Component versus superspace approaches to D = 4, N = 1 conformal supergravity

Transcription

1 Prog. Theor. Exp. Phys. 2016, 073B07 36 pages DOI: /ptep/ptw090 Component versus superspace approaches to D = 4, N = 1 conformal supergravity Taichiro Kugo 1, Ryo Yokokura 2,, and Koichi Yoshioka 3 1 Department of Physics and Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto , Japan 2 Department of Physics, Keio University, Yokohama , Japan 3 Osaka University of Pharmaceutical Sciences, Takatsuki , Japan ryokokur@rk.phys.keio.ac.jp Received March 31, 2016; Accepted May 16, 2016; Published July 25, The superspace formulation of N = 1 conformal supergravity in four dimensions is demonstrated to be equivalent to the conventional component field approach based on the superconformal tensor calculus. The detailed correspondence between the two approaches is explicitly given for various quantities: superconformal gauge fields, curvatures and curvature constraints, general conformal multiplets and their transformation laws, and so on. In particular, we carefully analyze the curvature constraints leading to the superconformal algebra and also the superconformal gauge fixing leading to Poincaré supergravity, since they look rather different between the two approaches.... Subject Index B11 1. Introduction N = 1 supergravity SUGRA in four dimensions has been important for giving a boundary theory around the unification scale for constructing viable phenomenological models beyond the standard model. It has also come to have increasing importance as a low-energy effective theory for superstrings and as a tool for analyzing supersymmetric gauge theories on curved backgrounds. However, various explicit calculations, e.g., the construction of the SUGRA Lagrangian, are complicated and nontrivial. The simplest and most convenient method is presumably the superconformal tensor calculus, which was developed by Kaku, Townsend, van Nieuwenhuizen, Ferrara, Grisaru, de Wit, van Holten, and Van Proeyen [1 6]. It is a set of rules for constructing invariant actions under local superconformal transformations, i.e., superconformal gauge fields including gravity and gravitinos and various types of matter multiplets, their transformation laws, multiplication rules, and superconformal invariant action formulas. The power of the superconformal tensor calculus comes from a larger symmetry than the usual Poincaré SUGRA. Indeed, its power as a practical computational tool was clearly demonstrated in Ref. [7] for computing the action for the general Yang Mills YM matter-coupled SUGRA system. Kugo and Uehara KU have presented [8] the superconformal tensor calculus in its most complete form, and discussed the spinorial derivative D α for the first time in the component field approach. They found that a special condition on an operand multiplet V Ɣ must be satisfied so that its spinorial derivative D α V Ɣ exists and gives a conformal multiplet. The condition depends on the spinor index The Authors Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Funded by SCOAP 3

2 α of D α and the Lorentz index Ɣ of the operand V Ɣ, and KU implicitly suspected that the superspace formulation might not exist for the conformal SUGRA. Nevertheless, Butter [9] has recently presented a superspace formalism of the conformal SUGRA. In contrast with the previous expectation, his formalism realizes a simpler algebra of covariant derivatives than any other superspace Poincaré SUGRA: { α, β }={ α, β }=0, { α, β }= 2i α β. 1.1 Requiring this algebra together with several constraints on curvatures in the vector spinor direction, he succeeded in constructing a superspace counterpart of the conformal SUGRA in the component approach. The covariant derivatives A = a, α, α can be freely applied on any superfield with no restriction and are identified with the transformations P A = P a, Q α, Q α of the superconformal group. The reason why KU s spinorial derivatives could not be freely applied turns out to be because KU required an extraneous condition that the derivative again give a primary multiplet. Since the superspace formalism manifests supersymmetry in a geometrically clear way, it gives a transparent and powerful means to treat systems in new situations, such as finding nonlinear realization, brane worlds, decomposition of higher-n supersymmetry, partial breaking of local supersymmetry, massive SUGRA, etc. On the other hand, one needs to write down the action explicitly in terms of component fields, which could be done most easily and efficiently with the tensor calculus. That is, we have two approaches to the conformal SUGRA; one is the superspace approach based on the conformal superspace and superfields, and the other is the component approach based on the superconformal tensor calculus. Both approaches have their own strong and weak points. In order to use the advantages of both approaches, it is desirable to see the correspondence between them. The purpose of this paper is to show the equivalence of the two approaches by making the detailed correspondences manifest. This paper is organized as follows. In Sect. 2, we recapitulate the essential parts, first of the superconformal tensor calculus in the component approach, and then of the conformal superspace approach. We use individual notation for each of these approaches and separately provide a dictionary to translate between them for the convenience of reading the references. In Sect. 3 we explicitly present the correspondences of various quantities. We first discuss gauge fields and curvatures in Sect. 3.1 and show how all the curvature constraints in the component approach are satisfied in the superspace approach, although the constraints look rather different from each other. The same superconformal transformation algebras are realized in both approaches under these curvature constraints. We then discuss the component fields and transformation rules for a conformal multiplet with arbitrary external Lorentz index in Sect. 3.2, and the chiral projection and the invariant action formulas in Sect We analyze in Sect. 3.4 the compensated or u-associated derivative that maps a primary superfield to another primary superfield. There we also discuss KU s restriction on the spinorial derivatives. In Sect. 4, we investigate the matter-coupled SUGRA system and the superconformal gauge fixing to Poincaré SUGRA, mainly from the superspace viewpoint. We discuss the superspace counterpart of KU s gauge fixing, which leads directly to the canonically normalized Einstein Hilbert EH and Rarita Schwinger RS terms. The correspondence to the component approach is nontrivial since the gauge invariance in the superspace approach is much larger than the component approach, and the gauge fixing written in terms of superfields give more fixing conditions than the component case. One remarkable fact is that the covariant spinor derivatives remaining after the gauge fixing automatically reproduce the complicated supersymmetry transformation in Poincaré SUGRA. The 2/36

3 final section is devoted to the summary. We add three appendices. The notations in the component and superspace approaches are summarized separately and the dictionary to translate between them is given in Appendix A. The standard form of the supersymmetry transformation law for the general conformal multiplet with arbitrary external Lorentz index is cited for convenience in Appendix B. We present in Appendix C some explicit computations that are necessary in deriving the results in the text. 2. Conformal SUGRA We first briefly review the component and superspace approaches for D = 4, N = 1 conformal SUGRA. In both approaches the conformal SUGRA is constructed as the gauge theory of the superconformal group. The Lie algebra of the superconformal group contains the following elements: translation P a, supersymmetry Q, Lorentz transformation M ab, conformal boost K a, supersymmetry of conformal boost S, dilatation D, and chiral rotation A Component approach In this subsection we review the component approach. For any part of component approach in this paper, we use the notations and conventions of Ref. [8], which are the same as those of Ref. [10] except for two-component spinors and the dual of antisymmetric tensors. The details of the notations are summarized in Appendix A. The superconformal algebra consists of 15 bosonic and 8 fermionic generators, which obey the following graded commutation relations: [M ab, M cd ]= M ad δ bc + M bd δ ac + M ac δ bd M bc δ ad, [M ab, P c ]= P a δ bc + P b δ ac, [M ab, K c ]= K a δ bc + K b δ ac, [D, P a ]=P a, [D, K a ]= K a, [K a, P b ]=2δ ab D + 2M ab, {Q, Q T }= 1 2 γ ac 1 P a, {S, S T }= 1 2 γ ac 1 K a, [M ab, Q] =σ ab Q, [M ab, S] =σ ab S, [D, Q] = 1 2 Q, [D, S] = 1 2 S, [A, Q] = 3 4 iγ 5Q, [A, S] = 3 4 iγ 5S, [K a, Q] = γ a S, [S, P a ]= γ a Q, {Q, S T }= 1 2 C 1 D σ ab C 1 M ab + iγ 5 C 1 A. 2.1 All other commutation relations vanish. The generators are generically denoted as X A and the above commutation relations are written as [X A, X B }= f AB C X C. 2.2 Note that these generators represent the active operators transforming fields, not the representation matrices. The commutation relations change signs if written for representation matrices instead of active operators. In the conformal SUGRA, the superconformal symmetry is treated as local 3/36

4 symmetry. The corresponding gauge fields and transformation parameters are given by h μ A X A = e μ a P a + ψ μ Q ω μ ab M ab + b μ D + A μ A + ϕ μ S + f μ a K a, 2.3 ɛ A X A = ξ a P a + εq λab M ab + ρd + θa + ζ S + ξ K a K a. 2.4 In the component approach, the Greek letters μ, ν,... denote curved vector indices and the Roman letters a, b,... flat Lorentz indices. The group transformation laws of the gauge fields under the superconformal symmetry are δ group B ɛ B h μ A = μ ɛ A + h μ B ɛ C f CB A. 2.5 The curvature of the superconformal algebra before the deformation below is R A μν = νh μ A μ h ν A + h ν B h μ C f CB A. 2.6 The P a translation is deformed so as to be related to the general coordinate GC transformation δ GC as δ P ξ a := δ GC ξ μ A =P δ A ξ μ h μ A, 2.7 where ξ μ = ξ a e a μ, and ξ a is a field-independent parameter. In order to have [δ Q, δ Q ] δ P, several constraints on the curvatures are imposed: R μν P a = 0, 2.8 R μν Qγ ν = 0, 2.9 R νλ M ab e aλ e b μ 1 2 R λμqγ ν ψ λ i R μν A = 0, 2.10 where R μν is the dual of R μν. By these constraints, the M ab -, S-, and K a -gauge fields ω ab μ, ϕ μ, and f a μ, respectively become dependent fields expressed by other independent gauge fields. The Q transformations δ Q ε of the dependent gauge fields are determined by those of independent gauge fields, and they deviate from the original group transformation δ group Q ε as δ Q ε = δ group Q ε + δ Q ε The deviation part δ Q ε is given by where δ Q εω μ ab = 1 2 Rab Qγ μ ε, δ Q εϕ μ = 1 4 iγ ν γ 5 R νμ A + R νμ Aε, δ Q εf μ a = 1 2 Rcov νμ Sσ aν ε 1 4 eaν R cov νμ Sγ 5ε, 2.12 R cov μν A = R μν A + δ Q ψ μh A ν δ Q ψ νh μ A /36

5 Note that the RHS of Eq are given by εe c μ f QP c X with X = M ab, S, K a. So they can be regarded as the deformation of the algebra by changing the structure constant of the [Q, P c ] commutator from originally zero to the nonvanishing f QPc X for X = M ab, S, K a. The resultant commutation relations are the same as the original ones [δ A ɛ A 1, δ Bɛ B 2 ]= C δ C ɛ A 1 ɛb 2 f BA C, 2.14 for all A and B,ifP a on the RHS of the Q Q commutator is understood to be P a : 1 [δ Q ε 1, δ Q ε 2 ]=δ P 2 ε 2γ a ε Moreover, the definition of P a transformation leads to [δ P ξ a, δ Q ε] = δ A ξ a δ Q εh a A = A=M,S,K [δ P ξ a 1, δ P ξ b 2 ]= A =P δ A ξ1 a ξ 2 b Rcov A ab, A=M,S,K δ A ξ a ε α f Qα P a A, f Pa P A b = R cov A ab 2.16 where α = α, α. The superconformally covariant derivative on fields carrying only flat Lorentz indices is defined through the P a transformation as ξ a D a φ := δ P ξ a φ = ξ a e a μ μ φ A = P δ A h a A φ Next, we introduce superconformal multiplets. A general conformal multiplet V Ɣ is a set of dim Ɣ complex fields, V Ɣ =[C Ɣ, Z αɣ, H Ɣ, K Ɣ, B aɣ, αɣ, D Ɣ ], 2.18 where Ɣ represents arbitrary spinor indices Ɣ = α 1,..., α m ; β 1,..., β n and dim Ɣ is the dimension of the Lorentz representation of Ɣ. The first component C Ɣ is defined as having the lowest Weyl weight in the multiplet so that its transformation law is given by δ Q εc Ɣ = 1 2 i εγ 5Z Ɣ, δd ρ + δ A θ C Ɣ = δ M λ ab C Ɣ = 1 2 λab ab Ɣ C =: 1 2 λab ab C Ɣ wρ inθ C Ɣ, δ S ζ C Ɣ = δ K ξ a K C Ɣ = Here ab is the representation matrix of the Lorentz generator that C Ɣ belongs to, and w and n are the Weyl and chiral weights of C Ɣ. The S and K a transformations must annihilate the lowestweight component C Ɣ since they lower the Weyl weights of operands. The Q transformation law δ Q εc Ɣ = 1 2 i εγ 5Z Ɣ simply defines the second component Z Ɣ. All the higher components in the multiplet and their superconformal transformation laws are determined by requiring the superconformal algebra to hold on them, aside from some arbitrariness in defining higher-component fields. The Q transformation laws of all component fields are summarized in B1, which also fixes the definition of higher-component fields. We call the transformation laws B1 the standard form. Since the first component C Ɣ specifies the whole multiplet, we denote the conformal multiplet V Ɣ using the first component as V Ɣ = C Ɣ /36

6 A constrained-type multiplet also exists as a conformal multiplet if some conditions are met on Weyl and chiral weights and also on its Lorentz representation. The chiral multiplet w,n Ɣ, for instance, exists only when the Weyl and chiral weights w, n satisfy w = n and the Lorentz index Ɣ is made of purely undotted spinor indices; then the chiral multiplet has dim Ɣ complex components denoted by w=n Ɣ=α 1 α l = [A Ɣ, P R χ Ɣ, F Ɣ ] These three components of a chiral multiplet are embedded into a general conformal multiplet in the form V Ɣ =[A Ɣ, ip R χ Ɣ, F Ɣ, if Ɣ, id a A Ɣ,0,0], so that their Q and S transformation laws are given by δ QS A Ɣ = δ Q ε + δ S ζ A Ɣ = 1 2 ε RP R χ Ɣ, δ QS P R χ Ɣ = 1 Ɣ γ a D a A Ɣ ε L + F Ɣ ε R + 2wA Ɣ ab A Ɣ σ ab ζ R, δ QS F Ɣ = 1 2 ε Lγ a D a P R χ Ɣ + ζ R 1 wp R χ Ɣ 1 2 σ ab ab P R χ Ɣ For the multiplet V w,n Ɣ with purely undotted spinor Ɣ satisfying w = n + 2, the chiral projection operator exists and [ w, n=w 2 1 V Ɣ = 2 H Ɣ ik Ɣ, ip R γ a D a Z Ɣ + Ɣ, 1 ] 2 D Ɣ + C Ɣ + id a B aɣ 2.23 gives a chiral multiplet with the Weyl and chiral weights w + 1, w + 1. Here = D a D a is the superconformal d Alembertian. The superconformal tensor calculus gives the superconformally invariant action in simple forms. The F-type invariant action formula is applied only to the chiral multiplet = [ A = 1 2 A + ib, P R χ, F = 1 2 F + ig ] satisfying w = n = 3 and carrying no external Lorentz index. The action is given by d 4 x [ w=n=3 ] F = d 4 xe F ψ a γ a χ ψ a σ ab A iγ 5 Bψ b The D-type invariant action formula is applied only to the real and Lorentz-scalar multiplet V = [ C, Z, H, K, Ba, λ, D ] with w = 2 and n = 0. The action is derived from the F-type formula with the chiral projection operator as d 4 x [ V w=2 ] D = d 4 x [ V w=2 ] F = d 4 xe D + C 1 2 i ψ a γ a γ 5 γ b D b Z + λ 1 2 ψ a σ ab H + iγ 5 Kψ b = d 4 xe D 12 i ψ a γ a γ 5 λ i ϕ a γ a γ 5 Z + 13 R C + 1e ψ μ ε μνρσ γ 5 γ ν ρ ψ σ + i4 ω ρ ab σ ab ψ σ iεabcd ψ a γ b ψ c B d A d C 1 2 ψ d Z /36

7 For the general YM matter-coupled SUGRA system, the action is given by L = 1 [ φs, Se 2V ] G c c 2 D + [ c 3 gs ] F 1 fαβ W 4[ α W β ] F = 1 [ φs, Se 2V ] G D + [ 0 3 ] F 1 fαβ W 4[ α W β ] F, 2.26 where S i = [z i, P R χ i, h i ] are the chiral matter multiplets with vanishing weights w = n = 0 and S i are their conjugate. In the first term, V G means the YM vector multiplet of internal symmetry. The field c is a chiral compensator carrying weights w, n = 1, 1. For the system possessing nonvanishing superpotential gs, it is convenient to redefine the compensator as c 0 = g 1/3 S c =[z 0, P R χ 0, h 0 ] so that φ becomes the combination of φ and superpotential: φs, Se 2V G = φs, Se 2V G gs 2/3. In the third term, f αβ is a holomorphic function of S i, symmetric under the exchange α β, and W α is the gaugino multiplet field-strength supermultiplet of internal symmetry. For the YM vector multiplet, the Wess Zumino WZ gauge is imposed, and then the gaugino multiplet is constructed by the Q transformation that preserves the WZ gauge. We denote such Q transformation as δ YM Q ε. To go down to the Poincaré SUGRA, we fix the extraneous D-, A-, S-, K a -gauge symmetries. The so-called improved gauge-fixing conditions adopted in Ref. [7] are D, A-gauge : z 0 = z 0 = 3φ 1 2 z, z, S-gauge : χ R0 = z 0 φ 1 φ i χ Ri, K a -gauge : b μ = 0, 2.27 where χ R0 = 1 2 P Rχ 0 and χ Ri = 2 1 P Rχ i. These gauge conditions set the first and second components of the vector multiplet φ 0 0 to 3 and 0, respectively, in the D-type action formula. As a result, the canonically normalized EH and RS terms are obtained directly. The relation between the Q transformation δq P ε in the resultant Poincaré SUGRA and the gaugefixed conformal Q transformation is given by δq P ε = δym Q ε + δ Aθε + δ S ζε + δ K ξ a ε, 2.28 where θε = i Gi ε R χ Ri G i ε L χ i L, 3 ζ R ε = 1 h 0 z h ig i ε R 1 G ij Gi G j ε R χ Rj + Gj i ε Lχ j L χ Ri 1 G i γ a a z i G i γ a a z i ε L iγ a A a ε L, ξ a ε = 1 ϕa ε ψ a ζε In this expression μ z i is the covariant derivative of the internal symmetry, and the G are given by G = 3 log 1 3 φz, z. The indices of G represent the differentiation with respect to z i and z i, e.g., G i j = 2 G/ z i z j Conformal superspace Next we review the conformal superspace approach [9]. In superspace, the supersymmetry transformation can be treated as a translation in the direction of the Grassmannian spinor coordinate on 7/36

8 the same footing as the usual translation P a. The anticommutation relations between the spinor covariant derivatives become complicated in Poincaré SUGRA, whereas, in conformal superspace, they are as simple as in global supersymmetry. In any part of the superspace approach in this paper, we use the notations and conventions of Butter [9], with a few exceptions, which will be explained below. The details of the notations are summarized in Appendix A. The superconformal algebra is the same as 2.1 given in the component approach, if we perform a suitable translation of the generators between the two approaches see Table 1. Here we refer to only a few characteristic commutation relations: {Q α, Q α }= 2iσ a α α P a, {S α, S α }=2iσ a α α K a, [S α, P a ]=iσ a α β Q β, {S α, Q β }=2D 3iAɛ αβ 2M αβ, [ S α, P a ]=i σ a αβ Q β, { S α, Q β }=2D + 3iAɛ α β 2M α β, with M αβ = σ ba ɛ αβ M ab, M α β = σ ba ɛ α β M ab Note that the normalizations of Q, S, A are different from the component approach. The gauge superfields corresponding to the superconformal group are denoted as h M A X A = E M A P A φ M ba M ab + B M D + A M A + f M A K A, 2.31 where we use the calligraphic index A for the total superconformal algebra, the Roman uppercase index A for the Lorentz vector and spinor set as P A = P a, Q α, Q α and K A = K a, S α, S α, and the index M for the set of curved indices, e.g., A M = A m, A μ, A μ. We assume that the vielbein E M A is invertible: E M A E A N = δ M N, E A M E M B = δ A B The gauged superconformal transformations are taken by real parameter superfields. These parameter superfields are denoted as ξ A X A = ξp A P A ξmab M ba + ξdd + ξaa + ξk A K A The gauge fields receive the superconformal transformation δ G ξ A X A as δ G ξ B X B h M A = M ξ B δ B A + h M C ξ B f B C A, 2.34 where the primed calligraphic index A means all the superconformal generators other than P A, namely, X A = P A, X A. Note that Ref. [9] used different notation in which X A was expressed as X A with no distinction from A for a, α, α, and our h M A and h M A were denoted by W M A and h M a, respectively. In the same spirit as the component approach, the P A transformation is defined as being related to the general coordinate transformation δ GC using the field-independent parameter superfield ξ A as δ G ξ A P A = δ GC ξ M := ξ A E A M δ G ξ M h M B X B, 2.35 where ξp A is abbreviated to ξ A. The P A transformation acting on a superfield with no curved index defines the covariant derivative as δ G ξ A P A = ξ A P A = ξ M M = ξ M M h M A X A /36

9 That is, P A = A = E A M M on superfields with flat indices. The curvature R MN A is defined as R MN A = M h N A N h M A E N C h M B E M C h N B f B C A h N C h M B f B C A Here and hereafter, we use the convention of implicit grading. In superspace, we generally treat both bosonic and fermionic quantities at the same time by the index A or M, and should be careful with the grading of fermionic objects such as X AB = ab+n E B N E A M X MN, [ A, B }= A B ab B A, and Z = a Y A A. The grading is uniquely determined if the standard order of indices is specified. For example, the standard order of X AB is AB and hence E B N E A M X MN should be accompanied by the grading factor ab+n since one jumps the index A over two indices B and N of E B N in order to recover the standard order AB. Implicit grading means to assume the omission of such unique grading factors from everywhere. In other words, we can treat the indices A, M as if they were bosonic ones. The same implicit grading convention is also used for the index A of superconformal generators. In the definition of curvatures, the commutation relation of P A is as follows: [P A, P B ]= R AB C X C = RP AB C P C 1 2 RM AB dc M cd RD AB D RA AB A RK AB C K C, 2.38 where R AB C = E B N E A M R MN C in terms of R MN C givenin2.37. In Ref. [9], RP AB C is expressed as T AB C, RM AB cd is R AB cd, RD AB is H AB, and RA AB is F AB. Several constraints are imposed on the curvature superfields to eliminate the redundant degrees of freedom. First, the constraints on R αβ are as follows: R αβ A = 0, R α β A = 0, RP α β c = 2iσ c α β, R α β A = 0 otherwise, 2.39 which guarantees that the commutation relations of covariant spinor derivatives take the simple form as in the global supersymmetry case: { α, β }=0, { α, β }=0, { α, β }= 2i α β Secondly, the following constraints on R αa are imposed: By solving the Bianchi identities RP γ b A = 0, RD βa = 0, RA βa = [ A, [ B, C ]]+[ B, [ C, A ]]+[ C, [ A, B ]] = under these constraints with implicit grading understood, one finds that all other nonvanishing curvatures can be expressed by a single superfield W αβγ with totally symmetric undotted spinor indices α, β, γ as seen below. The Bianchi identities with the first constraints 2.39 imply that the curvatures R αb and R ab can be expressed by a gaugino superfield W α, which is superconformal algebra valued, R α,β γ = [ α, β γ ]=2iɛ αβ W γ, R α, βγ = [ α, βγ ]=2iɛ α β W γ, 2.43 R α α,β β = ɛ α β { α, W β } ɛ αβ { α, W β }, /36

10 where R α,β γ = σ b β γ R αb, which is R αβ γ in Ref. [9]. The brackets on the indices imply symmetrization with the weight one, e.g., ψ α χ β = 1/2ψ α χ β + ψ β χ α. This algebra-valued superfield W α satisfies { α, W γ }={ α, W γ }=0, chirality 2.45 { α, W α }={ β, W β }, reality 2.46 [M bc, W α ]=σ bc α β W β, [D, W α ]= 3 2 W α, [A, W α ]=iw α, [K A, W α ]= The further input of the second constraints 2.41 implies that W α has no P A, D, A components, WP α A = WD α = WA α = 0, so that W α = 1 2 WM α bc M cb + WK α B K B With the help of the superconformal algebra, the chirality and reality conditions 2.45 and 2.46 lead to the final expression W α = ɛσ bc βγ W αβγ M cb + 1 γ β W γα S β W α = σ bc ɛ γ β W α β γ M cb 1 γ γ W 2 α β S β 1 2 with which the reality condition is written: γ β W γα β K β β, 2.49 γβ W α β γ K β β, 2.50 { α, W α }={ α, W α }= 1 2 α γ β W γα β K β β = 1 2 α γβ W γ α β K β β In this way, the gaugino superfield W α is expressed by the totally symmetric superfield W αβγ, which satisfies α W αβγ = 0, DW αβγ = 3 2 W αβγ, AW αβγ = iw αβγ, K A W αβγ = Owing to Eqs and 2.44, all the curvatures R AB can also be written in terms of W αβγ, its conjugate, and their covariant derivatives. In particular, the R ab component is expressed as R α α,β β = ɛ α β 2W γ αβ Q γ + γ W δ αβm δγ + γ W γαβ D 3 2 i γ W γαβ A W γ αβ S γ i γ γ W γαβ S γ α βγ Wγβ δ K δ β + ɛ αβ 2W α β γ Q γ + γ W δ α β M γ δ + γ W γ α β D i γ W γ α β A W β α δ S δ i γγ W γ α β S γ α γβ W γ β δ K β δ Now the concept of primary superfield is introduced to describe matter superfields, invariant action over the superspace, and so on. A primary superfield Ɣ is defined as the superfield on which the action of the superconformal group is M bc Ɣ = S bc Ɣ, D Ɣ = Ɣ, A Ɣ = iw Ɣ, K A Ɣ = 0, /36

11 where Ɣ and represent general Lorentz indices such as Ɣ = α 1,..., α n, β 1,..., β m, and S bc is the representation matrix of Lorentz algebra to which Ɣ belongs. The real constant numbers and w are called the Weyl and chiral weights, respectively. The last property K A Ɣ = 0 is most important for Ɣ being primary. This is generally violated for its derivative A Ɣ. As for W αβγ, Eqs imply that it is a primary chiral superfield with Weyl weight = 3/2 and chiral weight w = 1, where a chiral superfield means that it satisfies α = 0 as usual. It should be noted that this chirality condition is superconformally covariant. An invariant integral over the superspace is given by S D = d 4 xd 4 θ EV, 2.55 where E = dete M A. Here we are using implicit grading and omitting to write the superdeterminant sdet. The superconformal transformation law of the density E is δ G ξ A X A E = EE A M δ G ξ A X A E M A = EE A M h M C ξ B f B C A = Eξ B f B A A DE = 2 E, M ab E = AE = K A E = 0, 2.56 since the superconformal generators X B other than P A carry nonpositive Weyl weights so that the commutator [X B, X C ] yields a positive Weyl weight P A only when X C = P C and X B = D or A, in which case E A M h M C = E A M E M C = δ A C. From 2.56, the invariance conditions for the action S D become DV = 2 V, AV = M ab V = K A V = That is, V must be a, w = 2, 0 primary real superfield with no Lorentz index. The invariance of S D under the GC transformation in superspace is manifest and hence invariant under the P A transformation. Thus the action S D is fully superconformal invariant and is called D-type integration. The superconformal counterpart of the d 2 θ integral in global supersymmetry is S F = d 4 xd 2 θ EW The chiral density E is given by the superdeterminant of the vielbein in the chiral subspace with dotted spinor directions being omitted from E A M, i.e., E = det E a m with E a m = E a m, a = a, α, and m = m, μ.in2.58, W is a covariantly chiral superfield defined by α W = 0. The invariance of the action S F requires that W must be a, w = 3, 2 primary chiral superfield with no Lorentz index. Since the integral S F does not depend on θ, it is supposed to be executed at θ = 0, which is called F-type integration. Performing the d 2 θ integration in 2.58, we obtain the component expression of the F-type integration as d 4 xd 2 θ EW = d 4 xe W + i 2 ψ a α σ a αβ β W ψ a σ ab ψ b W θ= θ=0 The D-type integration is related to the F-type one as d 4 xd 4 θ EV = 1 d 4 xd 2 θ EP[V ]+ 1 d 4 xd 2 θ Ē P[V ], /36

12 where P[V ]= V 2.61 is the chiral projection operator. The component expression of the D-type integration is obtained using Eq The action of the matter-coupled SUGRA system is given in conformal superspace as S = 3 d 4 xd 4 θ E c c e K/3 + d 4 xd 2 θ E c 3 W + h.c., 2.62 where c is the compensator chiral superfield carrying Weyl and chiral weights, w = 1, 2/3. The Kähler potential K and the superpotential W are the functions of chiral matter superfields i with weights, w = 0, 0. In addition, K is a real function and W is holomorphic. The gaugefixing conditions leading to Poincaré SUGRA with the canonically normalized EH term are given in Ref. [9]: D, A-gauge : c = c = e K/6, K A -gauge : B M = The K A -gauge condition B M = 0 together with the curvature constraints imply the restriction of the form of the K A -gauge field f M A. Recall that the curvature RD AB is written as RD AB = E A M E B N M B N N B M + 2f AB a 2f BA b We see that the constraints RD αβ = 0 and B M = 0 constrain f αβ in the form The constraint RD αb = 0 implies f αβ = ɛ αβ R, f α β = ɛ α β R, f α β = f βα = 1 2 G α β f αb = f bα Further, the constraints RK αβ, γ = 0 and RK α γ β = 0 and their conjugates give 3.42, 3.43 of Ref. [9] 3if α,β β = 1 2 D αg β β + D βg α β + ɛ αβ D β R, if α,β β = 1 2 D α G β β + D β G β α + ɛ α β D αr, 2.68 where D A is the covariant derivative after K A -gauge fixing; D A =[ A + f A B K B ] BM =0. Finally, the constraint RK α β c = 0 means 3.49 in Ref. [9] f α α,β β = i 2 D αf α,β β + D α f α,β β + 2ɛ αβɛ α β R R G β αg α β Since the conformal curvatures are written in terms of W αβγ, the curvatures after gauge fixing are written in terms of R, G α β, W αβγ, and the derivative D A. 12/36

13 It is noted that the gauge-fixing conditions 2.63 also fix the A-gauge superfield A M. The covariantly chiral condition of c is 0 = α c = E αm M c B α c 2 3 ia α c, and further imposing the gauge conditions c = e K/6 and B α = 0 leads to A α = i 4 K i D α i, 2.70 where D α i = E αm M i and K i = K/ i. The chirality condition for matter superfields i is used; 0 = α i = E αm M i = D α i. In the same way, from α c = 0, A α is fixed as Similarly, from the relation α α c = 2i α α c, we obtain A α = i 4 K id α i A α α = i 4 K id α α i K i D α α i 3 2 G α α g ij D α i D α j, 2.72 where G α α = 2f α α and g ij = 2 K/ i j. 3. Correspondence between component and superspace approaches In this section we present the correspondence between the component and superspace formulations. The objects that we deal with are the superconformal algebra, gauge fields, curvatures and their constraints, conformal multiplets with external Lorentz indices, chiral projection, and invariant actions. Note that the notations and conventions are different in the two approaches and a dictionary to translate between them is given in Appendix A for spinors, vectors, gamma matrices, and tensors Superconformal algebra, gauge fields, and curvatures As discussed in Sect. 2.1, the Q and P a transformations in the component approach are deformed from the original group laws. In the following, we use only the final form of them and the deformed P a transformation is simply denoted as P a. Let us begin with the dictionary for the normalization of superconformal generators and Weyl and chiral weights. The correspondence is shown in Table 1. The correspondence of gauge parameters is set to satisfy ɛ A X A ξ A X A and is shown in Table 2. The vertical bar means the θ = θ = 0 projection, i.e., the lowest component of the superfield. Since gauge fields generators essentially represent the common quantity in both approaches, h μ A X A h m A X A, the correspondence of gauge fields appears with inverse normalizations of the generators shown in Table 3. In the table, the curved index μ of the component approach corresponds to the index m of superspace. Table 1. The generators and the Weyl and chiral weights. component superspace P a, 2Q α, 2 Q α P a, Q α, Q α = P A 4 M ab, D, 3 M ab, D, A K a, 2S α, 2 S α K a, S α, S α = K A w, 2 n, w 3 13/36

14 Table 2. The gauge transformation parameters. component superspace P, Q ξ a 1, 2 ξpa, ξp α, ξp α =ξp A M, D, A λ ab 3, ρ, 4 ξmab, ξd, ξa K, S ξk a, 1 2 ξk a, ξk α, ξk α =ξk A Table 3. The gauge fields. component superspace P, Q e a 1 μ, 2 μ E A m =E a m, E mα, E α m =e a m, 1 ψ 2 mα, 1 ψ α 2 m M, D, A ω ab 3 μ, b μ, 4 μ φ ab m =ω ab m, B m, A m K, S f a μ, 1 ϕ 2 μ f A m =f a m, f mα, f α m Table 4. The covariant curvatures. R cov component superspace R ab P c 1, R 2 abq RP c ab, RP γ ab, RP ab γ = C RPab ab M cd 3, R ab D, R 4 aba RM cd ab, RD ab, RA ab R cov ab K c, 1 2 Rcov ab S RK c ab, RK γ ab, RK ab γ = C RKab The curvature in superspace, R C mn, with curved tensor indices was defined in Eq The lowest component of the flat indexed curvature superfield R C ab is given by R C ab =E N b E M a R C MN =e m a e n b R C mn i ψ α a σ b 2 α β ψ b α σ a α β W βc + i ψ a α σ b αβ ψ b α σ a αβ W C β ψ a α ψ β b R C αβ. 3.1 Using the correspondence of gauge fields given in Table 3, we find that the curvatures coincide with the negative of the covariant curvatures with the algebra deformation of the component approach, up to the normalization of generators shown in Table 4. The covariantization is necessary only for the M, S, K a curvatures in the component approach, which correspond in superspace to the fact that the gaugino superfield W α has nonvanishing components only for the M, S, K a generators. In obtaining Table 4, we have used the relations of W α shown in Table 5 to the curvatures in the component approach. Note that these quantities stand for the spinor vector components of the superspace curvature R C αb because of the relations The correspondences of the curvatures Table 4 are summarized in a simple expression: R cov C ab X C component R ab C X C superspace. 3.2 We emphasize that such identification holds for the flat indexed curvatures, as it does for the curved indexed gauge fields: h μ C X C component h m C X C superspace /36

15 Table 5. W α and the curvatures. component superspace R ab Qγ 5 WM α ab, WM α,ab = 2 α RPab, RP ab, α i 4 σ ab WKα γ 5 R ab A β 0 0 WK α β = i σ ab β α σ ab α RA β 0 1 ab 1 4 Rcov bc Sγ c γ 5 WK α b WK αb β 0 σ RKbc RK bc, β c β α cov R bc Sγ c = i 2 σ c βα β 0 σ RKbc RK bc, β c β α 2 σ c βα 0 Table 6. W αβγ, W α β γ, and the curvatures RX A ab in superspace. curvatures RX aba in superspace 1 4 RM, βα, δγ, RD βα = 2 3 ira βα, RP ab c 0 RP γβ α, RP + γ β, α W γβ α, W γ β α RK γβ α, RK γβ a σ a α α, 1 4 RM+, + β α, δ γ δ W γβα, δw γ β α RD + β = 2 α 3 ira+ β 1 α 2 γ 1 W γβα, γ W 2 γ β α RK + γ β, α RK + γ β a σ a α α W α γβ, 1 2 γ δ αw δβα, W γ β α 1 2 γ α δ W δ β α For instance, the component approach counterpart of the flat indexed gauge field in superspace is found through the expression h a C =E a M h M C =e a m h m C 1 2 ψ a α h α C. 3.4 The constraints on curvatures also have a correspondence, though the constraints in superspace are directly imposed on the spinor spinor or spinor vector components of curvatures. The restricted form of the vector vector component R ab in superspace is derived from other constraints and is explicitly written in terms of the primary chiral superfield W αβγ. That is, Eq implies the expressions shown in Table 6 for the curvatures RX A ab in superspace. The chiral decomposition of the antisymmetric tensor is defined in Eq. A19. We can see the correspondence of curvature constraints using the fact that all the curvature components RX A ab with vector vector indices are expressed by W αβγ in superspace. First, the constraint 2.8 in the component approach is equivalent to R ab P c = 0 and hence corresponds to RP c ab =0 in superspace, as seen in Tables 4 and 6. Secondly, the constraint 2.9, equivalent to R ab Qγ b = 0, corresponds to the equation RP α ab σ b α δ = 0 and its conjugate in superspace. It is found from Table 6 that RP α ab has only a chiral component RP γ γ,β β α = 2ε γ β W γβ α so that RP ab α σ b α δ σ b ββ ε γ β W γβ α σ b α δ ε γ δ W γα α, /36

16 Table 7. The superconformal group transformations. component superspace δ P ξ a + δ Q ε δ G ξp a P a + δ G ξp α Q α = δ G ξp A P A δ M λ ab + δ D ρ + δ A θ δ G 1 2 ξmba M ab + δ G ξd D + δ G ξa A δ K ξk a + δ Sζ δ G ξk a K a + δ G ξk α S α = δ G ξk A K A which vanishes since W αβγ is a totally symmetric superfield. The final constraint 2.10 inthe component approach, which is equivalently rewritten as R cov ac M cb i R b aa = 0, 3.6 corresponds to the lowest of the relation between RM ab cd and RA ab as RM ac cb RAb a = This also follows from Table 6, which says that both RM cb ac and RA ab are given by β W βαγ and its conjugate. The correspondence of the superconformal group transformations is shown in Table 7. The correspondence of transformation parameters is given in Table 2. These can be shown by examining the commutation relations in both approaches. The correspondence is trivial for the M ab, D, A, S, K a transformations, but slightly nontrivial for the commutation relations of P A = P a, Q α, Q α.in particular, the supercharge Q α is treated differently in both approaches. In the superspace approach, it is the spinor part of the translation in superspace so that it is defined as a combination of the general coordinate and gauge transformations. In the component approach, the Q transformation is defined as the YM group law of the superconformal group though it is deformed by the curvature constraints. Let us examine the commutation relations of the P A transformation, which is defined in Eq. 2.35as δ G ξ A P A = δ GC ξ M := ξ A E A M δ G ξ M h M B X B. 3.8 We thus need the commutation relations between two GC transformations in superspace and the GC and group transformations X B other than P A. Noting that the field-independent pieces are the flat indexed parameters ξ A and η A, we find with a straightforward calculation the following commutation relations: [δ GC ξ B E B N, δ GC η C E C L ] = δ GC ξ N η L L E N A N E L A E A M, [δ G ξ A h A A X A, δ GC η A E A M ] = δ G η L L ξ N h N B X B + δ G η L ξ N L E N A h A B X B δ GC η L E L C ξ N h N B f B C D E D M, 16/36

17 [δ G ξ A h A A X A, δ G η B h B B X B ] = δ G ξ L η N h N B E L E h L B E N E f B E F h F A X A + δ G η N N ξ L ξ N N η L h L A X A + δ G η N ξ L R LN A X A. 3.9 Using these relations and the definition of the P A transformation, we obtain [δ G ξ A P A, δ G η B P B ]= δ G ξ A η B R AB C X C The parameter ξ A is either vector ξ a or spinor ξ α. When we take both ξ A and η B to be spinors, Eq implies the following Q Q commutation relation by using the constraints on R αβ : [δ G ξ α Q α, δ G η β Q β ]=2δ G η β 0 σ a β α ξ α η β i σ a βα 0 ξ α P a This agrees with the Q Q commutation relation in the component approach, [δ Q ε 1, δ Q ε 2 ]=δ P 1 2 ε 2γ a ε 1, 3.12 if 1 2 ε 1 ξ α ξ α and 1 2 ε 2 η β η β, as given in Table 2. Next, if we consider the vector parameter ξ a and the spinor parameter η β, Eq becomes [δ G ξ a P a, δ G η β Q β ] = δ 12 G ξ a η β RM dc aβ M cd + δg ξ a η β RK γ aβ S γ + δg ξ a η β RK c aβ K c Using the curvature expression R aβ = iσ a β γ W γ, this commutation relation corresponds to [δ P ξ a, δ Q ε] = δ A ξ b δ Q εh b A A=M,S,K 1 = δ M 2 ξ c R ab Qγ c ε 1 + δ S 4 iξ a γ b γ 5 R ba A + R ba Aε + δ K 1 2 ξ b R cov cb Sσ ac 1 4 ξ b δ ac R cov cb Sγ 5ε 3.14 in the component approach when 1 2 η ε = β η β. Finally, setting both ξ A and η B to be vectors, we have which reproduces [δ G ξ a P a, δ G η b P b ]= δ G ξ a η b R ab A X A, 3.15 [δ P ξ a 1, δ Pξ b 2 ]= A =P δ A ξ a 1 ξ b 2 RcovA ab 3.16 in the component approach with the correspondence ξ a 1 ξ a and ξ b 2 ηb. Note that both R ab P c in the component approach and RP ab c in superspace vanish. We comment on the geometrical meaning of the correspondence of commutation relations. In particular, the commutation relation [δ P, δ Q ], which is algebraically determined by some constraints in the component formulation, is understood as a vector spinor curvature in superspace. 17/36

18 Table 8. Conformal multiplet with the Weyl weight w and Lorentz index Ɣ. Weyl weight component superspace w C Ɣ Ɣ w + 1 i α Z Ɣ 2 Ɣ +i α Ɣ H Ɣ Ɣ + 2 Ɣ w + 1 K Ɣ i 4 2 Ɣ 2 Ɣ B aɣ 1 σ 4 a ββ [ β, β] Ɣ w Ɣ i 4 2 α Ɣ + 2 α Ɣ Wα + 2i W α Ɣ w + 2 D Ɣ 1 8 α 2 α Ɣ +W α α Ɣ = 1 8 α 2 α Ɣ W α α Ɣ 3.2. Conformal multiplet We have shown that the superconformal transformations in both approaches satisfy exactly the same algebra. Once the algebra is fixed, the transformation rule for a general conformal multiplet is uniquely determined in the component approach. That is, if the component with the lowest Weyl weight is specified, all other components in the multiplet and their transformation rules are found, up to some ambiguity in field definitions. So we are led to the exact correspondence of superconformal multiplets: Conformal multiplet V Ɣ in 2.18 Primary superfield Ɣ in In the component approach, the first component C Ɣ in V Ɣ is defined as having the lowest Weyl weight in the multiplet so that its S and K a transformations, which lower the Weyl weight, must vanish. In the superspace approach, a primary superfield is defined as being K A invariant. As discussed before, C Ɣ and Ɣ satisfy the same form of superconformal transformations, Eqs and 2.54, respectively. Further, if they have the same Weyl weight w = and chiral weight n = 3/2w as well as the same representation matrices for the Lorentz group ab = S ab, the multiplets in both approaches coincide with each other. The higher components are determined successively by Q transformations and some ambiguities in the field definitions are fixed by the standard form B1 in the component approach [8]. Thus, in the superspace approach, higher components in a superfield can be found by applying Q α = α successively and comparing them with the transformation laws in the component approach. The details are given in Appendix C.1, from which we find the superfield expressions shown in Table 8 for the correspondence of a conformal multiplet with the Weyl weight w and the Lorentz index Ɣ. In this correspondence, the overall factor is fixed by the identification of the first components C Ɣ Ɣ. In the last line, we have used an identity α 2 α α 2 α = 8 W α α + W α α +{ α, W α }, 3.17 which is the conformal superspace counterpart of the identity D α D 2 D α D α D 2 D α = 0 in global supersymmetry. The RHS in 3.17 comes from nonzero vector spinor curvatures and depends on 18/36

19 the gaugino superfield W α. Noticing that W α has only the M and K A components, and { α, W α }= { α, W α } have only the K a component see Eqs. 2.49, 2.50, and 2.51, the above superfield expressions of and D for a multiplet with no Lorentz index reduce to i 4 2 α + 2 α which are the same forms as in global supersymmetry., 1 D 8 α 2 α = 1 8 α 2 α, Chiral projection and invariant actions In this subsection we discuss the correspondences of chiral multiplets, the chiral projection, and the superconformally invariant actions. In the superspace approach, a primary chiral superfield Ɣ is defined as being a primary superfield satisfying the chirality condition α Ɣ = Since primary means the K A invariant, a consistency for such a multiplet to exist requires 0 ={ S α, β } Ɣ = 2D + 3iAɛ α β 2M α β Ɣ, 3.20 which requires Ɣ to have the Weyl and chiral weights, w satisfying 2 3w = 0 and to carry only undotted spinor indices Ɣ = α 1 α 2. These conditions for weights and Lorentz index are exactly the same as given in Eq in the component approach. The component fields in a conformal multiplet with the chirality condition are found from Table 8: CƔ, Z Ɣ, H Ɣ, K Ɣ, B aɣ, Ɣ, D Ɣ Ɣ, i α Ɣ, Ɣ, i 4 2 Ɣ, i a Ɣ,0,0, 3.21 by using the equations β α Ɣ ={ β, α} = 2i α β Ɣ, W α Ɣ = The last equation follows from M β γ Ɣ = K A Ɣ = 0 for a primary superfield Ɣ with purely undotted Ɣ. Comparing the expression 3.21 with the embedding formula referred to above Eq in the component approach, we find the following correspondence between a conformal chiral multiplet in the component approach and a primary chiral superfield Ɣ, [ A Ɣ, P R χ Ɣ, F Ɣ ] component [ Ɣ, α Ɣ, Ɣ ] superspace 3.23 The algebra { α, β }=0in Eq implies that the equation α 2 Ɣ = identically holds for any superfield Ɣ.So 2 Ɣ formally seems a chiral superfield. However, if 2 Ɣ is not primary, it still has to contain components, in contrast with the fact that a chiral superfield has only components. This odd property happens in the superconformal case since S α acts as an inverse operator of α. If 2 Ɣ is primary, it contains only components for a primary chiral superfield. For Ɣ with the Weyl and chiral weights, w, 2 Ɣ has + 1, w + 2 and becomes chiral and primary 19/36

20 if w + 2 = 0 and Ɣ is purely undotted. This means that 2 gives a chiral projection operator if it acts on a primary superfield Ɣ whose weights and index satisfy those conditions. This agrees with the conditions for the chiral projection operator in the component approach given in Eq Taking care with the coefficients, we find the correspondence between the chiral projection operators in the component approach and P in superspace: P = We show in Appendix C.2 that the component fields of a projected superfield P Ɣ are identified with those of V Ɣ in Eq in the component approach. In this identification, the following equations are useful: 2 2 = α 2 α + 8 a a 2i a σ a αα [ α, α ] 8W α α, 2 2 = α 2 α + 8 a a + 2i a σ a αα [ α, α ]+8W α α We remark that the sum of these yields α 2 α α 2 α = 16 a a + 8W α α 8W α α, 3.27 which is the conformal superspace counterpart of the global supersymmetry identity D 2 D 2 + D 2 D 2 2D α D 2 D α = Finally, we discuss the superconformally invariant actions. First is the correspondence of the F-type invariant action for the conformal chiral multiplet without external Lorentz index. The component expansion of the F-type integration 2.59 is coincident with the expression 2.24 of F-type invariant action in the component approach, if account is taken of the correspondences of gauge fields Table 3 and chiral multiplet components 3.23: d 4 x [ ] F d 4 xd 2 θ E + d 4 xd 2 θ Ē The other is the correspondence of the D-type invariant action for the general real conformal multiplet V without external Lorentz index. Since the D-type formula is obtained from the F-type one by using the chiral projection operator, the correspondences of 3.25 and 3.29 directly lead to the following correspondence of the D-type invariant actions 2.25 in the component approach and the D-type integral 2.60 in the superspace approach: d 4 x [ V ] D 2 d 4 xd 4 θ EV u-associated derivatives Restriction on the existence of conformal spinor derivatives? We first mention a historical puzzle about the conformal spinor derivative. In Ref. [8], KU constructed the spinor derivative in the component approach and claimed that such a spinor derivative D α exists only when some special conditions are met on an operand multiplet V Ɣ. On the other hand, Butter defined [9] in the superspace formalism the conformally covariant derivatives A, which can act on any superfield with no restriction. What is the difference? 20/36

21 The point is that KU defined in their component approach a conformal multiplet V Ɣ by its first component C Ɣ, denoting V Ɣ = C Ɣ, which has the lowest Weyl weight in the multiplet. Therefore, the S and K a transformations of C Ɣ must vanish, since S and K a lower the Weyl weight. In superspace terminology, such a multiplet is arranged in a primary superfield Ɣ : V Ɣ = C Ɣ Ɣ, δ S C Ɣ = δ K C Ɣ = 0 K A Ɣ = KU looked for the spinor derivative D α as a mapping of a conformal multiplet V Ɣ to another conformal multiplet whose first component is Z αɣ, which is the second component of V Ɣ : D α : V Ɣ = C Ɣ Dα V Ɣ = Z αɣ That is crucial and is the only difference from the superspace covariant derivatives A, which generally do not bring a primary superfield into primary. This freedom employed in the superspace formulation is consistent with the freedom of the Q transformation in the component approach, because the S transformation of Z αɣ is not generally required to vanish. Thus, the conformal covariant spinor derivative that corresponds to the Q transformation is α, not D α. Conversely speaking, once the image α Ɣ is required to be primary, S β α Ɣ = 0 leads to the same conditions for Ɣ as KU found in the component approach u-associated derivative We need the S and K a invariance of multiplets, for instance, in constructing the invariant actions by the D-type and F-type formulas. Reference [8] has shown that, if one has a compensating multiplet u or any conformal multiplet whose first component is guaranteed to be nonvanishing, like the compensator used for gauge fixing, the covariant derivative D α u is constructed, which maps a conformal multiplet into another conformal one without any restriction. Consider a conformal multiplet u with the Weyl and chiral weights w 0, n 0 and no external Lorentz index. The component fields are denoted as u = [ C u, Z u, H u, K u, B ua, u, D u ] Assuming that the first component C u is nonvanishing, we construct the following spinor: λ S := iz u w 0 + n 0 C u, 3.34 which is nonlinearly shifted under the S transformation as δ S ζ λ S = ζ. Then the u-associated spinor derivative D u α is defined by D u α V Ɣ = Z αɣ + iw + nλ S α C Ɣ σ ab β α λ S β ab C Ɣ, 3.35 where w and n are theweyl and chiral weights of C Ɣ. Since δ S ζ Z Ɣ = iw+nζ C Ɣ +σ ab ζ ab C Ɣ, the quantity in the brackets on the RHS is invariant under the S transformation, so that it defines a conformal multiplet D u α V Ɣ. The barred derivative D αu is given by D αu V Ɣ = D αu VƔ. Similarly, the u-associated vector derivative is constructed as follows. We define a vector V K a and a spinor χ S by Da C u V a K := 1 4w 0 C u + D ac u C u, χ S := 1 Zu iγ 5 2w 0 C u + Z u C u, /36