Component versus Superspace Approaches to D=4, N=1 Conformal Supergravity

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 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 Poincare supergravity since they look rather different between two approaches.

1 Introduction N = 1 supergravity (SUGRA) in four dimensions has been important as giving a boundary theory around the unification scale for constructing viable phenomenological models beyond the standard model. It has also become to have increasing importance as low-energy effective theory of superstring and as a tool for analyzing supersymmetric gauge theories on curved backgrounds.
However various explicit calculations, e.g., the construction of SUGRA Lagrangian, are complicated and non-trivial. 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; that is, superconformal gauge fields including gravity and gravitino 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 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-matter coupled SUGRA system. Kugo and Uehara (KU) have presented [8] the superconformal tensor calculus in the 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 Γ must be satisfied so that its spinorial derivative D α V Γ exists and gives a conformal multiplet. The condition depends on the spinor index α of D α and the Lorentz index Γ of the operand V Γ , 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. Contrary to the previous expectation, his formalism realizes a simpler algebra of covariant derivatives than any other superspace Poincaré SUGRA: 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 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 ,Qα) of superconformal group. The reason why KU's spinorial derivatives could not be freely applied turns out that 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 transparent and powerful means to treat the systems in new situations such as finding non-linear realization, brane world, 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 two approaches by making the detailed correspondences manifest. This paper is organized as follows. In section 2, we recapitulate the essential parts, first, of the superconformal tensor calculus in component approach, and then, of the conformal superspace approach. We use the individual notation for each of these approaches and separately give a dictionary between them for the convenience of reading the references.
In section 3 we explicitly present the correspondences of various quantities. We first discuss gauge fields and curvatures in Sec. 3.1 and show how all the curvature constraints in component approach are satisfied in 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 Sec. 3.2, and the chiral projection and the invariant action formulas in Sec. 3.3. We analyze in Sec. 3.4 the compensated (or u-associated) derivatives which map a primary superfield to primary one. There we also discuss the KU's restriction on the spinorial derivatives.
In section 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 the 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 non-trivial since the gauge invariance in 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 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 between them is given in appendix A. The standard form of 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 which are necessary in deriving the results in the text.

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 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 the component approach part 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 anti-symmetric tensors. The detail of the notations is summarized in appendix A. The superconformal algebra consists of 15 bosonic and 8 fermionic generators, which obey the following graded commutation relations: All other commutation relations vanish. The generators are generically denoted as X A and the above commutation relations are written as Note that these generators represent the active operators transforming fields, not the representation matrices. The commutation relations change the signs if written for representation matrices instead of active operators. In the conformal SUGRA, the superconformal symmetry is treated as local symmetry. The corresponding gauge fields and transformation parameters are given by In component approach, the Greek letters µ, ν, ... denote the curved vector indices and the Roman letters a, b, ... the flat Lorentz indices. The group transformation laws of the gauge fields under the superconformal symmetry are The curvature of the superconformal algebra (before the deformation below) is The P a translation is deformed so as to be related to the general coordinate (GC) transformation where ξ µ = ξ a e a µ , and ξ a is a field-independent parameter. In order to have [δ Q , δ Q ] ∼ δP , several constraints on the curvatures are imposed: whereR µν is the dual of R µν . By these constraints, the M ab , S and K a gauge fields (ω µ ab , ϕ µ and f µ a , 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 The deviation part δ ′ Q (ε) is given by Note that the RHS of Eq. (2.12) are given by εe c µ f Q Pc 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 [Q, P c ] commutator from (originally) zero to the non-vanishing f Q Pc X for X = M ab , S, K a . The resultant commutation relations are the same as the original ones for all A and B, if P a on the RHS of Q-Q commutator is understood to beP a : Moreover, the definition ofP a transformation leads to where α = (α,α). The superconformally covariant derivative on fields carrying only flat Lorentz indices is defined through theP a -transformation as Next, we introduce superconformal multiplets. A general conformal multiplet V Γ is a set of (8 + 8) × dim Γ complex fields, (2.18) where Γ represents arbitrary spinor indices Γ = (α 1 , ..., α m ;β 1 , ...,β n ) and dim Γ is the dimension of Lorentz representation of Γ. The first component C Γ is defined to have the lowest Weyl weight in the multiplet so that its transformation law is given by Here Σ ab is the representation matrix of Lorentz generator which C Γ belongs to, and w and n are the Weyl and chiral weights of C Γ . The S and K a transformations must annihilate the lowest weight component C Γ since they lower the Weyl weights of operands. The Q transformation law δ Q (ε)C Γ = 1 2 iεγ 5 Z Γ simply defines the second component Z Γ . All the higher components in the multiplet and their superconformal transformation laws are determined by demanding 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 (B.1), which also fix the definition of higher component fields. We call the transformation laws (B.1) the standard form. Since the first component C Γ specifies the whole multiplet, we denote the conformal multiplet V Γ using the first component as (2.20) 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 Σ (w,n) Γ , for instance, exists only when the Weyl and chiral weights (w, n) satisfy w = n and the Lorentz index Γ is made of purely undotted spinor indices; then the chiral multiplet has (2 + 2) × dim Γ complex components denoted by (2.21) These three components of a chiral multiplet are embedded into a general conformal multiplet , so that their Q and S transformation laws are given by For the multiplet V (w,n) Γ with purely undotted spinor Γ satisfying w = n + 2, the chiral projection operator Π exists and 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 Σ = 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 The D-type invariant action formula is applied only to the real and Lorentz-scalar multiplet V = C, Z, H, K, B a , λ, D with w = 2 and n = 0. The action is derived from the F-type formula with the chiral projection operator Π as For the general YM-matter coupled SUGRA system, the action is given by 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 Σ c is a chiral compensator carrying weights (w, n) = (1, 1). For the system possessing non-vanishing superpotential g(S), it is convenient to redefine the compensator as Σ c → Σ 0 = g 1/3 (S)Σ c = [z 0 , P R χ 0 , h 0 ] so that φ becomes the combination ofφ and superpotential: φ(S,Se 2V G ) =φ(S,Se 2V G )|g(S)| −2/3 . In the third term, f αβ is a holomorphic functions 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 [7] are where χ R0 = 1 2 P R χ 0 and χ Ri = 1 2 P R χ i . These gauge conditions set the first and second components of the vector multiplet φΣ 0Σ0 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 δ P Q (ε) in the resultant Poincaré SUGRA and the gauge-fixed conformal Q transformation is given by (2.29) In this expression ∇ µ z i is the covariant derivative of the internal symmetry, and G are given by G = 3 log 1 3 φ(z, 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 the same footing as the usual translation P a . The (anti-)commutation relations between the spinor covariant derivatives become complicated in Poincaré SUGRA, whereas in conformal superspace, they are as simple as in global supersymmetry. In the superspace approach part in this paper, we use notations and conventions of Butter [9] with a few exceptions which will be explained below. The detail of the notations is summarized in appendix A. The superconformal algebra is the same as (2.1) given in component approach, if we perform a suitable translation of generators between two approaches (see Table (3.1)). Here we refer to only a few characteristic commutation relations where we use the calligraphic index A for the total superconformal algebra, while the Roman uppercase index A for the set of Lorentz vector and spinor as P A = (P a , Q α ,Qα) and K A = (K a , S α ,Sα), and the index M is the set of curved indices, for example, A M = (A m , A µ , Aμ). We assume that the vierbein E M A is invertible: The gauged superconformal transformations are taken by real parameter superfields. These parameter superfields are denoted as The gauge fields receive the superconformal transformation 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] uses the different notation that X A was expressed as X A in 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 component approach, the P A transformation is defined to be related to the general coordinate transformation δ GC using field-independent parameter superfield ξ A as where ξ(P ) A is abbreviated to ξ A . The P A transformation acting on a superfield Φ with no curved index defines the covariant derivative as 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 for grading of fermionic objects such as 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 M N should be accompanied by the grading factor (−) a(b+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. The implicit grading means the understanding of omitting 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 used also for the index A of superconformal generators. In the definition of curvatures, the commutation relation of P A is as follows Several constraints are imposed on the curvature superfields to eliminate the redundant degrees of freedom. First, the constraints on R αβ are as follows which guarantees the commutation relations of covariant spinor derivatives to take the simple form (as in the global supersymmetry case) Secondly, the following constraints on R αa are imposed By solving the Bianchi identities under these constraints (with implicit grading understood), one finds that all other non-vanishing 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, [9]. The brackets ( ) on the indices imply the symmetrization with weight one, e.g., ψ (α χ β) = (1/2)(ψ α χ β + ψ β χ α ). This algebra-valued superfield W α satisfies The further input of the second constraints (2.41) implies that W α has no P A , D, A components, With the help of the superconformal algebra, the chirality and reality conditions (2.45) and (2.46) leads to the final expression In this way, the gaugino superfield W α is expressed by the totally symmetric superfield W αβγ which satisfies Owing to Eqs. (2.43) 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 (2.52) Now the concept of primary superfield is introduced to describe matter superfields, invariant action over the superspace, and so on. A primary superfield Φ Γ is defined as the superfield on which the action of superconformal group is where Γ and Σ represent general Lorentz indices such as Γ = (α 1 , . . . , α n ,β 1 , . . . ,β m ), and S bc is the representation matrix of Lorentz algebra which Φ Γ belongs to. The real constant numbers ∆ and w are called the Weyl and chiral weights, respectively. The last property K A Φ Γ = 0 is most important for Φ Γ being primary. That is generally violated for its derivative ∇ A Φ Γ . As for W αβγ , Eqs. (2.51) imply it is a primary chiral superfield with Weyl weight ∆ = 3/2 and chiral weight w = 1, where a chiral superfield means that it satisfies∇αΦ = 0 as usual. It should be noted that this chirality condition is superconformally covariant. An invariant integral over the superspace is given by Here we are using implicit grading and omitting to write the superdeterminant "sdet". The superconformal transformation law of the density E is since the superconformal generators X B ′ other than P A carry non-positive Weyl weights so that the commutator [X B ′ , X C ] yields positive Weyl weight P A only when X C = P C (and 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, called the D-type integration.
The superconformal counterpart of the d 2 θ integral in global supersymmetry is The chiral density E is given by the superdeterminant of vielbein in the chiral subspace with dotted spinor directions being omitted from , and m = (m, µ). In (2.57), W is a covariantly chiral superfield defined by∇αW = 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 onθ, it is supposed to be executed at θ = 0, which is called the F-type integration. Performing the d 2 θ integration in (2.57), we obtain the component expression of the F-type integration as (2.58) The D-type integration is related to the F-type one as is the chiral projection operator. The component expression of the D-type integration is obtained using the equation (2.59). The action of the matter coupled SUGRA system is given in conformal superspace as where Φ 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 Φ i with weights (∆, w) = (0, 0). In addition, K is a real function and W is holomorphic. The gauge-fixing conditions leading to Poincaré SUGRA with the canonically normalized EH term are given in Ref. [9]: From the gauge-fixing condition for K A -gauge, the D gauge field B M vanishes and the K A gauge field f M A loses the gauge freedom. So f M A drops out from the covariant derivative and curvatures. The derivative after gauge fixing is written as The constraints for R(D) αβ lead The constraints R(K) αβ,γ = 0 and R(K) αγ β = 0 and their conjugates give Finally the constraint R(K) αβ c = 0 means Since the conformal curvatures are written in terms of W αβγ , the curvatures after gauge fixing are written in terms of R, G αβ , W αβγ and the derivative D A . It is noted that the gauge-fixing conditions (2.62) also fix the A gauge superfield A M . The iAαΦ c , and further imposing the gauge conditions Φ c = e K/6 and Bα = 0 leads to The chirality condition for matter superfields Φ i are used; 0 =∇αΦ i = Eα M ∂ M Φ i =DαΦ i . In the same way, from ∇ αΦ c = 0, A α is fixed as

Correspondence between component and superspace approaches
In this section we present the correspondence between component and superspace formulations. The objects which 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 two approaches and the dictionary between them is given in appendix A for spinors, vectors, gamma matrices, and tensors.

Superconformal algebra, gauge fields and curvatures
As discussed in Sec. 2.1, the Q and P a transformation in 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 given by The correspondence of gauge parameters is set to satisfy ǫ A X A ↔ ξ A |X A and given by The vertical bar "|" means the θ =θ = 0 projection, i.e., the lowest component of superfield.
Since gauge fields × generators essentially represent the common quantity in both approaches, In the table, the curved index µ of component approach corresponds to the index m of superspace. The curvature in superspace, R mn C , with curved tensor indices was defined in Eq. (2.37). The lowest component of flat indexed curvature superfield R ab C is given by Using the correspondence of gauge fields given in (3.3), we find that the curvatures coincide with (the negative of) the covariant curvatures with the algebra deformation of component approach, up to the normalization of generators The 'covariantization' is necessary only for the M, S, K a curvatures in component approach, which correspond in superspace to the fact that the gaugino superfield W α has non-vanishing components only for the M, S, K a generators. In obtaining the correspondence table (3.5), we have used the following relations of W α to the curvatures in component approach Note that these quantities stand for the spinor-vector components of superspace curvature R αb C because of the relations (2.43).
The correspondences of the curvatures (3.5) are summarized in a simple expression We emphasize that such identification holds for the flat indexed curvatures, while it does for the curved indexed gauge fields For instance, the component approach counterpart of the flat indexed gauge field in superspace is found through the expression The constraints on curvatures also have the correspondence, though the constraints in superspace are directly imposed on the spinor-spinor or spinor-vector component of curvatures. The restricted form of the vector-vector component R ab in superspace is derived from other constraints and explicitly written in terms of the primary chiral superfield W αβγ . That is, Eq. (2.52) implies the following expressions for the curvatures R(X) ab A in superspace The chiral decomposition of anti-symmetric tensor is defined in Eq. (A.19).
We can see the correspondence of curvature constraints using the fact that all the curvature components R(X) ab A with vector-vector indices are expressed by W αβγ in superspace. First, the constraint (2.8) in component approach is equivalent to R ab (P c ) = 0 and hence corresponds to R(P ) ab c | = 0 in superspace, as seen in Table (3.5) and (3.10). Secondly, the constraint (2.9), equivalent to R ab (Q)γ b = 0, corresponds to the equation R(P ) ab α |(σ b ) αδ = 0 and its conjugate in superspace. This is found from (3.10) that R(P ) ab α has only chiral component R(P ) γγ,ββ α = 2ε˙γβW γβ α so that which vanishes since W αβγ is a totally symmetric superfield. The final constraint (2.10) in component approach, which is equivalently rewritten as This also follows from (3.10) which says that both R(M) ac cb and R(A) ab are given by ∇ β W βαγ and its conjugate.
The correspondence of the superconformal group transformations is as follows: 14) The correspondence of transformation parameters is given in (3.2). 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 non-trivial for the commutation relations of P A = (P a , Q α ,Qα). In particular, the supercharge Q α is treated differently in both approaches.
In superspace approach, it is the spinor part of the translation in superspace so that it is defined to be a combination of the general coordinate and gauge transformations. In component approach, the Q transformation is defined to be the YM group law of 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. (2.35) as 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: Using these relations and the definition of P A transformation, we obtain 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. (3.17) implies the following Q-Q commutation relation by using the constraints on R αβ , This agrees with the Q-Q commutation relation in component approach if 1 2ε 1 ↔ ξ αξα | and 1 2ε 2 ↔ η βηβ | as given in the correspondence table (3.2). Next, if we consider the vector parameter ξ a and the spinor parameter η β , Eq. (3.17) becomes Using the curvature expression R aβ = −i(σ a ) βγ Wγ, this commutation relation corresponds to in component approach when 1 2ε = η βηβ |. Finally, setting both ξ A and η B to be vectors, we have in component approach with the correspondence ξ a 1 ↔ ξ a | and ξ b 2 ↔ η b |. Note that both R ab (P c ) in component approach and R(P ) ab c in superspace vanish. We remark 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 component formulation, is understood as a vector-spinor curvature in superspace.

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 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 lead to the exact correspondence of superconformal multiplets Conformal multiplet V Γ in (2.18) ↔ Primary superfield Φ Γ in (2.53).
In component approach, the first component C Γ in V Γ is defined to have the lowest Weyl weight in the multiplet so that its S and K a transformations, which lower the Weyl weight, must vanish. In superspace approach, a primary superfield is defined to being K A invariant. As discussed before, C Γ and Φ Γ | satisfy the same form of superconformal transformations, Eq. (2.19) and Eq. (2.53), respectively. Further if they have the same Weyl weight w = ∆ and chiral weight n = (3/2)w as well as the same representation matrices for Lorentz group Σ ab = −S ab , the multiplets in both approaches coincide with each other. The higher components are determined successively by Q transformations and some ambiguity in field definitions are fixed by the standard form (B.1) in component approach [8].
Thus in superspace approach, higher components in a superfield can be found by applying Q α (= ∇ α ) successively and comparing them with the transformation laws in component approach. The detail is given in appendix C.1 from which we find the following superfield expressions for the correspondence of a conformal multiplet with the Weyl weight w and the Lorentz index Γ In this correspondence, the overall factor is fixed by the identification of the first components C Γ ↔ Φ Γ |. In the last line, we have used an identity which is the conformal superspace counterpart of the identity D αD2 D α −DαD 2Dα = 0 in global supersymmetry. The RHS in (3.25) comes from nonzero vector-spinor curvatures and depends on the gaugino superfield W α . Noticing that W α has only the M and K A components (see Eq. (2.48)), the above superfield expressions of Λ and D for a multiplet with no Lorentz index reduce to which are the same forms as in global supersymmetry.

Chiral projection and invariant actions
In this subsection we discuss the correspondences of chiral multiplets, the chiral projection and the superconformally invariant actions. In superspace approach, a primary chiral superfield Φ Γ is defined to be a primary superfield satisfying the chirality condition∇α Φ Γ = 0 . (3.27) Since "primary" means the K A invariant, a consistency for such multiplet to exist requires by using the equations The last equation follows from Mβ˙γΦ Γ = K A Φ Γ = 0 for a primary superfield Φ Γ with purely undotted Γ. Comparing the expression (3.29) with the embedding formula referred to above Eq. (2.22) in component approach, we find the following correspondence between a conformal chiral multiplet in component approach and a primary chiral superfield Φ Γ , The algebra {∇α,∇β} = 0 in Eq. (2.40) implies the equation identically holds for any superfield Ψ Γ . So∇ 2 Ψ Γ formally seems a chiral superfield. However, if ∇ 2 Ψ Γ is not primary, it still has to contain 8 + 8 components contrary to the fact that a chiral superfield has only 2 + 2 components. This odd property happens in the superconformal case sinceSα acts as an inverse operator of∇α. If∇ 2 Ψ Γ is primary, it contains only 2 + 2 components for a primary chiral superfield. For Ψ Γ with the Weyl and chiral weights (∆, w),∇ 2 Ψ Γ has (∆+1, w +2) and becomes chiral and primary if 2(∆ + 1) − 3(w + 2) = 0 and Γ is purely undotted. This means that∇ 2 gives a chiral projection operator if it acts on a primary superfield Ψ Γ whose weights and index satisfy those conditions. That agrees with the conditions for the chiral projection operator in component approach given in Eq. (2.23). Taking care of coefficients, we find the correspondence between the chiral projection operators Π in component approach and P in superspace We show in appendix C.2 that the component fields of a projected superfield PΨ Γ are identified with those of ΠV Γ in Eq. (2.23) in component approach. In this identification, the following equations are useful We remark that the sum of these yields which is the conformal superspace counterpart of the global supersymmetry identity 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 Ftype one by using the chiral projection operator, the correspondences of (3.33) and ( We first mention to a historical puzzle on the conformal spinor derivative. In Ref. [8], KU constructed the spinor derivative in component approach and claimed that such spinor derivative D α exists only when some special conditions are met on an operand multiplet V Γ . On the other hand, Butter defined [9] in superspace formalism the conformally covariant derivatives ∇ A which can act on any superfield with no restriction. What is the difference?
The point is that KU defined in their component approach a conformal multiplet V Γ by its first component C Γ , denoting V Γ = C Γ , which has the lowest Weyl weight in the multiplet. Therefore the S and K a transformations of C Γ must vanish since S and K a lower the Weyl weight. In superspace terminology, such a multiplet is arranged in a primary superfield Φ Γ : KU looked for the spinor derivative D α as a mapping of a conformal multiplet V Γ to another conformal multiplet whose first component is Z αΓ which is the second component of V Γ That is crucial and only difference from the superspace covariant derivatives ∇ A , which generally do not bring a primary superfield into primary. This freedom employed in superspace formulation is consistent with the freedom of Q transformation in component approach, because the S transformation of Z αΓ 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 Φ Γ is required to be primary, S β ∇ α Φ Γ = 0 leads to the same conditions for Φ Γ as KU found in 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. Ref. [8] has shown that, if one has a compensating multiplet u (or any conformal multiplet whose first component is guaranteed to be non-vanishing, 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 (3.41) Assuming that the first component C u is non-vanishing, we construct the following spinor which is non-linearly shifted under the S transformation as δ S (ζ)λ S = ζ. Then the u-associated spinor derivative D α (u) is defined by where w and n are the Weyl and chiral weights of C Γ . Since δ S (ζ)Z Γ = −i(w+n)ζC Γ +σ ab ζ (Σ ab C) Γ , the quantity in the bracket on the RHS is invariant under the S transformation, so that it defines a conformal multiplet D α Similarly, the u-associated vector derivative is constructed as follows. We define a vector V a K and a spinor χ S by so that V a K and χ S are shifted under the K a and S transformations, respectively, as δ K (ξ K )V a K = ξ K a and δ S (ζ)χ S = ζ. The S transformation of the vector V a K yields the spinor χ S as δ S (ζ)V a K = −1 4ζ γ a χ S . By adding appropriate terms containing these fields, the superconformally covariant derivative D a C Γ defined in (2.17) can be made S-invariant, and the u-associated vector derivative is defined by so that D a (u) V Γ is a conformal multiplet. We now show the superspace expression for the u-associated derivatives using the correspondences given in the previous section. First we introduce the primary superfield X u that corresponds to u u ↔ X u , (3.46) where X u has the Weyl and chiral weights (∆ 0 , w 0 ) = (w 0 , 2 3 n 0 ). From the correspondences of weights and component fields (3.24), λ S is identified as By reading the correspondence of D α (u) V Γ (3.43), we find the following superspace expression for the u-associated spinor derivative and similarly for the dotted spinor derivative. When X u is a real superfield with special weights ∆ 0 = 2 and w 0 = 0, this expression reduces to the compensated spinor derivatives discussed in Ref. [11]. So, (3.48) stands for the generalization to X u with arbitrary weights. We also construct the superspace expression for the u-associated vector derivative. For this purpose, we consider the real superfield Y u defined by * Y u = log X u + logX u . (3.49) Using the component field correspondence (3.24), we identify V a K and χ S as Then the superspace expression for the u-associated vector derivative is found by translating * Precisely speaking, this Y u itself is not a proper primary superfield unless ∆ 0 = w 0 = 0 since log X u has no definite values of Weyl and chiral weights. In the following expressions, however, only its derivative ∇ A Y u = ∇ A X u /X u + ∇ AXu /X u appears, which is a proper superfield with the Weyl and chiral weights of the operator ∇ A . (3.45): When X u is a real primary superfield X with the weights (∆ 0 , w 0 ) = (2, 0), this reduces to the compensated vector derivative with parameter λ = 1 given in Ref. [11], if we replace Y u → 2 log X.

Superconformal gauge fixing to Poincaré SUGRA
The superconformal group is larger than the super Poincaré and the extra D, A, S, K a gauge symmetry should be fixed to have Poincaré SUGRA, which is useful e.g. for phenomenological applications. In this section, we examine the superconformal gauge fixing of the matter-coupled conformal SUGRA to Poincaré SUGRA, and give the correspondence of gauge-fixing conditions between superspace and component approaches. In this paper we focus on the chiral superfield matter system. The gauge fixing for the system containing YM gauge fields of internal symmetry will be discussed elsewhere [12].

Gauge fixing in superspace approach
The matter superfields Φ i (i = 1, 2, . . . , n) are introduced to be primary and covariantly chiral with respect to the superconformal symmetry,∇αΦ i = 0. They have the Weyl and chiral weights (∆, w) = (0, 0). The matter-coupled SUGRA action in conformal superspace is given by where the Kähler potential K = K(Φ i ,Φ i * ) is a real function of matter superfields, and the superpotential W = W (Φ i ) is a holomorphic one. In the first term (the D-type action), the superconformal gauge invariance leads to the conditions that the compensator chiral superfield Φ c is primary and has the weights (∆, w) = (1, 2/3). In the second term (the F-type action), the compensator dependence is also fixed by the superconformal gauge invariance. Let us discuss the gauge fixing of superconformal symmetry to go down to Poincaré SUGRA. For a non-vanishing superpotential, it is useful to redefine the compensator Φ c as † (4.2) † The redefinition (4.2) is possible when W = 0. For W = 0, a convenient gauge choice may be Φ c = e K/6 (and B M = 0) which is the same condition as the one given in [9] and mentioned in section 2.2.
The new chiral compensator Φ 0 has the weights (∆, w) = (1, 2/3). The action in terms of Φ 0 is given by One of the virtues of using Φ 0 and G is revealed in introducing YM gauge fields, that is, Φ 0 and G are invariant under possible internal symmetry, while Φ c and K are not invariant. This invariant property of Φ 0 and G makes it simple to fix the superconformal gauge symmetry irrespectively of internal ones [12].
In component approach, Ref. [7] discussed the superconformal gauge-fixing conditions which realize the canonically normalized EH and RS terms and also give a real gravitino mass, given in (2.27). We find its superspace counterparts are The second condition is imposed by an appropriate K A gauge transformation of the D-gauge superfield: On the other hand, the first condition seems peculiar since the chiral superfield Φ 0 does not have enough numbers of independent components which can be set equal to the general real superfield e G/6 . It is however noticed that the gauge fixing (4.5) is given in conformal superspace where all gauge transformations have real superfield parameters. Therefore the finite D and A gauge transformations Φ 0 → e ξ(D)+ 2i 3 ξ(A) Φ 0 are possible with the real superfield parameters ξ(D) = G/6 − (1/2) ln(Φ 0Φ0 ) and ξ(A) = (3/4i) ln(Φ 0 /Φ 0 ) which brings Φ 0 to e G/6 .
The gauge-fixing conditions (4.5) imply several other equations for superfield components. We here focus on the chiral compensator Φ 0 and the A-gauge superfield A M . Recall that the covariant derivative takes the following form for a primary superfield Φ (∆,w) with the Weyl and chiral weights (∆, w) and no external Lorentz index: where the last term (2∆ + 3w)f Aα Φ (∆,w) stands for −f A β {K β , ∇ α }Φ (∆,w) and we have used the equation which is the covariant derivative in Poincaré SUGRA and different from the derivative in Ref. [9] (D A discussed in section 2.2). Plugging the gauge-fixing conditions (4.5) into the RHS of (4.6) and (4.7), we find the components of the chiral compensator superfield Φ 0 , Φ 0 = e G/6 , (4.9) iA α e G/6 , (4.10) iA α e G/6 − 4f α α e G/6 . (4.11) Note that ∇ α Φ 0 = ∇ α e G/6 but D P α Φ 0 = D P α e G/6 since the gauge-fixing condition Φ 0 = e G/6 violates the D and A symmetries but preserves M ab .
After the gauge fixing, the chirality condition of the compensator Φ 0 turns out to determine the A-gauge superfield. Applying (4.6) to∇αΦ 0 = 0 and using the gauge-fixing condition (4.5), we obtain where the field derivatives of G are denoted as G i = ∂G/∂Φ i and G i * = ∂G/∂Φ i * . Similarly, the condition ∇ αΦ 0 = 0 fixes A α as with which the components of the chiral compensator, (4.10) and (4.11), are rewritten as 14) We have used the relation f αβ = −ǫ αβR which comes from the curvature constraints after the gauge fixing. The equation (4.15) relates the compensator F component to the auxiliary fieldR, undetermined part of the S-gauge field f α β . It is noticed that superfield components are not given by the covariant derivative of Poincaré SUGRA (D P ) but should be defined by the conformal one (∇). For the matter superfields Φ i with vanishing weights (∆, w) = (0, 0), these two derivatives give same results for the first derivatives (spinor components), but different for the second ones (F components). For the comparison with component approach, we rewrite the above results with the conformally covariant derivative ∇. Eqs. (4.6), (4.7) and (4.13) imply D P We then find (4.14) and (4.15) are given by The chirality condition of the compensator also fixes the vector part of A-gauge field. The chirality condition and the algebra {∇ α ,∇β} = −2i∇ αβ implȳ After the gauge fixing, the vector derivative on the RHS becomes ∇ αβ Φ 0 = D P αβ Φ 0 − 2 3 iA αβ Φ 0 which is used to determine the vector part A αβ . Evaluating the LHS of (4.19) by using (4.7) with the gauge-fixing conditions (4.5) and Eq. (4.14), we find (4.20) In going to the second line, we have used D P α Φ i = ∇ α Φ i and f αβ = −G αβ /2 which reads from the curvature constraints after the gauge fixing. The second order derivative is modified by usinḡ which also follows from (4.7) and the gauge-fixing conditions. Eq. (4.20) is regarded as determining G αβ in terms of the auxiliary A-gauge field A a .

Correspondence to component approach
We here show the correspondence of superconformal gauge fixing between the superspace and component approaches. First, note that the correspondences of the potentials and compensators are as follows: The symbols in component approach are explained in section 2.1.
Let us see that the gauge-fixing conditions (4.5) in superspace are equivalent to the improved D, A, S, K a gauge conditions (2.27) in component approach. As discussed in section 3.3, the component correspondence between the compensator multiplet Σ 0 and the compensator superfield The gauge conditions (4.5) or its consequence (4.9) directly means the correspondence of the gauge-fixed lowest components For the spinor components, the S-gauge condition in component approach and Eq. (4.17) in superspace exactly agree with each other: The correspondence of the K a gauge is trivial. Note that the S gauge condition in superspace approach, B α = 0, is used in deriving the spinor component of Φ 0 (4.17), which leads to the correspondence (4.25). For the F components, the auxiliary field h 0 in Σ 0 is not gauge-fixed in component approach. This corresponds to the fact in superspace that the F component of Φ 0 contains the auxiliary partR after the gauge fixing, as given in (4.18).
The virtue of gauge fixing in conformal superspace is two fold. The first is that it leads to the superspace Poincaré SUGRA directly and easily. The second is that it finds the supersymmetry transformation in the resultant Poincaré SUGRA in a straightforward way. Namely, the remaining Poincaré supersymmetry is just given by the covariant spinor derivatives D P α andD Ṗ α . In component approach, however, the remaining supersymmetry is deformed from the original one by the requirement that it keeps the D, A, S, K a gauge conditions intact, and explicitly found in Ref. [7] by adding a complicated combination of the A, S, K a gauge transformations with non-trivial field-dependent parameters (2.28). We finally show this correspondence of the Poincaré supersymmetry after the gauge fixing, in particular, the covariant spinor derivative D P α reproduces the deformed supersymmetry in component approach.
The Poincaré spinor derivative (4.8) is related to the conformal one as D P α = ∇ α +A α A+f α A K A after the gauge fixing. The supersymmetry transformation in Poincaré superspace defined by η α D P α is then given by the following linear combination of superconformal A, K A transformations (4.26) Similarly, the Poincaré Q transformation after the gauge fixing is written as with the same parameters given in (4.26). We show this transformation is exactly the same as the Q transformation (2.28) in component approach by examining the correspondence between the transformation parameters (4.26) and (2.29). For the A transformation, the parameter in superspace is Noticing the parameter correspondences 3 4 θ ↔ ξ(A)| andε ↔ 2 η αηα | given in Table (3.2), we find (4.28) agrees with θ(ε) of (2.29) in component approach. For the S transformation, the above parameter ξ(K) ′ (η) α in superspace is rewritten by the other auxiliary fields with (4.18) and (4.20), and given by With this form at hand, the parameter ξ(K) ′ (η) a of (4.26) in superspace is rewritten as The RHS is same as ξ a (ε) of (2.29) in component approach with the correspondences of the S gauge field given in Table (3.3) and the S transformation parameter discussed above.
The action in superspace formalism has a huge number of gauge invariance than the component approach. Therefore the correspondence between two approaches should be clarified also for the gauge fixing conditions. We make comprehensible how to obtain Poincaré SUGRA and the remaining supersymmetry by fixing the superconformal gauge symmetry in the general mattercoupled SUGRA system.

A Notations
In the component approach part in the text, we use the notation of KU [8], which is the same as Ref. [10] except for two-component spinors and the dual of second rank anti-symmetric tensor. In the superspace approach part, we use the notation of Wess and Bagger [13].

A.1 Notation in component approach
We use Roman letters for flat Lorentz indices, Greek letters µ, ν, . . . for curved vectors, and Greek letters α, β, . . . for two-component spinors. We also use the Euclidian notation (the Pauli metric). The metric and the totally anti-symmetric tensor are given by 1, 1, 1), The gamma matrices satisfy and γ 5 and σ ab are defined as The relation between four-component and two-component spinors is where ǫ αβ is the anti-symmetric tensor with ǫ 12 = ǫ 12 = 1. The raising and lowering rules of spinor index are defined by The dual of anti-symmetric tensor F ab , and its self-dual and anti-self-dual parts are defined as Using the relationσ ab = −γ 5 σ ab , we find

A.2 Notation in superspace approach
We where ǫ αβ and ǫαβ are the second-order anti-symmetric tensors with ǫ 12 = ǫ 21 = 1. The hermitian conjugate of spinor is given by (ψ α ) † =ψα, and the hermitian conjugate rule for spinor product is The four-dimensional Pauli matrices σ a are defined as and their hermitian conjugates are (σ a )α β = εα˙γε βδ (σ a ) δγ = (σ a ) βα . (A.14) With these matrices, any flat Lorentz vector V a can be expressed as a mixed spinor V αβ and vice versa: The matrices σ ab andσ ab are defined as and satisfy the relations In two-component spinor notation, any anti-symmetric tensor F ab can be decomposed into chiral and anti-chiral parts: The dual of anti-symmetric tensor F ab is defined as The self-dual and anti-self-dual parts of F ab are which coincide with the chiral and anti-chiral parts, respectively:

A.3 Correspondence of notations
We summarize the correspondence of notations between component and superspace approaches:

B Q transformation of conformal multiplet
The supersymmetry Q transformation laws take the following form for the fields in a general conformal multiplet [C Γ , Z Γ , H Γ , K Γ , B aΓ , Λ Γ , D Γ ]:

Conformal multiplets with arbitrary Lorentz indices
In this subsection we explicitly derive the correspondences of conformal multiplets with arbitrary Lorentz index, that is, between V Γ in component approach (Eq. (2.18)) and Φ Γ in superspace approach (Eq. (2.53)). In the first place, the correspondence of the lowest components is obtained by the property of superconformal transformations. There is an ambiguity for overall constant factor, which is fixed by We then obtain the correspondences of higher components by operating the Q transformations in order. The action of Q transformation is given by the covariant spinor derivative since the fields have only Lorentz indices. As given in Table (3.2), the correspondence of Q transformation parameters isε ↔ 2 ξ(P ) αξ (P )α |. In the following, we simply denote the parameters in superspace as ξ αξα .
The correspondence of the second components is obtained by the Q transformations of the first components, namely, By comparing with δ Q (ε)C Γ in (B.1) and using the correspondence of γ 5 in Table (A.23), we find The correspondences of the other components are obtained in similar ways. The Q transformations of the second components are .

C.2 Chiral projection
In this subsection, we show the correspondence of the chiral projection between two approaches. In component approach, the chiral projection operator Π acts on a conformal multiplet V Γ with special weights and index, and gives a chiral multiplet ΠV Γ whose component expression is explicitly given in (2.23). In superspace approach, the chiral projection operator P is defined by the superconformal covariant derivative as P = −1 4∇ 2 and gives a chiral superfield PΦ Γ from a primary superfield Φ Γ with special weights and index. In particular, Γ should be made of purely undotted spinor indices. When one matches V Γ with Φ Γ , the correspondence of the chiral projection is In what follows, we show this correspondence explicitly by component level, namely, each component of the chiral superfield 1 4∇ 2 Φ Γ coincide with (2.23) in component approach.
For a general chiral superfield, its components which should match to those of the corresponding chiral multiplet are given in (3.31). First, the lowest component of 1 We find from (C.5) that the RHS just corresponds to 1 2 (H Γ − iK Γ ) in component approach, which is the lowest component of the chiral multiplet ΠV Γ as shown in (2.23).
The second component of chiral superfield is given by its covariant derivative, and for 1 where the identity (C.7) has been used. By comparing with the component correspondences (C.3) and (C.8), we find the RHS reads iP R (γ a D a Z Γ + Λ Γ ) in component approach, which is exactly the second component of ΠV Γ given in (2.23).