N=1 superfield description of vector-tensor couplings in six dimensions

We express supersymmetric couplings among the vector and the tensor multiplets in six dimensions (6D) in terms of N=1 superfields. The superfield description is derived from the invariant action in the projective superspace. The obtained expression is consistent with the known superfield actions of 6D supersymmetric gauge theory and 5D Chern-Simons theory after the dimensional reduction. Our result provides a crutial clue to the N=1 superfield description of 6D supergravity.


Introduction
The N = 1 superfield description 1 of higher dimensional supersymmetric (SUSY) theories is quite useful when we discuss phenomenological SUSY models with extra dimensions.
It makes the derivation of 4-dimensional (4D) effective theories easier since the Kaluza-Klein mode expansion can be performed keeping the N = 1 off-shell structure. Besides, the action is expressed compactly, and general setups can be treated. Since higher-dimensional SUSY theories have extended SUSY, the full off-shell formulations are complicated and less familiar, or do not even exist for theories higher than 6 dimensions (6D). In contrast, the N = 1 superfield description is always possible because it only respects part of the full offshell SUSY structure. Hence it is powerful especially when we describe interactions between sectors whose dimensions are different, such as the bulk-boundary couplings in 5D theories compactified on S 1 /Z 2 . For the above reasons, a lot of works along this direction have been published [1]- [9].
When we discuss a realistic extra-dimensional models, the moduli play important roles.
They have to be stabilized to finite values by some mechanism, and are often relevant to the mediation of SUSY-breaking to the visible sector. In order to treat the moduli properly, we need to consider supergravity (SUGRA). The N = 1 superfield description of 5-dimensional (5D) SUGRA is already obtained in Refs. [5]- [9]. Making use of it, the moduli dependence of the 4D effective action can systematically be derived [10]- [13]. Our aim is to extend the 5D superfield action to 6D. Since the minimal number of SUSY is the same in the 5D and 6D cases, the desired N = 1 description is expected to be similar to that of 5D theories. However, there is an obstacle to a straightforward extension of the 5D result. In contrast to the 5D case, the 6D superconformal Weyl multiplet contains an anti-self-dual antisymmetric tensor T − M N L (M, N, L: 6D Lorentz indices) [14]. This leads to a difficulty for the Lagrangian formulation, similar to that for type IIB SUGRA. This difficulty can be evaded by introducing a tensor multiplet, which contains an antisymmetric tensor B + M N whose field strength F + M N L ≡ ∂ [M B + N L] is subject to the self-dual constraint [14]. Combining this multiplet with the Weyl multiplet, we obtain a new multiplet 2 that contains an unconstrained antisymmetric tensor B M N , whose field strength is given by the sum of T − M N L and F + M N L . Namely, the off-shell formulation of 6D SUGRA requires the existence of the tensor field B M N , which is not a necessary ingredient in 5D SUGRA. 3 The off-shell action of 6D SUGRA is provided in Refs. [14,15]. In that action, the tensor field B M N is coupled to the vector fields. Thus, in this paper, we clarify how the vector-tensor couplings are expressed in terms of N = 1 superfields. Since we focus on these couplings, we do not consider the gravitational couplings in this paper. In this sense, this work is the generalization of Ref. [1] including the vector-tensor couplings. For our purpose, the projective superspace formulation [16,17,18] is useful. In fact, the N = 1 superfield description of 5D SUGRA can be derived from the action in the projective superspace [8].
As for 6D SUGRA, the off-shell action in this formulation is provided in Ref. [19]. 4 We derive the N = 1 superfield action from it.
The paper is organized as follows. In Sec. 2, we provide a brief review of N = 2 supersymmetric actions in the projective superspace. In Sec. 3, we decompose N = 2 superfields into N = 1 superfields, and express the vector-tensor couplings in terms of the latter. We also clarify the relation between our result and the known N = 1 superfield description of 6D SUSY gauge theory or 5D SUSY Chern-Simons theory through the dimensional reduction. Sec. 4 is devoted to the summary. In the appendices, we list our notations for spinors, and show explicit derivations of some of the results in the text.
The natural conjugate operation in the projective superspace is the combination of the complex conjugate and the antipodal map on CP 1 (ζ * → −1/ζ), which is called the smile conjugate denoted asΞ Then the N = 2 SUSY invariant action formula is given by [19,22] where C is a contour surrounding the origin ζ = 0, the "Lagrangian superfield" L(x, Θ, ζ) is a smile-real projective superfield (L = L), and The complex number η is chosen arbitrarily as long as 1 + ζη = 0. In fact, the action (2.5) is independent of η.

Explicit forms of Lagrangians
A 6D hypermultiplet is described by an arctic superfield Υ, which is a projective superfield that is non-singular at the north pole of CP 1 (ζ = 0). Namely, it is expanded as A 6D vector multiplet is described by a tropical superfield V , which is a smile-real projec- Namely, it is expanded as Using these projective superfields, the Lagrangian superfield L in the hypermultiplet sector is given by In the following, we consider Abelian gauge theories, for simplicity.
In contrast to the above multiplets, a 6D tensor multiplet is not described by a projective superfield. As first shown in Ref. [23], it can be described by a constrained real superfield Φ that satisfies or equivalently described by an SU(2)-Majorana-Weyl spinor superfield T iα constrained by where the parentheses denote the symmetrization for the indices. We can identify these superfields as but we can also regard them as independent tensor multiplets. From these two superfields, we can construct a projective composite superfield, where T [1]α ≡ −ζT 1α + T 2α . This certainly satisfies the condition D [1] α T = 0 due to the constraints (2.11) and (2.12). For an SU(2)-Majorana-Weyl spinor Ψ iα , a quantity Ψ [1]α ≡ −ζΨ 1α + Ψ 2α is transformed by the smile conjugation as where the overline denotes the covariant conjugation defined by (A.20), and we have used (A.23). Using this property, it is shown that T [2] is smile-real (T [2] = T [2] ), and thus it can be the Lagrangian superfield for the tensor multiplets.
L tensor = T [2] . (2.16) Besides the description by the tropical superfield, a 6D vector multiplet is also described by a superfield F iα subject to the same constraint as (2.12) if it is further constrained by As we will see later, this superfield contains the field strength of the gauge field, and thus gauge-invariant. Using F iα with Φ and V , we can construct the Lagrangian superfield for the vector-tensor couplings as where The action constructed from the above Lagrangian superfields (2.10), (2.16) and (2.18) is invariant under the following gauge transformations.
where the transformation parameters Λ and G iα are an arctic superfield and a constrained superfield that satisfies the same constraints as (2.12) and (2.17), respectively.

N = 1 superfield description
In this section, we express the N = 2 invariant action in the previous section in terms of N = 1 superfields. For this purpose, it is convenient to devide the bosonic coordinates x M into the 4D part x µ (µ = 0, 1, 2, 3) and the extra-dimensional part z ≡ 1 2 (x 4 + ix 5 ) andz. As for the fermionic coordinates Θ iα , they are decomposed into (θ α ,θα) that describes the N = 1 subsuperspace we focus on and the rest part (θ ′α ,θ ′α ) as shown in (A.28). We follow the notations of Ref. [24] for the 2-component spinor indices.

Superfield action formula
Since the action (2.5) is independent of the choice of η, we choose it as η = 0 in the following. Then, where D α andD ′α are defined in (A.30), and we have used (A.8) and (A.29). Since the Lagrangian superfield L is a projective superfield, it satisfies D [1] α L = 0. From (A.29), this is rewritten as where ∂ ≡ ∂ z = ∂ 4 − i∂ 5 , and we have used (A.32). Therefore, the action (2.5) becomes where a total derivative term is dropped, and the symbol | denotes the projection θ ′ = 0.
For a given projective superfield Ξ(x, Θ, ζ), its expansion coefficients Ξ n (x, Θ) in (2.2) satisfy which comes from the constraint (2.1). Note that Ξ n (x, Θ) is decomposed into the following The condition (3.5) provides constraints on these N = 1 superfields. The action formula (3.4) is expressed in terms of them. Although each projective superfield contains infinite number of N = 1 superfields, only a finite small number of them survive in the final expression of the action as we will see below.
As a simple example, let us consider a free hypermultiplet. The Lagrangian is given by Since the arctic superfield Υ does not have terms with negative power in ζ (i.e., Υ n = 0 for n < 0), the constraint (3.5) becomes Thus the constraints on Υ 0 | and Υ 1 | are isolated from the other N = 1 superfields.
We have used (A.32). Note that Υ n | (n ≥ 2) are unconstrained superfields. 7 Hence they can be easily integrated out and obtain This is further rewritten as where Φ ≡ Υ 0 | is a chiral superfield, and ξ and κ are unconstrained N = 1 superfields. In fact, integrating out κ andκ, this reduces to (3.10) with ξ = Υ 1 |. On the other hand, if we integrate out ξ andξ, we obtain [22] whereΦ ≡D 2 κ is another chiral superfield, up to total derivatives. This is consistent with However, since the latter does not appear in the action, the former can be regarded as an unconstrained superfield.

Decomposition into N = 1 superfields
The constraint (2.12) is rewritten as Since D [1] α , D where the prepotential P [−2] is a ζ-independent N = 2 superfield, which is a real scalar.
The overall ζ-dependence is determined so that T [1]α satisfy In the 2-component-spinor notation, (3.14) is rewritten as We have used (A.8) and (A.29).
Since T [1]α is a linear function of ζ, the prepotential P [−2] should satisfȳ From the linear and constant terms in ζ, we can read off the components T iα as Then, Φ constructed by (2.13) is calculated as The N = 2 superfield P [−2] is decomposed into the following N = 1 superfields.
Then, (3.17) is translated into the following constraints. 8 When the spinor derivatives D α andDα act on N = 1 superfields, they are understood From the second and the third constraints in (3.21), we obtain Namely, p α 4 is a complex anti-linear superfield and expressed as where q 4 is a complex scalar superfield. The fourth constraint in (3.21) indicates that is a chiral superfield. Thus χ α can be expressed as which indicates that is a complex linear superfield, i.e.,D 2 Z α = 0.

N = 1 description of tensor multiplet
From (3.18) and (3.20), we obtain (3.30) where Although A 0 cannot be expressed in terms of only Y α and Z α ,DαA 0 can. In fact, after some calculations, we obtain where we have used that Z α − 4∂Y α = D α X(= −iT 1α |). Thus, A 0 is expressed as where φ 0 is a chiral superfield that is determined so that A 0 is real.
In summary, the 6D tensor multiplet is described by the spinor superfields Y α and Z α , where the latter is constrained byD 2 Z α = 0.

N = 1 description of vector multiplet
As mentioned in Sec. 2.2, a 6D vector multiplet can also be described by the constrained superfield F iα . This is decomposed into N = 1 superfields in a similar way to the tensor multiplet.
where Z α F and φ F 0 are a complex linear and a chiral superfields, respectively. In contrast to the tensor multiplet, F iα is further constrained by (2.17), which indicates that This is regarded as the Bianchi identity, and solved as where V is an unconstrained real superfield. Then, Z α F is expressed as As mentioned in Sec. 2.2, F iα are invariant under the gauge transformation, where Λ is a chiral superfield. Especially, F 2α is proportional to the field strength superfield, Since φ F 0 is a chiral superfield and A F 0 is real, we find that φ F 0 = 8∂Σ. Namely, where 4 ≡ ∂ µ ∂ µ , and is the projection operator [24].
In summary, the 6D vector multiplet is described by a chiral superfield Σ and a real superfield V , which are independent of each other.

Vector-tensor couplings
Now we consider the vector-tensor couplings. Note that F [2] defined in (2.19) is an O(2) multiplet, i.e., where F [2] 0 is real. Since we can calculate F [2] 1 | and F [2] 0 | after some calculations by using the results in the previous subsections as Note that the real linear superfield Φ T and and the chiral superfield W α T are not independent. As shown in Appendix B, they are related through From these relations, we obtain where we have used that P T Φ T = Φ T andD 2 D 2 W α T = 16 4 W α T . Namely, Φ T and W α T are on-shell. This stems from the fact that the 6D tensor multiplet contains a self-dual tensor field B + µν . In the 6D global SUSY theories, the tensor multiplet cannot be described as offshell superfields, 9 and thus should be treated as external fields. As shown in Ref. [14,15], the off-shell description of the tensor multiplet becomes possible by combining the Weyl multiplet when the theory is promoted to SUGRA. Therefore, from (2.18), the Lagrangian in the vector-tensor sector is where V 0 and V 1 are the coefficient superfields in the tropical superfield (2.9). Using 2 and performing the partial integrals, the above Lagrangian is rewritten as The second line in (3.52) is further rewritten as where we have used (3.49). Thus, L VT becomes This is our main result. This contains the result in Ref. [1] as a special case: Φ T = 1 and W α T = 0, which corresponds to the case where the tensor multiplet is absent. In such a case, (3.56) becomes

Dimensional reduction to 5D
Here we consider the dimensional reduction of (3.56) to five dimensions by neglecting the x 5 -dependence of the N = 1 superfields. Then (3.49) becomes Since the right-hand-side of the first equation is now real, W α T satisfies the Bianchi identity D α W T α =DαWα T . Hence it is a field-strength superfield.
where V T is a real superfield. Substituting this into the second constraint in (3.58), we is a chiral part of ∂ 4 V T . Then, the first constraint in (3.58) is rewritten as Namely, the 6D tensor multiplet becomes an (on-shell) 5D vector multiplet after the dimensional reduction. 11 As shown in Appendix C, the Lagrangian (3.56) becomes the following expression after the dimensional reduction. This agrees with the 5D supersymmetric Chern-Simons terms [1,26].

Bilinear terms in tensor multiplets
In this subsection, we consider the Lagrangian terms that consist of only tensor multiplets.
It is given by (2.16). The Lagrangian is expressed as where T [2] = ζT [2] 1 + T (3.67) 10 Note that Φ T is a real linear superfield, i.e., Φ T = P T ∂ 4 V T . 11 Although there exists a 5D tensor field among the fields obtained from the 6D tensor field B + MN by the dimensional reduction, such a field is dual to a 5D vector field. The duality between the 5D tensor (gauge) multiplet and the 5D vector multiplet is explicitly shown in component fields in Ref. [25].
Using the expressions in Sec. 3.3, T [2] 0 | is calculated as Here we treat two tensor multiplets (Φ T , W α T ) originating from Φ and (X,Ỹ α ) originating from T iα as independent multiplets. The Lagrangian (3.66) is then expressed as We have used (3.49), and dropped total derivative terms. At the last step, we have used that This Lagrangian can be further rewritten as At the last step, we have used the fact that W α T andZ α are a chiral and a linear superfields. This Lagrangian vanishes due to the second constraint in (3.49). However we can relax that constraint if we regardỸ α as the Lagrange multiplier. In that case, the constraint is obtained as the equation of motion forỸ α . As shown in Appendix B, that constraint is necessary in order for F [2] defined in (2.19) to satisfy D [1] α F [2] = 0, which is relevant to the N = 2 SUSY invariance of the action. Thus, in such a case, the full N = 2 SUSY invariance of the vector-tensor coupling terms (3.56) is ensured only at the on-shell level. 12 Nevertheless, (3.72) is expected to play an important role when we promote the theory to SUGRA. It corresponds to (2.14) in Ref. [23], which is described in the harmonic superspace.

Identification of component fields
where the ellipses denote fermionic or auxiliary fields.
The 6D tensor multiplet contains a real scalar field σ and a self-dual tensor field B + M N , which satisfy [23] 4 + ∂∂ σ = 0, where the vector multiplet (Σ G , V G ) is the transformation parameter that satisfies the on-shell condition. Note that Φ T and W T α are invariant under this transformation. In components, this gauge transformation is expressed as where λ M are the transformation parameters. From the conditions D 2 Φ T =D 2 Φ T = 0 and (3.49), we find that σ and B + M N are embedded into Φ T and W T α as where C µ ≡ B + µ4 + iB + µ5 , and W T α is expressed in the chiral basis (x µ + iθσ µθ , z,z, θ,θ). Note that these expressions are invariant under (3.76).

Summary
We have derived the N = 1 superfield description of supersymmetric coupling terms among 6D tensor and vector multiplets from the projective superspace action provided in Ref. [19]. This is necessary to describe 6D SUGRA in terms of N = 1 superfields. Our result contains the result in Ref. [1] as a special case. It also reproduces the 5D supersymmetric Chern-Simons terms after the dimensional reduction.
The tensor multiplet is described by two complex spinor superfields Y α and Z α , where Z α is constrained asD 2 Z α = 0. They appear in the action in the forms of a real linear superfield Φ T and a chiral spinor superfield W T α defined by (3.48). These superfields are constrained by (3.49), which leads to the on-shell conditions. Thus they should be treated as external fields. This stems from the fact that the 6D tensor multiplet contains a self-dual tensor field B + M N . As shown in Ref. [14] in the component fields, the on-shell condition for the tensor multiplet can be relaxed when the theory couples to the gravity.
Our result (3.56) provides a good starting point to obtain the N = 1 superfield description of 6D SUGRA. We will discuss this issue in the subsequent paper.

A.2 Conjugation matrices
An 8-component Dirac spinorΨ is decomposed into 4-component Weyl spinors aŝ where the signs denote eigenvalues of Γ 7 . The Dirac conjugate ofΨ is defined aŝ whereÂ satisfiesÂ The explicit form ofÂ is given bŷ whereᾱ denotes a 4-component spinor index of the complex conjugate of the Weyl spinors. where These matrices satisfy The charge conjugation matrixĈ, which satisfieŝ is constructed fromÂ andB asĈ Thus the charge conjugation flips the 6D chirality.
The covariant conjugate of a spinorΨ is defined aŝ This operation is not a Z 2 transformation sincê For an SU(2)-doublet spinorΨ i (i = 1, 2), a Z 2 transformation is obtained by combining the covariant conjugation with lowering the SU(2) index, Thus we can impose the SU(2)-Majorana condition, Here the antisymmetric tensors ǫ ij and ǫ ij are chosen as ǫ 12 = ǫ 21 = 1. Since the covariant conjugation preserves the 6D chirality, we can impose this condition on 6D Weyl spinors.
Namely, the SU(2)-Majorana-Weyl condition is expressed in the 4-component-spinor notation as In the two-component-spinor notation, the SU(2)-Majorana-Weyl spinors are expressed as

A.3 Covariant spinor derivatives
We introduce the Grassmann coordinates Θ iα , which form an SU(2)-Majorana-Weyl spinor with the 6D chirality −. Then the covariant spinor derivatives are defined as In the 2-component-spinor notation, Θ iα are expressed as Then, the covariant spinor derivatives are expressed as and ∂ ≡ ∂ 4 − i∂ 5 . The algebra (A.27) is decomposed as We list some useful formulae following from this algebra. From the definition (3.48), We have used thatDαX =Zα − 4∂Ȳα and D 2Zα = 0.

C Derivation of 5D Lagrangian
We derive (3.64) from (3.56) after the dimensional reduction to 5D. By using (3.58) and (3.61), we can calculate We have also used D α W α =DαWα, and dropped total derivative terms. Thus, after the dimensional reduction to 5D, (3.56) becomes L (5D) We have dropped total derivative terms, and used that Using (A.31) and (A.32), we can show that up to total derivatives. Thus, (C.2) is rewritten as If we relabel (Σ, V ) and (Σ T , V T ) as (Σ 1 , V 1 ) and (Σ 2 , V 2 ), this is expressed as (3.64).