Full diffeomorphism and Lorentz invariance in 4D ${\cal N}=1$ superfield description of 6D SUGRA

We complete the four-dimensional ${\cal N}=1$ superfield description of six-dimensional supergravity. The missing ingredients in the previous works are the superfields that contain the sechsbein $e_4^{\;\;\underline{\nu}}$, $e_5^{\;\;\underline{\nu}}$, $e_\mu^{\;\;\underline{4}}$, $e_\mu^{\;\;\underline{5}}$ and the second gravitino. They are necessary to make the action invariant under the diffeomorphisms and the Lorentz transformations involving the extra dimensions. We find the corresponding superfield transformation laws, and show the invariance of the action under them. We also check that the resultant action reproduces the known superfield description of five-dimensional supergravity through the dimensional reduction.


Introduction
When we consider higher-dimensional supersymmetric (SUSY) theories, it is useful to describe the action in terms of N = 1 superfields [1]- [9] for various reasons. 1 It makes the expression of the action much more compact than the component field expression. In particular, the complicated spacetime index structures become much simpler. In higher than six dimensions (6D), however, the full superspace formulation is not known due to the extended SUSY structure. Even in such cases, the N = 1 superfield expression is still possible because only partial SUSY structure is respected. Such an expression is useful to discuss a system in which the spacetime is compactified to four dimensions (4D) and the N = 1 SUSY is preserved. We can derive the 4D effective action directly from the higher dimensional theory, keeping the N = 1 superspace structure manifest. Especially, when the system contains lower dimensional branes or orbifold fixed points in the compactified space, the bulk-brane interactions are described in a transparent manner because all the sectors are expressed on the common N = 1 superspace. Besides, the N = 1 superfield formalism is familiar to many researchers, and is easy to handle.
For global SUSY theories, the N = 1 superfield description of the action has been already provided in 5-10 dimensions [2]. We have to extend it to the supergravity (SUGRA) in order to discuss the moduli stabilization, the interactions to the moduli or the higher dimensional gravitational multiplet, and so on. However, such an extension is not straightforward. First, it is a nontrivial task to identify the component fields of the N = 1 superfields. It usually happens that the non-gravitational fields form the superfields with the help of the gravitational fields, such as the vierbein and the gravitini. Of course, these superfields should reduce to the ones in Ref. [2] if the gravitational fields are replaced with their background values in the flat spacetime. However, such an observation alone is not enough to identify the dependence of each component of the superfield on the gravitational fields. The complete identification can be achieved by requiring the invariance of the action under various symmetry transformations, such as the gauge transformations, the diffeomorphisms, the Lorentz transformations, etc. We should note that the diffeomorphisms and the Lorentz transformations have to be divided into the 4D parts and the extra-dimensional parts, and treated separately because we only respect the N = 1 SUSY.
The invariance under their 4D parts is obvious. In contrast, the invariances under the 1 "N = 1" denotes SUSY with four supercharges in this paper. diffeomorphism in the extra dimensions and the Lorentz transformations that mix the 4D index with the extra-dimensional one are less trivial, but they are also expressed as the N = 1 superfield transformations. Besides, we should also note that the N = 1 superconformal parameters 2 depend on the extra-dimensional coordinates, and that the desired superfield action involves the derivatives with respective to such coordinates. Therefore, we need to covariantize such derivatives. The corresponding connection superfields contain the "off-diagonal" components of the vierbein e n µ and e ν m , where {µ, ν} and {m, n} denote the 4D and the extra-dimensional indices, respectively.
The simplest background for the extra-dimensional models is the five-dimensional (5D) spacetime. The N = 1 description of the 5D SUGRA action is provided in Refs. [8,9]. These works specify the dependence of the action on the "modulus" superfields that contains the extra-dimensional component of the fünfbein e 4 4 . This superfield description makes it possible to derive the 4D effective action for various setups systematically [15,16,17,18]. However, the superfield action in Refs. [8,9] does not contain the "off-diagonal" components of the fünfbein e 4 µ , e ν 4 and their N = 1 SUSY partners. Thus, the action is not invariant under the diffeomorphism in the extra dimension and the Lorentz transformations that mix the 4D and the fifth dimensions. Those missing ingredients are incorporated at the linearized level in Ref. [19], and play an important role in the calculation of the one-loop effective potential [20,21,22].
In this paper, we focus on 6D SUGRA [23,24,25]. The 6D spacetime is the next simplest setup for the extra-dimensional models, and the minimal setup where the shape modulus for the extra-dimensional space appears. 6D SUGRA generically contains the Weyl multiplet as the gravitational multiplet, and n H hypermultiplets, n V vector multiplets and n T tensor multiplets as the mattter multiplets. From the anomaly cancellation condition, the numbers of the multiplets are constrained by 29n T + n H − n V = 273 [26,27,28]. In contrast to 5D SUGRA, the Weyl multiplet contains the anti-self-dual tensor field T − M N (M, N = 0, 1, · · · , 5), and a 6D tensor multiplet contains the self-dual tensor field B + M N . In general, the (anti-)self-dual condition is an obstacle to the Lagrangian formulation, similar to that of type IIB SUGRA. However, when n T = 1, this difficulty can be solved because we can construct an unconstrained tensor field B M N by combining T − M N with B + M N [25,29]. When n T = 1, the (anti-)self-dual conditions remain, and thus the theory cannot be described by the Lagrangian. Hence, we focus on the case of n T = 1 in this paper.
In our previous work [30], we found the N = 1 superfield description of the vector-tensor couplings in 6D global SUSY theories, which is derived from the invariant action [31] in the projective superspace [32,33,34]. 3 Then, we extend this result to 6D SUGRA in Ref. [38] by identifying the "moduli superfields" that contain the extra-dimensional components of the sechsbein e n m (m, n = 4, 5), and inserting them into the result in Ref. [30]. We have checked that the resultant action is invariant under the supergauge transformation, and reproduces the known 5D SUGRA action after the dimensional reduction. In this paper, we complete the N = 1 superfield description of 6D SUGRA by incorporating the missing ingredients, i.e., the "off-diagonal" components of the sechsbein e n µ and e ν m (m, n = 4, 5) and their N = 1 superpartners. The identification of the corresponding superfields and the dependence of the action on them are determined by the invariance under the full 6D diffeomorphisms. These newly incorporated superfields, which are the real superfields U m and the spinor superfields Ψ α m (m = 4, 5), are also necessary for the invariance under the Lorentz transformations that mix the 4D and the extra-dimensional indices. This work corresponds to the 6D extension of Ref. [19]. We will treat the 4D N = 1 SUGRA part at the linearized level for a technical reason. Due to this approximation, we can only determine the dependence of the action on Ψ α m at the linearized level. In contrast, we clarify the dependence on U m at the full order 4 because it is determined only by the invariance under diffeomorphisms in the extra dimensions, independently of the 4D diffeomorphism.
The paper is organized as follows. We provide a brief review of our previous work [38] in the next section. In Sec. 3, we require the invariance of the action under the diffeomorphisms in the extra dimensions, and introduce the connection superfields U m (m = 4, 5) that contain the "off-diagonal" components of the sechsbein. In Sec. 4, we covariantize the derivatives with respective to the extra-dimensional coordinates by introducing another connection superfields Ψ α m (m = 4, 5). In Sec. 5, we address the Lorentz transformations that mix the 4D and the extra-dimensional indices, and show the invariance of the action under them. In Sec. 6, we check that the resultant superfield action of 6D SUGRA reduces to the known 5D SUGRA action after the dimensional reduction. Sec. 7 is devoted to the summary. In Appendix A, we collect the results of Ref [14] that discusses the 4D linearized SUGRA and the superfield description of the N = 1 superconformal transformation. In Appendices B and C, we show the diffeomorphisms and the Lorentz transformations in the component field expression, and provide the correspondence to the superfield description.
3 6D projective superspace is also discussed in Refs. [35,36,37]. 4 Some of the U m -dependent terms are treated at the linearized level due to technical difficulties.
, V I contains a 6D vector field A I M , and T contains a real scalar field σ and an anti-symmetric tensor field B M N . The hypermultiplets H A are divided into the compensator multiplets A = 1, 2, · · · , n comp and the physical ones A = n comp + 1, · · · , n comp + n phys . The lowest bosonic components of the superfields The supergauge transformations are given by The anomaly cancellation conditions constrain the numbers of the multiplets (see the introduction) and the gauge group [26,27,28]. In this paper, we do not consider such constraints, and assume that the gauge groups are Abelian, for simplicity. 6 The factor i/2 was missing for the lowest component of Σ I in Ref. [38]. Besides, V Tm = −8X m (m = 4, 5) and Υ Tα = 8D 2 Y α in the notation of Ref. [38].
where the transformation parameters Λ I are chiral superfields, and The gauge-invariant field strength superfields are given by The SUSY extension of the tensor gauge transformation: where the transformation parameters V G and Σ G are a real and a chiral superfields respectively, which form a 6D vector multiplet V G .
The superfields other than T are neutral. The field strengths invariant under this transformation are where Namely, X T and Y Tα are real linear and chiral superfields, respectively. The tensor multiplet (Υ Tα , V Tm ) is subject to the constraints: (2.14) In the global SUSY limit, these constraints reduce to the superfield version of the self-dual condition: In fact, in the limit of S E → e −iπ/4 and V E → 1, (2.14) is reduced to The field strength superfield Y Tα becomes In the second line, we have used the first constraint in (2.16). From these expressions, we This and the second constraint in (2.16) contain the self-dual condition (2.15). Thus, the antisymmetric tensor B M N in Υ Tα and V Tm becomes the self-dual tensor B + M N in the global SUSY limit.
In the SUGRA case, the second constraint in (2.14) can be solved as follows. Using the first constraint in (2.14), Y Tα can be expressed as Thus the second constraint in (2.14) is rewritten as which can be solved as where Σ T is a chiral superfield. The lowest component of Σ T is identified as Eq. (2.21) indicates that the "volume modulus" superfield V E is expressed by Υ Tα , V Tm and Σ T , and is not an independent degree of freedom.

Invariant action
The N = 1 superfield description of (the U µ -independent part of) the 6D SUGRA action provided in Ref. [38] is the metric of the hyperscalar space that discriminates the compensator multiplets from the physical ones, 7 f IJ = f JI are real constants, and 8 The matrices t I are the generators for the Abelian gauge groups.
The above action is invariant under the gauge transformation: where Λ I are chiral superfields, and the other superfields are neutral. We should also note that (2.23) becomes the 5D SUGRA action in Refs. [8,9] with the norm function: N (X) = f IJ X I X J X T (the index T denotes the 5D vector multiplet originated from the 6D tensor multiplet) after the dimensional reduction.
We list the Weyl weights of the N = 1 superfields in Table I. 7 In contrast to 4D SUGRA, an arbitrary number of the compensators is possible in 5D and 6D SUGRAs.
When n comp > 1, the superconformal gauge-fixing conditions cannot eliminate all the degrees of freedom of the compensators. So some auxiliary multiplets are necessary to eliminate them. (See Ref. [39], for example.) The number n comp determines the geometry of the space spanned by the physical hyperscalars. 8 R E is denoted as U 2 E in Ref. [38].

Chiral superspace
First, we focus on the chiral superspace in the hypersector.
In the N = 1 chiral superspace, the transformation parameters ξ m are promoted to the chiral superfields as where a m are real functions. From (B.3), (B.11) and (B.13), the chiral superfields S E , H odd , H even and Σ I transform as where H = H odd , H even . Because the first terms in the right-hand sides correspond to the shift of the coordinates x m , they have the universal structure for all the chiral superfields.
In fact, noticing that we can show that the chiral superspace part of the action (2.23), i.e., the second line of

Full superspace
Next we consider the invariance in the full superspace. There, terms originating from the shift of x m in the δ Ξ -transformation should have a common form for all superfields.
However, those for the chiral and the anti-chiral superfields have different forms. In order to accommodate them, we introduce the real superfields U m (m = 4, 5), and introduce the operator P U that shifts x m by iU m .
Then, for a chiral superfield Φ (i.e., transforms as 9 if we assume that Since U m transform nonlinearly, these correspond to the gauge fields for the δ Ξtransformation. The components of U m are identified as For an anti-chiral superfieldΦ, transforms as which has the same form as (3.8).
With the P U operation, (3.2) becomes From (2.6) and (B.12), the δ Ξ -transformation of the vector superfield V I is found to be Therefore, the combination in the first line of L H transforms as As for the factor in front of L (2) H in L H , we should note that the combination V E R E transforms as which is consistent with (B.10). This transformation law is derived from (3.52) and (3.57) explained later.
Consider the Jacobian for P U , which is calculated as which satisfies for a chiral superfield Φ. After some calculations, we can show that J P transforms as Then, we obtain Combining these transformation laws, we find

Comment on P U
Here, we give a comment on the operator P U . Let us consider a chiral superfield Φ whose components are given by Namely, the operator P U replaces the derivative ∂ µ appearing in the components with We have dropped the fluctuation of e ν µ around the background δ ν µ , and terms beyond linear in the "off-diagonal" components of the sechsbein. Recall that the index of σ µ is the flat one. So the 4D indices contracted with it should also be the flat ones. In higherdimensional SUGRA, this means that terms involving the "off-diagonal" components of the vielbein must be incorporated, which are missing in the original superfield Φ. The operator P U provides such missing terms.
For later convenience, we "covariantize" the spinor derivatives D α andDα as Then, we can also see the same effect of P U in the N = 1 SUSY algebra.
where O(U 2 ) denotes terms beyond linear in U m .

Field strength superfields
From (3.14), we can show that 11 (3.28) Hence, if we modify the field strength superfield W I α in (2.9) aŝ it transforms as which is consistent with the component transformation. However, this is not gaugeinvariant under This stems from the fact that W I α should include the field strength F µν , and where the ellipsis denotes terms beyond the linear order in the "off-diagonal" components {e ν m , e n µ }, or terms involving the fluctuation of e ν µ . The superfield defined in (3.30) only contains the first term in (3.34). Thus, we have to modify (3.30) by adding terms that depend on U m and Σ I , in order to cancel the variation (3.33). The identification of the additional terms is left for the subsequent paper, in which the gauge group is extended to non-Abelian, but such correction terms should be determined so that the transformation law (3.31) is maintained.
Next we consider the tensor multiplet. The δ Ξ -transformations of Υ Tα , V Tm and Σ T are found from (B.1) and (B.14) as The definition of the field strength X T in (2.12) is modified as Then, it transforms as The second term in δ Ξ V Tm exists because V Tm has an external index m. Thus we extend the operator P U as follows. For a chiral superfield Φ m , we define the operator Q U as 12 Since Φ m has an external index m, its δ Ξ -transformation has a form of Then we can show that Note that this has the same form as δ Ξ V Tm in (3.35). Hence, it follows that 13 Making use of these properties, W Tmα in (2.13) should be modified as (3.44) 12 The operators P U and Q U are understood as e iLU , where L U is the Lie derivative along U m . 13 Specifically, Then, it transforms as Summing (3.46) and (3.47), we obtain the δ Ξ -transformation of Y Tα defined in (2.12) as We have used the constraint (2.14).
From (3.35), we also obtain Therefore, if we modify the definition of V T in (2.21) as we find that Recall that V E = V T /X T from (2.21). Thus, from (3.37) and (3.51), we obtain which is consistent with (B.9). However, this and (3.13) are not consistent with (3.17).
Hence, we modify the definition of R E given in (2.24) in such a way that V E R E transforms as (3.17). We modify R E as The higher order terms O(U 2 ) are determined so that J S and J S transform as As a result, R E transforms as From (3.52) and (3.57), we certainly obtain the transformation law (3.17).

Invariance of action
Let us first consider the δ Ξ -invariance of the first line of L VT in (2.23). If we define we find that This is the same transformation law as that ofΣ I . Similarly, ∂ P E D P α V I also has the same transformation law. Combining these properties with (3.48), we can show that we find that As for the third line of L VT , the combination V . (3.67) From (3.37) and (3.57), we obtain Therefore, we find that can see that (3.70) We have used the property (3.19), which also ensures that Using the results obtained in this section, we can modify the action in (2.23) so that it is δ Ξ -invariant up to total derivatives. We will provide the modified Lagrangian in Sec. 5.3.

Covariantization of ∂ E
So far, we have concentrated on the δ Ξ -transformation, i.e., a diffeomorphism in the extra dimensions. In this section, we argue the consistency with its 4D counterpart, i.e., the 4D N = 1 superconformal transformation. Notice that ∂ m does not preserve the proper transformation laws for the N = 1 superconformal transformation collected in Appendix A.2.
Thus we need to introduce the connection superfields Ψ α m that transform as δ L Ψ α m = −∂ m L α (L α is the N = 1 superconformal transformation parameter), and covariantize ∂ m .

Chiral superspace
On a chiral superfield, we define the covariant derivative ∇ m as where w is the Weyl weight. Then, ∇ m H (H = H odd , H even ) transforms as at the leading order in Ψ α . 14 This is the same law as δ L H. (See (A.6).) Hence, (3.5) is modified as where This is invariant under the δ L -transformation up to total derivatives.
Next we consider the δ Ξ -transformation. This should commute with the δ Ltransformation in order for the chiral property of the N = 1 chiral superfields to be preserved. From this requirement, the δ Ξ -transformation of Ψ α m is found to be In fact, we can see that  Then, ∇ m H transforms as where We have used that As a result, the δ Ξ -transformation of (4.3) becomes total derivatives.
Note that L H has the Weyl weight 3.

Full superspace
In the full superspace, ∇ m in (4.1) is modified as where n is the chiral weight (i.e., the U(1) A charge), and the operator R U is defined by Then, from the relation: and the transformation law: we find that which leads to where Besides, the δ Ξ -transformations (3.13), (3.14) and (3.35) are modified as where ∇ P m ≡ P U∇m P −1 U , ∇ m Ξ n ≡ P U (∇ m Ξ n ) .

Rotations that mix 4D and extra dimensions
Here we consider the Lorentz transformations that mix 4D and the extra dimensions. In order to simplify the discussion, we treat the "off-diagonal" superfields U m and Ψ α m at the linearized level in this section. Then, the corresponding superfield transformation laws are given by whereṼ E ≡ V E R E , and the transformation parameter N is a complex general superfield whose θθ-component is

Invariance in hyper sector
The invariance of the action under the δ N -transformation is less manifest than the δ L -and the δ Ξ -transformations because the cancellation between the d 4 θ-and the d 2 θ-integrals occurs in the δ N -transformation. Here, we show the invariance in the hyper sector to illustrate such cancellation.
From (5.1), the hatted superfields transform as After some straightforward calculations, we can see that L  4) and up to total derivatives. We have dropped the U m -and the Ψ α m -dependent terms in the right-hand-sides. The last line in δ N L (2) H can be rewritten as This can also be rewritten as (5.8) up to total derivatives. Therefore, we obtain We should also note that Making use of these, we can show that H + h.c. = 0, (5.11) up to total derivatives. We have used the relation d 2θ = − 1 4D 2 in the d 2 θ-integration.

Kinetic terms for U m and Ψ α m
Now we consider the kinetic terms for the gravitational superfields, which originate from the 6D Weyl multiplet. Among {U µ , U m , Ψ α m , V E , S E }, only V E and S E have nonvanishing background values. Here, we treat the superfields {U µ , U m , Ψ α m } and the fluctuation parts of V E and S E at the linearized order, and neglect terms beyond quadratic in them. As shown in Appendix A, the kinetic term for U µ , L N =1 E , is given by (A.12). There is an additional term that involves the "off-diagonal" component superfields U m and Ψ α m . We define the covariant derivatives of U µ as where σ µ αα = e µ ν σ ν αα . This has the Weyl weight 0, and is invariant under the δ Ltransformation. In order to construct the δ N -invariant term, we redefine the above covariant derivatives as where ∂ µ ≡ e µ ρ e ν τ η ρτ ∂ ν . Then, the combination: is δ L -and δ N -invariant at the linearized order.
Using this combination, we can construct the following δ L -and δ N -invariant Lagrangian term.
where Ω and W are real and holomorphic functions respectively, whose explicit forms will be given in Sec. 5.3. Recall that δ Ξ Ω = ∂ m ReΞ m Ω from the results in Sec. 3. Then, we where we have dropped total derivatives, and also dropped the fluctuation part of Ω. 15 Here, since Therefore, from the δ Ξ -invariance of the action, we find

6D SUGRA Lagrangian
Here we summarize our results. The 6D SUGRA Lagrangian is expressed as H + h.c. , (5.22) 15 The superconformal gauge-fixing condition Ω| θ=0 = −3M 4 6D must be imposed in order to obtain the Poincaré SUGRA. (M 6D is the 6D Planck mass.) where L E ≡ Ω HVT 3 and The covariant derivatives ∇ E , ∇ P E and ∇ P m are defined in Sec. 4, and the field strengths are given byŴ and J P , R E and J The U µ -dependence ofŴ I α is given by (A.10). The real superfield V E is expressed as 27) and the chiral superfield Υ Tα is subject to the constraint: Note that this contains Ψ α m (m = 4, 5). This constraint indicates that either Ψ α 4 or Ψ α 5 is a dependent superfield, i.e., it can be expressed in terms of the other superfields.

Dimensional reduction to 5D
We consider the situation that the two extra dimensions are compactified on a torus, i.e., invariance. As a result, we can replace ∇ E with 1 S E ∇ 4 (−S E ∇ 5 ) in this limit. Let us consider the limit |S E | → ∞ as an example. 16 In this case, we can neglect the x 5 -dependence of the superfields, and the only extra-dimensional coordinate is y ≡ x 4 .
Thus P U is understood as the operator that shifts y as y → y + iU 4 . 16 The procedure in the limit |S E | → 0 is similar if we use the relation (3.71).

Hyper sector
First, we consider the hyper sector. The covariant derivative ∇ E becomes H in (5.24) becomes where H (5D) As for the full superspace part, we obtain where The integrands (6.4) and (6.6) agree with those in Ref. [19] at the linearized order in U 4 .

Vector-tensor sector
Next consider the vector-tensor sector. Noting that where Therefore, we obtain The field strengths Y Tα and V T become Thus, we obtain and We have used the limit of J (1) As a result, the Lagrangian in the vector-tensor sector becomes , where the indicesĪ,J,K run over T, 0, 1, 2, · · · , and the completely symmetric constant tensor CĪJK is defined as C IJT = f IJ and the other components are zero. This agrees with the 5D result in Ref. [19] at the linearized order in Ψ α y and U 4 . At the last step in (6.17), we have used the relation (6.18) which can be shown in the same way as Appendix D in Ref. [38].

Gravitational sector
Finally, we consider the gravitational sector. Since C µ E in (5.14) becomes we find thatC where This agrees with the kinetic term for U 4 and Ψ α y in Ref. [19]. Finally, we give a comment on the independence of V (5D) E defined in (6.7). Notice that S E disappears in the 5D action, and Υ Tα appears only through X T in V E after the dimensional reduction. (The Υ T -dependence of Y Tα disappears as shown in (6.12).) Thus, although V E in the 6D SUGRA action is not an independent degree of freedom (see (5.27)), is independent in the 5D SUGRA action. Namely, the degrees of freedom of S E and Υ Tα are converted into that of V (5D) E .

Summary
In this paper, we have completed the N = 1 superfield description of 6D SUGRA. Specifically, we have clarified the dependence of the action on the N = 1 superfields that contain the "off-diagonal" components of the sechsbein e ν m , e n µ , which were missing in our previous work [38]. These superfields are necessary for the invariance of the action under the full 6D diffeomorphisms and the Lorentz transformations in the N = 1 superfield description.
The corresponding superfields U m and Ψ α m play roles of the gauge fields for those transformations. Although they do not have zero-modes in many extra-dimensional models, they can give significant effects on 4D effective theory when they are integrated out, as in the case of 5D SUGRA [17,18].
Our results are collected in Sec. 5.3. The superfields U m and Ψ α m appear in the action in a nontrivial manner, but the resultant action is consistent with the 6D diffeomorphisms, 6D Lorentz transformations and the transformation laws of the component fields. Besides, it reduces to the known 5D SUGRA action in Ref. [19]. These properties ensure the reliability of our result.
In this paper, Ψ α m are treated at the linearized level. This is because we have adopted the linearized 4D SUGRA formulation [14,41,42] to describe the 4D part of the 6D Weyl multiplet. In order to treat Ψ α m at full order, we need to use the complete conformal superspace formulation [13], which is technically more complicated.
Our 6D SUGRA description is useful to construct or analyze various setups for the braneworld models that contain lower-dimensional branes or the orbifold fixed points. Besides, it is also powerful for the systematic derivation of 4D effective action that keeps the N = 1 superspace structure.
We have focused on the case of the Abelian gauge group, for simplicity. In order to extend our result to the non-Abelian case, we need to include an additional term, which is the SUGRA counterpart of (3.9) in Ref. [1] or (2.23) in Ref. [2], to ensure the gauge invariance.
We will discuss these issues in a subsequent paper.
We have included e ν µ in the above expression in order to make the counting of the Weyl weight clear. This superfield has the Weyl weight 0.
We construct a chiral superfield from a (superconformal) chiral multiplet [φ, χ α , F ] as where w denotes the Weyl weight (i.e., the D charge) of this multiplet.

A.2 Superconformal transformation
With the above definitions of the superfields, the (linearized) superconformal transformations are expressed as 19 18 A complex scalar H is 1 2 (H + iK) in the notation of Ref. [12]. 19 We take the metric convention and the definitions of the spinor derivatives of Ref. [40], which are different from those in Ref. [14].
where the transformation parameter L α is an unconstrained complex spinor superfield.
The components of L α denoted as represent the transformation parameters for P , Q, M , D, U(1) A and S, respectively.
As we can see from (A.6), U µ transforms nonlinearly, and thus it corresponds to the gauge (super)field for the δ L -transformation. We should also note that this superfield transformation preserves the chirality condition:DαΦ = 0.

A.3 Invariant action
For a given global SUSY Lagrangian: where Ω is a real function, W and f are holomorphic functions, and W α ≡ − 1 4D 2 D α V , we can make it invariant under the δ L -transformation by inserting U µ in the following way.
Here, the operation of (1 + iU µ ∂ µ ) on Φ is understood as the embedding of the chiral multiplet into a general multiplet. The modified field strength superfield W U α is invariant under the gauge transformation: where Λ is a chiral superfield.
The kinetic term for U µ is given by 20 where the Weyl weight of U µ = e ρ µ e τ ν η ρτ U ν is −2. Using the above insertion of U µ , the N = 1 (linearized) SUGRA Lagrangian is obtained by choosing where Φ comp is the compensator chiral superfield, Φ is the physical chiral superfield, the real function K(Φ U , V ) is the Kähler potential, and the holomorphic function W SUGRA (Φ) is the superpotential.

B Diffeomorphism of component fields
Under the diffeomorphism, the coordinates and the fields transform as where the transformation parameters ξ M (x) are real functions. The 6D diffeomorphism δ ξ can be divided into the 4D part δ (1) ξ with ξ µ , and the extra-dimensional part δ (2) ξ with ξ m . In this section, we focus on the δ where S E | ≡ E 4 /E 5 .

C Lorentz transformations of component fields
In this section we see the Lorentz transformations of the component fields of the superfields.

C.1 Weyl multiplet
The sechsbein e N M transforms as where the transformation parameters λ N L are real, and λ N L = −λ LN . 22 In the following, we focus on the transformations by λ In the following, we neglect the "off-diagonal" components e ν m and e n µ in the right-hand 22 The flat indices M , N , · · · are raised and lowered by η M N and η M N , respectively.
which are consistent with the second and the third transformations in (5.1). Besides, since which are consistent with the transformations in the third line of (5.1).

C.3 Vector multiplet
We can also see that the last two transformations in (5.1) are consistent with the δ λtransformations of the component fields because δ λ A I µ = λ n µ A I n = λ 4 µ e 4 4 A I 4 + e 5 4 A I 5 + λ 5 µ e 4 5 A I 4 + e 5 5 A I