Topologically massive higher spin gauge theories

We elaborate on conformal higher-spin gauge theory in three-dimensional (3D) curved space. For any integer $n>2$ we introduce a conformal spin-$\frac{n}{2}$ gauge field $h_{(n)} =h_{\alpha_1\dots \alpha_n}$ (with $n$ spinor indices) of dimension $(2-n/2)$ and argue that it possesses a Weyl primary descendant $C_{(n)}$ of dimension $(1+n/2)$. The latter proves to be divergenceless and gauge invariant in any conformally flat space. Primary fields $C_{(3)}$ and $C_{(4)}$ coincide with the linearised Cottino and Cotton tensors, respectively. Associated with $C_{(n)}$ is a Chern-Simons-type action that is both Weyl and gauge invariant in any conformally flat space. These actions, which for $n=3$ and $n=4$ coincide with the linearised actions for conformal gravitino and conformal gravity, respectively, are used to construct gauge-invariant models for massive higher-spin fields in Minkowski and anti-de Sitter space. In the former case, the higher-derivative equations of motion are shown to be equivalent to those first-order equations which describe the irreducible unitary massive spin-$\frac{n}{2}$ representations of the 3D Poincar\'e group. Finally, we develop ${\cal N}=1$ supersymmetric extensions of the above results.

Simons-type action S (n) CS ∝ d 3 x e h α(n) C α(n) is Weyl and gauge invariant in any conformally flat space. The actions S (n) CS , which for n > 4 are higher-spin extensions of linearised conformal gravity, are used to construct gauge-invariant formulations for massive higher-spin fields in Minkowski and anti-de Sitter (AdS) space. As is known, the irreducible unitary massive spin-n 2 representations of the 3D Poincaré and AdS groups (for n > 1) are realised in terms of symmetric rank-n spinor fields obeying first-order differential equations. We demonstrate that these equations follow from the equations of motion in certain higher-derivative gauge theories. If n = 2s, our massive action is S

Introduction
A unique feature of three spacetime dimensions (3D) is the existence of topologically massive Yang-Mills and gravity theories. These theories are obtained by augmenting the usual Yang-Mills action or the gravitational action by a gauge-invariant topological mass term. Such a mass term coincides with a Chern-Simons functional in the Yang-Mills case [1,2,3,4,5] and with a Lorentz Chern-Simons term in the case of gravity [4,5]. The Lorentz Chern-Simons term is required to make the gravitational field possess nontrivial dynamics, for the pure gravity action propagates no local degrees of freedom. The Lorentz Chern-Simons term can be interpreted as the action for 3D conformal gravity [6,7]. 1 Topologically massive gravity possesses supersymmetric extensions. In particular, N = 1 topologically massive supergravity was constructed in [10,11]. The off-shell formulations for N -extended topologically massive supergravity theories were presented in [12,13] for N = 2, in [14] for N = 3, and in [14,15] for the N = 4 case. In all of these theories, the action functional is a sum of two terms, one of which is the action for pure N -extended supergravity (Poincaré or anti-de Sitter) and the other is the action for N -extended conformal supergravity. The off-shell actions for N -extended supergravity theories in three dimensions were given in [16] for N = 1, [17,18] for N = 2 and [17] for the cases N = 3, 4. The off-shell actions for N -extended conformal supergravity were given in [6] for N = 1, [19] for N = 2, [20] for N = 3, 4, 5, and in [21,22] for the N = 6 case. Refs. [20,22] made use of the off-shell formulation for N -extended conformal supergravity proposed in [23]. The on-shell formulation for N -extended conformal supergravity with N > 2 was given in [24]. On-shell approaches to N -extended topologically massive supergravity theories with 4 ≤ N ≤ 8 were presented in [25,26,27,28,29].
Topologically massive N = 1 supergravity, with or without a cosmological term, may be linearised about a maximally supersymmetric solution. The resulting linearised actions for the gravitino and the gravitational field contain higher derivatives. However, the genuine massive states prove to obey first-order differential equations. This paper is devoted to the description of higher-spin extensions of the linearised actions for topologically massive gravity and N = 1 supergravity. In particular, for every (half-)integer spin n/2, where n = 5, 6, . . . , we present a gauge-invariant higher-derivative action in Minkowski space that propagates a single massive state of helicity +n/2 or −n/2 on the mass shell. The action is of the form S massive = S massless + S CS . (1.1) Here S massless denotes the 3D massless spin-n 2 gauge action of the Fronsdal-Fang type [30,31], with no propagating degrees of freedom. The second term in the right-hand side of (1.1) is a conformal spin-n 2 gauge action [32,33] described by a Lagrangian of the schematic form L CS ∝ ϕ (n) ∂ n−1 ϕ (n) , where ϕ (n) stands for the conformal spin-n 2 field. We show that S massive propagates a single massive state described by the equations (2.1). We also present AdS extensions of the actions introduced, as well as their N = 1 supersymmetric generalisations.
In the case of Minkowski space, our actions (1.1) are in fact contained, at the component level, in the massive supersymmetric higher-spin models proposed in [34,35]. However, the analysis in [34,35] was carried out mostly in terms of superfields so that the component actions were not studied. All the massive higher-spin gauge models in AdS, which are presented in this paper, are new. This paper is organised as follows. In section 2 we review field realisations of the irreducible massive spin-n 2 representations (n = 2, 3 . . . ) of the 3D Poincaré and AdS groups. We also review the structure of on-shell massive higher-spin superfields for both 3D N = 1 Poincaré and AdS supersymmetry. In section 3 we introduce, for any integer n ≥ 2, a conformal spin-n 2 gauge field h (n) = h α 1 ...αn = h (α 1 ...αn) and argue that it possesses a Weyl primary descendant C (n) of dimension (1 + n 2 ) with the following properties: (i) is of the schematic form ∇ n−1 h (n) ; (ii) C (n) is divergenceless and gauge invariant in an arbitrary conformally flat space. These descendants C (n) are constructed in any conformally flat space. Making use of the primary fields C (n) , we propose Chern-Simons-type actions S (n) CS ∝ d 3 x e h α(n) C α(n) which are Weyl and gauge invariant in any conformally flat space, and which are higher-spin extensions of the linearised action for 3D conformal gravity. These conformal higher-spin actions are then used to construct massive higher-spin gauge theories in AdS, described by the actions (4.5a) and (4.5b). In section 4 we study the dynamics of the flat-space counterparts to the gauge theories (4.5a) and (4.5b).
Sections 5 and 6 are devoted to supersymmetric extensions of the results presented in sections 3 and 4. In section 5 we introduce conformal higher-spin gauge superfields H α(n) in curved N = 1 superspace. These conformal gauge superfields are argued to possess primary descendants W α(n) of dimension (1 + n 2 ) that are locally supersymmetric extensions of the linearised higher-spin super-Cotton tensors [33,35]. For any conformally flat superspace background, the primary superfields W α(n) are explicitly constructed, and are shown to be gauge invariant and conserved. Making use of H α(n) and W α(n) , we construct a higher-spin extension of the action for linearised N = 1 conformal gravity, S (n) SCS [H (n) ], which is given by eq. (5.21). We employ S (n) SCS [H (n) ] to construct massive higher-spin gauge actions in N = 1 AdS superspace, given by eqs. (5.39a) and (5.39b). Section 6 describes the component structure of the supersymmetric higher-spin theories introduced in section 5, with the analysis being restricted to the flat-superspace case. Concluding comments and discussion are given in section 7. The main body of the paper is accompanied by three appendices. Appendix A describes our notation and conventions. Appendix B reviews the Tyutin-Vasiliev action [36]. Appendix C provides two realisations for the higher-spin Cotton tensor C α(n) as a descendant of gauge-invariant field strengths corresponding to two different higher-spin massless models. 2

On-shell massive (super)fields
In this section we review the structure of irreducible massive higher-spin (super)fields in Minkowski space and in anti-de Sitter (AdS) space.

Massive fields
We first recall the definition of on-shell massive fields in Minkowski space. Given a positive integer n > 1, a massive field, φ α 1 ···αn =φ α 1 ...αn = φ (α 1 ···αn) , is a real symmetric rank-n spinor field which obeys the differential conditions [36] (see also [37]) with m being the mass of the field. In the spinor case, n = 1, eq. (2.1a) is absent, and the massive field is defined to obey the Dirac equation (2.1b). It is easy to see that (2.1a) and (2.1b) imply the mass-shell equation In the spinor case, n = 1, eq. (2.2) follows from the Dirac equation (2.1b). The helicity of φ α(n) is λ = n 2 σ, and the spin of φ α(n) is n/2. It should be remarked that the system of equations (2.1a) and (2.2) is equivalent to the 3D version of the Fierz-Pauli field equations [38]. The general solution to (2.1a) and (2.2) is a superposition of two massive states of helicity + n 2 and − n 2 , respectively. Twenty years ago, Tyutin and Vasiliev [36] proposed Lagrangian formulations for massive higher-spin fields that lead to the equations (2.1a) and (2.1b) on the mass shell. Their actions did not possess gauge invariance. In the present paper, we propose gauge-invariant formulations for massive higher-spin fields in Minkowski space that lead to the equations (2.1a) and (2.1b) on-shell.
In the case of AdS space, massive fields are defined by the following equations [39,40] (see also [41]) for some real mass parameter µ. Equation (2.3b) implies that where the parameter S is related to the AdS curvature via eq. (3.45a). Equation (2.4) can be rewritten in terms of the quadratic Casimir operator of the 3D AdS group SO(2, 2), with M γδ the Lorentz generators, see Appendix A.
Equations (2.3a) and (2.4) constitute the 3D AdS counterpart to the Fierz-Pauli field equations. They describe a reducible representation of the AdS isometry group. Gauge-invariant Lagrangian formulations for massive higher-spin fields in AdS, which lead to the equations (2.3a) and (2.4) on the mass shell, were developed in [42,43,44,45], including N = 1 supersymmetric extensions obtained by combining the bosonic and fermionic actions (on-shell supersymmetry). The formulations given in [42,43,44,45] are based on Zinoviev's gaugeinvariant approach [46] to describe massive higher-spin fields. In the present paper, we propose different gauge-invariant formulations for massive higher-spin fields in AdS that lead to the equations (2.3a) and (2.3b) on-shell.

Massive superfields
For n > 0, a massive superfield T α(n) is defined to be a real symmetric rank-n spinor, T α 1 ···αn =T α 1 ...αn = T (α 1 ···αn) , which obeys the differential conditions [47] (see also [35]) Here D 2 = D α D α , and D α is the spinor covariant derivative of N = 1 Minkowski superspace. It follows from (2.6a) that and thus T α(n) is an on-shell superfield, For the superhelicity of T α(n) we obtain We define the superspin of T α(n) to be n/2. The massive supermultiplet T α(n) contains two ordinary massive fields of the type (2.1), which are Their helicity values are n 2 σ and n+1 2 σ, respectively. The off-shell gauge-invariant formulations for massive higher-spin N = 1 supermultiplets in Minkowski superspace, which lead to the equations (2.6a) and (2.6b) on the mass shell, were constructed in [35].
In the case of N = 1 AdS supersymmetry, on-shell massive superfields are described by the equations [47] with µ a real mass parameter and D 2 = D α D α . Here D A = (D a , D α ) are the covariant derivatives of the N = 1 AdS superspace, see section 5 for the details. It can be shown that This differential operator, which is the square of the operator in the left-hand side of (2.12b), can be expressed via the the quadratic Casimir operator 4 of the 3D N = 1 AdS supergroup, 14) It is worth pointing out that the left-hand side of (2.12b) can be rewritten as where we have made use of (2.12a).
In this paper we propose off-shell gauge-invariant formulations for massive higher-spin supermultiplets in N = 1 AdS superspace that lead to the equations (2.12a) and (2.12b) on-shell.

Conformal higher-spin fields
The concept of conformal higher-spin field theory was introduced by Fradkin and Tseytlin in four dimensions [51]. (Super)conformal higher-spin field theories in three dimensions were discussed in [32,52]. In this section, our starting points will be (i) the description of conformal higher-spin gauge fields in Minkowski space given in [32,33]; and (ii) the approach advocated in [53].

Conformal gravity
The gravitational field may be described in terms of the torsion-free covariant derivatives Here M bc = −M cb denotes the Lorentz generators, e a m the inverse vielbein, e a m e m b = δ a b , and ω a bc the torsion-free Lorentz connection. Finally, R ab cd is the Riemann curvature tensor. In three dimensions, R ab cd is determined by the Ricci tensor R ab := η cd R cadb = R ba and the scalar The Weyl tensor is identically zero in three dimensions, which means The role of the Weyl tensor is played by the Cotton tensor W abc = −W bac , which is defined in terms of the 3D Schouten tensor P ab = R ab − 1 4 η ab R as follows Spacetime is conformally flat if and only if the Cotton tensor vanishes [50] (see [23] for a modern proof). The algebraic properties of the Cotton tensor are They imply that W ab := 1 2 ε acd W cd b is symmetric and traceless, It is also divergenceless, as a consequence of the Bianchi identity ∇ b R ab = 1 2 ∇ a R. The condition of vanishing torsion is invariant under Weyl (local scale) transformations of the form with the parameter σ(x) being completely arbitrary. In the infinitesimal case, the Weyl transformation laws of R ab and R are where ✷ = ∇ c ∇ c . The Cotton tensor is a Weyl primary field of weight +3, In what follows, we often convert every vector index into a pair of spinor ones using the wellknown correspondence: a three-vector V a can equivalently be realised as a symmetric spinor V αβ = V βα . The relationship between V a and V αβ is as follows: Associated with the traceless part of the Ricci tensor, R ab − 1 3 η ab R, and the Cotton tensor, W ab , are the following completely symmetric rank-4 spinors: The Weyl transformation of R αβγδ is

Conformal gauge fields
A real tensor field h α(n) := h α 1 ...αn = h (α 1 ...αn) is said to be a conformal spin-n 2 gauge field if (i) it is Weyl primary of some weight d n , (3.14) and (ii) it is defined modulo gauge transformations of the form with the real gauge parameter ζ α(n−2) being also Weyl primary. These conditions uniquely fix the Weyl weight of h α(n) to be Starting with h α(n) one can construct its descendant, C α(n) , defined uniquely, modulo a normalisation, by the following the properties: 1. C α(n) is of the form Ah α(n) , where A is a linear differential operator involving the covariant derivatives, the curvature tensors R α(4) and R and their covariant derivatives.
2. C α(n) is Weyl primary of weight (1 + n/2), 3. The gauge variation of C α(n) vanishes if the spacetime is conformally flat, where W (4) is the Cotton tensor.
4. C α(n) is divergenceless if the spacetime is conformally flat, Here and in (3.18), O W (4) stands for contributions containing the Cotton tensor and its covariant derivatives.
We now consider several examples. Given a conformal spin-1 gauge h αβ = h βα , the required Weyl primary descendant is C αβ = ∇ γ (α h β)γ and coincides with the gauge-invariant field strength, C ab = ∇ a h b − ∇ b h a , of the one-form h a . This implies that C α(2) is conserved, Next consider a conformal spin-3 2 gauge field h α(3) (i.e. conformal gravitino), The required Weyl primary descendant is Its gauge transformation is Computing its divergence gives Our last example is a conformal spin-2 gauge field h α(4) (i.e. conformal graviton), The required Weyl primary descendant of h α (4) is Its gauge transformation is The divergence of C α(4) may be shown to be Suppose that the spacetime under consideration is conformally flat, Then the tensor C α(n) is gauge invariant and conserved, (3.31b) These properties and the Weyl transformation law (3.17) tell us that the action is gauge and Weyl invariant, Here ⌊x⌋ denotes the floor function; it coincides with the integer part of a real number x ≥ 0. The above action is actually Weyl invariant in an arbitrary curved space. Condition (3.30) is required to guarantee the gauge invariance of S for n > 2. It follows from the Weyl transformation law (3.17) that ∇ βγ C βγα(n−2) is a primary field, This property means that the conservation equation (3.31b) is Weyl invariant.

Higher-spin Cotton tensor in Minkowski space
In Minkowski space, the linearised higher-spin Cotton tensor C α(n) (h), with n ≥ 2, is given by the expression [33] C α(n) (h) := 1 It is a descendant of the conformal field h α(n) defined modulo gauge transformations of the form The field strength is invariant under these gauge transformations, and obeys the Bianchi identity The higher-spin Chern-Simons action is conformal and invariant under (3.36).
In the case of even rank, n = 2s, with s = 1, 2, . . . , the field strength (3.35) can be shown to coincide with the bosonic higher-spin Cotton tensor given originally by Pope and Townsend [32]. It reduces to the linearised Cotton tensor for n = 4, and to the Maxwell field strength for n = 2. The fermionic case, n = 2s + 1, with s = 2, . . . , was not considered in [32]. It was presented for the first time in [33].
It should be pointed out that the conformal spin-3 case, n = 6, was studied for the first time in [54]. The spin-3/2 case, n = 3, was considered in [55]. The field strength C α(3) is the linearised version of the Cottino vector spinor [10,56].
The normalisation of C α(n) (h) defined by (3.35) can be explained as follows. The gauge freedom (3.36) allows us to impose a gauge condition Under this gauge condition, the field strength (3.35), with s = 1, 2, . . . , takes the form as a consequence of the identity The field strength (3.35) proves to be the general solution to the conservation equation (3.38). This result has recently been proved in [57] in the bosonic case, n = 2s, and the proof given is quite nontrivial (see also [58]). An alternative proof, which is valid for arbitrary integer n > 1 and is based on supersymmetry considerations, was given in [33].

Higher-spin Cotton tensor in conformally flat spaces
Consider a conformally flat spacetime M 3 . Its covariant derivatives ∇ a are related to the flat-space ones by for some scale factor σ. The linearised higher-spin Cotton tensor C α(n) in M 3 is related to the flat-space one, eq. (3.35), by the rule In general, it is a difficult technical problem to express C α(n) in terms of the covariant derivatives ∇ a and the gauge prepotential h α(n) = e (2−n/2)σ h α(n) . As an example, let us consider the case of AdS space, whose geometry is described by covariant derivatives satisfying the algebra or in two-component notation Here the parameter S is related to the AdS scalar curvature as R = −24S 2 . The Cotton tensor (3.44) for the cases n = 3, 4, 5 and 6 proves to be where Q is the quadratic Casimir of the 3D AdS group, SO(2, 2), given by eq. (2.5). Each of the tensors C α(n) given above can be written as C α(n) (h (n) ) = Ah α(n) , where the linear differential operator A is symmetric in the sense that for arbitrary prepotentials g α(n) and h α(n) . This means that it suffices to prove one of the two properties in (3.31), and then the second property follows.
4 Massive higher-spin actions in maximally symmetric spaces The conformal higher-spin actions in conformally flat spaces, eq. (3.32), are formulated in terms of the gauge fields h α(n) . The same gauge field can be used to construct massless Fronsdal-Fang-type actions [30,31,59,60] in maximally symmetric spaces. Such actions however, will involve not only h α(n) but also some compensators.
Here we describe these massless higher-spin gauge actions in AdS and then use them to construct gauge-invariant models for massive higher-spin fields.

Massive higher-spin actions in AdS space
There are two types of the higher-spin massless actions, first-order and second-order ones. Given an integer n ≥ 4, the first-order model is described by real fields h α(n) , y α(n−2) and y α(n−4) which are defined modulo gauge transformations of the form The Fang-Fronsdal-type gauge-invariant action, S (n) Here ⌈n/2⌉ stands for the ceiling function, which is equal to s for n = 2s and s+1 for n = 2s+1, with s ≥ 0 an integer.
Given an integer n ≥ 4, the second-order model is described by real fields h α(n) and y α(n−4) defined modulo gauge transformations of the form Our action (4.2) is a unique gauge-invariant extension to AdS space of the flat-space action given by Tyutin and Vasiliev [36], see Appendix B for a review. When n is odd, n = 2s + 1, (4.2) is the unique gauge-invariant 3D counterpart to the Fang-Fronsdal action in AdS 4 [60]. 5 When n is even, n = 2s, our action (4.4) is the unique gauge-invariant 3D counterpart to the Fronsdal action in AdS 4 [59]. The Fronsdal action [59] can also be generalised to d-dimensional AdS backgrounds [68,69]. Such an action in AdS d is formulated in terms of a symmetric double-traceless field and it is fixed by the condition of gauge invariance. 6 Each of the gauge-invariant actions (3.32), (4.2) and (4.4) proves to describe no propagating degrees of freedom. We claim that the following models describe irreducible massive fields in AdS. Here the parameter λ is dimensionless, while µ has dimension of mass. Since we do not have a closed form expression for C α(n) in AdS 3 , for arbitrary n, our analysis below will be restricted to the case of Minkowski space, M 3 . 5 It is worth pointing out that the Fang-Fronsdal action for a massless spin-(s + 1 2 ) field [31] is also described in terms of a triplet of fermionic gauge fields, Ψ α(s+1)α(s) , Ψ α(s−1)α(s) and Ψ α(s−1)α(s−2) and their conjugates, if one makes use of the two-component spinor notation, see section 6.9 of [61]. More generally, there exist bosonic and fermionic higher-spin triplet models in higher dimensions [62,63,64,65,66]. On-shell supersymmetric formulations for the generalised triplets in diverse dimensions have recently been given in [67]. 6 The dynamical equations for massless higher-spin fields in AdS d were studied by Metsaev [70,71,72,73].
For alternative descriptions of massless higher-spin dynamics in AdS d , see [74,75].

Massive higher-spin actions in Minkowski space: The fermionic case
In this section we study the dynamics of the flat-space counterparts to the gauge theories (4.5a) and (4.5b). In fact, the resulting flat-space actions are contained at the component level in the massive supersymmetric higher-spin models proposed in [34,35]. However, the analysis in [34,35] was carried out mostly in terms of superfields so that the component actions were not studied in detail.
We first analyse the flat-space limit of the fermionic model (4.5a). It is described by the action where the massless sector is The action (4.6) is invariant under the following gauge transformations: The equations of motion corresponding to the model (4.6) are We now demonstrate that the model (4.6) indeed describes an irreducible massive spin-(s + 1 2 ) field on the equations of motion. The gauge transformation (4.8c) tells us that y α(2s−3) can be completely gauged away, that is, we are able to impose the gauge condition y α(2s−3) = 0 . (4.10) Then, the residual gauge freedom is described by ζ α(2s−1) constrained by In the gauge (4.10), the equation of motion (4.9c) becomes the condition for y α(2n−1) to be divergenceless, Due to (4.12), the gauge transformation (4.8b) becomes Since y α(2s−1) and ζ α(2s−1) have the same functional type, we are able to completely gauge away the y α(2s−1) field, y α(2s−1) = 0 . (4.14) In accordance with (4.13) and (4.14), the residual gauge freedom is described by the parameter ζ α(2s−1) constrained by In the gauge (4.14), the equation of motion (4.9b) tells us that h α(2s+1) is divergenceless, So far the above analysis has been identical to that given in Appendix B of [34] for the massless model (4.7).

Massive higher-spin actions in Minkowski space: The bosonic case
Our next goal is to analyse the flat-space limit of the bosonic model (4.5b). It is described by the action where the second term is The action (4.27) is invariant under the gauge transformations The equations of motion corresponding to (4.27) are We will now show that on-shell, the model S massive describes a massive spin-s field which propagates a single degree of freedom. As follows from the gauge transformation (4.29b), it is possible to completely gauge away y α(2s−4) , y α(2s−4) = 0 . (4.31) Then, the residual gauge freedom is described by a parameter ζ α(2s−2) constrained by In the gauge (4.31), the equation of motion (4.30b) becomes According to (4.29a), the divergence of h α(2s) transforms as where we have made use of (4.32). Since ζ α(2s−2) and ∂ β(2) h α(2s−2)β(2) have the same functional type, it is possible to completely gauge away the divergence of h α(2s) , Under the gauge conditions imposed, there still remains some residual gauge freedom described by a gauge parameter constrained by (4.32) and ✷ζ α(2s−2) = 0. So far the above analysis has been identical to that given in Appendix B of [34] for the massless model (4.28).

Conformal higher-spin gauge superfields
Conformal higher-spin gauge superfields in N = 1 Minkowski superspace were introduced in [33,35], as a by-product of the N = 2 approach of [34]. In this section we start by generalising this concept to the case of N = 1 supergravity, building on the ideas advocated in [53].

Conformal supergravity
Consider a curved N = 1 superspace, M 3|2 , parametrised by local real coordinates z M = (x m , θ µ ), with m = 0, 1, 2 and µ = 1, 2, of which x m are bosonic and θ µ fermionic. We introduce a basis of one-forms E A = (E a , E α ) and its dual basis E A = (E a , E α ), which will be referred to as the supervielbein and its inverse, respectively. The superspace structure group is SL(2, R), the double cover of the connected Lorentz group SO 0 (2, 1). The covariant derivatives have the form: where is the Lorentz connection.
The covariant derivatives are characterised by graded commutation relations where T AB C and R AB cd are the torsion and curvature tensors, respectively. To describe supergravity, the covariant derivatives have to obey certain torsion constraints [16] such that the algebra (5.4) takes the form [17] {D α , D β } = 2iD αβ − 4iSM αβ , Here the scalar S and the symmetric spinor C αβγ = C (αβγ) are real. The dimension-2 Bianchi identities imply that We use the notation D 2 := D α D α .
The algebra of covariant derivatives is invariant under the following super-Weyl transformations [76,77,78] with the parameter σ being a real unconstrained superfield, provided the torsion superfields transform as The N = 1 supersymmetric extension of the Cotton tensor (3.3) was constructed in [79]. It is given by the expression The super-Weyl transformation of W αβγ proves to be It can be shown [23] that the curved superspace is conformally flat if and only if W αβγ = 0.
Starting with H α(n) one can construct its descendant, W α(n) (H), defined uniquely, modulo a normalisation, by the following the properties: 1. W α(n) is of the form AH α(n) , where A is a linear differential operator involving D A , the torsion tensors C αβγ and S and their covariant derivatives.
4. W α(n) is divergenceless if the superspace is conformally flat, Here O W (3) stands for contributions containing the super-Cotton tensor and its covariant derivatives.
As a simple example, we consider a U(1) vector multiplet coupled to supergravity, which corresponds to the n = 1 case. This multiplet is described by a real spinor prepotential H α which is super-Weyl primary of weight 1/2 and is defined modulo gauge transformations δ λ H α = iD α λ, where the gauge parameter λ is an unconstrained real superfield. The required super-Weyl primary descendant of weight 3/2 is given by and proves to be gauge invariant, The field strength obeys the Bianchi identity For n > 1 the right-hand sides of (5.14) and (5.15) are non-vanishing.
Suppose that our background curved superspace M 3|2 is conformally flat, Then the tensor superfield W α(n) is gauge invariant and conserved, These properties and the super-Weyl transformation laws (5.11) and (5.13) imply that the action 7 is gauge and super-Weyl invariant, We now turn to constructing the linearised higher-spin super-Cotton tensors W α(n) on such a conformally flat superspace.

Higher-spin super-Cotton tensor in Minkowski superspace
In Minkowski superspace, M 3|2 , the higher-spin super-Cotton tensor [33,35] is The fact that W α 1 ...αn defined by (5.23) is completely symmetric, is a corollary of the identities The normalisation in (5.23) is explained as follows. The gauge freeform (5.24) allows us to impose a gauge condition D β H βα(n−1) = 0 , (5.27) under which the expression for the super-Cotton tensor simplifies, This result can be fine-tuned to where s > 0 is an integer.
For completeness, we also give another representation for the higher-spin super-Cotton tensor derived in [33,35]: The following higher-spin action [33,35] S (n) is N = 1 superconformal. It is clearly invariant under the gauge transformations (5.24).

Higher-spin super-Cotton tensor in conformally flat superspaces
Consider a conformally flat spacetime M 3|2 . Its covariant derivatives D A are related to the flat-space ones by for some scale factor σ. In accordance with (5.13), the higher-spin super-Cotton tensor W α(n) in M 3|2 is related to the flat-space one, eq. (5.23) or equivalently (5.29), by the rule In general, it is a difficult technical problem to express W α(n) in terms of the covariant derivatives D A and the gauge prepotential H α(n) = e (1−n/2)σ H α(n) . As an example, we only give expressions for the supersymmetric photino W α and Cottino W α(2) tensors in AdS superspace. The geometry of AdS 3|2 is encoded in the following algebra of covariant derivatives: with S being a non-zero real parameter. The tensors W α and W α (2) are expressed in terms of the operator with the properties These properties follow from the identity The expressions for W α and W α(2) are:

Massive supersymmetric higher-spin theories in AdS superspace
Massive supersymmetric higher-spin actions in AdS involve different massless sectors depending on the value of superspin.

(5.40)
A supersymmetric gauge-invariant action of lowest order in derivatives is The gauge invariance of S (n) FO follows from the identity (5.37). Our action (5.41) is a higherspin AdS extension of the model for the massless gravitino multiplet (n = 2) in Minkowski superspace proposed by Siegel [1] (see also [16]).
When n is even, n = 2s, this action is the unique gauge-invariant AdS extension of the massless integer superspin action of [35].

Second-order massless actions
The massless half-integer superspin action in AdS is where Q is the quadratic Casimir operator (2.14). One can express Q in the form The action (5.45) is invariant under gauge transformations The action (5.45) is the unique gauge-invariant AdS extension of the massless half-integer superspin action of [35].

From AdS superspace to AdS space
To conclude this section, we briefly discuss the key aspects of component reduction for supersymmetric field theories formulated in AdS superspace, AdS 3|2 . In general, the action functional of such a theory is given by where the Lagrangian L is a scalar superfield. In accordance with the general formalism described in section 6.4 of [61], the isometry transformations of AdS 3|2 are generated by the Killing vector fields ξ A E A which are defined to obey the master equation [80] ξ for some Lorentz superfield parameter Λ bc = −Λ cb . An infinitesimal isometry transformation acts on a tensor superfield T as The action (5.48) is invariant under the isometry group of AdS 3|2 .
As shown in [80], the parameters in (5.49) obey the following Killing equations: Equation (5.52a) tells us that ξ a is a Killing vector, while (5.52c) means that ξ α is a Killing spinor. The component form of the action (5.48) is computed using the formula (see also [79]) Here and in what follows, the θ-independent component T | θ=0 of a superfield T (x, θ) will simply be denoted T |. To complete the formalism of component reduction, we only need the following relation where ∇ a is the standard torsion-free covariant derivative of AdS space. Making use of the AdS transformation law δ ξ L = ξL in conjunction with the identities (5.51) and (5.52), one may check that the action (5.53) is invariant under arbitrary isometry transformations of the AdS superspace.

Supersymmetric higher-spin actions in components
In this section we will describe the component structure of the supersymmetric higher-spin theories introduced in the previous section. Our analysis will be restricted to the flat-superspace case. As in [35], the integration measure 8 for N = 1 Minkowski superspace is defined as follows:

Superconformal higher-spin action
We start by reducing the superconformal higher-spin action (5.30) to components. The gauge freedom (5.24) can be used to impose a Wess-Zumino gauge In this gauge, there remain two independent component fields Due to the conservation equation (5.25), the higher-spin super-Cotton tensor (5.29) also has two independent components, which we define as The field strengths C α(n) and C α(n+1) are given in terms of the gauge potentials h α(n) and h α(n+1) , respectively, according to eq. (3.35). To prove this statement for C α(n+1) , one has to use the identity n 2j + n 2j + 1 = n + 1 2j + 1 .
Reducing the action (5.30) to components gives where the conformal higher-spin action S (n) CS [h (n) ] is defined by eq. (3.39). In the gauge (6.2), the residual gauge freedom is characterised by the conditions At the component level, the remaining independent gauge transformations are generated by ζ α(n−1) ∝ λ α(n−1) and ζ α(n−2) ∝ i n D β λ βα(n−2) .
The gauge freedom allows us to choose a Wess-Zumino gauge Then, the residual gauge freedom is characterised by the conditions These conditions imply that there remains only one independent gauge parameter at the component level. We define it as We define the component fields as Their gauge transformation laws are +4(n − 1)y βα 1 ...α n−2 ∂ β γ y γα 1 ...α n−2 + 2(n − 2)(n + 1) n y α 1 ...α n−3 ∂ βγ y βγα 1 ...α n−3 The fields h α(n) and Z α(n−2) appear in the action without derivatives. This action can be rewritten in the form is the flat-space version of (4.2), eg. (B.5), with n replaced by (n + 1).
The above component analysis clearly demonstrates that the model (6.17) describes a single massive supermultiplet subject to the equations (2.6a) and (2.6b) with n = 2s on the mass shell. The superfield proof was provided in [35].

Massless second-order model
Finally we consider the massless half-integer superspin model describe by the action [35] . To preserve these conditions, the residual gauge symmetry has to be constrained by The gauge transformation laws of h α(2s+2) and h α(2s+1) can be shown to be We now define the component fields of X α(2s−2) as follows: The gauge transformation laws of y α(2s−2) , y α(2s−1) and y α(2s−3) are as follows: In principle, we do not need to derive the gauge transformation of F α(2s−2) since this field turns out to be auxiliary.

Massive half-integer superspin action
We now have all ingredients at our disposal to read off the component form of the massive half-integer superspin action that is obtained from (5.39b) in the flat-superspace limit, (6.34) Here the symbol '≈' indicates that the auxiliary field has been eliminated.

Concluding comments
All massive higher-spin theories in Minkowski space, which have been presented in this paper, were extracted from off-shell supersymmetric field theories. As shown in section 6, all the theories studied in section 4 are contained at the component level in the N = 1 supersymmetric massive higher-spin theories proposed in [35]. The latter models were obtained from the N = 2 supersymmetric massive higher-spin theories of [34] by carrying out the N = 2 → N = 1 superspace reduction. Furthermore, the off-shell structure of the massless 3D N = 2 supersymmetric higher-spin actions of [34], which constitute one of the two sectors of the N = 2 massive actions, were designed following the pattern of the gauge off-shell formulations for massless 4D N = 1 higher-spin supermultiplets developed in the early 1990s [81,82].
Our supersymmetric massive higher-spin theories, which are formulated in AdS 3|2 superspace and are described by the actions (5.39a) and (5.39b), contain two different models for a massive integer-spin field in AdS at the component level. One of them is the gauge-invariant model (4.5a). The second model is described by the action which does not possess gauge invariance and which is the AdS uplift of the model (6.19). The action (7.1) leads to the equation of motion The action (7.1) can be turned into a gauge-invariant one by making use of the Stückelberg trick. An interesting feature of the model (7.1) is that it is well-defined in an arbitrary conformally flat space.
The models (4.5b) and (7.1) are higher-spin analogues of the two well-known equivalent models for a massive vector field (see [83,84] and references therein) with Lagrangians New duality transformations were introduced in [33] for theories formulated in terms of the linearised higher-spin Cotton tensors C α(n) and their N = 1 supersymmetric counterparts W α(n) . These duality transformations can readily be generalised to arbitrary conformally flat backgrounds, with C α(n) and W α(n) replaced with C α(n) and W α(n) , respectively.
In the present paper, we have been unable to obtain closed-form expressions for C α(n) and W α(n) in terms of the covariant derivatives of AdS (super)space for arbitrary n. These are interesting open problems.
The field strengths C α(n) and W α(n) are the higher-spin extensions of the linearised Cotton and super-Cotton tensors, respectively. The actions (3.32) and (5.21) are the higher-spin extensions of the linearised actions for conformal gravity and supergravity, respectively. An intriguing question is: Do nonlinear higher-spin extensions exist? Within the approach initiated in [85,86], Linander and Nilsson [87] constructed the full nonlinear spin-3 Cotton equation coupled to spin-2. They made use of the frame field description and the Chern-Simons formulation for 3D (super)conformal field theory due to Fradkin and Linetsky [52]. The construction of the nonlinear spin-3 Cotton tensor [87] requires an elimination of certain auxiliary fields, a procedure that becomes extremely difficult for s > 3. However, so far this is unexplored territory. There exist nonlinear formulations for the massless spin-3 theory [88,89], and the generalisation from s = 3 to s > 3 is shown in [89] to be trivial within the formulation developed. These results indicate that it is possible to construct a nonlinear topologically massive higher-spin field theory. The fundamental results by Prokushkin and Vasiliev [90,91] should be essential of course. Any attempt to construct a supersymmetric interacting higher-spin theory should inevitably be an extension of the conformal superspace approach [23,20]. tr[γ a γ b γ c γ d ] = 2η ab η cd − 2η ac η db + 2η ad η bc .
(A.5b) Given a three-vector x a , it can be equivalently described by a symmetric second-rank spinor x αβ defined as In the 3D case, an antisymmetric tensor F ab = −F ba is Hodge-dual to a three-vector F a , specifically Then, the symmetric spinor F αβ = F βα , which is associated with F a , can equivalently be defined in terms of F ab : These three algebraic objects, F a , F ab and F αβ , are in one-to-one correspondence to each other, F a ↔ F ab ↔ F αβ . The corresponding inner products are related to each other as follows: −F a G a = 1 2 F ab G ab = 1 2 F αβ G αβ .

C Higher-spin Cotton tensor as a descendent of gaugeinvariant field strengths
The Cotton tensor is defined in terms of the Ricci tensor according to (3.3). The latter determines the equations of motion corresponding to the Einstein-Hilbert action. In this appendix we show that analogous properties hold for the linearised higher-spin Cotton tensor defined by eq. (3.35).