Abelian tensor hierarchy and Chern-Simons actions in 4D N=1 conformal supergravity

We consider Chern-Simons actions of Abelian tensor hierarchy of $p$-form gauge fields in four-dimensional ${\cal N}=1$ supergravity. Using conformal superspace formalism, we solve the constraints on the field strengths of the $p$-form gauge superfields in the presence of the tensor hierarchy. The solutions are expressed by the prepotentials of the $p$-form gauge superfields. We show the internal and superconformal transformation laws of the prepotentials. The descent formalism for the Chern-Simons actions is exhibited.


Introduction
The superstring theory is a candidate for the unified theory of the fundamental interactions including quantum gravity. There are strings and branes in the superstring theory. Our universe can be described by the strings and branes in a unified way at the low energy limit. The stability of the branes is preserved by supersymmetry (SUSY) and conserved Ramond-Ramond charges. We can avoid unstable tachyons by SUSY. Further, the conserved charges guarantee the number of branes. Antisymmetric tensor (p-form) gauge fields are coupled to the conserved charges of the branes.
One of the most important issues in the superstring theory is to construct realistic four-dimensional (4D) effective theories. Since the superstring theory is a ten-dimensional theory, 4D effective theories are obtained by compactifying extra six dimensions. 4D N = 1 supergravity (SUGRA) is a candidate for the effective theories. This theory consists of chiral fermions as well as gravity. Thus, we can embed the standard model particles into the theory. Further, the stability of the theory is ensured by SUSY.
Thus, it is important to consider p-form gauge fields in 4D N = 1 SUGRA [1][2][3][4][5][6][7][8][9][10][11]. In particular, we study the p-form gauge fields which can be regarded as dimensionally reduced ones from higher dimensions. Such p-form gauge fields differ from those of just defined in 4D. Because the original gauge transformation laws of the p-form gauge fields are given in higher dimensions, the gauge transformations of the p-form gauge fields should contain different rank forms in 4D. The structure of the transformations is called a tensor hierarchy [12][13][14][15].

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In 4D N = 1 global SUSY, Becker et al. constructed such Abelian tensor hierarchy in superspace [16]. They showed Chern-Simons (CS) actions. 1 The CS actions are constructed by integrands which are proportional to p-form gauge fields. Since the different ranked tensors are related each other by the tensor hierarchy, each of the integrands is not independent. The internal gauge invariance requires the relations between the integrands. They showed the relations in a systematic manner, which is called descent formalism. The descent formalism relates the integrands each other by derivatives. The CS actions are important because they are related to the anomaly cancellation in 4D [18][19][20].
In this paper, we embed the CS actions of Abelian tensor hierarchy obtained in ref. [16] into 4D N = 1 SUGRA. We use 4D N = 1 conformal superspace formalism [21]. This formalism has larger gauge symmetries than superconformal tensor calculus [22][23][24][25][26][27][28][29][30] and Poincaré superspace formalism [31,32]. Superconformal tensor calculus and Poincaré superspace formalism are obtained from the conformal superspace formalism by using their correspondences [21,33,34]. The CS actions are constructed by the prepotentials of the p-form gauge superfields in the presence of the tensor hierarchy. 2 We obtain the prepotentials by using so-called covariant approach, which are shown in our previous paper [35]. In the covariant approach, we introduce p-form gauge superfields and their field strength superfields in the superspace. The field strength superfields have some constraints, since they have superfluous degrees of freedom. We obtain the prepotentials as the solutions to the constraints. The CS actions in 4D N = 1 SUGRA would be useful to discuss the roles of the p-form gauge fields, e.g. in cosmology [36,37].
In the conformal superspace, the derivations of the solutions to the constraints are mostly the same as the case of the global SUSY in ref. [2]. This is because superconformally covariant spinor derivatives satisfy the same anti-commutation relations as those of global SUSY. Moreover, we can naturally extend the descent formalism of the CS actions in global SUSY [16] into the conformal superspace, since the relation between D-and F-term integrations in the conformal superspace are quite similar to the global SUSY case. This paper is organized as follows. In section 2, we briefly review the covariant approach to Abelian tensor hierarchy in 4D N = 1 conformal superspace. The prepotentials of p-form gauge superfields are obtained in section 3. We show the internal gauge transformation laws of the prepotentials. Section 4 is devoted to constructing the CS actions of the tensor hierarchy. In particular, the descent formalism in the conformal superspace is discussed. Finally, we conclude this paper in section 5. Throughout this paper, we use the terms "gauge superfields", "field strengths superfields", and "gauge parameter superfields" are simply written as "gauge fields", "field strengths", and "gauge parameters", respectively.

Review of the covariant approach
We briefly review so-called covariant approach to Abelian tensor hierarchy in 4D N = 1 conformal superspace discussed in ref. [35]. Covariant approach is an approach to constructing supersymmetric theories of p-form gauge fields in superspace. 1 They also showed CS actions in the case of non-Abelian tensor hierarchy [17]. 2 In this paper, we use the term "prepotentials" to refer to superfields which consist of the bosonic gauge fields and field strengths as well as their superpartners.

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We use the notations and conventions of ref. [35] except the normalizations of the superfields Y I 3 and L I 2 , which are the same as G S and H M in ref. [16], respectively.

Conformal superspace
We firstly review conformal superspace formalism to construct SUGRA [21]. Superspace is space which is spanned by the ordinary spacetime coordinates x m and the Grassmannian coordinates (θ µ ,θμ). Here, the indices m, n, . . . are used to refer to curved vector indices. The indices µ, ν, . . . andμ,ν, . . . denote curved undotted and dotted spinor indices, respectively. In the superspace, SUSY transformations are understood as the translations to Grassmannian coordinates. Thus, we simply denote these coordinates at the same time: Conformal superspace is superspace where the superconformal symmetry is introduced as a gauge symmetry. The generators of the superconformal symmetry are spacetime translations P a , SUSY transformations (Q α ,Qα), Lorentz transformations M ab , dilatation D, chiral rotation A, conformal boosts K a , and conformal SUSY transformations (S α ,Sα).
Here, Roman letters a, b, . . . denote flat vector indices. Greek letters α, β, . . . andα,β, . . . denote flat spinor indices. All of the generators of the superconformal symmetry are denoted as X A , where we use calligraphic indices A, B, . . . to refer to the generators of the superconformal symmetry. In the conformal superspace, both of P a and (Q α ,Qα) are understood as the translations. Thus, we simply express P a and (Q α ,Qα) at the same time: P A := (P a , Q α ,Qα). Here, capital Roman letters A, B, . . . are used for both of flat vector and spinor indices. Similarly, we denote both of K a and (S α ,Sα) as K A := (K a , S α ,Sα). The (anti-)commutation relations of the generators are summarized in ref. [21].
The gauge fields of the superconformal symmetry are given by where we assume that the vielbein E M A is invertible, and the inverse of the vielbein is Note that the gauge fields h M A are also expressed by differential forms on the conformal superspace as Here, we use the convention of ref. [31] for the differential forms. The differential forms dz M = (dx m , dθ µ , dθμ) are bases of the superforms on the conformal superspace. The gauge transformation parameters are denoted as We denote infinitesimal superconformal transformations as δ G (ξ A X A ). The transformation laws of the gauge fields h M A under the superconformal transformations other than P A are given by

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Here, primed calligraphic indices A ′ , B ′ , . . . are used to refer to the generators of the superconformal symmetry other than P A : X A ′ = (M ab , D, A, K A ). The coefficients f CB A are the structure constants of the superconformal symmetry: [X C , X B ] = −f CB A X A , where we use the convention of "implicit grading" [21]. We define SUSY transformations and spacetime translations in the conformal superspace. In the conformal superspace, SUSY transformations are regarded as translations to the Grassmannian coordinates. Using field-independent parameters ξ A , we relate infinites- transformations on a superfield without curved indices Φ define superconformally covariant derivatives ∇ A :

Covariant approach to Abelian tensor hierarchy
Next, we introduce p-form gauge fields in the conformal superspace, where p runs over p = −1, 0, 1, 2, 3, 4. We assume that (−1)-forms are zero as in ordinary differential geometry. The p-form gauge fields are denoted as Here, I p are indices of internal degrees of freedom, which run over I p = 1, . . . , dim V p . The ranks of the differential forms are represented as [p]. The X A ′ -transformations of the p-form gauge fields are defined as The explicit transformation of the vielbein is summarized in ref. [35]. The infinitesimal internal gauge transformations δ T (Λ) of the p-form gauge fields are given by Here, d denotes the exterior derivative in the conformal superspace, and Λ is the set of the real gauge parameter superforms: Λ = (Λ I 1 [0] , . . . , Λ I 4 [3] ). We assume that Λ ..M 1 are field dependent parameters. Ordinary Abelian gauge transformations are expressed by the first JHEP12(2016)092 term in eq. (2.10). Shifts of the gauge fields are represented by the second term due to the tensor hierarchy. q (p) are real linear maps from the vector space V p+1 to the vector space [p] . Note that q (p) can be understood as the exterior derivative on the extra dimensions [16].
The P A -transformations are redefined with respect to the internal gauge transformations in the presence of the tensor hierarchy. The redefinitions are given by and ι ξ is a interior product In the presence of the tensor hierarchy, the field strengths of the p-form gauge fields are given by using the exterior derivative and q's. The definitions of the field strengths of the p-form gauge fields are given as follows: (2.14) The field strengths are transformed under the internal gauge transformations as The invariances of the field strengths under the internal transformations require conditions on the q's as The covariant derivatives on the field strengths with Lorentz indices are given by We summarize the explicit forms of the gauge fields, field strengths and Bianchi identities in table 1.
We impose some constraints on the field strengths to eliminate degrees of freedoms because there are superfluous degrees of freedoms in the field strengths in the superspace. The constraints are the same ones as the case without the tensor hierarchy [2,3,32]. We exhibit the constraints in table 2. In this table, the indices α, β, . . . denote both undotted and dotted spinor indices: α = (α,α). Note that the constraints are covariant under both superconformal and internal gauge transformations.

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form gauge field field strength Bianchi identity form constraints We solve the Bianchi identities under the constraints. As a result, the field strengths are expressed by the irreducible superfields. The irreducible superfields of the 2-and 0-form gauge fields are L I 2 and Ψ I 0 in table 2. We find the irreducible superfields of 3-and 1-form gauge fields Y I 3 and W I 1 α as follows, respectively: Note that the Weyl weights ∆ and chiral weights w of the irreducible superfields are as follows: Hereafter, we use the term "conformal weights" to refer to " Weyl and chiral weights".

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The tensor hierarchy deforms the properties of the irreducible superfields such as the linear multiplet conditions for L I 2 and reality conditions for W I 1 α : (2.23) Note that the derivatives ∇ A on the superfields Y I 3 , L I 2 , W I 1 α , and Ψ I 0 are superconformally covariant and internally invariant derivatives because of the properties in eq. (2.17).

Prepotentials
In this section, we construct the prepotentials of the p-form gauge fields in the presence of the tensor hierarchy. The prepotentials and their gauge transformation laws are needed to construct CS actions. The prepotentials are obtained by solving the constraints on the field strengths in certain gauge-fixing conditions. The relations between the prepotentials and the irreducible superfields are also obtained by the relations of the gauge fields and field strengths in eq. (2.14). The gauge transformations of the prepotentials are determined by the gauge transformations which leave the gauge-fixing conditions invariant.

Gauge-fixing conditions for the p-form gauge fields
We solve the constraints on the field strengths. Since the constraints in table 2 are gauge covariant, we solve the constraints under the gauge-fixing conditions where some components of the gauge fields are gauged away by using the definitions of the field strengths and the internal gauge transformation laws of the gauge fields Here, ∇ A are covariant with respect to only the superconformal symmetry, and T CB A are the coefficients of torsion 2-form defined by

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form conditions on the gauge fields Table 3. The gauge-fixing conditions on the gauge fields. The gauge-fixing conditions are imposed in order of 4-, 3-, 2-and 1-form gauge fields.
The gauge-fixing conditions take the same form as the case of global SUSY without the tensor hierarchy [2] because of the following three reasons. First, the constraints on the following components of the torsion are the same as those of global SUSY (see ref. [21]): Second, as announced in section 1, the superconformally covariant spinor derivatives obey the same anti-commutation relations as those of global SUSY: Third, if we impose the gauge-fixing conditions and solve the constraints in order of 4-, 3-, 2, and 1-form, the gauge-fixing conditions are not deformed from the case of the absence of the tensor hierarchy in ref. [2]. For example, we discuss the gauge-fixing conditions for C I 3 γβα . Since the field strengths of the 4-form gauge fields are the same as the case of the absence of the tensor hierarchy, we fix some of the 4-form gauge fields, e.g., U I 4 δγβα = 0. Under the gauge-fixing conditions U I 4 δγβα = 0, the field strengths of the 3-form gauge fields Σ I 3 δγβα are written as Σ I 3 δγβα = ∇ δ C I 3 γβα + ∇ γ C I 3 δβα + ∇ β C I 3 γδα + ∇ α C I 3 δγβ . We find that the terms (q (3) · U δγβα ) I 3 do not appear in the field strengths Σ I 3 δγβα in this gauge. Thus, we impose the same gauge-fixing conditions as the case of global SUSY without the tensor hierarchy: C I 3 γβα = 0, which are derived from the constraints Σ I 3 δγβα = 0. Therefore, the gauge-fixing conditions take the same ones as the case in which the tensor hierarchy does not exist in global SUSY. The explicit forms are summarized in table 3. In this table, X I 3 and V I 1 are real superfields, which are the prepotentials of the 3-and 1-form gauge fields, respectively.

Prepotentials: the solutions to the constraints
In this subsection, we show the prepotentials for the p-form gauge fields. Under the gaugefixing conditions and the constraints on the field strengths, the gauge fields are expressed in terms of the prepotentials. We remark that the gauge-fixing conditions of p-form gauge fields in table 3 have the same spinor structure as the constraints on the field strengths of (p−1)-form gauge fields in table 2. Thus, we solve the constraints by the same procedure as the Bianchi identities for the field strengths [35]. The conformal weights of the prepotentials JHEP12(2016)092 are also determined by using eq. (2.9). We exhibit the expressions of the gauge fields in terms of the prepotentials as follows.
The 4-form gauge fields. The solutions to the gauge-fixing conditions and constraints for the field strengths are the same as the case of the absence of the tensor hierarchy. The prepotentials of the 4-form gauge fields are given as the 2-spinor/2-vector components: The prepotentials Γ I 4 are primary superfields with conformal weights (∆, w) = (3, 2), which are derived from the superconformal transformation laws of U I 4 δγba in eq. (2.9). The prepotential Γ I 4 andΓ I 4 are chiral and anti-chiral superfields, respectively: The other components the 4-form gauge fields are expressed in terms of the prepotentials The 3-form gauge fields. We find the prepotentials of the 3-form gauge fields X I 3 in the 2-spinor/1-vector component, where X I 3 are real primary superfields with conformal weights (∆, w) = (2, 0). The derivatives of the prepotentials give the other components of the gauge fields as (3.11) The 2-form gauge fields. The prepotentials of the 2-form gauge fields are primary superfields Σ I 2 α and their conjugatesΣ I 2 α . The prepotentials are found in the spinor/vector components: Here, Σ I 2 α are primary superfields with conformal weights (∆, w) = (3/2, 1). The prepotential Σ I 2 α andΣ I 2 α are chiral and anti-chiral superfields, respectively: The 2-vector components are as follows: (3.14) The 1-form gauge fields. As in ordinary super QED case, the spinor components of 1-form gauge fields are given by real primary superfields V I 1 in table 3. The conformal weights of V I 1 are (∆, w) = (0, 0). The vector components are expressed by We assume that V I 1 are primary sueprfields: K A V I 1 = 0. This assumption and conformal weights of V I 1 are consistent with the K A -invariances of A I 1 α [29].
The 0-form gauge fields. The constraints on the field strengths of the 0-form are satisfied if the gauge fields are real parts of chiral superfields Φ I 0 , which are the prepotentials of 0-form gauge fields: Here, the conformal weights of Φ I 0 are (∆, w) = (0, 0), and Φ I 0 are assumed to be primary superfields.
The relations between the prepotentials and the irreducible superfields. We then find the relations between the prepotentials and the irreducible superfields. The relations are found as follows. On the one hand, the irreducible superfields are given by the components of the field strengths Σ I 3 δγba , Σ I 3 δγba , H I 2 γβa , F I 1 βa , and g I 0 a . On the other hand, the field strengths are expressed by the derivatives of the gauge fields in eq. (2.14), which are now written in terms of the prepotentials. In addition, the field strengths of (−1)-form gauge fields ω I −1 are given by the 0-form gauge fields f I 0 as in table 1: Since the 0-form gauge fields are expressed by the prepotential Φ I 0 , the field strengths ω I −1 are now given by the real parts of chiral superfields J I −1 = −(q (−1) · Φ) I −1 : Thus, we find the relations by using the definitions of the field strengths in terms of gauge fields (2.14), the definitions of the superfields in eqs. (2.19), (2.20) and table 2.
The results are summarized in table 4. Note that the irreducible superfields for p-form gauge fields are expressed by the prepotentials of p-and (p + 1)-form gauge fields due to the tensor hierarchy.

The gauge transformation laws of the prepotentials
In this subsection, we show internal transformation laws of the prepotentials. The transformation laws are important when we construct CS actions. We have solved the gauge fields in terms of the prepotentials under the set of the gauge-fixing conditions. Although

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it seems that the gauge parameters are exhausted to fix the gauge fields, there are remaining gauge parameters which preserve the gauge-fixing conditions in table 3 invariant. The  remaining gauge transformation laws are determined by the conditions for the gauge fields  which are gauged away in table 3: We denote the remaining parameters as Θ = (Θ I 1 , Θ I 2 , Θ I 3 α , Θ I 4 ). We determine the properties of Θ's and the gauge transformation laws of the prepotentials as follows.
The 4-form gauge fields. The gauge parameters are determined by the conditions so that the gauge-fixing conditions in table 3 are invariant: The gauge transformations which preserve the gauge-fixing conditions are given by (3.20) Here, Θ I 4 are real superfields. The prepotentials Γ I 4 andΓ I 4 are transformed by Θ I 4 as which are determined by the gauge transformation laws of U I 4 δγba and U I 4 δγba , respectively. We can impose Wess-Zumino (WZ) gauge for the prepotentials Γ I 4 by using Θ I 4 as follows: Here, the symbol of "|" means θ =θ = 0 projection.
The 3-form gauge fields. We determine the remaining gauge parameters so that the gauge-fixing conditions in table 3 are invariant as follows: The invariances are preserved by the following conditions for the gauge parameters:

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Here, Θ I 3 α andΘ I 3 α are chiral and anti-chiral superfields, respectively: The gauge transformation laws of the prepotential X I 3 are determined by those of C I 3 γβa : We find that X I 3 are also transformed by the remaining gauge parameters Θ I 4 due to the tensor hierarchy. The WZ gauge conditions for the prepotentials X I 3 can be imposed by the parameters Θ I 3 α as follows: Note that the WZ conditions in eq. (3.28) are imposed under the WZ gauge conditions for the prepotentials of 4-form gauge fields in eq. (3.22).
The 2-form gauge fields. The remaining parameters are found by the conditions so that the gauge-fixing conditions in table 3 are invariant: By using the gauge invariances, we determine the remaining parameters Θ I 2 as where Θ I 2 are real superfields. Again, Θ I 3 α do not affect the gauge-fixing conditions in eq. (3.29) in the presence of the tensor hierarchy. The gauge transformation laws of the prepotential Σ I 2 α are given by (3.31) Under the conditions in eqs. (3.22) and (3.28), we can go to the WZ gauge conditions for Σ I 2 α : The 1-form gauge fields. The gauge transformations for the 1-form gauge fields are the same as in ordinary super QED case except the shifts due to the tensor hierarchy. We find that the gauge transformations which leave the gauge-fixing conditions in table 3 invariant are given by Here, Θ I 1 andΘ I 1 are chiral and anti-chiral superfields, respectively:

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The gauge transformations of the 1-form prepotentials are given by the imaginary parts of Θ I 1 and the shifts by the gauge parameters of 2-form gauge fields Θ I 2 : We can impose the WZ gauge conditions for the prepotentials V I 1 under the conditions in eqs. (3.22), (3.28) and (3.32): The 0-form gauge fields. The gauge transformation laws of the prepotentials of 0-form are given by the chiral shifts by the gauge parameters Θ I 1 : Again, the shifts come from the tensor hierarchy.

Chern-Simons actions
In this section, we construct CS actions in the conformal superspace. The CS actions of the tensor hierarchy is related to anomaly cancellations in low energy effective theories. The construction of the CS actions in the conformal superspace is quite similar to the global SUSY case [16]. CS actions are constructed by the combinations of the prepotentials and irreducible superfields Y I 3 , L I 2 , W I 1 α , Ψ I 0 , J I −1 and their conjugates. To construct the CS actions, we use the descent formalism. This formalism systematically gives the CS actions from the internal transformation laws of the prepotentials. We show that the descent formalism that was given in ref. [16] is straightforwardly extended in the case of the conformal superspace.
Descent formalism in global SUSY. We briefly review the descent formalism in global SUSY in ref. [16]. The descent formalism in global SUSY is given by the combinations of the prepotentials and irreducible field strengths as Here, c's are polynomials of the irreducible superfields Y I 3 , L I 2 , W I 1 α , Ψ I 0 , J I −1 and their conjugates. The superfields c I 1 and c I 3 are real superfields, and c I 0 , c I 2 α , and c I 4 are chiral superfields. The internal gauge invariance requires that c's are related each other as − 1 4D 2 c I 1 = (q (0) ) I 0 I 1 c I 0 , 1 2i D α c I 2 α −Dαcα I 2 = −(q (1) ) I 1 I 2 c I 1 , − 1 4D 2 D α c I 3 = (q (2) ) I 2 I 3 c I 2 α , 1 2i (c I 4 −c I 4 ) = −(q (3) ) I 3 I 4 c I 3 .
Descent formalism in the conformal superspace. We now discuss the descent formalism in the conformal superspace. The descent formalism in the conformal superspace is given by a natural extension of global SUSY case as S CS = d 4 xd 4 θE(V I 1 c I 1 − X I 3 c I 3 ) + Re i d 4 xd 2 θE(Φ I 0 c I 0 + Σ I 2 α c I 2 α + Γ I 4 c I 4 ) , (4.4) where E and E are the density of the whole superspace and chiral subspace, respectively. The integrations d 4 xd 4 θE and d 4 xd 2 θE are called D-and F-term integration, respectively [21]. The superfields c's are polynomials of the irreducible superfields Y I 3 , L I 2 , W I 1 α , Ψ I 0 , J I −1 and their conjugates. Again, c I 1 and c I 3 are real superfields, and c I 0 , c I 2 α , and c I 4 are chiral superfields. The c's have two type of conditions. One is the condition that is required by the superconformal invariance. The conditions are that all the c's are primary superfields, and the conformal weights of them are as follows:  The other is the condition that is required by the internal gauge invariance of the tensor hierarchy as in the global SUSY case. The internal gauge invariance requires the same conditions as those of ref. [16]: ∇ α c I 2 α −∇αcα I 2 = −(q (1) ) I 1 I 2 c I 1 , (c I 4 −c I 4 ) = −(q (3) ) I 3 I 4 c I 3 . (4.6) The internal gauge invariances are obtained by superspace partial integrations of the integrands. In the conformal superspace, the relation between F-term and D-term actions is

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The CS actions in 4D N = 1 SUGRA, in particular the action in eq. (4.8), would be useful to discuss phenomenology such as inflation of the early universe [36,37]. It would be interesting to embed the approach which was proposed in ref. [38] into the conformal superspace.