Direct relation of linearized supergravity to superconformal formulation
aa r X i v : . [ h e p - t h ] N ov November 28, 2011 KEK-TH-1475
Direct relation of linearized supergravityto superconformal formulation
Yutaka Sakamura ∗ KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,Tsukuba, Ibaraki 305-0801, JapanDepartment of Particles and Nuclear Physics,The Graduate University for Advanced Studies (Sokendai),Tsukuba, Ibaraki 305-0801, Japan
Abstract
We modify the four-dimensional N = 1 linearized supergravity in a way thatcomponents in each superfield are completely identified with fields in the full super-conformal formulation. This identification makes it possible to use both formulationsof supergravity in a complementary manner. It also provides a basis for an extensionto higher-dimensional supergravities. ∗ e-mail address: [email protected] Introduction
The superconformal formulation of supergravity (SUGRA) is a powerful and systematicmethod for constructing various SUGRA actions [1, 2, 3, 4]. Most of the known off-shellSUGRA actions are reproduced by this formulation. It has also been extended to the five-dimensional (5D) case [5, 6], which is useful to discuss the brane-world scenario based ongeneral 5D SUGRA. Although the actions are obtained in a systematic way, their explicitexpressions are lengthy and awkward due to a number of auxiliary fields. Especially thecouplings between the matter and the SUGRA fields ( i.e. , the vierbein, the gravitino, etc.)are complicated.Linearized supergravity [7, 8] is easier to deal with because it is described in terms ofsuperfields on the ordinary superspace. It is powerful for some calculations because theordinary superfield techniques are applicable just as in the global supersymmetry (SUSY)case. An extension to 5D case for the minimal set-up was done in Ref. [9], and it makesit possible to calculate the SUGRA loop contributions in the 5D brane-world models [10],keeping the N = 1 SUSY off-shell structure. On the other hand, we cannot use thisformalism for calculations beyond the linearized order in the SUGRA fields. The fullSUGRA formulation, such as the superconformal formulation, is necessary for them.Therefore it will be useful to combine the two formulations in a complementary man-ner. In fact, it is pointed out in Ref. [7, 8] that the linearized SUGRA transformationscontain some of the superconformal transformations at the linearized level. Although bothformulations are self-consistent, an explicit relation between them has not been providedso far. This is the main obstacle to the complementary use of them.In this paper, we will modify the linearized SUGRA formulation and provide a com-plete identification of component fields in each superfield with fields in the superconformalformulation developed in Ref. [4]. This identification also provides a basis for an extensionto higher-dimensional SUGRA.The paper is organized as follows. In Sec. 2, we consider superfield transformationswhich are identified with the linearized superconformal transformations. In Sec. 3, wetranslate such transformation laws into those for component fields, and identify the fieldsand the transformation parameters with those in the superconformal formulation of Ref. [4].In Sec. 4, we construct the invariant action formulae in terms of the superfields, which areconsistent with those in Ref. [4]. Sec. 5 is devoted to the summary. In Appendix A, weprovide explicit component expressions of some superfields in the text, and in Appendix B,we collect the invariant action formulae in Ref. [4] in our notations.2 Superfield transformations
In this section, we consider the transformation laws of N = 1 superfields. We assumethat the background geometry is a flat 4D Minkowski spacetime. Basically we use thespinor notations of Ref. [11], except for the metric and the spinor derivatives. We take thebackground metric as η µν = (1 , − , − , −
1) so as to match it to that of Ref. [4], and wedefine the spinor derivatives D α and ¯ D ˙ α as D α ≡ ∂∂θ α − i (cid:0) σ µ ¯ θ (cid:1) α ∂ µ , ¯ D ˙ α ≡ − ∂∂ ¯ θ ˙ α + i ( θσ µ ) ˙ α ∂ µ , (2.1)which satisfy (cid:8) D α , ¯ D ˙ α (cid:9) = 2 iσ µα ˙ α ∂ µ . We begin with a brief review of the formulation developed in Ref. [8]. (A compact reviewis also provided in Ref. [9].) First we consider the super-diffeomorphism transformationacting on a chiral superfield Φ. It is expressed as δ Φ = Λ α D α Φ + Λ µ ∂ µ Φ , (2.2)where Λ α and Λ µ are a spinor and a vector superfields, respectively. We require that thistransformation δ preserves the chiral condition,¯ D ˙ α Φ = 0 . (2.3)Then we obtain ¯ D ˙ α δ Φ = ¯ D ˙ α Λ α D α Φ + (cid:0) − i Λ α σ µα ˙ α + ¯ D ˙ α Λ µ (cid:1) ∂ µ Φ = 0 . (2.4)We have used (2.3). Thus we find that¯ D ˙ α Λ α = 0 , − i Λ α σ µα ˙ α + ¯ D ˙ α Λ µ = 0 . (2.5)The most general solution to these conditions can be parametrized byΛ α = −
14 ¯ D L α , Λ µ = − iσ µα ˙ α ¯ D ˙ α L α − Ω µ , (2.6) In the notation of Ref. [11], the spinor derivatives satisfy (cid:8) D α , ¯ D ˙ α (cid:9) = − iσ µα ˙ α ∂ µ . Then, the (global)SUSY generators Q α and ¯ Q ˙ α , which anticommute with D α and ¯ D ˙ α , satisfy (cid:8) Q α , ¯ Q ˙ α (cid:9) = 2 iσ µα ˙ α ∂ µ . Thisleads to the SUSY algebra with an opposite sign to the usual one. (See Chapter IV of Ref. [11].) L α is a general complex spinor superfield and Ω µ is a chiral superfield. In terms of L α and Ω µ , the transformation of a chiral superfield Φ is rewritten as δ Φ = −
14 ¯ D L α D α Φ − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ Φ= −
14 ¯ D ( L α D α Φ) − Ω µ ∂ µ Φ . (2.7)Similarly, we can find the transformation acting on an anti-chiral superfield ¯Φ as δ ¯Φ = − D ¯ L ˙ α ¯ D ˙ α ¯Φ − (cid:0) iσ µα ˙ α D α ¯ L ˙ α + ¯Ω µ (cid:1) ∂ µ ¯Φ= − D (cid:0) ¯ L ˙ α ¯ D ˙ α ¯Φ (cid:1) − ¯Ω µ ∂ µ ¯Φ . (2.8)This preserves the anti-chiral condition D α ¯Φ = 0.Next we consider the transformation of a product of a chiral and an anti-chiral su-perfields Φ and ¯Φ . In order to define the transformation acting on the product that isconsistent with (2.7) and (2.8), we introduce a real superfield U µ that transforms inhomo-geneously as δU µ = 12 σ µα ˙ α (cid:0) ¯ D ˙ α L α − D α ¯ L ˙ α (cid:1) − i (cid:0) Ω µ − ¯Ω µ (cid:1) , (2.9)and insert it into the product as¯Φ Φ → V ( ¯Φ Φ ) ≡ ¯Φ (cid:16) iU µ ↔ ∂ µ (cid:17) Φ , (2.10)where A ↔ ∂ µ B ≡ A∂ µ B − ( ∂ µ A ) B . As we will see in the next section, U µ contains theSUGRA fields. The inserted product V ( ¯Φ Φ ) transforms as δ V ( ¯Φ Φ ) = δ ¯Φ Φ + i ¯Φ δU µ ↔ ∂ µ Φ + ¯Φ δ Φ = (cid:26) −
14 ¯ D L α D α − D ¯ L ˙ α ¯ D ˙ α − i σ µα ˙ α (cid:0) ¯ D ˙ α L α + D α ¯ L ˙ α (cid:1) ∂ µ − (cid:0) Ω µ + ¯Ω µ (cid:1) ∂ µ (cid:27) (cid:0) ¯Φ Φ (cid:1) = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ + h . c . (cid:27) V ( ¯Φ Φ ) . (2.11)Here and henceforth, we neglect the U µ -dependent terms in the right-hand sides of thetransformation laws because they are irrelevant to an invariance of the action at the lin-earized order in U µ . A general superfield has the same transformation law as (2.11). Notethat this law preserves the reality condition. We keep both L α and Ω µ as the transformation parameters while the latter is set to zero in Ref. [9].In our formulation, the degrees of freedom in Ω µ will be eliminated by the constraints (3.5) or absorbedinto ˜ ξ I in (3.8). M Q translation local Lorentz SUSY D U (1) A S K dilatation R-symmetry conformal SUSY conformal boostTable I: 4D N = 1 superconformal transformations It is mentioned in Ref. [8] that the transformation δ discussed in the previous subsectioncontain some of the superconformal transformations in addition to the super-Poincar´eones. (The 4D N = 1 superconformal transformations are listed in Table I.) However itis unclear how those transformations are related to those of Ref. [4]. For example, the δ -transformation law of the conformal compensator superfield is essentially different fromthat of a matter chiral superfield [9]. On the other hand, they transform in the same way inthe superconformal formulation [4], except for the Weyl weight w and the chiral weight n ,which are the charges of D and U (1) A .In order to incorporate these weights explicitly into the superfield transformations, wemodify the transformation acting on a general superfield Ψ as δ sc Ψ ≡ δ Ψ + ( w + n ) ΛΨ + ( w − n ) ¯ΛΨ , (2.12)where w and n are the Weyl and the chiral weights of Ψ, and Λ is a chiral superfield to bedetermined later. (See eq.(3.13).) This modified transformation δ sc preserves the (anti-)chiral condition or the reality condition, and satisfy the Leibniz rule since both weights areadditive quantum numbers. Because w = n and w = − n for a chiral and an anti-chiralsuperfields Φ and ¯Φ, (2.7) and (2.8) are modified as δ sc Φ = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ + 2 w Λ (cid:27) Φ ,δ sc ¯Φ = (cid:26) − D ¯ L ˙ α ¯ D ˙ α − (cid:0) iσ µα ˙ α D α ¯ L ˙ α + ¯Ω µ (cid:1) ∂ µ + 2 w ¯Λ (cid:27) ¯Φ . (2.13)A real general superfield V ( i.e. , n = 0) transforms as δ sc V = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ + w Λ + h . c . (cid:27) V. (2.14) The Weyl and the chiral weights of a superfield denote those of the lowest component in the superfield. U µ does not change from (2.9), δ sc U µ = 12 σ µα ˙ α (cid:0) ¯ D ˙ α L α − D α ¯ L ˙ α (cid:1) − i (cid:0) Ω µ − ¯Ω µ (cid:1) . (2.15)The manner of inserting the connection superfield U µ in (2.10) is generalized to aproduct of arbitrary superfields Ψ , Ψ , · · · , Ψ n asΨ Ψ · · · Ψ n → V (Ψ Ψ · · · Ψ n ) ≡ (cid:16) iU µ ˆ ∂ µ (cid:17) (Ψ Ψ · · · Ψ n ) , (2.16)where ˆ ∂ µ ≡ ∂ µ (on chiral superfields)0 (on general superfields) − ∂ µ (on anti-chiral superfields) (2.17)Then the transformation δ sc acts on this product as δ sc V (Ψ Ψ · · · Ψ n ) = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ + h . c . + ( w + n ) Λ + ( w − n ) ¯Λ (cid:9) V (Ψ Ψ · · · Ψ n ) , (2.18)where w and n are the Weyl and the chiral weights for the product Ψ Ψ · · · Ψ n . Noticethat V defined in (2.16) is regarded as an embedding into a general superfield. Now we will see the transformation laws for component fields. First we consider theconnection superfields U µ , whose components are defined as U µ = u µ + θχ Uµ + ¯ θ ¯ χ Uµ + θ a µ + ¯ θ ¯ a µ + (cid:0) θσ ν ¯ θ (cid:1) ˜ e νµ + ¯ θ (cid:16) θ ˜ ψ µ (cid:17) + θ (cid:16) ¯ θ ¯˜ ψ µ (cid:17) + θ ¯ θ d µ , (3.1)where u µ , ˜ e νµ and d µ are real. Note that ˜ e νµ is neither symmetric nor anti-symmetric forthe indices. The transformation parameter superfields are expanded as L α = l α + θ α v + ( σ µν θ ) α w µν − (cid:0) σ µ ¯ θ (cid:1) α ξ µ + θ ζ α + ¯ θ ǫ α + θ α (cid:0) η µ σ µ ¯ θ (cid:1) + 12 ¯ θ θ α ϕ −
12 ¯ θ ( σ µν θ ) α λ µν + θ (cid:0) σ µ ¯ θ (cid:1) α κ µ − θ ¯ θ ρ α , Ω µ = ω µ + θζ µ Ω + θ F µ Ω − i (cid:0) θσ ν ¯ θ (cid:1) ∂ ν ω µ − i θ (cid:0) ¯ θ ¯ σ ν ∂ ν ζ µ Ω (cid:1) − θ ¯ θ (cid:3) ω µ , (3.2)6here (cid:3) ≡ η µν ∂ µ ∂ ν , and w µν and λ µν are real anti-symmetric, while the others are com-plex. The transformation laws for the component fields of U µ are read off from (2.15)as δ sc u µ = ξ µ R + ω µ I ,δ sc χ Uµα = − (cid:18) σ µ ¯ ǫ − σ µ ¯ σ ν η ν − i σ ν ¯ σ µ ∂ ν l + i ζ µ Ω (cid:19) α ,δ sc a µ = − κ µ + i ∂ µ v − i ∂ ν w νµ − ǫ µνρτ ∂ ν w ρτ − i F µ Ω ,δ sc ˜ e µν = − δ µν ϕ R + λ µν + 12 (cid:0) ∂ µ ξ I ν + ∂ ν ξ µ I − δ µν ∂ ρ ξ ρ I + ǫ µνρτ ∂ ρ ξ τ R (cid:1) − ∂ ν ω µ R ,δ sc ˜ ψ µα = (cid:18) σ µ ¯ ρ + i σ ν ¯ σ µ ∂ ν ǫ − i σ ν ∂ µ ¯ η ν + 14 σ ν ∂ ν ¯ ζ µ Ω (cid:19) α ,δ sc d µ = − ∂ µ ϕ I + 14 ǫ µνρτ ∂ ν λ ρτ − (cid:3) ω µ I , (3.3)where the subscript R and I denote the real and imaginary parts, respectively. By usingthe freedom of Ω µ , we can set u µ = χ Uµα = a µ = 0 . (3.4)This is analogous to the Wess-Zumino gauge for a gauge superfield. This gauge is preservedif the transformation parameters satisfy the following relations. ω µ I = − ξ µ R ,ζ µ Ω α = (2 iσ µ ¯ ǫ − iσ µ ¯ σ ν η ν + σ ν ¯ σ µ ∂ ν l ) α ,F µ Ω = 2 iκ µ + ∂ µ v − ∂ ν w νµ + i ǫ µνρτ ∂ ν w ρτ . (3.5)We further impose an additional condition, ξ µ R = 0 . (3.6)Then the transformation laws for the residual symmetries reduce to δ sc ˜ e µν = − δ µν ˜ ϕ R + ˜ λ µν + ∂ ν ˜ ξ µ I ,δ sc ˜ ψ µα = (2 σ µ ¯˜ ρ + iσ ν ¯ σ µ ∂ ν ǫ ) α ,δ sc d µ = − ∂ µ ϕ I + 14 ǫ µνρτ ∂ ν ˜ λ ρτ , (3.7)where ˜ ϕ R ≡ ϕ R + 12 ∂ µ ξ µ I , ˜ λ µν ≡ λ µν + 12 ( ∂ µ ξ I ν − ∂ ν ξ I µ ) , ˜ ξ µ I ≡ ξ µ I − ω µ R , ˜ ρ α ≡ ρ α + i ∂ ν η µ σ µ ¯ σ ν ) α + 18 (cid:3) l α . (3.8)7n fact, we can identify ˜ ξ I µ , ǫ α , ˜ λ µν , ˜ ϕ R , − ϕ I and ˜ ρ α with the transformation parametersfor P , Q , M , D , U (1) A and S , respectively. They are also expressed as˜ ξ µ I = − Re (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1)(cid:12)(cid:12) , ǫ α = −
14 ¯ D L α (cid:12)(cid:12)(cid:12)(cid:12) , ˜ λ µν = −
12 Re n ( σ µν ) αβ D α ¯ D L β o(cid:12)(cid:12)(cid:12)(cid:12) , ˜ ϕ R = Re (cid:18) D α ¯ D L α (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , − ϕ I = Im (cid:18) − D α ¯ D L α (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , ˜ ρ α = − D ¯ D L α (cid:12)(cid:12)(cid:12)(cid:12) , (3.9)where the symbol | denotes the lowest component of a superfield.With the above identification of the parameters, the transformations in (3.7) agree withthose for the Weyl multiplet in Ref. [6], which corresponds to the SUGRA multiplet, if wespecify the components of U µ as˜ e µν = e µν − δ µν , ˜ ψ µα = i ( σ ν ¯ σ µ ψ ν ) α ,d µ = 34 A µ − ǫ µνρτ ∂ ν ˜ e ρτ , (3.10)where e µν , ψ µ and A µ are the vierbein, the gravitino and the U (1) A gauge field. Namely,˜ e µν is the fluctuation of the vierbein since h e µν i = δ µν , and the (linearized) transformationlaws of ψ µ and A µ are given by [6] δ sc ψ µα = ∂ µ ǫ α + i ( σ µ ¯˜ ρ ) α , δ sc A µ = − ∂ µ ϕ I . (3.11)In the subsequent subsections, we compare the transformation laws for componentfields in each superfield with those in the superconformal formulation [4], and identify thecomponent fields. The transformation laws in Ref. [4] are compactly summarized in Sec. 3of Ref. [6]. Hence we basically use the notations of Ref. [6] as the component fields in eachmultiplet. Now we consider the transformation laws of a chiral superfield Φ. In this subsection, wework in the chiral coordinate y µ ≡ x µ − iθσ µ ¯ θ . (Recall our definition of ¯ D ˙ α in (2.1).) Then,it is expanded as Φ = φ + θχ + θ F. (3.12) The U (1) A gauge field A µ is an auxiliary field [4]. L α and Ω µ . A choice of Λ = − ¯ D D α L α reproduces the correct U (1) A transformation,but there are extra terms for other superconformal transformations. Fortunately such extraterms are summarized in the form of a chiral superfield. Thus there exists a choice of Λthat realizes the correct superconformal transformations for a chiral multiplet. It is givenby Λ = − (cid:0) ¯ D D α L α + 4Ξ (cid:1) , (3.13)whereΞ ≡ ( − ϕ R + ∂ µ ξ µ I ) + 8 θ (cid:18) ˜ ρ − i σ ν ¯ σ µ ∂ ν η µ + 18 (cid:3) l (cid:19) + 2 iθ ∂ µ (cid:18) κ µ − i ∂ µ v (cid:19) . (3.14)Then, the transformation laws of the component fields are read off as δ sc φ = ˜ ξ µ I ∂ µ φ + ǫχ + w ˜ ϕ R φ − iw ϕ I φ,δ sc χ α = ˜ ξ µ I ∂ µ χ α + 12 ˜ λ µν ( σ µν χ ) α + 2 ǫ α F − i ( σ µ ¯ ǫ ) α ∂ µ φ + (cid:18) w + 12 (cid:19) ˜ ϕ R χ α − i (cid:18) w − (cid:19) ϕ I χ α − w ˜ ρ α φ,δ sc F = ˜ ξ µ I ∂ µ F − i ¯ ǫ ¯ σ µ ∂ µ χ + ( w + 1) ˜ ϕ R F − i w − ϕ I F + 2 ( w −
1) ˜ ρχ. (3.15)These transformations agree with those in Ref. [6].Let us comment on a chiral superfield Φ in the full superspace integral R d θ . Unlikethe global SUSY case, moving the bases from the chiral coordinate y µ to the original one x µ is not enough. In fact, Φ must appear in the form of V (Φ) = (1 + iU µ ∂ µ ) Φ . (3.16)This is regarded as the embedding of the chiral multiplet into a general multiplet as men-tioned below (2.18). The embedded superfield has the following components. V (Φ) = φ + θχ + θ F − i (cid:0) θσ µ ¯ θ (cid:1) (cid:0) e − (cid:1) νµ ∂ ν φ − i θ n ¯ θ ¯ σ µ (cid:0) e − (cid:1) νµ ∂ ν χ − (cid:16) ¯ θ ¯˜ ψ µ (cid:17) ∂ µ φ o + i ¯ θ (cid:16) θ ˜ ψ µ (cid:17) ∂ µ φ − θ ¯ θ n g µν ∂ µ ∂ ν φ + 2 i ˜ ψ µ ∂ µ χ − id µ ∂ µ φ o , (3.17)where ( e − ) νµ ≡ δ νµ − ˜ e νµ and g µν ≡ η µν − ˜ e µν − ˜ e νµ are the inverse matrices of the vierbeinand the metric, respectively. 9 .3 Real general multiplet Next we consider a real general superfield V . From (2.14) with (3.13), its transformationlaw is given by δ sc V = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ − w (cid:0) ¯ D D α L α + 4Ξ (cid:1) + h . c . (cid:27) V. (3.18)Each component of V is defined as V = C ′ + iθζ ′ − i ¯ θ ¯ ζ ′ − θ H ′ − ¯ θ ¯ H ′ − (cid:0) θσ µ ¯ θ (cid:1) B ′ µ + iθ (cid:0) ¯ θ ¯ λ ′ (cid:1) − i ¯ θ ( θλ ′ ) + 12 θ ¯ θ D ′ , (3.19)where C ′ , B ′ µ and D ′ are real. By expanding (3.18) in terms of the components, we obtaintheir transformation laws. They do not agree with the transformation laws in Ref. [6] asis. To reproduce the latter laws, we need to redefine the components as C ≡ C ′ , ζ α ≡ ζ ′ α , H ≡ H ′ ,B µ ≡ B ′ µ + ζ ′ ψ µ + ¯ ζ ′ ¯ ψ µ + w C ′ A µ ,λ α ≡ λ ′ α + i n σ µ (cid:0) e − (cid:1) νµ ∂ ν ¯ ζ ′ o α + ( σ µ ¯ σ ν ψ µ ) α B ′ ν + w (cid:0) σ µ ¯ ζ ′ (cid:1) α A µ ,D ≡ D ′ + 12 g µν ∂ µ ∂ ν C ′ + (cid:18) ¯ λ ′ ¯ σ µ ψ µ − i ∂ ν ζ ′ σ µ ¯ σ ν ψ µ − i∂ µ ζ ′ ψ µ − iw ζ ′ σ µν ∂ ν ψ µ + h . c . (cid:19) − (cid:16) d µ − w A µ (cid:17) B ′ µ . (3.20)The explicit form of V in terms of these redefined components is shown in (A.3). Then weobtain δ sc C = ˜ ξ µ I ∂ µ C + iǫζ − i ¯ ǫ ¯ ζ + w ˜ ϕ R C,δ sc ζ α = ˜ ξ µ I ∂ µ ζ α + 12 ˜ λ µν ( σ µν ζ ) α + 2 iǫ α H + ( σ µ ¯ ǫ ) α ( iB µ − ∂ µ C )+ (cid:18) w + 12 (cid:19) ˜ ϕ R ζ α + i ϕ I ζ α + 2 iw ˜ ρ α C,δ sc H = ˜ ξ µ I ∂ µ H − i ¯ ǫ ¯ λ − ¯ ǫ ¯ σ µ ∂ µ ζ + ( w + 1) ˜ ϕ R H + iϕ I H − i ( w −
2) ˜ ρζ ,δ sc B µ = ˜ ξ ν I ∂ ν B µ + ˜ λ µν B ν − iǫσ µ ¯ λ − i ¯ ǫ ¯ σ µ λ − ǫ∂ µ ζ − ¯ ǫ∂ µ ¯ ζ + ( w + 1) ˜ ϕ R B µ − i ( w + 1) (cid:0) ˜ ρσ µ ¯ ζ + ¯˜ ρ ¯ σ µ ζ (cid:1) ,δ sc λ α = ˜ ξ µ I ∂ µ λ α + 12 ˜ λ µν ( σ µν λ ) α + iǫ α D − ( σ µν ǫ ) α ( ∂ µ B ν − ∂ ν B µ ) + (cid:18) w + 32 (cid:19) ˜ ϕ R λ α + iw ∂ µ ˜ ϕ R (cid:0) σ µ ¯ ζ (cid:1) α − i ϕ I λ α − iw ( σ µ ¯˜ ρ ) α B µ + 2 iw ˜ ρ α ¯ H + w ( σ µ ¯˜ ρ ) α ∂ µ C,δ sc D = ˜ ξ µ I ∂ µ D + 2 ∂ µ ˜ λ µν ∂ ν C + ( w + 2) ˜ ϕ R D + w∂ µ ˜ ϕ R ∂ µ C − (cid:8) ¯ ǫ ¯ σ µ ∂ µ λ + 2 iw ˜ ρλ + w ˜ ρσ µ ∂ µ ¯ ζ + h . c . (cid:9) . (3.21)10hese transformation laws agree with those in Ref. [6] at the linearized level, except forthe following two points. • The second term in δ sc D is absent in Ref. [6]. However, this term is harmless whenwe consider the invariance of the action because D is the highest component and thisterm is a total derivative. • The terms proportional to ∂ µ ˜ ϕ R in δ sc λ α and δ sc D also seem to be extra terms thatare absent in Ref. [6], at first glance. Actually, they indicate that the D -gauge field b µ is set to zero in our linearized SUGRA. Its superconformal transformation (at thelinearized level) is δ sc b µ = ∂ µ ˜ ϕ R − ξ Kµ , (3.22)where ξ Kµ is the transformation parameter for K . Thus, keeping the condition b µ = 0requires ξ Kµ = ∂ µ ˜ ϕ R . In fact, after the replacement: ∂ µ ˜ ϕ R → ξ Kµ , (3.21) repro-duces the correct transformations. Here we consider a gauge multiplet, which corresponds to a real general multiplet with w = 0. From (3.20), such a real general superfield V is expressed as V = C + iθζ − i ¯ θ ¯ ζ − θ H − ¯ θ ¯ H − (cid:0) θσ µ ¯ θ (cid:1) (cid:0) e − (cid:1) νµ ˆ B ν + iθ ¯ θ (cid:26) ¯ λ − i σ µ (cid:0) e − (cid:1) νµ ∂ ν ζ − i ¯˜ ψ µ ˆ B µ (cid:27) − i ¯ θ θ (cid:26) λ − i σ µ (cid:0) e − (cid:1) νµ ∂ ν ¯ ζ + i ˜ ψ µ ˆ B µ (cid:27) + 12 θ ¯ θ (cid:26) D − g µν ∂ µ ∂ ν C + (cid:18) − i λ ¯ σ µ ˜ ψ µ + ∂ µ ζ ˜ ψ µ + h . c . (cid:19) + 2 d µ ˆ B µ (cid:27) , (3.23)where ˆ B µ ≡ (cid:0) δ νµ + ˜ e νµ (cid:1) B ν − ζ ψ µ − ¯ ζ ¯ ψ µ , (3.24)is interpreted as a gauge field. This definition of the gauge field is consistent with that ofRef. [4].The gauge transformation can be defined just in a similar way to the global SUSY caseas V → V + V (Σ) + V ( ¯Σ) , (3.25) The complex scalar H should be understood as ( H + iK ) in the notation of Ref. [6]. We consider an abelian gauge multiplet for simplicity. An extension to the nonabelian case is straight-forward. φ Σ + θχ Σ + θ F Σ (in the coordinate y µ = x µ − iθσ µ ¯ θ ) is a chiral superfield.Note that Σ must be embedded into a general multiplet by V in order to be added to V .We can move to the Wess-Zumino gauge by choosing Σ asRe φ Σ = − C, χ Σ α = − iζ α , F Σ = H . (3.26)In this gauge, V is written as V WZ = − (cid:0) θσ µ ¯ θ (cid:1) (cid:0) e − (cid:1) νµ ˆ B ′ ν + iθ ¯ θ (cid:16) ¯ λ − i ¯˜ ψ µ ˆ B ′ µ (cid:17) − i ¯ θ θ (cid:16) λ + i ˜ ψ µ ˆ B ′ µ (cid:17) + 12 θ ¯ θ (cid:26) D + (cid:18) − i λ ¯ σ µ ˜ ψ µ + h . c . (cid:19) + 2 d µ ˆ B ′ µ (cid:27) , (3.27)where ˆ B ′ µ ≡ ˆ B µ − ∂ µ Im φ Σ is the gauge-transformed gauge field. We can move to thisgauge only when w = 0. The set of the components [ ˆ B ′ µ , λ α , D ] form a gauge multiplet inRef. [4, 6].Next we construct a field strength superfield W α that is gauge-invariant from the gaugesuperfield V . A naive definition of W α , W naive α ≡ −
14 ¯ D D α V, (3.28)is not invariant under (3.25). If we define X ≡ (cid:18) U µ ¯ σ ˙ ββµ (cid:2) D β , ¯ D ˙ β (cid:3)(cid:19) V, (3.29)its gauge transformation becomes simpler, X → X + Σ + ¯Σ . (3.30)Hence, − ¯ D D α X becomes gauge-invariant. However, this is not the only way to constructa gauge-invariant quantity. We can define the following quantity by adding the second termthat is also gauge-invariant.ˆ W α ≡ −
14 ¯ D D α X + c ¯ D (cid:16) U µ σ ˙ ββµ D α D β ¯ D ˙ β V (cid:17) , (3.31)where c is a constant to be determined later. The second term in (3.31) does not contributeto the lowest component, and we find thatˆ W α = − i (cid:18) λ + 12 ˜ e νµ σ µ ¯ σ ν λ (cid:19) α + O ( θ ) . (3.32)12his indicates that we have to multiply ˆ W α by a superfield Z βα = δ βα − ˜ e νµ ( σ µ ¯ σ ν ) βα + O ( θ )in order to obtain the desired field strength superfield whose lowest component is − iλ α . Thehigher components of Z βα and the constant c in (3.31) are determined so that W α ≡ Z βα ˆ W β has the correct components. The result is W α = − Z βα ¯ D (cid:26) D β (cid:18) V + 14 U µ ¯ σ ˙ γγµ (cid:2) D γ , ¯ D ˙ γ (cid:3) V (cid:19) − U µ ¯ σ ˙ γγµ D β D γ ¯ D ˙ γ V (cid:27) = − Z βα ¯ D (cid:26) D β V + 14 D β U µ ¯ σ ˙ γγµ (cid:2) D γ , ¯ D ˙ γ (cid:3) V − iU µ ∂ µ D β V (cid:27) ,Z βα ≡ δ βα −
12 ˜ e νµ ( σ µ ¯ σ ν ) βα − (cid:16) σ µ ¯˜ ψ µ (cid:17) α θ β , (3.33)where Z βα is expressed in the chiral coordinate y µ . Each component of W α is calculatedas W α = − iλ α + θ α D + i ( σ µν θ ) α (cid:0) e − (cid:1) ρµ (cid:0) e − (cid:1) τν ˆ F ρτ − θ n σ µ (cid:0) e − (cid:1) νµ D ν ¯ λ o α , (3.34)where ˆ F µν ≡ ∂ µ ˆ B ν − ∂ ν ˆ B µ + i (cid:0) ψ µ σ ν ¯ λ − ψ ν σ µ ¯ λ (cid:1) + i (cid:0) ¯ ψ µ ¯ σ ν λ − ¯ ψ ν ¯ σ µ λ (cid:1) , (cid:0) D µ ¯ λ (cid:1) ˙ α ≡ (cid:26)(cid:18) ∂ µ − ω νρµ σ νρ + 3 i A µ (cid:19) ¯ λ (cid:27) ˙ α + (cid:0) ¯ σ νρ ¯ ψ µ (cid:1) ˙ α ˆ F νρ + i ¯ ψ ˙ αµ D. (3.35)Here ω νρµ is the spin connection, and expressed at the linearized level as ω νρµ = − ∂ µ (˜ e νρ − ˜ e ρν ) + 12 (cid:8) ∂ ν (cid:0) ˜ e ρµ + ˜ e ρµ (cid:1) − ∂ ρ (cid:0) ˜ e νµ + ˜ e νµ (cid:1)(cid:9) . (3.36)The field strength ˆ F µν and the covariant derivative D µ ¯ λ defined in (3.35) agree with thosein Ref. [6]. We have used the identification (3.10). F -term action formula First we consider the F -term invariant action, which consists of only chiral multiplets. Itreduces to the chiral superspace integral in the global SUSY limit, S gl F [ W ] ≡ Z d x Z d θ W + h . c ., (4.1)13here W is a chiral superfield, which is referred to as the superpotential. From (2.13) with(3.13), this transforms as δ sc S gl F [ W ] = Z d x Z d θ (cid:26) −
14 ¯ D ( L α D α W ) − Ω µ ∂ µ W − (cid:0) ¯ D D α L α + 4Ξ (cid:1) W (cid:27) + h . c . = Z d x Z d θ (cid:26) −
14 ¯ D D α ( L α W ) − Ω µ ∂ µ W − Ξ W (cid:27) + h . c . = Z d x (cid:20)Z d θ D α ( L α W ) − Z d θ (Ξ − ∂ µ Ω µ ) W (cid:21) + h . c .. (4.2)We have assumed that w = n = 3 for W . In the last equation, we performed the partialintegrals. In order to make the action invariant, we introduce a chiral superfield ˜ E whosetransformation law is given by δ sc ˜ E = Ξ − ∂ µ Ω µ = (cid:16) − ϕ R + ∂ µ ˜ ξ µ I (cid:17) + 8 θ (cid:18) ˜ ρ − i σ µ ∂ µ ¯ ǫ (cid:19) , (4.3)in the coordinate y µ , and modify the action formula as S F [ W ] ≡ Z d x Z d θ (cid:16) E (cid:17) W + h . c .. (4.4)From (3.7) and (4.3), we identify ˜ E as˜ E = ˜ e µµ − θσ µ ¯˜ ψ µ = ˜ e µµ − iθσ µ ¯ ψ µ . (4.5)Therefore, the factor (1+ ˜ E ) in (4.4) corresponds to the chiral density multiplet in Ref. [11]. The action S F [ W ] is now invariant under δ sc at the linearized level. Note that it isinvariant only when the Weyl weight of W is 3. This is consistent with the F -term actionformula in Ref. [4], which is shown in (B.1) in our notations. We can explicitly see that(4.4) reproduces (B.1). D -term action formula Next we consider the D -term invariant action. It reduces to the full superspace integral inthe global SUSY limit, S gl D [ K ] ≡ Z d x Z d θ K, (4.6) Note that det (cid:0) e νµ (cid:1) = 1 + ˜ e µµ at the linearized level. The factor 2 is necessary to match the normalization of the D -term action formula in Ref. [4]. K is a real general superfield, which is referred to as the K¨ahler potential. From(2.14) with (3.13), this transforms as δ sc S gl D [ K ] = 2 Z d x Z d θ (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ − w (cid:0) ¯ D D α L α + 4Ξ (cid:1) + h . c . o K (4.7)= 2 Z d x Z d θ (cid:26) − w
24 ¯ D D α L α − (cid:0) iσ µα ˙ α ∂ µ ¯ D ˙ α L α − ∂ µ Ω µ (cid:1) − w . c . (cid:27) K, where w is the Weyl weight of K . We have performed the partial integral in the secondequality. Here we define a real scalar superfield ˜ E from the connection superfield U µ as˜ E ≡
14 ¯ σ ˙ ααµ (cid:2) D α , ¯ D ˙ α (cid:3) U µ . (4.8)Then it transforms as δ sc ˜ E = −
12 ¯ D D α L α + 3 i σ µα ˙ α ∂ µ ¯ D ˙ α L α − ∂ µ Ω µ + h . c .. (4.9)By using ˜ E and ˜ E defined in (4.5), we modify the action formula as S D [ K ] ≡ Z d x Z d θ (cid:26) (cid:16) ˜ E + ˜ E + ¯˜ E (cid:17)(cid:27) K, (4.10)so that its transformation becomes δ sc S D [ K ] = 2 Z d x Z d θ (cid:26) − w
24 ¯ D D α L α + 2 − w . c . (cid:27) K. (4.11)Therefore, S D [ K ] is now δ sc -invariant when w = 2, and can be identified with the D -termaction formula in Ref. [4], which is shown in (B.2) in our notations. Since the prefactor of K in (4.10) is expanded as (see (A.5))1 + 13 (cid:16) ˜ E + ˜ E + ¯˜ E (cid:17) = 1 + ˜ e µµ + O ( θ ) , (4.12)this corresponds to the density multiplet in the full superspace [11]. We can explicitly showthat (4.10) reproduces (B.2), except for the kinetic terms for the SUGRA fields, which willbe discussed in Sec. 4.4. Notice that the “chiral density superfield” ˜ E defined in (4.5) is redundant because it cannotbe expressed in terms of U µ and the SUGRA fields are already contained in the latter. In15act, we can eliminate ˜ E from the action formulae (4.4) and (4.10) by the following superfieldredefinition. ˆΦ ≡ (cid:16) w E (cid:17) Φ , ˆ V ≡ n w (cid:16) ˜ E + ¯˜ E (cid:17)o V, (4.13)where Φ and V are a chiral and a real general superfields, respectively. The redefinedsuperfields transform as δ sc ˆΦ = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ − w (cid:0) ¯ D D α L α + 4 ∂ µ Ω µ (cid:1)(cid:27) ˆΦ , (4.14) δ sc ˆ V = (cid:26) −
14 ¯ D L α D α − (cid:0) iσ µα ˙ α ¯ D ˙ α L α + Ω µ (cid:1) ∂ µ − w (cid:0) ¯ D D α L α + 4 ∂ µ Ω µ (cid:1) + h . c . (cid:27) ˆ V .
Now the transformations are expressed only in terms of L α and Ω µ . In terms of theseredefined superfields, the action formulae are expressed as S F [ W ] = Z d x Z d θ ˆ W + h . c .,S D [ K ] = 2 Z d x Z d θ (cid:18) E (cid:19) ˆ K = 2 Z d x Z d θ (cid:18) σ ˙ ααµ (cid:2) D α , ¯ D ˙ α (cid:3) U µ (cid:19) ˆ K. (4.15)As for the gauge kinetic term, the ˜ E -dependence automatically cancels with anotherredundant superfield Z βα in (3.33) because W α = Z βα ˆ W β = ˆ W α −
12 ˜ e νµ (cid:16) σ µ ¯ σ ν ˆ W (cid:17) α − (cid:16) θ ˆ W (cid:17) (cid:16) σ µ ¯˜ ψ µ (cid:17) α , W α W α = n − ˜ e µµ + θσ µ ¯˜ ψ µ o ˆ W α ˆ W α = (cid:16) − ˜ E (cid:17) ˆ W α ˆ W α . (4.16)Thus we obtain S gaugekin [ V ] ≡ S F (cid:20) − W α W α (cid:21) = Z d x Z d θ (cid:16) E (cid:17) (cid:18) − W α W α (cid:19) + h . c . = Z d x Z d θ (cid:18) −
14 ˆ W α ˆ W α (cid:19) + h . c ., (4.17)where ˆ W α = −
14 ¯ D (cid:18) D α ˆ V + 14 D α U µ ¯ σ ˙ ββµ (cid:2) D β , ¯ D ˙ β (cid:3) ˆ V − iU µ ∂ µ D α ˆ V (cid:19) . (4.18)Note that ˆ V = V since the Weyl weight of the gauge superfield is zero.16 .4 Kinetic terms for SUGRA fields Here we discuss the kinetic terms for the SUGRA superfield U µ . In the superconformalformulation, the corresponding terms are contained in the D -term action formula in Ref. [4]as follows. (See eq.(B.2).) S D [Ω] = e (cid:20) D + · · · + C (cid:8) R ( ω ) + 4 ǫ µνρτ (cid:0) ψ µ σ τ ∂ ν ¯ ψ ρ + h . c . (cid:1)(cid:9)(cid:21) , (4.19)where e is the determinant of the vierbein, R ( ω ) is the scalar curvature constructed fromthe spin connection, and Ω = [ C, · · · , D ] is a real general multiplet with the Weyl weight 2.The Einstein-Hilbert term is obtained by imposing the D -gauge fixing condition, C = − . Since the above kinetic terms are quadratic in the SUGRA fields, it seems that we need toextend the D -term action formula in (4.15) as S D [Ω] = 2 Z d x Z d θ (cid:26) (cid:16) ˜ E + ˜ E (cid:17)(cid:27) ˆΩ , (4.20)where ˜ E is quadratic in U µ . The quadratic part ˜ E is specified by requiring the invarianceof the action up to linear order in the SUGRA fields. However, information on higher ordercorrections to (4.9) and (4.14) in the SUGRA fields is necessary for this procedure, whichis beyond the linearized SUGRA. Fortunately, it is possible to extend (4.15) to includethe kinetic terms for U µ without information on the higher order corrections, as we willexplain below.Recall that the SUGRA kinetic terms are proportional to the lowest component of ˆΩ,which will be set to a constant Ω = − after the D -gauge fixing. Thus we expand ˆΩ asˆΩ = Ω + ˜ˆΩ , (4.21)and focus on ˜ E Ω out of ˜ E ˆΩ in (4.20). Namely, we consider S D [Ω] = 2 Z d x Z d θ (cid:26)(cid:18) E (cid:19) (cid:16) Ω + ˜ˆΩ (cid:17) + 13 ˜ E Ω (cid:27) = 2 Z d x Z d θ (cid:26)
13 ˜ E Ω + (cid:18) E (cid:19) ˜ˆΩ (cid:27) , (4.22)as an extension of (4.15). We have used a fact that ˜ E is a total derivative at the secondequality. For example, when ˆΩ is given byˆΩ = − | ˆΦ C | e − ˆ K/ , (4.23) We have taken the unit of the Planck mass, M Pl = 1. K is quadratic in matter fields and ˆΦ C = 1 + · · · is a chiral compensator superfield,the action (4.22) is written as S D [Ω] = Z d x Z d θ (cid:26) − ˜ E + (cid:18) E (cid:19) (cid:16) ˆ K + · · · (cid:17)(cid:27) , (4.24)where the ellipsis denotes higher order terms.We require the action (4.22) to be invariant up to linear order in the SUGRA fields forΩ -dependent terms while up to zeroth order in the SUGRA fields for ˜ˆΩ-dependent terms.Up to this order, Z d x Z d θ (cid:26) (cid:16) δ sc ˜ E (cid:17) ˜ˆΩ + δ sc ˜ˆΩ (cid:27) = 0 , (4.25)as shown in Sec. 4.2. Hence the variation of (4.22) becomes δ sc S D [Ω] = 2 Z d x Z d θ (cid:26) (cid:16) δ sc ˜ E (cid:17) Ω + 13 ˜ E δ sc ˜ˆΩ (cid:27) . (4.26)In order to discuss the invariance of the action up to the order under consideration, weonly need a field-independent part of δ sc ˜ˆΩ. From (4.14), it is read off as δ sc ˜ˆΩ = δ sc ˆΩ = − Ω (cid:0) ¯ D D α L α + 4 ∂ µ Ω µ + h . c . (cid:1) + · · · , (4.27)where the ellipsis denotes field-dependent terms. We have used that the Weyl weight ofΩ is 2. Since the above field-independent part of δ sc ˜ˆΩ is not affected by including higherorder corrections in the SUGRA fields, the variation (4.26) becomes δ sc S D [Ω] = 2 Z d x Z d θ Ω (cid:26) δ sc ˜ E −
112 ˜ E (cid:0) ¯ D D α L α + 4 ∂ µ Ω µ + h . c . (cid:1)(cid:27) . (4.28)After some calculations, we can show that δ sc (cid:26) − U µ D α ¯ D D α U µ + 13 ˜ E − ( ∂ µ U µ ) (cid:27) = 16 ˜ E (cid:0) ¯ D D α L α + 4 ∂ µ Ω µ + h . c . (cid:1) , (4.29)where total derivatives are dropped. Therefore, ˜ E is identified as˜ E = − U µ D α ¯ D D α U µ + 16 ˜ E −
12 ( ∂ µ U µ ) . (4.30)This has a similar form to the counterpart of Ref. [9]. The first term of the second line in(4.22) with (4.30) is the kinetic terms for U µ .Finally we comment on the relation of the superfield action (4.22) to the componentexpression (4.19). In order to reproduce the quadratic part of the SUSY Einstein-Hilbertterms L SGquad in (B.3), we also need to count the SUGRA fields contained in the redefined18uperfields, in addition to the kinetic terms for U µ . By including higher order correctionsin the SUGRA fields, the redefinition of a real general superfield in (4.13) is extended asˆΩ ≡ (1 + Y + Y ) Ω = (1 + Y + Y ) (cid:16) Ω + ˜Ω (cid:17) , (4.31)where Y ≡ (cid:16) ˜ E + ¯˜ E (cid:17) and Y is quadratic in the SUGRA fields. Thus, from (4.22) and(4.31), the Ω -dependent part of the action is expressed as S D [Ω] = 2 Z d x Z d θ (cid:18)
13 ˜ E + 13 ˜ E Y + Y (cid:19) Ω + · · · , (4.32)where the ellipsis denotes terms beyond quadratic order in the SUGRA fields or dependingon the matter fields. This corresponds to L SGquad in (B.3).
We have modified the 4D N = 1 linearized SUGRA, and provided a complete identificationof component fields in each superfield with fields in the superconformal formulation ofSUGRA developed in Ref. [4]. The results of our work makes it possible to use bothformulations in a complementary manner.In our modified linearized SUGRA, (anti-) chiral superfields and real general superfieldsshould be understood as the redefined ones defined in (4.13) whose components are identi-fied with the fields in Ref. [4] through (3.12), (3.19), (3.20) and (4.5). The components ofthe connection superfield U µ are identified with the SUGRA fields in the Weyl-multipletas (3.10). The invariant action formulae are expressed in terms of the redefined superfieldsas S F [ W ] = Z d x Z d θ ˆ W + h . c .,S D [Ω] = 2 Z d x Z d θ (cid:26) Ω E + (cid:18) E (cid:19) ˆΩ (cid:27) ,S gaugekin [ V ] = Z d x Z d θ (cid:18) −
14 ˆ W α ˆ W α (cid:19) + h . c ., (5.1)where ˜ E and ˜ E are defined in (4.8) and (4.30), and the field strength superfield ˆ W α is defined in (4.18). Ω is a constant part of ˆΩ, which is set to − / D -term action formula, a chiral multiplet ˆΦ must19e embedded into a general multiplet. Such embedding is provided at the linearized orderin the SUGRA fields by V ( ˆΦ) ≡ (1 + iU µ ∂ µ ) ˆΦ . (5.2)The gauge transformation of the gauge multiplet ˆ V is given byˆ V → ˆ V + V ( ˆΣ) + V ( ¯ˆΣ) , (5.3)where ˆΣ is a chiral superfield. The field strength superfield ˆ W α is invariant under thistransformation.Our work will also be useful to discuss higher-dimensional SUGRA. When we considerit in the context of the brane-world scenario, it is convenient to express the action in termsof N = 1 superfields, keeping only N = 1 SUSY that remains unbroken at low energiesmanifest. The authors of Ref. [9] construct minimal version of 5D linearized SUGRAalong this direction. Although their formulation is powerful to calculate SUGRA loopcontributions and is self-consistent, it is not clear how the component fields are related tofields in other off-shell formulations of 5D SUGRA. Especially, it is obscure how to extendtheir result to more general 5D SUGRA. The superconformal formulation of 5D SUGRAhas been developed in Refs. [5, 6], and its N = 1 superfield description was provided inRef. [12, 13]. Hence, by combining these results with our current work, it is possible toobtain the linearized SUGRA formulation of general
5D SUGRA. Furthermore, our workis a good starting point to construct an N = 1 description of 6D or higher-dimensionalSUGRA because the linearized SUGRA formulation is based on the ordinary superspaceand thus is easier to handle than the full supergravity. These issues are left for our futureworks. Acknowledgements
This work was supported in part by Grant-in-Aid for Young Scientists (B) No.22740187from Japan Society for the Promotion of Science.
A Component expressions
Here we collect component expressions of some superfields appearing in the text.20he coefficient superfields of the differential operators in (2.13) and (2.14) are writtenas −
14 ¯ D L α = ǫ α + 12 θ α ˜ ϕ −
12 ( θσ µν ) α ˜ λ µν − θ ˜ ρ α − i (cid:0) θσ µ ¯ θ (cid:1) ∂ µ ǫ α − i θ n(cid:0) ¯ θ ¯ σ µ (cid:1) α ∂ µ ˜ ϕ − (cid:0) ¯ θ ¯ σ ρ σ µν (cid:1) α ∂ ρ ˜ λ µν o − θ ¯ θ (cid:3) ǫ α ,iσ µα ˙ α ¯ D ˙ α L α + Ω µ = − ˜ ξ µ I + 2 iθσ µ ¯ ǫ + 2 i ¯ θ ¯ σ µ ǫ − i (cid:0) θσ ν ¯ θ (cid:1) (cid:18) η µν ˜ ϕ + ˜ λ µν − ∂ ν ˜ ξ µ I + i ǫ µνρτ ˜ λ ρτ (cid:19) − iθ (cid:26) ¯ θ ¯ σ µ ˜ ρ + i (cid:0) ¯ θ ¯ σ ν σ µ ∂ ν ¯ ǫ (cid:1)(cid:27) − ¯ θ ( θσ ν ¯ σ µ ∂ ν ǫ ) − θ ¯ θ ∂ ν (cid:18) η µν ˜ ϕ + ˜ λ µν − ∂ ν ˜ ξ µ I + i ǫ µνρτ ˜ λ ρτ (cid:19) , (A.1)where ˜ ϕ ≡ ˜ ϕ R + iϕ I , ˜ λ µν , ˜ ξ I and ˜ ρ α are defined in (3.8). An explicit expression of Λ definedin (3.13) is Λ = − (cid:0) ¯ D D α L α + 4Ξ (cid:1) = 12 (cid:18) ˜ ϕ R − i ϕ I (cid:19) − θ ˜ ρ − i (cid:0) θσ µ ¯ θ (cid:1) ∂ µ (cid:18) ˜ ϕ R − i ϕ I (cid:19) + iθ (cid:0) ¯ θ ¯ σ µ ∂ µ ˜ ρ (cid:1) − θ ¯ θ (cid:3) (cid:18) ˜ ϕ R − i ϕ I (cid:19) . (A.2)We have taken the gauge (3.4).For a real general multiplet [ C, ζ α , H , B µ , λ α , D ], each component is embedded into areal superfield that transforms by a law (2.14) as V = C + iθζ − i ¯ θ ¯ ζ − θ H − ¯ θ ¯ H − (cid:0) θσ µ ¯ θ (cid:1) (cid:16) B µ − ζ ψ µ − ¯ ζ ¯ ψ µ − w CA µ (cid:17) + iθ ¯ θ (cid:26) ¯ λ − i σ µ (cid:0) e − (cid:1) νµ ∂ ν ζ − (cid:0) ¯ σ µ σ ν ¯ ψ µ (cid:1) B ν + w σ µ ζ ) A µ (cid:27) − i ¯ θ θ (cid:26) λ − i σ µ (cid:0) e − (cid:1) νµ ∂ ν ¯ ζ − ( σ µ ¯ σ ν ψ µ ) B ν − w (cid:0) σ µ ¯ ζ (cid:1) A µ (cid:27) + 12 θ ¯ θ (cid:26) D − g µν ∂ µ ∂ ν C − (cid:18) w − A µ + 12 ǫ µνρτ ∂ ν ˜ e ρτ (cid:19) B µ + (cid:18) − ¯ λ ¯ σ µ ψ µ − i∂ µ ζ σ µν ψ ν + i∂ µ ζ ψ µ + 2 iw ζ σ µν ∂ ν ψ µ + h . c . (cid:19)(cid:27) . (A.3)21he real superfield ˜ E defined in (4.8) is expressed as˜ E = 14 ¯ σ ˙ ααµ (cid:2) D α , ¯ D ˙ α (cid:3) U µ = ˜ e µµ + θσ µ ¯˜ ψ µ − ¯ θ ¯ σ µ ˜ ψ µ + (cid:0) θσ µ ¯ θ (cid:1) (2 d µ − ǫ µνρτ ∂ ν ˜ e ρτ )+ i θ (cid:16) θσ µ ¯ σ ν ∂ ν ˜ ψ µ (cid:17) − i θ (cid:16) ¯ θ ¯ σ µ σ ν ∂ ν ¯˜ ψ µ (cid:17) + 14 θ ¯ θ (cid:0) (cid:3) ˜ e µµ − ∂ µ ∂ ν ˜ e µν (cid:1) = ˜ e µµ + 2 i (cid:0) θσ µ ¯ ψ µ + ¯ θ ¯ σ µ ψ µ (cid:1) + 32 (cid:0) θσ µ ¯ θ (cid:1) ( A µ − ǫ µνρτ ∂ ν ˜ e ρτ ) − θ ( θ∂ µ ψ µ ) − θ (cid:0) ¯ θ∂ µ ¯ ψ µ (cid:1) + 14 θ ¯ θ (cid:0) (cid:3) ˜ e µµ − ∂ µ ∂ ν ˜ e µν (cid:1) . (A.4)Thus the density superfield is calculated as1 + 13 (cid:16) ˜ E + ˜ E + ¯˜ E (cid:17) = (cid:0) e µµ (cid:1) + 12 (cid:0) θσ µ ¯ θ (cid:1) ( A µ − ǫ µνρτ ∂ ν ˜ e ρτ ) −
13 ¯ θ { θ (2 η µν + σ µ ¯ σ ν ) ∂ µ ψ ν } − θ (cid:8) ¯ θ (2 η µν + ¯ σ µ σ ν ) ∂ µ ¯ ψ ν (cid:9) − θ ¯ θ (cid:0) (cid:3) ˜ e µµ + 2 ∂ µ ∂ ν ˜ e µν (cid:1) . (A.5) B Invariant action formulae
Here we collect the invariant action formulae in Ref. [4] in our notations. For a chiralmultiplet Φ = [ φ, χ α , F ] with weight ( w, n ) = (3 , F -term action formula is given by S F [Φ] = = Z d x e (cid:0) F − i ¯ ψ µ ¯ σ µ χ + h . c . + · · · (cid:1) = Z d x (cid:8)(cid:0) e µµ (cid:1) F − i ¯ ψ µ ¯ σ µ χ + h . c . + · · · (cid:9) , (B.1)where e ≡ det (cid:0) e νµ (cid:1) , and the ellipsis denotes terms beyond the linear order in the SUGRAfields.For a real general multiplet Ω = [ C, ζ α , H , B µ , λ α , D ] with weights ( w, n ) = (2 , D -term action formula is given by S D [Ω] = Z d x e (cid:20) D − ¯ ψ µ ¯ σ µ λ + ψ µ σ µ ¯ λ + 4 i (cid:0) ζ σ µν ∂ µ ψ ν − ¯ ζ ¯ σ µν ∂ µ ¯ ψ ν (cid:1) + C (cid:8) R ( ω ) + 4 ǫ µνρτ (cid:0) ψ µ σ τ ∂ ν ¯ ψ ρ − ¯ ψ µ ¯ σ τ ∂ ν ψ ρ (cid:1)(cid:9) + · · · (cid:21) = Z d x (cid:20)(cid:0) e µµ (cid:1) D + (cid:18) ψ µ σ µ ¯ λ + 4 i ζ σ µν ∂ µ ψ ν + h . c . (cid:19) + 2 ˜ C (cid:0) ∂ µ ∂ ν ˜ e νµ − (cid:3) ˜ e µµ (cid:1) + L SGquad + · · · , (B.2)22here R ( ω ) is the scalar curvature constructed from the spin connection, ˜ C ≡ C − Ω where Ω is a constant to which C will be set by the D -gauge fixing. Thus ˜ C will vanishafter the D -gauge fixing. The quadratic part in the SUGRA fields L SGquad is given by L SGquad = Ω (cid:8) ˜ e µµ (2 ∂ ν ∂ ρ ˜ e ρν − (cid:3) ˜ e νν ) − ˜ e µν (cid:0) ∂ ν ∂ ρ ˜ e ( µρ ) + ∂ µ ∂ ρ ˜ e ( νρ ) − (cid:3) ˜ e ( µν ) (cid:1) +4 ǫ µνρτ (cid:0) ψ µ σ τ ∂ ν ¯ ψ ρ + h . c . (cid:1)(cid:9) , (B.3)where ˜ e ( µν ) ≡ (˜ e µν + ˜ e νµ ), and the ellipsis denotes terms beyond the linear order in theSUGRA fields. References [1] M. Kaku, P. K. Townsend, P. van Nieuwenhuizen, Phys. Rev.
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