Off-shell superconformal higher spin multiplets in four dimensions

We formulate off-shell N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 superconformal higher spin multiplets in four spacetime dimensions and briefly discuss their coupling to conformal supergravity. As an example, we explicitly work out the coupling of the superconformal gravitino multiplet to conformal supergravity. The corresponding action is super-Weyl invariant for arbitrary supergravity backgrounds. However, it is gauge invariant only if the supersymmetric Bach tensor vanishes. This is similar to linearised conformal supergravity in curved background.


Introduction
The role of conformal field theories as cornerstones for the exploration of more general quantum field theories, which are connected to them via renormalization group flows, has been appreciated since a long time ago. Higher spin gauge theories [1][2][3][4][5][6][7] have an even longer history and have attracted considerable interest recently. It is quite natural to combine the two symmetry principles and to study conformal higher spin theories [8]. A further symmetry which is compatible with conformal and higher spin symmetry is supersymmetry. This leads to superconformal higher spin theories, first advocated in [9], which are the main focus of this note. More specifically, we introduce off-shell N = 1 superconformal higher spin multiplets in four dimensions and analyse in some detail the problem of lifting such supermultiplets to curved backgrounds. Our main technical tool, as far as the supersymmetry and supergravity aspects are concerned, is superspace and we refer to [10] for a thorough introduction to this formalism.
We first study superconformal higher spin theories in flat superspace. In section 2 we review superconformal transformations and the important notion of superconformal primaries. In section 3 we construct off-shell superconformal higher spin multiplets: starting from prepotentials and their transformation laws under higher spin gauge transformations

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and under superconformal transformations, we construct field strengths and invariant actions. The two cases of half-integer and integer superspin as well as the superconformal gravitino multiplet have to be treated separately. The component fields of these multiplets are totally symmetric traceless tensor and tensor-spinor fields. More general fields will be briefly discussed in the last part of section 3. In section 4 we couple the superconformal higher spin multiplets to conformal supergravity where the notion of superconformal transformations is replaced by that of super-Weyl transformations. While super-Weyl invariance is easy to achieve, gauge invariance requires non-minimal couplings. We explicitly discuss the gravitino multiplet, but defer the general case to the future. Section 5 contains concluding comments, including the explicit expressions for conserved higher spin current multiplets that correspond to the superconformal higher spin prepotentials. The main body of the paper is accompanied by two technical appendices. Appendix A contains those results concerning the Grimm-Wess-Zumino superspace geometry [11,12], which are important for understanding the supergravity part of this paper. Appendix B contains the essential information about the super-Weyl transformations [13].
There are different ways to describe N = 1 conformal supergravity in superspace. 1 The simplest option is to make use of the superspace geometry of [11,12], which underlies the Wess-Zumino approach [14] to the old minimal formulation for N = 1 supergravity developed independently in [15,16]. Another option is to work with the U(1) superspace proposed by Howe [17,18]. Finally, one can make use of the so-called conformal superspace [19]. The three superspace approaches to N = 1 conformal supergravity are equivalent, although each of them has certain advantages and disadvantages (see [19] for a detailed discussion of the relationship between these formulations). In this paper we make use of the oldest and simplest superspace setting [11,12].

Superconformal transformations
In this section we briefly recall the structure of N = 1 superconformal transformations in Minkowski superspace M 4|4 , see [10] for more details. We denote by z A = (x a , θ α ,θα) the Cartesian coordinates for M 4|4 , and use the notation D A = (∂ a , D α ,Dα) for the superspace covariant derivatives.
The vector components of ξ A obey the equations as well as the ordinary conformal Killing equation Another consequence of (2.1) is The most general conformal Killing supervector field proves to be where we have introduced the complex four-vector along with the complex bosonic coordinates x a + = x a + iθσ aθ of the chiral subspace of M 4|4 . The constant bosonic parameters in (2.8) correspond to the spacetime translation (aα α ), Lorentz transformation (K β α ,Kαβ), special conformal transformation (b αβ ), and combined scale and R-symmetry transformations (σ = τ − 2 3 iϕ). The constant fermionic parameters in (2.8) correspond to the Q-supersymmetry (ǫ α ) and S-supersymmetry (η α ) transformations. The constant parameters K αβ and σ are obtained from K αβ [ξ] and σ[ξ], respectively, by setting z A = 0.
A tensor superfield T (with its indices suppressed) is said to be superconformal primary of weight (p, q) if its superconformal transformation law is for some parameters p and q. The dimension of T is (p + q), while (p − q) is proportional to its R-symmetry charge. If T is superconformal primary and chiral,DαT = 0, then T cannot possess dotted indices, i.e.MαβT = 0, and it must hold that q = 0. In the chiral case, it suffices to say that T is superconformal primary of dimension p. Given a real scalar L, which is superconformal primary of weight (1,1), is invariant under superconformal transformations. Given a chiral scalar L c , which is superconformal primary of dimension +3, is invariant under superconformal transformations.

Off-shell superconformal multiplets in flat space
In this section we introduce off-shell superconformal higher spin multiplets. We first consider the half-integer and integer superspin cases, and then give some generalisations of the constructions proposed. Strictly speaking, the notion of superspin is defined only for super-Poincaré multiplets. The rationale for our use of this name in the superconformal framework is that our superconformal multiplets will be described solely in terms of the gauge prepotentials corresponding to the off-shell massless higher spin multiplets constructed in [20,21]. Each of these massless multiplets also involves certain compensator superfields, in addition to the gauge prepotential.
The superconformal transformation law of H α(s)α(s) is This transformation law is uniquely determined if one requires both the gauge superfield H α(s)α(s) and the gauge parameter Λ α(s)α(s−1) in (3.1) to be superconformal primary (see also [25,26]). It follows from (3.1) that the chiral symmetric spinor is gauge invariant [20]. 2 Our crucial observation is that W α(2s+1) is superconformal primary of dimension 3/2. We conclude that the gauge-invariant action is superconformal. In the s = 1 case, it coincides with the action for linearised conformal supergravity [22]. One may check that We briefly comment on the component structure of the superconformal theory (3.4). The gauge parameterD (α 1 Λ α 1 ...αsα 2 ...αs) in (3.1) may be represented as where H 0 = θσ aθ ∂ a and all component gauge parameters are complex. The parameters χ β,α(s)α(s) and f β,α(s)α(s−1) transform in the representation 2 ⊗ (2s + 1) of SL(2, C) with respect to their undotted indices. It follows from (3.1) and (3.6) that a Wess-Zumino gauge may be chosen of the form
The superconformal transformation law of Ψ α(s)α(s−1) is postulated to be It follows from (3.10) that the following chiral descendants of the prepotentials are gauge invariant. 3 As before, the crucial observation is that the field strengths W α(2s) and Z α(2s) are superconformal primaries of dimension 1 and 2, respectively. This allows us to construct a superconformal and gauge-invariant action One checks that We now comment on the component structure of (3.13). One may choose a Wess-Zumino gauge of the form where the bosonic fields h α(s)α(s) and B α(s+1)α(s−1) are complex. In the Wess-Zumino gauge chosen, the bosonic fields h α(s)α(s) and B α(s+1)α(s−1) and the fermionic fields ψ α(s+1)α(s) and ψ α(s)α(s−1) are defined modulo gauge freedom of the type (3.9). 4 More specifically, the field B α(s+1)α(s−1) belongs to a more general family of conformal fields than those described by the gauge transformation laws (3.9). The point is that one may consider conformal higher spin fields φ α(m)α(n) , where m and n are integers such that m > n > 0. Since m = n, the field φ α(m)α(n) is complex. Postulating the gauge transformation law δφ α 1 ...αmα 1 ...αn = ∂ (α 1 (α 1 λ α 2 ...αm)α 2 ...αn) (3.16) and requiring both the field φ α(m)α(n) and the gauge parameter λ α(m−1)α(n−1) to be primary, the dimension of φ α(m)α(n) is fixed to be equal to 2 − 1 2 (m + n). We can define two gaugeinvariant field strengthŝ The field strength (3.12a) was introduced in [21]. 4 The case s = 1 is not considered here. Its special feature is that B αβ is not a gauge field.

Superconformal gravitino multiplet
In the s = 1 case, the gauge transformation law (3.10) has to be replaced with This gauge transformation was given in ref. [28], which proposed the off-shell formulation for the massless gravitino multiplet in terms of the gauge spinor prepotential Ψ α in conjunction with two compensators, an unconstrained real scalar and a chiral scalar. The prepotential Ψ α is required to be superconformal primary of weight (−1, 0), which is a special case of (3.11). The superconformal primary superfields (3.12) for s = 1 are obviously invariant under the gauge transformations (3.19).

Off-shell superconformal multiplets in supergravity
We now turn to exploring whether the superconformal higher spin multiplets introduced in the previous section may be consistently lifted to curved superspace backgrounds.

General considerations
Just as in the non-supersymmetric setting, where conformal invariance in Minkowski space is replaced by Weyl invariance, in a curved background geometry, superconformal invariance is replaced by super-Weyl invariance. In other words, super-Weyl invariance in curved superspace implies superconformal invariance in Minkowski superspace. A tensor superfield T (with its indices suppressed) is said to be super-Weyl primary of weight (p, q) if its super-Weyl transformation law is for some parameters p and q. Similar to the rigid supersymmetric case (2.10), we will refer to (p + q) as the dimension of T . Given a covariantly chiral tensor superfield T defined on a general supergravity background,DαT = 0, it may carry only undotted indices, MαβT = 0, as a consequence of (A.3b). If T is covariantly chiral and super-Weyl primary, eq. (4.1), then q = 0. An example is provided by the super-Weyl tensor W αβγ with the transformation law (B.2c). In appendix B we also collect the transformation properties of various other geometric quantities under super-Weyl transformations.
As reviewed in appendix A, the curved superspace geometry of [11,12] does not possess torsion tensors of dimensions 1/2. This means that the gauge transformation (3.1) is uniquely extended to curved superspace as It is compatible with the following super-Weyl transformation of the prepotential: The chiral field strength (3.3) may uniquely be lifted to curved superspace as a covariantly chiral superfield of the general form with the super-Weyl transformation law The ellipsis in (4.4a) stands for terms involving the super-Ricci tensor G αα and its covariant derivatives. Such terms can always be found. A systematic construction is to start with conformal superspace [19], where G αα appears as a connection, and then to implement the so-called de-gauging procedure in order to arrive at the ordinary curved superspace geometry of [11,12]. 5 Details of the construction will be given elsewhere, but 5 In conformal superspace, the required primary chiral field strength W α(2s+1) has a minimal form W α(2s+1) = − 1 4∇ 2 ∇ (α 1β 1 · · · ∇α sβ s ∇α s+1 H α s+2 ...α 2s+1 )β 1 ...βs , where ∇A = (∇a, ∇α,∇α) denotes the corresponding covariant derivatives [19].
We may now consider a minimal extension of (3.4) to curved superspace given by where E is the chiral integration measure. It follows from (4.5) that this functional is super-Weyl invariant. However, for non-vanishing background super-Weyl tensor, W αβγ = 0, the field strength W α(2s+1) and, therefore, the action (4.6) are not gauge invariant. In general, the gauge variation δ Λ W α(2s+1) is proportional to the background super-Weyl tensor W αβγ , its conjugateWαβγ and their covariant derivatives. 6 The action (4.6) needs to be completed to include non-minimal terms which contain W αβγ ,Wαβγ and their covariant derivatives. An example will be given in section 4.3, where we discuss the gravitino supermultiplet. Let us first discuss the simplest case of W α(2s+1) , s = 1, which is linearised conformal supergravity. The linearised super-Weyl tensor is The important point is that each term in δ Λ W αβγ involves either the background super-Weyl tensor or its covariant derivative. The variation vanishes if the background superspace is conformally flat, W αβγ = 0. In this case the functional (4.6) is the required superconformal gauge-invariant action. Here 'superconformal' means that the action is invariant under arbitrary superconformal isometries of the background superspace. As another example of W α(2s+1) , we consider the case s = 2. The field strength W α(5) is uniquely determined to be It is a tedious exercise to check that W α(5) is super-Weyl primary.

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the gauge-invariant chiral field strength W α(2s+1) was found in [32]. It is The curved-superspace extension of the gauge transformation (3.10) is It is compatible with the following super-Weyl transformation of the prepotential

Superconformal gravitino multiplet
Our next example is the superconformal gravitino multiplet. It is characterised by the gauge freedom δΨ α = D αΛ + ζ α ,Dβζ α = 0 , (4.14) and the super-Weyl transformation The following covariantly chiral field strengths are super-Weyl primary of dimension +1 and +2, respectively. These superfields are not invariant under the gauge transformations (4.14). One finds the following non-vanishing variations of W αβ and Z αβ : Consider the action Here E and E denote the chiral measure and the full superspace measure, respectively. The action S GM is super-Weyl invariant, The second and third terms on the right of (4.18) are fixed by requiring S GM to be invariant under the ζ-transformation (4.14),

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Finally, a lengthy calculation gives Here B αα denotes the N = 1 supersymmetric extension of the Bach tensor, with the super-Weyl transformation One can rewrite B αα is a manifestly real form [10,33] We recall that the super-Bach tensor may be introduced (see [10,33] for the technical details) as a functional derivative of the conformal supergravity action [34,35], with respect to the gravitational superfield, specifically with ∆H αα the covariantised variation of the gravitational superfield defined in [36,37]. The super-Bach tensor obeys the conservation equation which expresses the gauge invariance of the conformal supergravity action. It follows from (4.20) and (4.21) that the action (4.18) is gauge invariant if the background super-Bach tensor is equal to zero, This holds, e.g., for all Einstein superspaces, which are characterised by

Linearised conformal supergravity
The condition (4.29) is also required to define an off-shell superconformal multiplet of superspin 3/2 in curved superspace. The point is that (4.29) is the equation of motion for conformal supergravity, since varying the action (4.26) with respect to the gravitational superfield gives 7 The gauge-invariant action for the superconformal superspin-3 2 multiplet in curved background is obtained by linearising the conformal supergravity action (4.26) around its arbitrary stationary point, B αα = 0. In accordance with [36,37] (see also [10] for a review), the linearised gauge transformation of the prepotential is given by (4.2) with s = 1. The linearised conformal supergravity action is automatically invariant under the gauge and super-Weyl transformations. Its explicit structure will be described elsewhere.

Concluding comments
In this paper we constructed the off-shell N = 1 superconformal higher spin multiplets in four dimensions 8 and also sketched the general scheme of coupling such multiplets to conformal supergravity. Our work opens two new approaches to interacting conformal higher spin theories. Firstly, every conformal higher spin field may be embedded into an off-shell superconformal multiplet (the latter actually contains several bosonic and fermionic conformal fields). Instead of trying to couple the original conformal field to gravity, we can look for a consistent interaction of the superconformal multiplet with conformal supergravity. Since the gravitational field belongs to the conformal supergravity multiplet (also known as the Weyl multiplet), this will automatically lead to a consistent coupling of the component conformal fields to gravity.
The second avenue to explore is a superfield extension of the effective action approach to conformal higher spin fields advocated in [41][42][43]. One may start with a free massless chiral scalar superfield Φ,DαΦ = 0, and couple it to an infinite tower of background superconformal higher spin prepotentials H α(s)α(s) (source superfields) by the rule Here J α(s)α(s) denotes a composite primary superfield, which describes a conserved current multiplet when Φ is on-shell. Then it is natural to consider the generating functional for correlation functions of these conserved higher spin supercurrents defined by The two terms in the right-hand side of (4.26) differ by a total derivative related to the Pontryagin invariant [10,33]. 8 In three dimensions, the off-shell superconformal higher spin multiplets have recently been described in [38,39] and [40] for the N = 1 and N = 2 cases, respectively.

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Similar to the non-supersymmetric analysis of [44], one may show that the action (5.1), properly deformed by terms nonlinear in H α(s)α(s) , has an exact non-Abelian gauge symmetry (associated with the conformal higher-spin superalgebra shsc ∞ (4|1) described in [9]) which reduces to (4.2) at lowest level in the superfields H α(s)α(s) , Φ andΦ. Here we restrict our discussion to giving the explicit expressions for J α(s)α(s) . We turn to describing the structure of the conserved current multiplets J α(s)α(s) . In order for the source term The s = 1 case corresponds to the superconformal version [22] of the Ferrara-Zumino supercurrent [45]. Extension to the other cases s > 1 was given in [46] (building on [47]). The authors of [46] also postulated the prepotential H α(s)α(s) as the source to generate the Noether coupling (5.3), as well as the gauge transformation (3.1) as the transformation of H α(s)α(s) which leaves (5.3) invariant. However, no higher spin extensions of linearised conformal supergravity were given. Consider a free on-shell massless chiral scalar Φ, which is superconformal primary of dimension +1. By analogy with the construction of [48], the conserved current multiplets J α(s)α(s) , with s = 1, 2, . . . , can be obtained as unique composites of Φ andΦ of the form where one should keep in mind that It is an instructive exercise to check that the conservation equations (5.4) are satisfied. Choosing s = 1 in (5.6) gives the well-known supercurrent [45] J αα = D α ΦDαΦ + 2i(Φ ∂ ααΦ − ∂ αα ΦΦ) .

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The higher spin supercurrent (5.6) may be compared with the 3D N = 2 result reported in [49]. Similar to the bosonic superfield prepotentials H α(s)α(s) , one may define conserved higher spin current supermultiplets associated with the fermionic superfield prepotentials Ψ α(s)α(s−1) . Consider a source term of the form In order for S In the s = 1 case, the conservation equations are [50] D β J β = 0 ,D 2 J α = 0 , (5.10) as a consequence of (3.19). Conserved current multiplets J α(s)α(s−1) , with s = 1, 2, . . . , may be constructed from two free massless chiral superfields Φ + and Φ − , One may check that the following composite obeys the conservation equations (5.9) for s > 1. One may also check that J α(s)α(s−1) is antisymmetric with respect to the interchange Φ + ↔ Φ − . Choosing s = 1 in (5.12) gives the composite which obeys the conservation equations (5.10). The superfield J α contains a conserved fermionic current j αββ = j βαβ that corresponds to the second supersymmetry current. 9
In the case of a free N = 2 hypermultiplet described in terms of two N = 1 chiral scalars Φ + and Φ − , the action (5.1) should be replaced with and describes a general coordinate transformation generated by the supervector field ξ = ξ B E B as well as a local Lorentz transformation generated by the antisymmetric tensor K bc .

B Super-Weyl transformations
The algebra of covariant derivatives (A.3) preserves its functional form under super-Weyl transformations [13] δ σ D α = σ − Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.