The massless integer superspin multiplets revisited

We propose a new off-shell formulation for the massless ${\cal N}=1$ supersymmetric multiplet of integer superspin $s$ in four dimensions, where $s =2,3,\dots$ (the $s=1$ case corresponds to the gravitino multiplet). Its gauge freedom matches that of the superconformal superspin-$s$ multiplet described in arXiv:1701.00682. The gauge-invariant action involves two compensating multiplets in addition to the superconformal superspin-$s$ multiplet. Upon imposing a partial gauge fixing, this action reduces to the one describing the so-called longitudinal formulation for the massless superspin-$s$ multiplet. Our new model is shown to possess a dual realisation obtained by applying a superfield Legendre transformation. We present a non-conformal higher spin supercurrent multiplet associated with the new integer superspin theory. This fermionic supercurrent is shown to occur in the Fayet-Sohnius model for a massive ${\cal N}=2$ hypermultiplet. We also give a new off-shell realisation for the massless gravitino multiplet.


Introduction
In N = 1 supersymmetric field theory in four dimensions, a massless multiplet of (half) integer superspinŝ > 0 describes two ordinary massless fields of spinŝ andŝ + 1 2 . Such a supermultiplet is often denoted (ŝ,ŝ + 1 2 ). The three lowest superspin values,ŝ = 1 2 , 1 and 3 2 , correspond to the vector, gravitino and supergravity multiplets, respectively. It follows from first principles that the sum of two actions for free massless spin-ŝ and spin-(ŝ + 1 2 ) fields should possess an on-shell supersymmetry. This means that there is no problem of constructing on-shell massless higher superspin multiplets, withŝ > 3 2 , for it is only necessary to work out the structure of supersymmetry transformations. The latter task was completed first by Curtright [1] who made use of the (Fang-)Fronsdal actions [2,3], and soon after by Vasiliev [4] who employed his frame-like reformulation of the (Fang-)Fronsdal models pioneered in [4]. Applications of the on-shell higher spin supermultiplets presented in [1,4] are rather limited. In particular, they do not allow one to construct supermultiplets containing conserved higher spin currents that have to be off-shell, like the so-called supercurrent multiplet [5] containing the energy-momentum tensor and the supersymmetry current. To obtain such higher spin supercurrents, offshell realisations for the massless higher superspin multiplets are required, and these are nontrivial to construct. 1 The problem of constructing gauge off-shell formulations for the massless higher superspin multiplets was solved in the early 1990s in the case of Poincaré supersymmetry [9,10]. 2 For each superspinŝ > 3 2 , half-integer [9] and integer [10], these publications provided two dually equivalent off-shell actions formulated in N = 1 Minkowski superspace. At the component level, each of the two superspin-ŝ actions [9,10] reduces, upon imposing a Wess-Zumino-type gauge and eliminating the auxiliary fields, to a sum of the spin-ŝ and spin-(ŝ + 1 2 ) actions [2,3]. The massless higher superspin theories of [9,10] were generalised to the case of anti-de Sitter supersymmetry in [8].
The non-supersymmetric higher spin theories of [2,3] and their supersymmetric counterparts of half-integer superspin [9] share one common feature. For each of them, the action is formulated in terms of a (super)conformal gauge (super)field coupled to certain compensators. Such a description does not yet exist for the massless supermultiplets of integer superspinŝ ≥ 2. One of the goals of this paper is to provide such a formulation by properly generalising the off-shell supersymmetric actions given in [10]. We now make these points more precise.
The gauge freedom of Ψ α 1 ...αsα 1 ...α s−1 is chosen to coincide with that of the superconformal superspin-s multiplet [14], which is The longitudinal linear superfield is invariant under the ζ-transformation (2.5a) and varies under the V-transformation as It may be checked that the following action is invariant under the gauge transformations (2.5). By construction, the action is also invariant under (2.4).
The V-gauge freedom (2.5) may be used to impose the condition In this gauge, the action (2.8) reduces to that describing the longitudinal formulation for the massless superspin-s multiplet [10]. The gauge condition (2.9) does not fix completely the V-gauge freedom. The residual gauge transformations are generated by with the parameter L α(s)α(s−1) being an unconstrained superfield. With this expression for , the gauge transformations (2.5a) and (2.5b) coincide with those given in [10]. Our consideration implies that the action (2.8) indeed provides an off-shell formulation for the massless superspin-s multiplet .
Instead of choosing the condition (2.9), one can impose an alternative gauge fixing In accordance with (2.5b), in this gauge the residual gauge freedom is described by The action (2.8) includes a single term which involves the 'naked' gauge field Ψ α(s)α(s−1) and not the field strength G α(s)α(s) , the latter being defined by (2.6) and invariant under the ζ-transformation (2.5a). This is actually a BF term, for it can be written in two different forms The former makes the gauge symmetry (2.4) manifestly realised, while the latter turns the ζ-transformation (2.5a) into a manifest symmetry.
Making use of (2.13) leads to a different representation for the action (2.8). It is . (2.14)

Dual formulation
The theory with action (2.14) possesses a dual formulation that can be obtained by applying the duality transformation introduced in [9,10]. In general, it works as follows. Suppose we have a supersymmetric field theory formulated in terms of a longitudinal linear superfield G α(t)α(s) and its conjugateḠ α(s)α(t) , and the action has the form where L(G,Ḡ) is an algebraic function of its arguments. We now associate with this theory a first-order model of the form where U α(t)α(s) is a complex unconstrained superfield, and the Lagrange multiplier Γ α(t)α(s) is transverse linear. Varying S first-order with respect to the Lagrange multiplier gives U α(t)α(s) = G α(t)α(s) , and then S first-order reduces to the original action (3.1). On the other hand, we can consider the equation of motion for U α(t)α(s) , which is we assume that (3.3) can be solved to express U β(t)β(s) in terms of Γ α(t)α(s) and its conjugate. Plugging this solution back into (3.2) gives a dual action In the t = s = 0 case, the above duality transformation coincides with the so-called complex linear-chiral duality [16] which plays a fundamental role in the context of off-shell supersymmetric sigma models with eight supercharges [17,18].
We now associate with our theory (2.14) the following first-order action 7 The first-order action (3.5) is also invariant under the gauge V-transformation (2.5b) and (2.5c), which acts on U α(s)α(s) and Γ α(s)α(s) as Eliminating the auxiliary superfields U α(s)α(s) andŪ α(s)α(s) from (3.5) leads to The specific normalisation of the Lagrange multiplier in (3.5) is chosen to match that of [8,10].
where we have defined We point out that Γ α(s)α(s) is invariant under the gauge transformations (2.4) and (3.7b).
In accordance with (2.5c), the gauge V-freedom may be used to impose the condition Z α(s−1)α(s−1) = 0 . In this gauge the action (3.9) reduces to the one defining the transverse formulation for the massless superspin-s multiplet [10]. The gauge condition (3.11) is preserved by residual local V-and ξ-transformations of the form Making use of the parametrisation (2.10), the residual gauge freedom is which is exactly the gauge symmetry of the transverse formulation for the massless superspin-s multiplet [10].

Higher spin supercurrent multiplets
We now make use of the new gauge formulation (2.8), or equivalently (2.14), for the integer superspin-s multiplet to derive non-conformal higher spin supercurrents.
In order for S We see that the superfields J α(s)α(s−1) and T α(s−1)α(s−1) are transverse linear and longitudinal linear, respectively. Finally, requiring S (s) source to be invariant under the Vtransformation (2.5) gives the following conservation equation and its conjugate As a consequence of (4.3), from (4.4a) we deduce The equations (4.2) and (4.5) describe the conserved current supermultiplet which corresponds to our theory in the gauge (2.9).
Taking the sum of (4.4a) and (4.4b) leads to The equations (4.2), (4.3) and (4.6) describe the conserved current supermultiplet which corresponds to our theory in the gauge (2.11). As a consequence of (4.3), the conservation equation (4.6) implies As in [21], it is useful to introduce auxiliary complex variables ζ α ∈ C 2 and their conjugatesζα. Given a tensor superfield U α(p)α(q) , we associate with it the following field on C 2 which is homogeneous of degree (p, q) in the variables ζ α andζα. We introduce operators that increase the degree of homogeneity in the variables ζ α andζα, We also introduce two nilpotent operators that decrease the degree of homogeneity in the variables ζ α andζα, specifically 5 Higher spin supercurrents in a massive chiral model Consider the Fayet-Sohnius model [19,20] for a free massive hypermultiplet where the superfields Φ ± are chiral,DαΦ ± = 0, and the mass parameter m is chosen to be positive.
In the massless case, m = 0, the conserved fermionic supercurrents J α(s)α(s−1) were constructed in [14]. In our notation they read Making use of the massless equations of motion, D 2 Φ ± = 0, one may check that J (s,s−1) obeys, for s > 1, the conservation equations We will now construct fermionic higher spin supercurrents corresponding to the massive model (5.1). Assuming that J (s,s−1) has the same functional form as in the massless case, eq. (5.2), and making use of the equations of motion we obtain It can be shown that the massive supercurrent J (s,s−1) also obeys (4.11).
We now look for a superfield T (s−1,s−1) such that (i) it obeys the longitudinal linear constraint (4.12); and (ii) it satisfies (4.14), which is a consequence of the conservation equation (4.13). We consider a general ansatz Condition (i) implies that the coefficients must be related by while for k = 1, 2, . . . s − 2, condition (ii) gives the following recurrence relations: Condition (ii) also implies that The above conditions lead to simple expressions for c k and d k : where k = 1, 2, . . . s−1. Now that we have already derived an expression for the trace multiplet T (s−1,s−1) , the superfield S (s−1,s−1) can be computed using the conservation equation (4.13). This gives One may verify that S (s−1,s−1) is a real superfield.

Concluding comments
To conclude this work, we make several final comments.
The formulation proposed in section 2 can naturally be lifted to the case of anti-de Sitter supersymmetry to extend the results of [8].
In the massive case, m = 0, we deal with the N = 2 Poincaré supersymmetry with a constant central charge on the mass shell, and the story becomes pretty subtle. In our previous work [21], we observed that the higher spin supercurrents J α(s)α(s) in the massive chiral model exist only for odd values of s. The same conclusion was also reached in a revised version (v3, 26 Oct.) of Ref. [23]. However, the conserved fermionic supercurrents J α(s)α(s−1) constructed in the present paper are realised for all values of s > 1.
The longitudinal and transverse actions for the massless integer superspin-s multiplet [10] are well defined for s = 1, in which case they describe two off-shell formulations for the massless gravitino multiplet. However, the action (2.8) is not defined in the s = 1 case. The point is that the gauge transformation law (2.5a) is not defined for s = 1. The gauge freedom in the superconformal gravitino multiplet model [14] is This transformation law of Ψ α coincides with the one occurring in the off-shell model for the massless gravitino multiplet proposed in [25]. In addition to the gauge superfield Ψ α , this model also involves two compensators, a real scalar H and a chiral scalar Φ,DαΦ = 0, with the gauge transformation laws δH = V +V , (6.3b) The gauge invariant action of [25] can be written in the form [11] S (I) where S (1, 3 2 ) [Ψ,Ψ, H] denotes the longitudinal action for the gravitino multiplet, which is obtained from (2.8) by choosing the gauge (2.9) and setting s = 1. At the component level, this manifestly supersymmetric model is known to describe the Fradkin-Vasiliev-de Wit-van Holten formulation for the gravitino multiplet [26,27].
There exists a dual formulation for (6.4) that is obtained by performing a superfield Legendre transformation [28]. The dual action given in [28] is where G =Ḡ is a real linear superfield,D 2 G = D 2 G = 0. The gauge freedom in this theory is given by eqs. (6.3a), (6.3b) and δG = −D α ζ α −Dαζα , (6.6) in accordance with [29]. It may be used to impose two conditions H = 0 and G = 0. Then we end up with the Ogievetsky-Sokatchev formulation for the gravitino multiplet [30] (see section 6.9.5 [11] for the technical details).
Actually, there exists one more dual formulation for (6.4) that is obtained by performing the complex linear-chiral duality transformation. It leads to where Σ is a complex linear superfield constrained byD 2 Σ = 0. The gauge freedom in this theory is given by eqs. (6.3a), (6.3b) and δΣ = −D α ζ α . (6.8) This gauge freedom does not allow one to gauge away Σ off the mass shell. To the best of our knowledge, the supersymmetric gauge theory (6.7) is a new off-shell realisation for the massless gravitino multiplet.
The transverse formulation for the massless gravitino multiplet, which was introduced in [10], is quite mysterious in the sense that it is not contained in any known off-shell formulation for N = 2 supergravity.