Duality-invariant (super)conformal higher-spin models

We develop a general formalism of duality rotations for bosonic conformal spin-$s$ gauge fields, with $s\geq 2$, in a conformally flat four-dimensional spacetime. In the $s=1$ case this formalism is equivalent to the theory of $\mathsf{U}(1)$ duality-invariant nonlinear electrodynamics developed by Gaillard and Zumino, Gibbons and Rasheed, and generalised by Ivanov and Zupnik. For each integer spin $s\geq 2$ we demonstrate the existence of families of conformal $\mathsf{U}(1)$ duality-invariant models, including a generalisation of the so called ModMax Electrodynamics ($s=1$). Our bosonic results are then extended to the $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric cases. We also sketch a formalism of duality rotations for conformal gauge fields of Lorentz type $(m/2, n/2)$, for positive integers $m $ and $n$.

Building on the seminal 1981 work by Gaillard and Zumino [1], the general theory of U(1) duality-invariant models for nonlinear electrodynamics in four dimensions was developed in the mid 1990s [2][3][4][5] and the early 2000s [6][7][8]. The formalism of [2][3][4][5] has been generalised to formulate general U(1) duality-invariant N = 1 and N = 2 globally [9,10] and locally [11][12][13] supersymmetric theories. In particular, Ref. [9] put forward the constructive perturbative scheme to compute N = 2 superconformal U(1) dualityinvariant actions for the N = 2 vector multiplet. Moreover, extending the earlier proposal of [14], the first consistent perturbative scheme to construct the N = 2 supersymmetric Born-Infeld action was given in [10]. The formalism of nonlinear realisations for the partial N = 4 → N = 2 breaking of supersymmetry advocated in [15] reproduced [16] the results of [10]. Further progress toward the construction of the N = 2 supersymmetric Born-Infeld action has been achieved in [17][18][19]. 1 Within the original bosonic formulation [2][3][4][5] and its supersymmetric extensions [9,10], U(1) duality invariance of a model for nonlinear (super) electrodynamics is equivalent to the condition that the Lagrangian satisfies a nonlinear self-duality equation. General solutions of such equations are difficult to find. Ivanov and Zupnik [6][7][8] provided a reformulation of nonlinear electrodynamics which makes use of certain auxiliary variables in such a way that (i) the self-interaction depends only on the auxiliary variables; and (ii) U(1) duality invariance is equivalent to the manifest U(1) invariance of the self-interaction. Supersymmetric extensions of the Ivanov-Zupnik approach were given in [26,27]. In particular, the N = 2 supersymmetric formulation of [26] has been used to obtain the 1 It should be pointed out that the N = 1 supersymmetric Born-Infeld action [20] is the first nontrivial U(1) duality-invariant supersymmetric theory [21]. Its remarkable property is that it is a Goldstone multiplet action for partial N = 2 → N = 1 supersymmetry breaking in Minkowski space [22,23], as well as in the maximally supersymmetric backgrounds [24] discovered in [25]: (i) R × S 3 ; (ii) AdS 3 × R; and (iii) a supersymmetric plane wave.
closed-form expression for a superconformal U(1) duality-invariant model proposed to describe the low-energy effective action for N = 4 super-Yang-Mills theory [28].
In this paper we will demonstrate that the known results for U(1) duality-invariant nonlinear electrodynamics (spin s = 1) can naturally be generalised to develop a general formalism of U(1) duality rotations for bosonic conformal spin-s gauge fields, with s ≥ 2, and their N = 1 and N = 2 supersymmetric cousins in a conformally flat four-dimensional background.
Our paper is organised as follows. In section 2 we introduce the notion of U(1) dualityinvariant conformal higher-spin (CHS) theories and present examples of such models, including higher-spin generalisations of the so-called "ModMax electrodynamics" [29] (see also [30]). This purely bosonic study is extended in section 3 to the case of N = 1 superconformal higher-spin (SCHS) multiplets. We present a one-parameter self-dual SCHS action, which generalises the N = 1 superconformal U(1) duality-invariant electrodynamics [28,31]. In section 4 we uplift the technical machinery of the previous two sections to N = 2 superspace and derive the self-duality equation for N = 2 SCHS models. In section 5 we provide concluding comments and sketch the formalism of duality rotations for CHS gauge fields of arbitrary rank. The main body of this paper is accompanied by four technical appendices. Appendix A reviews the salient details of conformal geometry in four dimensions pertinent to this work. Appendices B and C review the relevant aspects of N = 1 and N = 2 conformal superspace, respectively. In appendix D we derive a class of duality-invariant CHS models via the auxiliary field approach. In appendix E we provide arguments to fix the overall signs of the free (super)conformal higher-spin actions.

Duality-invariant CHS models
In this section we develop a formalism of duality rotations for CHS fields and propose some duality-invariant models. Consider a dynamical system describing the propagation of a conformal spin-s field h α(s)α(s) := h α 1 ...αsα 1 ...αs = h (α 1 ...αs)(α 1 ...αs) , with s ≥ 1, in curved spacetime. Its action functional S (s) [C,C] is assumed to depend on a field strength C α(2s) and its conjugatē Cα (2s) , with C α(2s) being defined as where ∇ a denotes the conformally covariant derivative, see appendix A. The real unconstrained prepotential h α(s)α(s) is a primary field, K b h α(s)α(s) = 0, where K b is the special conformal generator. It is defined modulo gauge transformations of the form with the gauge parameter ζ α(s−1)α(s−1) also being primary. This transformation law is conformally invariant provided where D is the dilatation generator. The field strength (2.1) is primary in a generic gravitational background, However, the gauge transformations (2.2) leave C α(2s) invariant only when s = 1, δ ζ C α(2) = 0. For s ≥ 2 gauge invariance holds only if the background is conformally flat, where C α(4) is the self-dual part of the background Weyl tensor, see appendix A. For the remainder of this section we will assume such a geometry.
We point out that C α(2) is Maxwell's field strength and C α(4) is the linearised Weyl tensor. We will refer to C α(2s) for s > 2 as the linearised spin-s Weyl tensor.

U(1) duality-invariant models
It is important to note that the field strength (2.1) obeys the Bianchi identity Now, we assume that S (s) [C,C] is extended to be a functional of an unconstrained field C α(2s) and its conjugate. We introduce where we have defined Varying S (s) [C,C] with respect to the prepotential h α(s)α(s) yields A crucial feature of our analysis above is that the functional form of the equation of motion (2.9) mirrors that of the Bianchi identity (2.6). Consequently, the union of equations (2.6) and (2.9) is invariant under the SO(2) ∼ = U(1) duality transformations: where λ is a constant, real parameter. One may then obtain two equivalent expressions for the variation of S (s) [C,C] with respect to (2.10) as a generalisation of similar derivations in nonlinear electrodynamics [4,5,10]. This implies the self-duality equation which must hold for an unconstrained field C α(2s) and its conjugate. In (2.11) we have employed the notational shorthand T 2 = T α(m) T α(m) (similarlyT 2 =Tα (m)Tα (m) ). The simplest solution of the self-duality equation (2.12) is the free CHS model (E.1), which was introduced in [32][33][34] in the case of Minkowski space and extended to arbitrary conformally flat backgrounds in [35]. 2 In the s = 1 case, the self-duality equation (2.12) was originally derived by Bialynicki-Birula [36], but unfortunately this work was largely unnoticed. 3 It was independently re-discovered by Gibbons and Rasheed in 1995 [2]. Two years later, it was re-derived by Gaillard and Zumino [4] with the aid of their formalism developed back in 1981 [1] but originally applied only in the linear case.
As is known, all gravity-matter theories allow for a Weyl-invariant formulation [37,38] in which the gravitational field is described in terms of two gauge fields. One of them is the inverse vielbein e a m and the other is a conformal compensator Ψ, the latter being a 2 For the free CHS model (E.1), one can also consider scale transformations in addition to the U(1) duality ones (2.10), which is similar to the case of electrodynamics discussed, e.g., in [10]. 3 We thank Dmitri Sorokin for bringing Ref. [36] to our attention.
nowhere vanishing scalar field. In this setting the gravity gauge group also includes Weyl transformations, which act on the gravitational fields as follows e a m → e σ e a m , Ψ → e σ Ψ . (2.13) Truly conformal theories, such as conformal gravity, do not depend on the compensator.
In the approach of [39], the gauge group is further enlarged to local conformal transformations, and the compensator is a primary dimension-1 scalar field, (2.14) If we allow for the action S (s) [C,C] to depend on the compensator, then the family of U(1) duality-invariant theories (2.12) is very large. For instance, the following U(1) duality-invariant model is a higher-spin generalisation of Born-Infeld electrodynamics [40]. The latter is obtained from (2.15) for s = 1 by making use of local scale transformations to impose a gauge condition Ψ 2 = g −1 = const. Owing to the dependence of S (s) BI [C,C; Ψ] on the compensator Ψ, it is clear that (2.15) is not conformal. As another solution of the self-duality equation (2.12), we propose a one-parameter duality-invariant extension of (2.15) S (s)

Self-duality under Legendre transformation
In the case of nonlinear (super) electrodynamics, U(1) duality invariance implies selfduality under Legendre transformations, see [10] for a review. This remarkable property proves to extend to the higher-spin case, as will be shown below.
We start by describing a Legendre transformation for a generic theory with action S (s) [C,C]. For this we introduce the parent action Here C α(2s) is an unconstrained field and C D α(2s) takes the form where h D α(s)α(s) is a Lagrange multiplier field. Indeed, upon varying (2.17) with respect to h D α(s)α(s) one obtains the Bianchi identity (2.6), and its general solution is given by eq. (2.1), for some real field h α(s)α(s) . As a result the second term in (2.17) becomes a total derivative, and we end up with the original action S (s) [C,C]. Alternatively, if we first vary (2.17) with respect to C α(2s) , the equation of motion is which we may solve to express C α(2s) as a function of C D α(2s) and its conjugate. Inserting this solution into (2.17), we obtain the dual model . is invariant under (2.10). The latter may be exponentiated to obtain the finite U(1) duality transformations Performing such a transformation with λ = π 2 on (2.22) yields Upon inserting this expression into (2.20), we obtain (2.21).
In the above analysis, we made use of the fact that the general solution of the Bianchi identity (2.6) is given by (2.1). To justify this claim, it suffices to work in Minkowski space. Let C α(2s) be a field subject to the equation (2.6), with ∇ a = ∂ a . Introduce its descendant defined by which is automatically transverse, The Bianchi identity (2.6) tells us that h ⊥ α(s)α(s) is real, h ⊥ α(s)α(s) = h ⊥ α(s)α(s) . Now we can express C α(2s) in terms of (2.25a), where ✷ = ∂ a ∂ a . In the final relation the real field h α(s)α(s) is not assumed to be transverse. This field is related to ✷ −s h ⊥ α(s)α(s) by a gauge transformation with a real gauge parameter ζ α(s−1)α(s−1) . Our consideration may be extended to the N = 1 and N = 2 supersymmetric cases studied in the next sections.

Auxiliary variable formulation
As a generalisation of the Ivanov-Zupnik [6][7][8] approach, here we will introduce a powerful formalism to generate duality-invariant models that makes use of auxiliary variables.

Consider the following action functional
Here we have introduced the auxiliary variable ρ α(2s) which is chosen to be an unconstrained primary dimension-2 field, The functional S (s) int [ρ,ρ], by definition, contains cubic and higher powers of ρ α(2s) and its conjugate. The equation of motion for ρ α(2s) is where F(x) is a real analytic function of a real variable. However, such models are not conformal if the action does depend on Ψ. The condition of conformal invariance imposes additional nontrivial restrictions.
As a warm-up example, let us consider the following nonlinear conformal action Requiring this action to obey the self-duality equation (2.12), we obtain the constraint Thus, the nonlinear theory is a one-parameter conformal U(1) duality-invariant extension of the free CHS action (E.1). In the s = 1 case our model coincides with ModMax electrodynamics.
In order to construct more general models, it is advantageous to make use of the auxiliary variable formulation described above. We introduce algebraic invariants of the symmetric rank-(2s) spinor ρ α(2s) , which has the same algebraic properties as the linearised spin-s Weyl tensor C α(2s) : If s is odd, all invariants ρ 2n+1 , with n a non-negative integer, vanish.
For simplicity, we restrict our analysis to the conformal graviton, s = 2. In this case the family of invariants (2.38) contains only two functionally independent invariants [42], ρ 2 and ρ 3 . In particular, one may show that s = 2 : Now we choose the self-interaction in (2.28) to be of the form where β and κ are real coupling constants. The resulting model is clearly conformal and U(1) duality-invariant. For κ = 0, elimination of the auxiliary variables ρ α(4) and ρα (4) does not result in a simple action like (2.37). In particular, such an elimination, to quadratic order in the couplings, yields the following self-dual model The ellipsis in (2.41) denotes additional contributions to the full nonlinear theory which are cubic or higher order in the coupling constants. We emphasise that for the special case κ = 0 the above action yields (2.37). A proof of this result is given in appendix D.
For s > 2 the number of algebraic invariants of ρ α(2s) grows, and therefore one can define families of conformal U(1) duality-invariant models.

N = 1 duality-invariant SCHS models
The purely bosonic study undertaken in the previous section can be generalised to the supersymmetric case. To this end, we consider a dynamical system describing the propagation of a conformal superspin-(s + 1 2 ) gauge multiplet H α(s)α(s) , s > 0, in N = 1 curved superspace [35,43]. The prepotential H α(s)α(s) is a real unconstrained superfield being defined modulo the gauge transformations with a primary unconstrained gauge parameter ζ α(s)α(s−1) . This gauge transformation law is superconformal provided where K B = (K b , S β ,Sβ) are the special conformal generators, see appendix B. The action functional of the dynamical system is required to take the form S (s) [W,W], where W α(2s+1) is the following chiral descendant of H α(s)α(s) : This field strength proves to be primary in a generic supergravity background [35,43], However, the gauge transformations (3.1) leave W α(2s+1) invariant only if the supergravity background is conformally flat, where W α(3) is the background super-Weyl tensor, see appendix B. For the remainder of this section we will assume such a geometry.
It should be pointed out that the prepotential H αα encodes the linearised conformal supergravity multiplet and its field strength W α(3) is the linearised super-Weyl tensor [44,45]. We will refer to W α(2s+1) for s > 1 as the linearised superspin-(s + 1 2 ) Weyl tensor, or simply as a higher-spin super-Weyl tensor.
With the exception of the gauge transformation law (3.1), the above results are also valid in the s = 0 case corresponding to the massless vector multiplet. As is well known, its prepotential H is defined modulo the gauge transformations [46,47] with the chiral scalar χ being primary and dimensionless. Unlike the s ≥ 1 case considered above, the vector multiplet field strength for an arbitrary supergravity background.

U(1) duality-invariant models
It is important to note that the field strength (3.3) obeys the Bianchi identity 4 [48] Moreover, the real superfield proves to be primary, and may be called the linearised superspin-(s + 1 2 ) Bach tensor. For s > 1 B α(s)α(s) is also referred to as a higher-spin super-Bach tensor.
We assume that S (s) [W,W] is consistently defined as a functional of a general chiral superfield W α(2s+1) and its conjugate. This allows us to introduce the symmetric spinor where the variational derivative is defined by and E is the chiral integration measure. It follows that M α(2s+1) is a primary chiral superfield, Varying the action with respect to the prepotential H α(s)α(s) yields the equation of motion Here the real superfield proves to be primary. The analysis above indicates that the functional form of (3.13) coincides with that of (3.7). It then follows that the system of equations (3.7) and (3.13) is invariant under the U(1) duality rotations: where λ is a constant, real parameter. One may then obtain two equivalent expressions for the variation of S (s) [W,W] with respect to (3.14) as a generalisation of similar derivations in nonlinear N = 1 supersymmetric electrodynamics [9][10][11]. 5 This implies the self-duality equation which must hold for a general chiral superfield W α(2s+1) and its conjugate. Every solution S (s) [W,W] of the self-duality equation describes a U(1) duality-invariant theory. In the s = 0 case, the equation (3.16) was originally derived in [9] in Minkowski superspace and extended to supergravity in [11]. The simplest solution of the self-duality equation (3.16) is the free N = 1 SCHS model (E.5), which was introduced in [43] for the cases of Minkowski and AdS backgrounds and later was generalised to arbitrary conformally flat backgrounds in [35].
The above results allow one to prove, in complete analogy with the non-supersymmetric analysis conducted in section 2.2, that the U(1) duality-invariant theories are self-dual under Legendre transformations.

Auxiliary variable formulation
As a generalisation of the auxiliary variable formalism sketched in section 2.3, here we will develop a reformulation of the U(1) duality-invariant systems introduced in the previous subsection. In the s = 0 case, it will reduce to the auxiliary superfield approach for U(1) duality-invariant supersymmetric electrodynamics introduced in [26,27].
Consider the action functional Here we have introduced the new dimension-3/2 multiplet η α(2s+1) , which is required to be primary and covariantly chiral, By definition, the functional S (s) int [η,η], contains cubic and higher powers of η α(2s+1) and its conjugate.
Employing perturbation theory, equation (3.19) allows one to express η α(2s+1) as a functional of W α(2s+1) and its conjugate. This means that (3.17) is dual to a SCHS theory with action Thus, (3.17) and (3.20) provide two equivalent realisations of the same model.
The power of this formulation is most evident when the self duality equation (3.16) is applied. A routine computation reveals that this constraint is equivalent to Thus, self-duality of the action (3.17) is equivalent to the requirement that S Within the superconformal approach to supergravity-matter dynamical systems [39], every theory of Einstein supergravity coupled to supersymmetric matter is realised as a coupling of the same matter multiplets to conformal supergravity and a superconformal compensator, see, e.g., [49]. Truly superconformal theories are independent of any compensator. Different off-shell formulations for Einstein supergravity correspond to different choices of the compensating multiplet. The most famous compensators are: (i) a chiral compensator S 0 of the old minimal supergravity; and (ii) a real linear compensator L = L of the new minimal supergravity. Both S 0 and L are required to be nowhere vanishing, such that (S 0 ) −1 and L −1 exist, and they satisfy the following superconformal properties: If we allow the action (3.20) to depend on a superconformal compensator, either S 0 or L, we can generate families of U(1) duality-invariant theories which satisfy the condition (3.22). For simplicity, unless otherwise stated, here we will restrict our attention to the following interaction and the compensator Υ has one of the two realisations: either Υ = S 0S0 or Υ = L. Applying (3.22), we find that our model is U(1) duality invariant provided   Employing (3.19), we arrive at where γ is a real coupling constant. For s > 0 the resulting model is (3.28) In both cases we have made use of the shorthand We note that (3.28) is invariant under the following rescaling of the conformal compensator which implies that the dependence of (3.28) on Υ is purely superficial. Consequently, the action is superconformal.
It is important to note that, at the component level, the purely bosonic sector of the interaction (the κ-term) present in (3.28) identically vanishes. 6 Thus, these actions describe different duality-invariant models than those presented in section 2.4. However, one may construct a supersymmetric duality-invariant model that contains, for instance, the bosonic theory (2.37) at the component level.

Consider the following supersymmetric duality-invariant model
The numerator in the κ-term in (3.28) contains a product of 4(s + 1) fermionic superfields.
The bosonic component fields of η α(2s+1) are: Action (3.31) can be reduced to components using the standard reduction rules Since we are interested only in the bosonic sector, it suffice to approximate Further, it may be checked that the interaction (the β-term) in (3.31) does not contain contributions which are (i) linear in ρ α(2s) and its conjugate; and (ii) linear in ρ α(2s+2) and its conjugate. Therefore, if we switch off the spin-s field, To conclude this section, we derive a supersymmetric extension of the cubic interaction present in (2.40). For this purpose, we will fix s = 1 in (3.17). The relevant interaction is constructed in terms of the two scalar primary descendants of η α(3) : 37) and the conformal compensator Υ. Using these, we construct the following action .

(3.38)
We note that it is not superconformal as it lacks invariance under (3.30), though we will show that, at the component level, it contains the cubic interaction present in (2.40). To demonstrate this, we restrict our attention to the bosonic sector of (3.38), which allows us to approximate Further, the interaction (the κ-term) present in (3.38) contains no terms linear in either ρ α(4) or ρ α (2) . Hence, if we switch off the spin-1 field, C α(2) = 0, then ρ α(2) = 0 solves the corresponding equations of motion. Under these conditions, the resulting bosonic action takes the form In particular, we note that it is locally conformal as it is independent of the compensator. Thus, (3.38) is a supersymmetric extension of the cubic interaction of (2.40). In closing, we emphasise that if one instead attempts to consider the branch C α(4) = 0 and C α(2) = 0, it is not legal to set ρ α(4) = 0. Such a choice would imply w| θ=0 ≈ 0, making the corresponding component action of (3.38) undefined.

N = 2 duality-invariant SCHS models
This section is devoted to a brief study of duality-invariant models for N = 2 conformal superspin-(s + 1) gauge multiplets, with s ≥ 0, in conformally flat backgrounds. 8 Our approach will generalise the general theory of U(1) duality-invariant models for N = 2 supersymmetric nonlinear electrodynamics (superspin-0) developed in [9,10,13,26]. The essential information about N = 2 conformal superspace, which is used throughout this section, is collected in appendix C.
We start with a review of superconformal higher-spin (SCHS) gauge multiplets [52]. Given a non-negative integer s, the conformal superspin-(s + 1) gauge multiplet is described by a real unconstrained prepotential H α(s)α(s) , which is a primary superfield, K B H α(s)α(s) = 0, defined modulo gauge transformations of the form with the gauge parameters ζ α(s)α(s−1)i and ζ ij being primary and complex unconstrained.
Here we have defined the second-order operators The gauge transformation laws (4.1) are superconformal provided where we have introduced the chiral projection operator ij∇ ij (4.5) and the second-order operators It can be shown that W α(2s+2) is primary in an arbitrary supergravity background, However, the gauge transformations (4.1) leave W α(2s+2) invariant, δ ζ W α(2s+2) = 0, only if the background curved superspace is conformally flat, where W α(2) is the background super-Weyl tensor, see appendix C. Throughout this section, we will restrict ourselves to such a geometry.
We should point out that the gauge prepotential H describes the linearised N = 2 conformal supergravity multiplet, and its field strength W α(2) is the linearised super-Weyl tensor. For s > 0 we will refer to the fields strengths W α(2s+2) as the linearised higher-spin super-Weyl tensors.
The chiral field strength defined by (4.4) carries at least two spinor indices. A chiral scalar field strength W corresponds to the massless N = 2 vector multiplet [53]. It can be described in terms of the curved superspace analogue of Mezincescu's prepotential [54] (see also [55]), V ij = V ji , which is an unconstrained real SU(2) triplet, V ij = V ij = ε ik ε jl V kl . The expression for W in terms of V ij was found in [56] to be The field strength W defines a primary reduced chiral superfield of dimension +1, and is invariant under gauge transformations of the form [57] with ζ α kij being primary and unconstrained modulo the algebraic condition given. In the flat-superspace limit, the gauge transformation law (4.11) reduces to that given in [54,55]. It is important to emphasise that W is invariant under the gauge transformations (4.11) in an arbitrary supergravity background.
The formalism we introduce below for nonlinear SCHS models generalises the existing framework for U(1) duality-invariant theories of the vector multiplet. The latter have been studied extensively in [9,10,26,28], thus we omit this special case from our considerations.

U(1) duality-invariant models
We consider a dynamical system describing the propagation of the superconformal gauge multiplet H α(s)α(s) , s ≥ 0, in a conformally flat superspace. We require its action functional to take the form S (s) [W,W], where the gauge-invariant chiral field strength W α(2s+2) is given by eq. (4.4).
It is important to note that the field strength (4.4) obeys the Bianchi identity We then assume that S (s) [W,W] can be defined as a functional of a general chiral superfield W α(2s+2) and its conjugate. This allows us to introduce the symmetric spinor which defines an arbitrary variation of the action where E is the N = 2 chiral integration measure. 9 It follows that M α(2s+2) is a primary chiral superfield, Varying the action with respect to the prepotential H α(s)α(s) yields the equation of motion This analysis above indicates that the functional form of (4.16) coincides with that of (4.12). As a result, the system of equations (4.12) and (4.16) is invariant under U(1) duality rotations: where λ is a constant, real parameter. One may then obtain two equivalent expressions for the variation of S (s) [W,W] with respect to (4.17) as a generalisation of similar derivations in nonlinear N = 2 supersymmetric electrodynamics [9,10,13]. This implies the self-duality equation which must hold for a general chiral superfield W α(2s+2) and its conjugate. Every solution S (s) [W,W] of the self-duality equation describes a U(1) duality-invariant theory. Equation (4.19) allows one to prove, in complete analogy with the non-supersymmetric analysis conducted in section (2.2), that the U(1) duality-invariant theories are self-dual under Legendre transformations. The simplest solution of the self-duality equation (4.19) is the free N = 2 SCHS model (E.8) proposed in [52], including the action for linearised conformal supergravity (s = 0).

Auxiliary variable formulation
This subsection is aimed at generalising the auxiliary variable formalism employed in section 3.2 to construct self-dual N = 1 SCHS models to an N = 2 setting. 10 To this end, we consider the action functional where we have introduced the auxiliary dimension-1 multiplet η α(2s+2) , which is primary and covariantly chiral, By construction, the self-interaction S (s) int [η,η] contains cubic and higher powers of η α(2s+2) and its conjugate. Varying (4.20) with respect to this superfield yields Thus, (4.20) and (4.23) provide two equivalent realisations of the same model.
We now analyse the self duality equation (4.19). It is easily verified that this constraint is equivalent to If we allow the action (4.20) to depend on a superconformal compensator, it is trivial to construct interactions satisfying the condition (4.25), for instance Here F(x) is a real analytic function of a real variable, and W 0 is the chiral field strength of a compensating vector multiplet.
It is not difficult to construct explicit examples of N = 2 superconformal and U(1) duality-invariant higher-spin theories, as a generalisation of the N = 2 vector multiplet models presented in [28]. For simplicity, here we restrict our attention to N = 2 Minkowski superspace. Let us consider the self-interaction

On N = 2 super ModMax theory
So far, an N = 2 superconformal extension of the ModMax theory has not been constructed. Perhaps it does not exist (see the discussion in [28]), the main reason being the fact that the full N = 2 superspace measure is dimensionless. However, if the action functional is allowed to depend on compensators, an N = 2 supersymmetric extension of the ModMax theory can be introduced. Specifically, let us consider the following N = 2 vector multiplet model 11 : where κ ∈ R is the coupling constant. We are going to show that this model incorporates N = 1 superconformal ModMax theory, eq. (3.27), in the sense described below.
The field strength W of the N = 2 vector multiplet contains two independent N = 1 chiral components, Φ and W α , defined by Here W α is the chiral field strength of the N = 1 vector multiplet, ∇ α W α =∇αWα. The auxiliary N = 2 chiral superfield η contains three independent N = 1 chiral components, χ, η α and Ψ, defined by Using the standard N = 2 → N = 1 reduction rules we may reduce the action (4.28) to N = 1 superspace. Here we restrict our attention to the N = 1 vector multiplet and switch off the N = 1 scalar multiplet, Φ = 0. Then it may be shown that setting χ = 0 and Ψ = 0 gives a solution of the equations of motion for these auxiliary superfields. 12 The resulting N = 1 superspace action is Our normalisation of the quadratic part of the vector multiplet action follows [28] 12 This follows form the fact that the N = 1 superspace reduction of (4.28) does not have terms linear in χ, Ψ and their conjugates provided Φ = 0.

Conclusion
In this paper we have generalised the general theory of U(1) duality-invariant nonlinear electrodynamics [3][4][5][6][7][8] and its N = 1 and N = 2 supersymmetric extensions [9-13, 26, 27] to the cases of bosonic conformal spin-s gauge fields, s ≥ 2, and their N = 1 and N = 2 superconformal cousins. These self-dual higher-spin theories share several important features of duality-invariant electrodynamics, such as self-duality under Legendre transformations. The crucial difference between the U(1) duality-invariant models for electrodynamics and their higher-spin counterparts is that the former are consistently defined on arbitrary gravitational backgrounds, while the latter are formulated in a conformally flat space.
In the non-supersymmetric case, we presented several families of self-dual models. The simplest of these is (2.37), which is a higher spin generalisation of ModMax electrodynamics [29]. We also introduced a two-parameter family of models for the conformal graviton (2.40) and provided insights on the construction of new families of multi-parameter duality invariant models for conformal higher spin fields. Finally, in the supersymmetric case, we generalised the constructon of N = 1 superconformal duality invariant electrodynamics [28,31] to higher spins (3.28). A distinguishing feature of this family of models is that, at the component level, their purely bosonic sectors vanish, which may be verified via a component reduction.
It should be emphasised that the literature on duality is huge and, unfortunately, it is not possible to mention all publications here. In this paper we were interested in duality as a continuous symmetry of the equations of motion. There exists a different approach to duality as a manifest symmetry of the action. The latter was advocated in [60][61][62][63][64][65][66]. Another important approach was inspired by early examples of dual field theories [67][68][69][70][71][72][73] and resulted in various dual formulations for massless, massive and partially massless higher-spin field theories, see [74][75][76][77][78][79][80] and references therein. This approach corresponds to discrete dualities.
There may be several generalisations of our results that are modelled on similar properties of duality-invariant theories of spin-1 gauge fields, see [81] for a review. In particular, we recall that, given a U(1) duality-invariant model for nonlinear electrodynamics, its compact duality group U(1) can be enhanced to the non-compact SL(2, R) group by coupling the electromagnetic field to the dilaton and axion [3][4][5]. It may be shown that this construction naturally extends to the higher-spin case. We should point out that a generalisation of the method of [6][7][8] to the case of SL(2, R) duality was given by Ivanov, Lechtenfeld and Zupnik [82].
Fundamental to our analysis in this paper has been the formalism of conformal (super) space, which trivialises calculations in backgrounds with vanishing (super) Weyl tensor. For the purpose of applications, however, it is often useful to work with Lorentz covariant derivatives as opposed to their conformally covariant counterparts. The process of translating results expressed in terms of the latter to the former is known as degauging. The general prescription for such an analysis is well-known (see e.g. [35,83,84]), however such calculations are often highly non-trivial on generic curved backgrounds. If the background geometry is restricted by turning off several components of the curvature, the resulting computations are greatly simplified. To this end, we now provide a dictionary to translate our main results to the AdS (super)space. 13 We begin with the non-supersymmetric story. The geometry of AdS 4 is encoded within the Lorentz covariant derivatives D a , which obey the algebra where µ is a nonzero complex constant with the dimension of mass encoding the curvature of the spacetime and µμ is proportional to the scalar curvature. It may be shown that in this geometry, equation (2.1) takes the form Further, C α(2s) satisfies the Bianchi identity (2.6) The conformal flatness of N -extended AdS superspace in four dimensions was established in [85] and further elaborated in [86].
Next, the geometry of the N = 1 AdS superspace, AdS 4|4 , is described by the covariant derivatives D A = (D a , D α ,Dα), which obey the algebra (see, e.g., [87]) where µ is a nonzero constant having the dimension of mass. In this supergeometry the field strengths W α(2s+1) take the form [88] W and obey the Bianchi identity Finally, we consider AdS 4|8 , the N = 2 AdS superspace. Recall that its geometry is encoded within the covariant derivatives D A = (D a , D i α ,Dα i ), which obey the algebra (see [86] for the technical details): where S ij is a nonzero covariantly constant iso-triplet, S ji = S ij , satisfying the integrability condition [S, S † ] = 0, with S = (S i j ). 14 Here, the field strengths W α(2s+2) take the form and satisfy the Bianchi identity Throughout this work we have restricted our attention to the analysis of models described by real gauge prepotentials. However, our construction readily extends itself to the complex case. In particular, one may consider a complex CHS gauge prepotential φ α(m)α(n) , with m, n ≥ 1 and m = n [35,43,89]. It is a primary field, where the gauge parameter ℓ α(m−1)α(n−1) is also primary. Conformal invariance of (5.10) uniquely fixes the dimension of the gauge field, Dφ α(m)α(n) = (2 − 1 2 (m + n))φ α(m)α(n) . The action functional describing its dynamics is required to take the form S (m,n) [Ĉ,Č,C,C], where we have made the definitionŝ The descendants introduced above are primary in generic backgrounds, and gauge-invariant in all conformally flat ones, It is important to note that the field strengths (5.11) also obey the Bianchi identity It is clear from the discussion above that the system of equations (5.14) and (5.18) are invariant under the U(1) duality rotations δ λĈα(m+n) = λM α(m+n) , δ λČα(m+n) = λM α(m+n) , (5.19a) δ λMα(m+n) = −λĈ α(m+n) , δ λMα(m+n) = −λČ α(m+n) . (5.19b) One may then perform similar analyses to those undertaken in section 2 and construct U(1) duality-invariant nonlinear models for such fields. 15 They satisfy the self-duality equation U(1) duality-invariant actions of the type considered above naturally arise at the component level of supersymmetric self-dual theories, such as those discussed in sections 3 and 4. Such considerations are beyond the scope of the current paper and present an interesting avenue for further work. It would also be interesting to study duality-invariant models of the higher-depth (super)fields of [35,91].

A Conformal space
This appendix reviews the salient details of conformal geometry in four dimensions [92] pertinent to this work. 16 We adopt the spinor conventions of [87], which are similar to those of [94]. We consider a curved spacetime M 4 parametrised by local coordinates x m . The structure group is chosen to be SU(2, 2), whose Lie algebra is spanned by the translation P a , Lorentz M ab , dilatation D and the special conformal K a generators. The covariant derivatives ∇ a then take the form where e a m is the inverse vielbein, ω a bc the Lorentz spin connection, b a the dilatation connection and f a b the special conformal connection. The commutation relations of ∇ a with the generators M ab , D and K a are obtained from those of P a with the same generators by replacing P a → ∇ a .
The covariant derivatives (A.1) obey the commutation relations Here C αβγδ andCαβ˙γδ are the self-dual and anti self-dual parts of the Weyl tensor C abcd , and are primary. We remind the reader that a field ϕ (with suppressed indices) is said to be primary if it obeys K αα ϕ = 0 . and we recall that the Lorentz generators act on vectors and Weyl spinors as follows: In this appendix we review the elements of the N = 1 conformal superspace approach to off-shell conformal supergravity relevant to this work. For more details we refer the reader to the original paper [83] (see also appendix A of [48]).
Consider a curved N = 1 superspace M 4|4 parametrised by local coordinates z M = (x m , θ µ ,θμ). The structure group is chosen to be SU(2, 2|1). Its corresponding superalgebra is spanned by the translation P A = (P a , Q α ,Qα), Lorentz M ab , dilatation D, R-symmetry Y and the special conformal K A = (K a , S α ,Sα) generators. The covariant derivatives ∇ A then have the form where E A M denotes the superspace inverse vielbein, Ω A bc the Lorentz connection, Φ A the U(1) R connection, B A the dilatation connection, and F AB the special superconformal connection.
The covariant derivatives (B.1) obey the algebra Here W αβγ is the N = 1 super-Weyl tensor and is subject to the constraints: as well as the Bianchi identity where the primary superfield B αα is the super-Bach tensor and was introduced in [95], see also [35,43,87].
We remind the reader that a tensor superfield Ψ (with suppressed indices) is said to be primary and of dimension ∆ Ψ and U(1) R charge q Ψ if the following conditions hold: Of particular importance are primary chiral superfields, which satisfy Requiring consistency of these constraints with the superconformal algebra yields highly non-trivial implications. Specifically, it must take the form Ψ = Ψ α(m) , and its U(1) R charge and dimension are related as follows: The algebra (B.2) is to be accompanied by the following (anti-)commutation relations: the U(1) R , dilatation and special conformal generators obey while the algebra of K A and ∇ B takes the form where all other graded commutators vanish.
The chiral and full superspace integrals are related according to the rule where U is a primary real superfield of dimension +2.

C N = 2 conformal superspace
This appendix reviews N = 2 conformal superspace, a formulation for off-shell N = 2 conformal supergravity developed by Butter [84] and then reformulated in [96].
We consider a curved N = 2 superspace M 4|8 parametrised by local coordinates z M = (x m , θ µ ı ,θ ıμ ). The structure group is chosen to be SU(2, 2|2). The corresponding superalgebra is spanned by the Lorentz M ab , translation P A = (P a , Q i α ,Qα i ), dilatation D, R-symmetry Y and J ij , and the special conformal K A = (K a , S α i ,S iα ) generators. The covariant derivatives ∇ A = (∇ a , ∇ i α ,∇α i ) then have the form As compared with (B.1), we have introduced Φ A ij , the SU(2) R connection. The corresponding generator acts on isospinors as follows: The covariant derivatives (C.1) obey the algebra Here W αβ is the N = 2 super-Weyl tensor and is subject to the constraints: We also find that W αβ obeys the Bianchi identity where the primary superfield B is the N = 2 super-Bach tensor. We remind the reader that a superfield Φ (with suppressed indices) is said to be primary of dimension ∆ Ψ and U(1) R charge q Ψ if the following conditions hold: Of particular importance are primary chiral superfields, which satisfy The consistency of these constraints with the superconformal algebra leads to highly nontrivial implications. In particular, it can carry no isospinor or dotted spinor indices, Ψ = Ψ α(m) , and its U(1) R charge and dimension are related as follows: Further, we note that for any primary tensor superfield U α(m) with the property q U = −2∆ U , the following object is both primary and chiral in conformally flat backgrounds [84,97].
The chiral and full superspace integrals are related according to the rule d 4 xd 4 θd 4θ E U = d 4 xd 4 θ E∇ 4 U , where U is a primary real dimension-0 superfield.
For further details regarding N = 2 conformal superspace, we refer the reader to the original work [84], as well as [52,96].

D Elimination of auxiliary variables
This appendix is devoted to a derivation of the conformal U(1) duality-invariant CHS models (2.37) via the auxiliary variable formalism introduced in section 2.3. We emphasise that the latter is a higher-spin generalisation of the Ivanov-Zupnik approach [6][7][8].
Consider the following action functional it is clear that (D.3) coincides with (2.37), which concludes our analysis. It is important to note that, in the s = 1 case, this computation was first performed in [28].
In this appendix we show that the overall signs of the free CHS actions [32][33][34] S where C ab = ∇ a h b − ∇ b h a . Moreover, similar arguments allow us to correctly fix the overall signs of the free N = 1 and N = 2 SCHS actions.
The overall sign in (E.1) is also fixed by identifying the action S (s) [C,C] with the induced one obtained by computing the logarithmically divergent part of the effective action of a conformal scalar field coupled to background conformal higher-spin fields [98].
where the ellipses denote the fermionic sector, which is irrelevant to our analysis. For the overall signs of the component actions present in (E.4) to agree with those of (E.1), we require z s = (−1) s . Therefore, the N = 1 SCHS actions take the form [ where the ellipses denotes the sector containing integer superspin field strengths, which is irrelevant to our analysis. For the overall signs of the N = 1 actions present in (E.7) to agree with those of (E.5), we require z s = (−1) s . Therefore, the N = 2 SCHS actions take the form [