New superconformal multiplets and higher derivative invariants in six dimensions

Within the framework of six-dimensional ${\cal N}=(1,0)$ conformal supergravity, we introduce new off-shell multiplets ${\cal O}{}^{*}(n)$, where $n=3,4,\dots,$ and use them to construct higher-rank extensions of the linear multiplet action. The ${\cal O}{}^{*}(n)$ multiplets may be viewed as being dual to well-known superconformal ${\cal O}(n)$ multiplets. We provide prepotential formulations for the ${\cal O}(n)$ and ${\cal O}{}^{*}(n)$ multiplets coupled to conformal supergravity. For every ${\cal O}{}^{*}(n)$ multiplet, we construct a higher derivative invariant which is superconformal on arbitrary superconformally flat backgrounds. We also show how our results can be used to construct new higher derivative actions in supergravity.


Introduction
Recently, there has been some interest in higher derivative superconformal invariants in six dimensions (6D) [1,2,3], e.g. in the context of general relations between the conformal and chiral anomaly coefficients in 6D N = (1, 0) superconformal theories. One example of such higher derivative models is the N = (1, 0) superconformal vector multiplet theory. It was formulated in Minkowski superspace in [4], and coupled to conformal supergravity in [5]. 1 In this paper, we propose new off-shell superconformal multiplets and use them to construct higher derivative invariants that are quadratic in the fields of the new multiplets and naturally generalise the superconformal vector multiplet action.
We recall that the superconformal vector multiplet action is the supersymmetric extension of an F ✷F term, where F is the two-form field strength of a gauge one-form. Its construction starts with the supersymmetric BF invariant 2 which is, schematically, the product of a vector multiplet with a linear multiplet. It may be written as a full superspace integral in N = (1, 0) Minkowski superspace as follows 3 where the superfield L ij = L (ij) satisfies the defining differential constraint for the linear multiplet [7,8], The superfield W αi satisfies the differential constraints appropriate for a vector multiplet, see [8,9] and references therein. The real iso-triplet U ij in (1.1) is the 6D counterpart of the Mezincescu prepotential [10] for the vector multiplet, and ρ αi is the prepotential [11] for the linear multiplet. The prepotentials ρ αi and U ij play the role of gauge fields, while L ij and W αi are the gauge-invariant field strengths of the linear and vector multiplets, respectively. One can construct the supersymmetric F ✷F action by building a linear multiplet from the fields of the vector multiplet. The appropriate composite linear multiplet is (1.4) 1 Its component Lagrangian in conformal supergravity was worked out in [6]. 2 The notation refers to the six-form which is the product of the field strength F and the four-form potential B contained in the linear multiplet. It is also known as the linear multiplet action. 3 The second form of the invariant in (1.1) follows from the expressions for the covariant field strengths W αi and L ij in terms of their prepotentials.
The F ✷F action is the BF invariant (1.1) with the replacement L ij → L ij : The above action turns out to be superconformal 4 and equivalent to the harmonic superspace action given in [4].
In order to formulate generalisations of the above construction, we will consider O(n) multiplets as higher-rank cousins of the linear multiplet, O(2), and uncover a new family of multiplets akin to the vector multiplet. These multiplets will allow us to generalise the BF action in such a way that it is now, schematically, the product of the O(n) multiplet with the new multiplet. Using this invariant we will show how to generate a whole family of superconformal higher derivative invariants bilinear in the new multiplets.
In what follows, we will use the superspace formulation for conformal supergravity in [5] and refer the reader there for our notation and conventions. The results in Minkowski superspace are obtained by setting the super-Weyl tensor W αβ to zero and replacing the conformal superspace covariant derivatives with those of Minkowski superspace, In this paper, all (super)fields are chosen to transform in irreducible representations of the R-symmetry group SU(2) so that they are symmetric in their SU(2) spinor indices, for instance L i 1 ···in = L (i 1 ···in) .

The O(n) multiplets and their prepotentials
Given an integer n ≥ 1, the O(n) multiplet is a primary superfield L i 1 ···in of dimension 2n that satisfies the differential constraint 5 The contraint (2.1) is primary since the S-supersymmetry generator S β j annihilates the left hand side (without requiring the constraint). This condition actually fixes the dimension of L i 1 ···in . The complex conjugate of L i 1 ...in , which is defined by , the higher-derivative linear multiplet (1.4) is a primary superfield. 5 The O(n) multiplets are well known in the literature on supersymmetric field theories with eight supercharges in diverse dimensions. For 4D N = 2 Poincaré supersymmetry, general O(n) multiplets, with n > 2, were introduced in [12,13,14]. The case n = 4 was first studied in [15]. The terminology "O(n) multiplet" was coined in [16]. As 6D N = (1, 0) superconformal multiplets, their complete description was given in [17] following the earlier approaches in four and five dimensions [18,19,20,21].
is also an O(n) multiplet. If n is even, n = 2m, one can consistently define real O(2m) multiplets by imposing the condition L i 1 .
The n = 1 case corresponds to the 6D version of the Fayet-Sohnius hypermultiplet [22], which is necessarily on-shell in six dimensions (the constraint ∇ (i α L j) = 0 implies ✷L i = 0) and, therefore, it will not be considered in what follows. The case n = 2 is the linear multiplet, which can be formulated in conformal superspace in terms of an unconstrained prepotential ρ i α as follows 6 where we have introduced ∇ αijk : δ and the covariant projection operator 7 The prepotential is defined up to the following gauge transformations where τ and τ α β ij are dimensionless primary superfields. These gauge transformations leave L ij invariant. In the flat-superspace limit, eq. (2.3) reduces to the one given in [11].
The field content of the O(n) multiplets follows from taking spinor covariant derivatives of the superfield L i 1 ···in and analysing the consequences of the constraint (2.1). The independent component fields are summarised in Table 1 from which it can be seen that the number of off-shell degrees of freedom is 8(n − 1) + 8(n − 1). The cases n ≤ 3 are special since the component fields for which the number of SU (2) indices become negative are truncated away. Note that, as the Dirac matrices are anti-symmetric, two antisymmetric Lorentz spinor indices are equivalent to a vector index, while three antisymmetric lower Lorentz spinor indices are equivalent to a single raised Lorentz spinor index by making use of the Levi-Civita symbol ε αβγδ . The n = 2 case is exceptional in the sense that the linear multiplet is a gauge multiplet because G a = 1 4 (γ a ) αβ G αβ may be identified with the dual of the field strength of a four-form gauge field since G a is divergenceless. For n ≥ 3 the supermultiplets do not possess similar restrictions at the component level and are therefore not gauge multiplets.
Since we will be concerned with the cases for which prepotential formulations exist, we restrict ourselves to the class of multiplets with n ≥ 2. Indeed, in addition to n = 2, which was discussed above, prepotential formulations can also be given for n ≥ 3. As n = 3 and n > 3 need separate treatment, we will discuss them in turn.

n = 3
We can solve the defining constraint for the O(3) multiplet in terms of an unconstrained prepotential as follows where the prepotential V α is defined up to the gauge transformations One can check that L ijk is primary using the fact that V α is a primary superfield of dimension 7/2.

n ≥ 4
For these cases, the solution of (2.1) is which is invariant under the gauge transformations of the unconstrained prepotential The O(n) multiplet (2.8) is primary. This is a consequence of the fact that the prepotential V i 1 ···i n−4 is a primary superfield of dimension 2n − 2. The analogue of the above prepotential formulation in 5D appeared in appendix G of [23].

New superconformal multiplets and their gauge prepotentials
We will now introduce new multiplets that generalise the vector multiplet. To do this we focus on a key property of the vector multiplet, which is that an invariant can Table 2: Field content of the vector multiplet be constructed by multiplying its prepotential with an O(2) multiplet as in eq. (1.1). This implies that the index structure of the prepotential for the vector multiplet is such that it can be contracted with L ij to form an SU(2) singlet. In generalising this property we seek to find multiplets that have prepotentials of the form U i 1 ···in that can be used to build a gauge invariant expression with an O(n) multiplet as an integral over full superspace. In this way the new multiplets will be 'dual' to the O(n) multiplets.
The field content of the vector multiplet is summarised in Table 2 Table 1 shows that every component field of the vector multiplet can be contracted with a component field of the O(2) multiplet to give a scalar of dimension 6. The analogous property also holds for the component fields of the new multiplets. We have to distinguish three cases, depending on the number of SU(2) indices of the prepotential. Note that while the vector multiplet is a gauge multiplet, the new multiplets are not.

n = 3
Starting from a primary superfield T α of dimension −3/2 we impose the differential constraint This constraint can be solved in terms of a primary dimension −4 unconstrained prepotential U ijk as follows where the prepotential is defined up to the gauge transformations The field content of this new multiplet is summarised in Table 3.

n = 4
We can define another supermultiplet by a primary dimension −4 scalar superfield T satisfying the constraint One can solve this in terms of an unconstrained prepotential U ijkl as follows 8 where U ijkl is primary of dimension −6. It is straightforward to check that T is primary given U ijkl is primary. One can also check that U ijkl is defined up to the gauge transformations To check the invariance of T , the following identities are useful: The field content of this multiplet is summarised in Table 4 with n = 4.

n ≥ 5
As a generalisation of the scalar superfield we used in the previous subsection, we can introduce superfields T i 1 ···i n−4 of dimension 4 − 2n satisfying the constraint Its solution in terms of an unconstrained prepotential is where U i 1 ···in is defined up to the gauge transformations The field content of these multiplets is summarized in Table 4. The form of the constraints (3.8) is 'transverse-like' as opposed to the 'longitudinal-like' constraints (2.1) of the O(n) multiplets and for this reason we may refer to the new multiplets as transverse multiplets. Although in this sense it might be natural to refer to the O(n) multiplets as longitudinal multiplets, we will stick to the well-established nomenclature for them. As for the transverse multiplets, we will denote them by O * (n) as a reminder that they were constructed to be 'dual' to the O(n) multiplets in the sense that one will be able to use them to construct an invariant as a product of it with an O(n) multiplet. For convenience, we also define O * (2) to be the vector multiplet, O * (3) to be the multiplet described by the superfield T α and O * (4) to be the multiplet described by the scalar superfield T . We have not defined O * (1) here but we will introduce it in section 6.

New superconformal invariants
We will now use the new multiplets and their prepotentials to generalise the BF invariant (1.1).

Generalised BF invariants
With the help of the results of the previous sections, one can immediately write down a supersymmetric invariant where E = Ber(E M A ) and, as usual, E A = dz M E M A is the supervielbein. Taking into account the differential constraint on the O(n) multiplet, it is straightforward to check that I (n) is invariant with respect to the gauge transformations of the prepotential U i 1 ···in . In the n = 2 case the invariant (4.1) corresponds to the standard BF invariant (1.1), while for n ≥ 3 we obtain new generalisations. 9 In the flat case, the supersymmetric invariant (4.1) can be seen to naturally originate in harmonic superspace, see Appendix A.
Note that one can equivalently define the generalised BF invariants by explicitly using the prepotential for the O(n) multiplets. For n = 3 eq. (4.1) is equivalent to while for n ≥ 4 it is equivalent to However, as one can see from the above, the fact that the prepotential obtains a Lorentz index for the n ≤ 3 cases means that we cannot treat all cases ubiquitously in this form.

Superform realisation of the invariants
It is worth mentioning that a manifestly gauge-invariant form of the invariant (4.1) for n ≥ 3 can be given by making use of the superform construction in [24,5]. It is based on the existence of a closed six-form in superspace that describes a locally superconformal invariant. It is expressed entirely in terms of a basic primary superfield A α ijk of dimension 9/2 which satisfies the differential constraint ∇ (i (α A β) jkl) = 0.
The superfield A α ijk plays the role of a supersymmetric Lagrangian similar to the chiral Lagrangian in 4D. Its component structure was elaborated in [6].
The invariant (4.1) can be equivalently described by a composite superfield A α ijk which is, roughly, the product of an O * (n) multiplet and the corresponding O(n) multiplet. More specifically, for n = 3 one has while for n ≥ 4, one takes A α ijk ∝ (1 + n)T i 1 ···i n−4 ∇ αl L ijkli 1 ···i n−4 + (5 + n)(∇ αl T i 1 ···i n−4 )L ijkli 1 ···i n−4 . (4.5) The above representations are democratic in the sense that they put both multiplets on the same footing. 10 The above results are useful in elaborating the component structure since they make the connection with the component action more direct through the results of [5,6]. Note, however, that while a manifestly gauge-invariant description exists for n ≥ 3, an analogous description in terms of the superfield A α ijk does not exist for n = 2.

Higher derivative superconformal invariants
With the results obtained so far, it is possible to construct composite O(n) multiplets and use them to construct higher derivative actions for the O * (n) multiplets. In this section we will restrict ourselves to a superconformally flat geometry, where the super-Weyl tensor vanishes, W αβ = 0.
One can construct a composite O(3) multiplet out of the superfield T α as follows The power of the d'Alembertian operator, ✷ := ∇ a ∇ a , is chosen to ensure that L ijk has the right dimension and is primary in the superconformally flat geometry. To check that it is primary, the following identities are useful: A higher derivative invariant can now be described by plugging in L ijk into the generalised BF invariant (4.1), giving This invariant is the supersymmetric extension of a λ α ✷ 4 ∂ αβ λ β term.
We can also describe higher derivative invariants for the O * (n) multiplets with n ≥ 4. We begin by building a composite O(n) multiplet from the superfield T i 1 ···i n−4 : 10 Any full superspace invariant d 6|8 z E L, with the Lagrangian L being a primary scalar superfield of dimension +2, can equivalently be described by (4.5) with a composite O(4) multiplet where T is chosen to be nowhere vanishing. Its dependence on T is artificial since the invariant is independent of T modulo a total derivative.
One can show that in a superconformally flat geometry L i 1 ···in is primary, using again eqs. (5.2) and the following identities: The higher derivative invariants are now described by the generalised BF invariant (4.1) with L i 1 ···in , i.e.

Discussion
A family of higher derivative superconformal actions was proposed in [2,3]. Upon appropriate identification of the fields, one can see that the actions sketched in [2,3] correspond to the higher derivative superconformal invariants presented in this paper. Note that analogous higher derivative invariants realised in terms of O(n) multiplets is not possible since all its fields have dimension 2n or higher. As a result, the only case when a local superconformal action, that is quadratic in the fields, could exist is n = 1, but this corresponds to an on-shell hypermultiplet. This tells us that such higher derivative invariants require the use of the O * (n) multiplets.
The question whether the higher derivative invariants of the previous section can be lifted to conformal supergravity is a separate problem. As a matter of fact, it appears not to be the case. The point is that (5.6) should include the termφ✷ 2n−1 φ once one truncates to N = 0 and it is known that there does not exist a locally conformal completion for n ≥ 3 [25]. The same is expected for the higher derivative invariant (5.3). In conformally flat backgrounds the obstruction for their existence is, however, absent [26].
The results in this paper may have interesting consequences in the construction of new higher derivative supergravity theories. To construct Poincaré supergravity theories within a superconformal framework, one must use a conformal compensating multiplet, which is a real scalar superfield Φ that we choose, without loss of generality, to have dimension one. In this framework it becomes apparent that the construction of composite O(n) multiplets from the O * (n) multiplets can be made in many more ways using the results of this paper. For instance, we can construct O(n) multiplets with n ≥ 4 as follows Furthermore, we can choose the compensator as Φ −4(n−2) = T i 1 ···i n−4 T i 1 ···i n−4 leading to a four-derivative invariant for the O * (n) multiplet once one plugs the composite into the generalised BF invariant (4.1). We will explore the supercurrents of such theories in a forthcoming paper [27].
It is interesting to note that supersymmetric invariants built out of O * (n) for n ≥ 4 may be mapped to invariants of the O(2) multiplet for even n = 2m. One only needs to make use of an O(2) multiplet L ij such that L 2 := L ij L ij is nowhere vanishing. Given the O(2) multiplet, we can choose the O * (2m) multiplet to be 11 In this way the resulting invariant will be expressed entirely in terms of the O(2) multiplet.
It is also possible to construct composite O * (n) multiplets out of O(n) multiplets leading to superconformal invariants for the O(n) multiplets. For instance, for n ≥ 4 we can take the composite We can choose the compensator to be Φ 4n = L i 1 ···in L i 1 ···in and then plugging the result into the invariant (4.3) gives a four-derivative invariant for the O(n) multiplet. Many more invariants can be constructed by iterating eqs. (6.1) and (6.3) in various ways, and using the fact that there is an obvious multiplication defined on the space of O(n) multiplets and the space of O * (n) multiplets. Similar constructions appeared in 4D for the O(2) multiplet in [28].
In this paper, we have defined the O * (n) multipelts for n ≥ 2. The O * (n) multiplet has the same number of degrees of freedom as the corresponding O(n) multiplet as follows from the generalised BF invaraint (4.1). We have ignored the n = 1 case since the O(1) multiplet is on-shell and does not have a prepotential formulation. However, it is interesting to ask whether there exists another on-shell multiplet possessing the same number of degrees of freedom as that of the O(1) multiplet that one could naturally add to the class of O * (n) multiplets. Such a superconformal multiplet is described by a primary superfield T α of dimension −1/2 satisfying the constraint It might be appropriate to define this multiplet as the O * (1) multiplet.
It is worthwhile mentioning that the new multiplets and constructions introduced in this paper imply the existence of analogues in lower spacetime dimensions. We leave their exploration for future work.

A Harmonic superspace construction
In this appendix we present a harmonic superspace origin of the supersymmetric invariant (4.1) in the flat case. Our construction is an extension of the known procedure to read off the Mezincescu prepotential from the analytic prepotential [30,31].
Given an O(n) multiplet L i 1 ...in (z) in Minkowski superspace, with n ≥ 2, we associate with it a harmonic superfield which is analytic and obeys the harmonic shortness constraint Then we can define a supersymmetric invariant in which the integrand involves an analytic potential Ω (4−n) . The integration in (A.4) is carried out over the analytic subspace of the harmonic superspace. It is defined by for any analytic Lagrangian L (4) . Here the u-integral denotes the integration over the group manifold SU(2) defined as in [30,31]. Here the prepotential U (−n) (z, u) is an unconstrained harmonic superfield which is defined modulo gauge transformations δU (−n) = D + α ξ (−n−1) α , (A. 8) with the gauge parameter ξ (−n−1) α (z, u) being an unconstrained harmonic superfield.
The analyticity constraint on the gauge parameter in (A.6) can be solved similarly to the representation (A.7) for Ω ( The unconstrained harmonic superfields U (−n) and ρ (−n−2) can be represented by convergent harmonic series It follows from these expressions that the gauge symmetry (A.10) allows us to impose a gauge condition U (−n) (z, u) = U i 1 ...in (z)u − i 1 . . . u − in , (A. 12) which completely fixes the gauge freedom (A.10). In this gauge, we still have the freedom to perform those gauge transformations (A.8) which are generated by parameters of the form 13 The resulting gauge transformation of the prepotential U i 1 ...in (z) coincides with the flat-superspace version of (3.10).
In the gauge (A.12), the supersymmetric invariant (A.4) can be represented as an integral over Minkowski superspace by making use of the identity dζ (−4) (D + ) 4 L(z, u) = d 6|8 z du L(z, u) , (A.14) and computing the relevant harmonic integral. This reduces the supersymmetric invariant (A.4) to the flat-superspace version of (4.1) modulo a numerical factor.
In the n = 2 case, the supersymmetric invariant (A.4) is the harmonic superspace realisation of the linear multiplet action.