Higher spin super-Cotton tensors and generalisations of the linear-chiral duality in three dimensions

In three spacetime dimensions, (super)conformal geometry is controlled by the (super) Cotton tensor. We present a new duality transformation for N-extended supersymmetric theories formulated in terms of the linearised super-Cotton tensor or its higher spin extensions for the cases N=2, 1, 0. In the N=2 case, this transformation is a generalisation of the linear-chiral duality, which is known to provide a dual description in terms of chiral superfields for general models of self-interacting N=2 vector multiplets in three dimensions and N=1 tensor multiplets in four dimensions. For superspin-1 (gravitino multiplet), superspin-3/2 (supergravity multiplet) and higher superspins s>3/2, the duality transformation relates a higher-derivative theory to one containing at most two derivatives at the component level. In the N=1 case, we introduce gauge prepotentials for higher spin superconformal gravity and construct the corresponding super-Cotton tensors, as well as the higher spin extensions of the linearised N=1 conformal supergravity action. Our N=1 duality transformation is a higher spin extension of the known superfield duality relating the massless N=1 vector and scalar multiplets. In the non-supersymmetric (N=0) case, the gauge prepotentials for higher spin conformal geometry (both bosonic and fermionic) and the corresponding Cotton tensors can be obtained from their N=1 counterparts by carrying out N=1 -->N=0 reduction. In the bosonic sector, this reduction is shown to lead to the Cotton tensors for higher spins constructed by Pope and Townsend, and by Damour and Deser in the spin-3 case. Our N=0 duality transformation is a higher spin extension of the vector-scalar duality.


Introduction
It is pertinent to recall the definition of the linear-chiral duality in the 3D N = 2 case we are interested in. We start from a self-interacting vector multiplet model with action It remains to point out that the constraint (1.1) is solved in the 3D case by where the real prepotential H is defined modulo gauge transformations of the form δH = λ +λ ,D α λ = 0 . (1.7) It is worth recalling one more type of duality that is naturally defined in the 3D N = 1 and 4D N = 2 cases, the so-called complex linear-chiral duality [11,12] (see also [13] for a review). It provides a dual description in terms of chiral superfields for general models of self-interacting complex linear superfields Γ and their conjugatesΓ. The complex linearchiral duality plays a fundamental role in the context of off-shell supersymmetric sigma models with eight supercharges [10,14]. The complex linear superfield Γ is defined by the only constraintD 2 Γ = 0 . (1.8) The complex linear-chiral duality works as follows. Consider a 3D N = 2 supersymmetric field with action We associate with it a first-oder action of the form Here the dynamical superfields comprise a complex unconstrained scalar V , a chiral scalar Ψ and their conjugates. Varying (1.10) with respect to the Lagrange multiplier Ψ gives V = Γ, and then the second term in (1.10) drops out as a consequence of the identities and its conjugate. The latter equations allow us to express the auxiliary superfields V andV in terms of Ψ andΨ. Then (1.10) turns into the dual action It should be mentioned that the complex linear-chiral duality can also be introduced in the reverse order, by starting with a chiral model and then applying a Legendre transformation to S[Ψ,Ψ] in order to result in a model described by a complex linear superfield Γ and its conjugateΓ. This makes use of the first-order action In the 4D N = 1 case, the first-order action (1.15) with K(U,Ū) = UŪ was considered for the first time by Zumino [20]. However, he did not realise the fact that this construction leads to a new off-shell description for the scalar multiplet, which was an observation made in [11,12].
The higher-spin generalisation of the complex linear-chiral duality has been given in [15,16,17,19]. For every integer 2s = 3, 4, . . . , it relates the two off-shell formulations for the massless superspin-s multiplet in four dimensions constructed 1 in [15,16,17] and for the massive superspin-s multiplet in three dimensions presented in [19]. The goal of this paper is to give higher-spin generalisations of the 3D N = 2 linear-chiral duality in three dimensions and its N = 1 and N = 0 cousins. This paper is organised as follows. The N = 2 duality transformation is presented in section 2. In section 3 we introduce gauge prepotentials for higher spin superconformal geometry, construct the corresponding super-Cotton tensors, and present the higher spin extension of the linearised N = 1 conformal supergravity action. Our N = 1 duality transformation is also described in this section. Finally, section 4 is devoted to the nonsupersymmetric (N = 0) higher spin duality transformation.

N = duality
Let n be a positive integer. We recall the higher-spin N = 2 superconformal field strength, where ⌊x⌋ denotes the floor (or the integer part) of a number x. The field strength W α(n) is a descendant of the real unconstrained prepotential H α(n) defined modulo gauge transformations of the form where the complex gauge parameter g α(n) is an arbitrary longitudinal linear superfield, The field strength is invariant under the gauge transformations (2.2), and obeys the Bianchi identitȳ As demonstrated in [19], W α(n) is a primary superfield of dimension (1 + n/2) if the prepotential H α(n) is chosen to be primary of dimension (−n/2). Associated with W α(n) is the N = 2 superconformal Chern-Simons action [19] which is invariant under the gauge transformations (2.2). In the n = 0 case, the Bianchi identity (2.4) should be replaced with (1.1).
In the n = 2 case, the field strength W αβ (H) coincides with the linearised version [21,19] of the N = 2 super-Cotton tensor [4,6]. Thus the field strength (2.1) for n > 2 is the higher-spin extension of the super-Cotton tensor.
For n = 0 the Bianchi identity (2.4) is not defined, and is instead replaced with its corollary (2.5). In this case, the expression (2.1) reduces to the vector multiplet field strength (1.6), and the gauge invariance (2.2) turns into (1.7). Finally, for n = 0 the action (2.6) reduces to the topological mass term for the Abelian vector multiplet [22,23,24].
Of crucial importance for our analysis is the fact that the expression (2.1) is the general solution to the constraint (2.4). This observation is the key to introducing a new type of duality. Let us consider a higher-derivative N = 2 superconformal theory with action where the superconformal compensator Φ is a nowhere vanishing primary superfield of dimension 1/2. The origin of Φ is not important for us. In particular, Φ may be frozen to a constant value, and then we result with a theory described solely in terms of the higher-spin gauge superfield H α(n) . We can associate with (2.7) the following first-order model The general solution of this constraint is for some unconstrained complex prepotential ζ α(n−1) . Varying (2.8) with respect to G α(n) and its conjugate gives In describing our duality transformation, we assumed n > 0. It is easy to see that it reduces to the linear-chiral duality in the n = 0 case.

N = 1 duality
The results in the previous section can be used to obtain a new type of N = 1 duality by carrying out the N = 2 → N = 1 superspace reduction sketched in [19]. Applying this reduction to (2.1) leads to the higher-spin N = 1 superconformal field strength 2 which is real, W α(n) =W α(n) . The field strength W α(n) is a descendant of the real unconstrained prepotential H α(n) defined modulo gauge transformations of the form δH α(n) = i n D (α 1 ζ α 2 ...αn) ,ζ α(n−1) = ζ α(n−1) .

(3.2)
The field strength is invariant under the gauge transformations (3.2), and obeys the Bianchi identity It may be shown that W α(n) is a primary superfield of dimension (1 + n/2), in the sense of [26,27], if the prepotential H α(n) is primary of dimension (1 − n/2). Associated with W α(n) is the N = 1 superconformal Chern-Simons action which is invariant under the gauge transformations (3.2). The superconformal invariance of S CS [H] is discussed in detail in [25]. The action (3.5) coincides for n = 1 with the topological mass term for the Abelian vector multiplet [22]. In the n = 3 case, (3.5) proves to be the linearised action for N = 1 conformal supergravity [13,5].
For n = 1 the field strength (3.1) is as a consequence of the anti-commutation relation The final expression for W α in (3.6) coincides with the gauge-invariant field strength of a vector multiplet [13]. The Bianchi identity D α W α = 0 is a corollary of For n = 2 the field strength (3.1) can be seen to coincide with the gravitino field strength [13]. Finally, for n = 3 the field strength (3.1) is the linearised version [28] of the N = 1 super-Cotton tensor [5,6]. This is why (3.1) can be called the higher spin super-Cotton tensor.
It should be pointed out that (3.1) is the general solution of the constraint (3.4). The simplest way to prove this is the observation that the field strength (3.1) may be recast in the form It is completely symmetric, W α 1 ...αn = W (α 1 ...αn) , as a consequence of (3.8).

(3.12)
Varying S[W, G] with respect to the Lagrange Ψ α(n−1) leads us back to the original action (3.10), and therefore the models (3.10) and (3.11) are equivalent. On the other hand, we can integrate out the auxiliary superfield W α(n) from (3.11), which leads us to a dual action of the form Unlike the original action (3.10), which is a higher-derivative theory for n > 1, its dual S D [G] is free of higher derivatives. In the n = 1 case, the duality transformation described corresponds to the standard duality between the N = 1 scalar and vector multiplets in three dimensions [9].

N = 0 duality
The N = 1 higher spin super-Cotton tensor and the N = 1 duality transformation described in section 3 were obtained by performing the N = 2 → N = 1 superspace reduction, in analogy with the earlier results for extended supersymmetric nonlinear sigma models [29,30]. Actually one can continue this process one step further and carry out the N = 1 → N = 0 reduction. This gives the higher-spin conformal field strength The field strength (4.1) is invariant under these gauge transformations, and obeys the Bianchi identity It may be shown that C α(n) is a primary field of dimension (1 + n/2) if the prepotential h α(n) is primary of dimension (2 − n/2). Associated with C α(n) is the conformal Chern-Simons action [31] S which is invariant under the gauge transformations (4.2).
In the case of even rank, n = 2s, with s = 1, 2, . . . , the field strength (4.1) coincides with the bosonic higher spin Cotton tensor given originally by Pope and Townsend [31]. It reduces to the linearised Cotton tensor for n = 4, and to the Maxwell field strength for n = 2. It should be pointed out that the conformal spin-3 case, n = 6, was studied for the first time in [32]. In the case of odd rank, n = 2s + 1, eq. (4.1) describes fermionic higher spin conformal field strengths. They did not appear in [31]. The spin-3/2 case, n = 3, was considered in [33], where the field strength C α(3) was called the Cottino tensor.
The field strength (4.1) proves to be the general solution to the conservation equation (4.4). This result has recently been proved in [34] in the bosonic case, n = 2s, and the proof given is quite nontrivial. There is an alternative proof based on supersymmetry considerations. The point is that the higher spin Cotton tensor C α(n) may be imbedded into the N = 1 super-Cotton tensor W α(n) as its lowest (θ-independent) component. The latter obeys the constraint (3.4), which has the general solution (3.1) or, equivalently, (3.9). The fact that (3.9) is the general solution to (3.4), is a corollary of the N = 1 identities (3.8).
An important feature of the spin-n 2 Cotton tensor (4.1) for n > 1 is that it can be represented as a linear superposition of the equations of motion for the massless spin-n 2 Fronsdal action in three dimensions [35,36] (see [19] for a review), with the coefficients being linear higher-derivative operators. A similar result in the N = 2 supersymmetric case was spelt out in detail in [19], which is why here our consideration will be restricted to the bosonic case, n = 2s, with s = 2, 3, . . . The massless spin-s action is described by two real gauge fields ϕ i = h α(2s) , h α(2s−4) defined modulo the gauge transformations The field h α(2s) is the conformal spin-s prepotential, while the other field h α(2s−4) is a gauge compensator. Associated with the gauge fields h α(2s) and h α(2s−4) are the following gauge-invariant field strengths which are related to each other by the Bianchi identity In terms of these field strengths, the equations of motion for the massless spin-s field are F α(2s) = 0 and F α(2s−4) = 0 (compare with eq. (B.4) in [19]). We state that the spin-s Cotton tensor C α(2s) can be expressed in terms of F α(2s) and F α(2s−4) , in particular for the spin-2 and spin-3 cases we have The general result for s > 3 may be deduced, e.g., from the relation (6.25) in [19] by reducing it to components. There is a simple explanation why C α(2s) can be expressed in terms of F α(2s) and F α(2s−4) . It is based on the well-known result that on the equations of motion F α(2s) = 0 and F α(2s−4) = 0 the fields h α(2s) and h α(2s−4) can be completely gauged away (see [19] for a review and proof), and hence C α(2s) = 0. This implies that C α(2s) is a descendant of F α(2s) and F α(2s−4) . Eq. (4.9a) is equivalent to the standard linearised expression for the Cotton tensor in terms of the Schouten tensor. In [37] a spin-3 analog of the Schouten tensor was introduced as a potential for C α (6) . This spin-3 result of [37] readily follows from (4.9b). In our opinion, the higher spin generalisation of the spin-2 relation (4.9a) is the expression for C α(2s) in terms of F α(2s) and F α(2s−4) , see [19] for the N = 2 supersymmetric case.
Let us turn to introducing the aforementioned N = 0 duality transformation. We consider a higher-derivative conformal theory with action S[C] = d 3 x ϕ 6 L C α(n) ϕ n+2 , (4.10) where ϕ is a real conformal compensator of dimension 1/2. To obtain a dual description for the model, we can associate with (4.10) the following first-order model S[C, G] = d 3 x ϕ 6 L C α(n) ϕ n+2 − i n G α(n) C α(n) , (4.11) where C α(n) is an unconstrained field, and the Lagrange multiplier is G α(n) = ∂ (α 1 α 2 Ψ α 3 ...αn) . (4.12) Varying S[C, G] with respect to Ψ α(n−2) implies C α(n) = C α(n) , and then S[C, G] reduces to the original action (4.10). On the other hand, we can first integrate out the auxiliary field C α(n) from (4.11), which leads to a dual action of the form S[G] = d 3 x ϕ 6 L ϕ n−4 G α(n) . (4.13) In the n = 2 case, the duality transformation described corresponds to the standard vector-scalar duality in three dimensions.