Abelian tensor hierarchy in 4D ${\cal N}=1$ conformal supergravity

We consider Abelian tensor hierarchy in four-dimensional ${\cal N}=1$ supergravity in the conformal superspace formalism, where the so-called covariant approach is used to antisymmetric tensor fields. We introduce $p$-form gauge superfields as superforms in the conformal superspace. We solve the Bianchi identities under the constraints for the superforms. As a result, each of form fields is expressed by a single gauge invariant superfield. The action of superforms is shown with the invariant superfields. We also show the relation between the superspace formalism and the superconformal tensor calculus.


Introduction
The superstring theory is regarded as a candidate for the unified theory of fundamental interactions including gravity. The theory contains strings/branes, which can describe gravity, gauge fields and matter ones in the low energy limit. For stable branes to describe our universe, anti-symmetric tensor fields are coupled to their preserved charges because they are extended objects. Further, unstable tachyons are avoided in the presence of the supersymmetry (SUSY). As a consequence, one may take supergravity (SUGRA), in which there exist such tensor fields on top of gravity, as a plausible low energy effective theory of the superstring theory.
Four-dimensional (4D) effective action is obtained through a compactification of extra dimensions. Hence, 4D N = 1 SUGRA with tensor fields is a possible starting point to construct the effective description of the superstring theory, because N = 1 SUGRA is a chiral theory, which has to contain the Standard Model of particle physics.
For these reasons, our interest is to construct the effective models of the superstring including p-form gauge fields within 4D N = 1 SUGRA [1][2][3][4][5]. For the effective description of the superstring theory, we need to respect the structure of ten-dimensional antisymmetric tensors. In 4D effective theory, such antisymmetric tensors and their gauge transformations are described by the 4D form fields, whose transformations inevitably contain form fields with different ranks. Such a structure is called a tensor hierarchy [6][7][8][9], and is related to the anomaly cancellation conditions in the string theory. Therefore, the construction of a tensor hierarchy in 4D N = 1 SUGRA is desirable in the context of string models.
In this paper, we consider the tensor hierarchy in 4D N = 1 SUGRA. In particular, we focus on the construction which is inspired by structures of the geometries of extra dimensions. Becker et al. did such a construction in 4D N = 1 global SUSY [8]. Our 4D SUGRA description can be applicable to discuss the roles of antisymmetric tensors, e.g. in cosmology [10][11][12][13][14][15] and SUSY breaking [16][17][18].
We will use the conformal superspace formalism [19], which is a superspace formalism of conformal supergravity. It has larger gauge symmetries than the superconformal tensor calculus [20][21][22][23][24][25][26][27][28] and Poincaré superspace [29,30]. The symmetries will be useful to construct the SUGRA system coupled to the tensors and matters. We can straightforwardly reproduce the corresponding system in terms of the Poincaré superspace and also the superconformal tensor calculus due to their correspondences [19,31].
We will adopt the so-called covariant approach [4,29,30,32,33]. In this approach, we regard bosonic tensors as components of differential superforms in superspace. This makes the local SUSY properties of the tensors manifest. In particular, it is straightforward to obtain gauge invariant superfields including bosonic field strengths. This paper is organized as follows. In Sec. 2, we review 4D N = 1 conformal superspace briefly. Then, Abelian tensor hierarchy is introduced to the conformal superspace. Section 3 is devoted to impose constraints on field strength, and to show the solutions to the Bianchi identities. In Sec. 4, we present the component formalism, which is written by superconformal tensor calculus. In Sec. 5, a superconformally invariant action for p-form gauge superfields is proposed. We conclude this paper in Sec. 6. In appendix A, our notations are summarized. We present the explicit derivations of the solutions of Bianchi identities in appendix B. We show the explicit forms of bosonic field strengths in appendix C. Throughout this paper, the terms "form", "gauge field" and "field strength" are used to refer "superform", "gauge superfield" and "field strength superfield", respectively. We use the conventions of Ref. [31] for conformal superspace except the notation of torsion. We use T CB A to refer the torsion, which is equal to R(P ) CB A in Ref. [31]. We also use the convention of Ref. [30] for superforms, exterior derivatives and interior products.

Abelian tensor hierarchy in conformal superspace
In this section, we introduce the Abelian tensor hierarchy into conformal superspace. We begin with a brief review of conformal superspace. Abelian tensor hierarchy is then introduced into conformal superspace.

Conformal superspace
Conformal SUGRA is one of the most convenient formulation of SUGRA thanks to its larger gauge symmetries. Conformal superspace is a superspace approach to formulate conformal SUGRA. In the conformal superspace, we can formulate conformal SUGRA in a geometrical manner.
Conformal SUGRA is constructed as the gauge theory of superconformal group. We formulate conformal SUGRA in a superspace [19]. Superspace is a space where the coordinates are spanned by ordinary bosonic spacetime coordinates x m and fermionic coordinates (θ µ ,θμ). Here, m, n, ... are used for curved vector indices, µ, ν, ... for undotted spinor indices, andμ,ν, ... for dotted spinor indices. In the superspace, we can deal with bosonic translations and SUSY transformations at the same time, since SUSY transformations are understood as fermionic translations. Thus, we denote both bosonic and fermionic coordinates as z M = (x m , θ µ ,θμ), where capital Roman letters M, N, ... express the sets of curved vector and spinor indices.
Conformal superspace is a superspace where gauge fields of the superconformal symmetry are introduced. The generators of the superconformal group are the following elements: spacetime translations P a , SUSY transformations Q α , Lorentz transformations M ab , dilatation D, chiral rotation A, conformal boosts K a and conformal SUSY transformations S α . Here, Roman letters a, b, ... denote flat vector indices, Greek letters α, β, ... andα,β, ... express undotted and dotted flat spinor indices, respectively. α, β, ... denote both spinor indices α = (α,α). In the superspace, we can deal with bosonic translations and SUSY transformations at the same time. Therefore, we simply write P A and K A as P A = (P a , Q α ,Qα) and K A = (K a , S α ,Sα), respectively. Here, we use Roman capital indices A, B, ... for the sets of Lorentz vectors and spinors A = (a, α,α). All the generators of superconformal group are denoted X A = (P A , M ab , D, A, K A ), where the calligraphic letters A, B, ... are used to refer the indices of the generators of the superconformal group.
The gauge fields of the superconformal symmetry are defined as Using a differential form [30], the gauge fields are expressed as The gauged superconformal transformations are generated by δ G (ξ A X A ). Here, ξ A are real parameter superfields, and ξ A X A denotes where primed calligraphic indices A ′ , B ′ , ... mean all the superconformal generators except for We shall define SUSY transformations and spacetime translations. In the superspace approach, we can deal with SUSY transformations and spacetime translations at the same time, namely P A -transformations. We relate P A -transformations to the general coordinate transformation δ GC by using field-independent parameter superfields ξ A as where ξ(P ) A are abbreviated to ξ A , and ξ M are defined by ξ M := ξ A E A M . The P A -transformations acting on a superfield Φ with no curved index define the superconformally covariant derivatives Let us consider the curvatures associated with the superconformal symmetry, which appear in the Bianchi identities. They are defined by Here, we use the convention of "implicit grading" [19]. Using differential forms, they are expressed as In particular, the curvatures associated with P A are the torsion twoforms T A . The torsions are given by explicitly which appear in the Bianchi identities for p-form gauge fields discussed later. The curvatures are expressed also in terms of the (anti-)commutation relations of the superconformally covariant derivatives The curvatures are constrained so that (anti-)commutation relations of the covariant derivatives are given by 14) and the parentheses for indices mean symmetrizations of spinor indices: ψ (α χ β) = 1 2 (ψ α χ β + ψ β χ α ). W αβγ are chiral primary superfields with Weyl weight 3/2 and chiral weight 1, and their indices are totally symmetric. Here, a primary superfield is a superfield that is invariant under the K A -transformations: K A W αβγ = 0. In particular, T αβ c are given by which is used to solve Bianchi identities in appendix B.

Abelian tensor hierarchy in conformal superspace
We introduce antisymmetric tensor gauge fields into conformal superspace. Antisymmetric tensor gauge fields are expressed in terms of p-form gauge fields. p-form gauge fields are transformed under Abelian internal gauge transformations using (p − 1)-form parameter superfields. In addition, p-form gauge fields are shifted using p-form parameter superfields. This structure of gauge transformation of the tensors is called an Abelian tensor hierarchy. We explain the hierarchy concretely. The p-form Here, the indices I p denote the indices of internal space of p-form V p , which are assumed to be real vector spaces. I p run over 1, ..., dim V p . We denote infinitesimal internal gauge transformations of p-forms as δ T (Λ), where Λ is a set of real p-form parameter superfields Λ I p+1 [p] : Λ = (Λ I 1 [0] , ..., Λ I 4 [3] ). The gauge transformation laws of C

Ip
[p] are given by 17) * (−1)-forms are defined to be zero as in the ordinary differential geometry.
where q (p) are matrices which map V p+1 to V p . (q (p) · Λ [p] ) Ip are given by explicitly Field strengths of p-form gauge fields are defined by (2.20) They are transformed under the internal gauge transformations The invariances of the field strengths require that The SUSY transformations and spacetime translations are redefined with respect to δ T transformations of C Ip [p] . The redefinitions are the same as the case that the tensor hierarchy does not exist [30]. P A -transformations are redefined by Here, Λ(ξ) is defined by Λ(ξ) = (ι ξ C I 1 [1] , ..., ι ξ C I 4 [4] ), (2.24) and ι ξ is a interior product In particular, the P A -transformations of C

Ip
[p] are given by The P A -transformation laws of superfields which are invariant under δ T transformations are not changed. Note that we obtain SUSY transformations of p-form gauge fields if we choose ξ A = ξ α . Field strengths obey the following Bianchi identities: The existence of the tensor hierarchy deforms the Bianchi identites: The tensor hierarchy relates the exterior derivatives on the (p + 1)-form field strengths to the (p + 2)-form field strengths. The Bianchi identities play an important role in the next section. Explicitly, we denote the p-form gauge fields, the field strengths and the Bianchi identities in table 1. In table 1, 4-form gauge fields appear. The bosonic component of gauge fields U I 4 qpnm exist in principle, but the bosonic components of the field strengths are zero: G I 4 rqpnm = 0. This is because G I 4 rqpnm is the fifth rank antisymmetric tensor, which must be zero in 4D. Thus, we impose by hand that field strengths G I 4 are equal to zero as in Ref. [32]. There are also 0-form "field strengths" in principle, but (−1)-form gauge field does not exist. Thus 0-form field strengths are defined by dω From the higher-dimensional view point [8], V p can be understood as spaces of differential forms on extra dimensions. The matrices q (p) are understood as the exterior derivative with respect to extra dimensions.

Constraints and Bianchi identities
In this section, constraints on the field strengths are imposed to construct irreducible superfields. We solve the Bianchi identities under these constraints. As a result, each of field strengths is expressed in terms of the corresponding gauge invariant superfields straightforwardly.

Constraints
In the previous section, we have defined the field strengths of the p-form gauge fields. The field strengths have redundant degrees of freedom, and we eliminate them by imposing constraints. We will take the constraints as the same ones without the tensor hierarchy [30,33]. The constraints are explicitly given by table 2. Here, L I 2 and Ψ I 0 are real superfields. In addition, we have imposed that Ψ I 0 are primary superfields in table 2.
We solve the Bianchi identities under these constraints. On the one hand, the field strengths obey the Bianchi identities. On the other hand, we impose the constraints on the field strengths. The consistency between Bianchi identities and the constraints leads to new relations of the field strengths. Since the constraints are imposed Lorentz covariantly, it is convenient to express form constraints 4-form the Bianchi identities (2.27) by flat indices rather than curved indices:

Solutions to the Bianchi identities
In this subsection we summarize the solutions to the Bianchi identities of Table 1 under the constraints of Table 2. The details of the derivations are discussed in appendix B. Each of the field strengths is expressed by a single gauge invariant superfield (Y I 3 , L I 2 , W I 1 α , Ψ I 0 ). We find the Weyl weights and chiral ones (∆, w) of the gauge invariant superfields. We also find that these gauge invariant field strengths are primary superfields (Ψ I 0 are imposed to be primary as in table 2). The weights and K A -invariance play an important role in constructing superconformally invariant actions.

3-form gauge fields
For 3-form gauge fields, all the components of the field strengths are expressed in terms of chiral superfields Y I 3 and their complex conjugatesȲ I 3 . They appear in the 2-spinor/2-vector components as They mean that Y I 3 are primary superfields and the weights are Similarly,Ȳ I 3 obey Other Bianchi identities lead to which mean that Y I 3 andȲ I 3 are chiral and anti-chiral superfields, respectively. Furthermore, 1-spinor/3-vector components are expressed in terms of spinor derivatives of Y I 3 and their conjugates: (3.7) Σ I 3 dcba are identified as the imaginary parts of ∇ 2 Y I 3 : We can understand the non-dynamical 4-form field strength in terms of the F -component of Y I 3 by the θ =θ = 0 projection of both hand sides of this equation, where the θ =θ = 0 projection is the projection from the superspace to the bosonic spacetime. The solutions to the Bianchi identities for 3-form gauge fields are the same as the case without tensor hierarchy [4,30]. Note that our normalization of Y I 3 is equivalent to 8G S in Ref. [8].

2-form gauge fields
The field strengths of 2-form gauge fields are expressed by real superfields L I 2 . We list the solutions to the Bianchi identities.

1-form gauge fields
The solutions to 1-form gauge fields are mostly the same as an ordinary Abelian case. The field strengths are expressed in terms of the gaugino superfields W I 1 α and their conjugates. The explicit solutions to the Bianchi identities are as follows.
• 1-spinor/2-vector components (3.14) • Chirality conditions∇β • 2-vector components (3.16) • Deformed reality conditions • D-, A-, K A -transformation laws Then, we find the weights of W I 1 α : The reality conditions of W I 1 α are deformed by the tensor hierarchy, i.e., q (1) : L I 2 appear in the imaginary parts of ∇ α W I 1 α in the presence of the tensor hierarchy. The deformed reality conditions reduce to the ordinary reality conditions ∇ α W I 1 α =∇αW I 1α if the tensor hierarchy does not exist. Note that our normalization of W I 1 α is equivalent to that of Ref. [8].

0-form gauge fields
The field strengths of 0-form gauge fields are expressed in terms of real primary superfields Ψ I 0 .
The solutions to the Bianchi identities are as follows.
• Vector components (3.20) • Modified higher component conditions We find the weights of Ψ I 0 : The conditions in Eq. (3.21) are a bit peculiar in the presence of tensor hierarchy q (0) : In the case of the absence of the tensor hierarchy, the Bianchi identities lead to the constraints ∇ 2 ∇ α Ψ I 0 = 0 and ∇ 2∇α Ψ I 0 = 0. We can find the expression of Ψ I 0 which can be consistent with the constraints. We can use chiral and anti-chiral primary superfields. Chiral primary superfields S I 0 as well as anti-chiral primary superfieldsS I 0 satisfy∇ 2 ∇ α S I 0 =∇ 2 ∇ αS I 0 = 0. The field strengths of 0-form gauge fields Ψ I 0 can be defined as the imaginary part of the chiral superfields: which are consistent with the constraints and the solutions to Bianchi identities. Note that S I 0 can be understood as the prepotentials for the 0-form gauge fields. Now we consider the case of the existence of the tensor hierarchy. Ψ I 0 are deformed to where V I 1 are the prepotentials for 1-form gauge fields. Using The results are consistent with Eq. (3.21). Note that S I 0 are not gauge invariant in the presence of tensor hierarchy.

Component formalism
In this section we show the correspondence between the conformal superspace and superconformal tensor calculus [27] using the results in Ref. [31]. Superconformal tensor calculus is presumably the most practically useful formalism. We focus on the correspondence of the superfields Y I 3 , L I 2 , W I 1 α and Ψ I 0 , which characterize the corresponding field strengths.
4.1 Components of the superfields Y I 3 , L I 2 , W I 1 α and Ψ I 0 We express Y I 3 , L I 2 , W I 1 α and Ψ I 0 within the superconformal tensor calculus. In the superconformal tensor calculus, we denote the components of a general complex multiplet V Γ with arbitrary Lorentz indices Γ as (4.1) The components of V Γ are expressed by corresponding primary superfields Φ Γ as in table 3.
In this table, the symbol of "|" means the θ =θ = 0 projection. As already appeared in Sec. 3, the θ =θ = 0 projection is the projection from the superspace to the bosonic spacetime. Component fields are obtained by θ =θ = 0 projections of superfields.  [27].
We also denote chiral conformal multiplets T Γ as T Γ are embedded into the general complex multiplets as 3) The components of gauge invariant superfields Y I 3 , L I 2 , W I 1 α and Ψ I 0 are given in tables 4, 5, 6 and 7, respectively. In these tables, note that the tensor hierarchy deforms the higher components of the L I 2 , W I 1 α and Ψ I 0 in the presence of q's.   Table 6: The components of the chiral primary superfields W I 1 α . The spinor index α is used for the external Lorentz index of W I 1 α .

Bosonic field strengths
In the previous subsection, the lowest component of the field strengths g I 0 a |, F I 1 ab |, H I 2 abc | and Σ I 3 abcd | appear. They are covariantly transformed under SUSY transformations, because they have only Lorentz indices. The lowest components of them are related to the lowest components of bosonic p-form gauge fields C Ip mp...m 1 |. In SUGRA, they are also related to vierbein e m a and gravitino ψ m α . Thus, we express the lowest components of field strengths in term of the lowest components of the bosonic p-form gauge fields, vierbein, and gravitino.
The expressions are obtained by the so-called "double bar projection" [30,35]. The double bar projections of the p-form gauge fields are defined as The symbol of "||" means the projection from superforms to forms on the bosonic spacetime: ction: dθ µ = θ µ = 0. However, there still exist fermion parts through contractions between indices as shown below. Explicitly, the double bar projections of the p-form gauge fields are as follows: (4.5) The double bar projections of the field strengths systematically lead to the expressions of the bosonic field strengths. We consider the simplest case. For the 1-form field strength of 0-form gauge field, the double bar projections of g I 0 are This relation gives rise to the component expression Here, we used The same procedure can be applied to higher forms. The results are summarized in appendix C.

A superconformally invariant action
In this section, we present a superconformally invariant action which contains the kinetic terms of the p-form gauge fields. In contrast to global SUSY case, superconformally invariant actions require the conditions for Weyl and chiral weights (∆, w) of integrands. The integrands of Fand D-type actions must have the weights (∆, w) = (3, 2) and (∆, w) = (2, 0), respectively. We compensate the weights when we construct invariant actions from Y I 3 , L I 2 , W I 1 α and Ψ I 0 . Such a procedure can be done by a so-called chiral compensator superfield Φ c . Chiral compensator is a chiral primary superfield with the weights (∆, w) = (1, 2/3). The compensations are needed for Y I 3 and L I 2 . We can rescale the weights of the superfields to (∆, w) = (0, 0) as follows.
Then y I 2 and l I 2 are weight (0, 0) primary superfields. Recall that the former are chiral superfields and the latter are the real ones. The compensations agree with those of [15], in which the tensor hierarchy does not exist. An invariant action is given by are the kinetic potential (rather than the Kähler potential) [30], the superpotential, and the gauge kinetic function respectively. All functions have the weights of (∆, w) = (0, 0). Note that K is a primary real superfield, both W and g I 1 J 1 are primary chiral superfields. Further, K and W are gauge invariant, whereas g I 1 J 1 is a function such that g I 1 J 1 W I 1 α W I 1 α is gauge invariant. Such actions for tensors were also discussed in Ref. [30]. For example, in the case of quadratic kinetic terms, K is given as where g I 0 J 0 , g I 2 J 2 and g I 3 J 3 are real constants. We may Taylor-expand W as where λ I 3(1) I 3(2) ···I 3(n) are complex constants. The expansion of this action will be done elsewhere [36]. We can reproduce the results in Ref. [30] by imposing the superconformal gauge fixing conditions. The conditions are the same as those of Ref. [19,25]. We impose the conditions for the compensator Φ c and dilatation gauge fields B M as follows:

Conclusions
In this paper, we have considered a way to couple the Abelian tensor hierarchy to 4D N = 1 conformal supergravity. We have used the conformal superspace formalism and covariant approach. The constraints on the field strengths have been imposed. The constraints are the same as the case that Abelian tensor hierarchy does not exist. We have solved the Bianchi identities under the constraints. Each of the field strengths is expressed in terms of the single superfield and its conjugate. The linearity conditions which appear in 2-form gauge fields are deformed by the tensor hierarchy. The reality conditions which appear in 1-form gauge fields are also deformed. Furthermore, we have obtained nontrivial conditions of superconformal transformation laws. We have also presented a superconformally invariant action.
There are remaining issues. The action of tensor superfields in terms of their components should be considered. Such an action would be needed for phenomenological applications. One can think also the Chern-Simons couplings of tensor fields. To introduce these terms, we need to reconstruct the tensor hierarchy with the so-called prepotential approach. Further, when there exist (non-Abelian ) gauge/matter fields, one has to take chiral anomalies into account. Then the prepotential S I 0 will appear also in the superpotential or gauge kinetic functions. We will address these issues elsewhere [36].

A Notations
In this section, we summarize our notations. The notations are the same as those of Wess-Bagger [29]. The Minkowski metric and the totally antisymmetric tensor are is given by The standard contractions of two-component spinors are given by ξ α ψ α andξαψα. The raising and lowering rules of indices are defined by where ǫ αβ , ǫ αβ , ǫαβ and ǫαβ are antisymmetric tensors that satisfy ǫ 12 = ǫ 21 = +1. The Hermitian conjugate of a spinors is defined as (ψ α ) † =ψ α . Hermitian conjugate reverses the order of the product of spinors: Pauli matrices are defined as Their Hermitian conjugates are given by Pauli matrices satisfy Lorentz vectors are expressed as mixed Lorentz spinors and vice versa: The matrices σ ab andσ ab are given by Any anti-symmetric tensor F ab can be decomposed into chiral and anti-chiral parts: In spinor notations, the equation is rewritten as

B.1 The Bianchi identities for 3-form gauge fields
Firstly, we solve the Bianchi identities for 3-form gauge fields. The Bianchi identities are given by Explicitly they are written by The constraints for 3-form are given in table 2. Under these constraints, we solve the Bianchi identities.
2) is written as This is solved as 17) or equivalently expressed as in Eq. (3.7). Similarly, Σ I 3 δcba are expressed in terms of∇δȲ I 3 : or Eq. (3.7).
Using (3.7), and contracting spinors, we obtain They are equivalently written as in Eq. (3.8). There is no more non-trivial Bianchi identity from constraints.

B.2 The Bianchi identities for 2-form gauge fields
Next, we solve the Bianchi identities for 2-form gauge fields. The Bianchi identities are written as The constraints on the field strengths of 2-form gauge fields are given in  Therefore, we obtain which is equivalent to Eq. (3.9). Similarly, for D =δ, C =γ, B = β, a = αα, we obtain which is equivalent to Eq. (3.9). For D = δ, C =γ, B = b, A = a, Eq. (B.21) is written as This equation implies Using Eqs. (3.2) and (3.9), we obtain Eq. (3.11). Similarly, for D =δ, C =γ, B = b, A = a, we find Eq. (3.11).

B.3 The Bianchi identities for 1-form gauge fields
Thirdly, we solve the Bianchi identities for the field strengths of 1-form gauge fields [29]: This means symmetric part of undotted spinors in F I 1 γβ,α is equal to zero. Then, we can write Similarly, for C = γ, B =β, A =α, we obtain

B.4 The Bianchi identities for 0-form gauge fields
Finally, we solve the Bianchi identities for 0-form gauge fields. The Bianchi identities are given by Furthermore, the actions of ∇ β on both hand sides lead to Similarly, for B =β, A =α, we obtain These consequences suggest that we may impose the constraints where we took λ as a complex constant, and Ψ I 0 are real primary superfields.
If we take λ = i, this equation reproduces the results in Ref. [8]. In this choice, g I 0 αα are written as Note that the degrees of freedom between bosons and fermions in Ψ I 0 are matched. If the tensor hierarchy does not exist, Eq. (3.21) means that the higher fermion components of Ψ I 0 vanish:

B.5 D-, A-, K A -transformation laws
We present the derivations of the D-, A-, K A -transformation laws of (Y I 3 , L I 2 , W I 1 α , Ψ I 0 ). The transformation laws of the superfields follow from those of F The D-, A-and K A -transformation laws of the vielbein are obtained as follows.
• D-transformations (B.57) • All the K a -transformations of the vielbein are equal to zero.
Using these equations, the D-, A-and K A -transformation laws of (Y I 3 , L I 2 , W I 1 α , Ψ I 0 ) are determined. Note that the M-transformation laws of (Y I 3 , L I 2 , W I 1 α , Ψ I 0 ) are obtained by their spinor indices.

B.5.1 3-form gauge fields
We show the D-, A-, K A -transformation laws of Y I 3 . Y I 3 are given in terms of Σ I 3δγ ba as in Eq. (3.2). The D-, A-, K A -transformations of Σ I 3δγ ba are determined as follows.
Here, we used the constraints Σ I 3δγ bǫ = Σ I 3δγ bǫ = 0 in the last lines of S α andSα transformation laws. These equations lead to the superconformal transformation laws of Y in Eq. (3.3). Those ofȲ are obtained similarly.

B.5.2 2-form gauge fields
The D-, A-, K A -transformation laws of L I 2 are obtained by the same procedure as the case of 3-form gauge fields. We summarize the results.

B.5.3 1-form gauge fields
The D-, A-, K A -transformation laws of W I 2 α are the same as the case that the tensor hierarchy does not exist. The results are as follows. We need to check the consistency between the weights of Ψ I 0 and the S α -invariances of Ψ I 0 .

C The explicit forms of bosonic field strengths
In this appendix, we summarize the explicit forms of the bosonic field strengths. For 3-form gauge fields, the double bar projections of Σ I 3 lead to the following relations (C.1) We expand this relation, and obtain e m a e n b e p c e q d Σ I 3 dcba | = ∂ q C I 3 pnm | + (−1) 1 ∂ p C I 3 qnm | + (−1) 2 ∂ n C I 3 qpm | + (−1) 3 ∂ m C I 3 qpn | − (q (3) · U qpnm |) I 3 We We obtain the expressions of the bosonic field strengths (C.6) The above expressions are basic building blocks in the constructions of component field actions.