2 r 2 ρ nr n Z 1,,Z Z M j = λx µ j, λ1 2Θ a j, λ1 2 Θja, n 2, 4.44c H ρ 1 r1 ρ 2 r 2 ρ nr n Z 1,,Z n 2 = e iκ 1+ +κ nω H ρ 1 r1 ρ 2 r 2 ρ nr n Z 1,,Z Z M j = X µ j, ei1 2 Ω Θ a j, e i1 2 Ω Θja. n 2, 4.44d Thus, in general n-point functions reduce to one unspecified n 2-point function which must transform homogeneously under the rigid transformations, 34

36 SL2,C SUN U1 D. Proof 9 Without loss of generality, using the supertranslational invariance we can put the n-point i function with z s k1 zk1, k 2 as Ψ ρ 1 1 r 1 z 1 Ψ ρn n r n z n = H ρ 1 r1 ρ 2 r 2 ρ nr n z 21, z 31,, z n1 n k=2 I ρ k ρ k ˆ x 1kI r k r k ˆvz 1k 4.45 x 2 1k 1 2 η k 4 N 1κk x 2 k η k+ 4 N 1κ k The superconformal invariance of the correlation function4.11, using eqs.3.67a, 3.85, 4.16a, implies H ρ 1 r1 ρ 2 r 2 ρ n rn z 21, z 31,, z n1 n k=2 D ρ k ρ k ˆ Lz1 ;g = Ωz 1 ;g η 1+η 2 + +η n Υz 1 ;g κ 1+κ 2 + +κ n H ρ 1 r 1 ρ 2r 2 ρnr n z 21, z 31,, z n n ρ D ρ 1 1 ˆLz 1+ ;g D r k r k ûz 1 ;g. k=1 Now we consider a superconformal transformation, z gu z, defined by G T z = G T z 1 G Tz s z ;u, G T z = G T z 1 1 G T z, 4.47 where z s z ;u is a special superconformal transformation given in eq.2.58 and z 1 arbitrary. Since can be G T z k1 = G TuG T z k1 for k 2, ˆLz 1+ ;g u = ˆ Lz1 ;g u = 1, Ωz 1 ;g u = Υz 1 ;g u = 1, 4.48 H ρ 1 r1 ρ 2 r 2 ρ nr n z 21, z 31,, z n1 possesses a supertranslational invariance H ρ 1 r1 ρ 2 r 2 ρ nr n z 21, z 31,, z n1 = H ρ 1 r1 ρ 2 r 2 ρ nr n z 21, z 31,, z n The key idea in this proof first appeared in [24]. 35

37 Thus we can write H ρ 1 r1 ρ 2 r 2 ρ nr n z 21, z 31,, z n1 = H ρ 1 r1 ρ 2 r 2 ρ nr n Z 11,,Z 1n 2, G T Z 1j 1 = G T z n1 1 G T z j1. With the transformation rule for Z , eq.4.50 completes our proof. Q.E.D We note that, in the case of n = 2, H ρ 1 r1 ρ 2 r 2 is independent of z 1,z 2 and eqs.4.44c,d show that two-point functions vanish if η 1 η 2 or κ 1 κ 2. Furthermore, if the representation is irreducible then H = 1 by Schur s Lemma. 4.5 Selection Rule & Superconformal Invariants We begin with fields, ψ I x, depending on x R 4 which are obtained by letting the Grassmann coordinates inside quasi-primary superfields, Ψ I x,θ a, θ a, be zero ψ I x Ψ I x,0, They are the lowest order term appearing in the power series expansions of superfields in Grassmann coordinates. The superconformal invariance under U1 transformations4.44d implies for arbitrary Ω S 1 ψ I 1 1 x 1 ψ In n x n = e iκ 1+ +κ nω ψ I 1 1 x 1 ψ In n x n, 4.52 hence, if the the sum of the R-symmetry charge, κ i, is not zero then the correlation function must vanish as exhibited in eq.1.2 ψ I 1 1 x 1 ψ In n x n = 0 if n κ i i=1 This selection rule can be generalized further to all the other component fields in the power series expansions of superfields Ψ I x,θ a, θ a = ψ I x+ψ I aα xθaα + θ α a Ia ψ α x If we define the R-symmetry charge of the component fields, ψ I x, ψaα I Ia x, ψ α x, etc. as κ, κ+ 1, κ 1, etc. respectively, then the invariance under U1 transformation, 2 2 Ψ I 1 1 x 1,θ a 1, θ 1a = e iκ 1+ +κ nω Ψ I 1 1 x 1,e i1 2 Ω θ a 1,e i1 2 Ω θ1a,

38 implies that the selection rule 4.53 holds for all the component fields. The existence of this kind of selection rule in N = 4 super Yang-Mills theory was previously predicted by Intriligator within the context of AdS/CFT correspondence, as the dual IIB supergravity contains a corresponding U1 symmetry [1]. Therefore our results provide a supporting evidence for the Maldacena conjecture, as the selection rule here is derived by purely considering the symmetry on CFT side without referring to the string side. Essentially, for N 4 case, the selection rule exists since the four-dimensional N 4 superconformal group includes the U1 factor inevitably. However in N = 4 case, as verified in subsection 2.5, the corresponding superconformal group is isomorphic to a semidirect product of U1 and a simple Lie supergroup so that it can be reduced to the simple Lie subgroup by breaking the U1 symmetry. In this case, the selection rule will not be applicable to the corresponding N = 4 superconformal theory, since the U1 representation becomes trivial, Υz;g = 1, and the R-symmetry charge is not defined. Now, we consider correlation functions of quasi-primary scalar superfields, Ψz. From eqs.4.44a,bhz 11,,Z 1n 2 mustbesl2,c SUNinvariantandhenceitisafunction of SL2,C SUN invariants. According to [42], these invariants can be obtained by contracting in all possible ways the spinorial indices of ǫ αβ, ǫ 1αβ, ǫ α β, ǫ 1 α β, X 1i α α, Θ aα 1i Θ α 1ja, ǫ a1 a N Θ a 1α 1 1i 1 Θa Nα N 1i N, ǫa 1 a N Θ α 1 1i 1 a 1 Θ α N 1iN a N. On the other hand if we write 4.56 X µ 1I = ˆX µ 1i, ˆΘ a 1j σµˆ Θ1ka, 1 I n 1n 2, 4.57 where Ẑ M 1j = ˆX µ 1j, ˆΘ a 1j, ˆ Θ1ja are normalized Z M 1j ˆX µ 1j = X µ 1j X X , ˆΘa 1j = Θ a 1j X X , ˆ Θ1ja = Θ 1ja X X then using eq.a.20a one can show that X 1I X 1J are all the invariants for SL2,C SUN U1 D and hence invariants for the whole N-extended superconformal group. Note that from eq.3.93b, some of them are pseudo invariants under superinversion. Explicitly, we may reproduce the invariants depending on three points 3.97a as ˆX = X2 11+, ˆX11+ X ˆX = X 11+ X 11, 4.59 X X ,