AdS superprojectors
aa r X i v : . [ h e p - t h ] F e b January, 2021Revised version: February, 2021
AdS superpro jectors
E. I. Buchbinder, D. Hutchings, S. M. Kuzenko and M. Ponds
Department of Physics M013, The University of Western Australia35 Stirling Highway, Perth W.A. 6009, Australia
Email: [email protected], [email protected],[email protected], [email protected]
Abstract
Within the framework of N = 1 anti-de Sitter (AdS) supersymmetry in fourdimensions, we derive superspin projection operators (or superprojectors). For atensor superfield V α ( m ) ˙ α ( n ) := V ( α ...α m )( ˙ α ... ˙ α n ) on AdS superspace, with m and n non-negative integers, the corresponding superprojector turns V α ( m ) ˙ α ( n ) into a mul-tiplet with the properties of a conserved conformal supercurrent. It is demonstratedthat the poles of such superprojectors correspond to (partially) massless multiplets,and the associated gauge transformations are derived. We give a systematic discus-sion of how to realise the unitary and the partially massless representations of the N = 1 AdS superalgebra osp (1 |
4) in terms of on-shell superfields. As an example,we present an off-shell model for the massive gravitino multiplet in AdS . We alsoprove that the gauge-invariant actions for superconformal higher-spin multipletsfactorise into products of minimal second-order differential operators. ontents so (3 ,
2) and osp (1 |
4) . . . . . . . . . . . . . . . 52.2 On-shell fields in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Partially massless and massive fields . . . . . . . . . . . . . . . . . . . . . 9 Conclusion 41A AdS superspace toolkit 42B Partially massless gauge symmetry 45
B.1 The non-supersymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . 45B.2 The supersymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Superprojectors [1–7] are superspace projection operators which single out irreduciblerepresentations of supersymmetry. Various applications of such operators have appearedin the literature, including the following constructions: (i) superfield equations of motion[8,9]; (ii) gauge-invariant actions in four dimensions [10,11]; and (iii) off-shell N -extendedsuperconformal actions in three dimensions [7]. Of special interest are those superpro-jectors which single out the highest superspin of tensor superfields V α ( m ) ˙ α ( n ) ( x, θ, ¯ θ ) := V α ...α m ˙ α ... ˙ α n ( x, θ, ¯ θ ) = V ( α ...α m )( ˙ α ... ˙ α n ) ( x, θ, ¯ θ ), since they may be viewed as supersym-metric extensions of the Behrends-Fronsdal spin projection operators [13, 14]. We recallthat a tensor field ϕ α ( m ) ˙ α ( n ) ( x ) of Lorentz type ( m/ , n/
2) is mapped by the correspondingBehrends-Fronsdal projector to a transverse field ϕ T α ( m ) ˙ α ( n ) ( x ) such that ∂ β ˙ β ϕ T βα ( m −
1) ˙ β ˙ α ( n − = 0 . (1.1)This constraint is the characteristic feature of a conserved current j α ( m ) ˙ α ( n ) .The four-dimensional results of [1–6] correspond to the Poincar´e supersymmetry. Notmuch is known about the structure of superprojectors corresponding to the AdS super-symmetry OSp (1 | N = 1 AdS superspace Ref. [7] was a natural extension of the work [12], where the spin projection operators in three dimen-sions were constructed and used to obtain simple expressions for the higher-spin Cotton tensors. Refs. [13, 14] made use of the four-vector notation in conjunction with the four-component spinorformalism, which resulted in rather complicated expressions for the spin projection operators. However,switching to the two-component spinor formalism leads to remarkably simple and compact expressionsfor these projectors [3, 6]. | and presented the projectors which single out such supermultiplets. We recall thata complex tensor superfield Γ α ( m ) ˙ α ( n ) on AdS | is said to be transverse linear if it satisfiesthe constraint¯ D ˙ β Γ α ( m ) ˙ β ˙ α ( n − = 0 ⇐⇒ ( ¯ D − n + 2) µ )Γ α ( m ) ˙ α ( n ) = 0 , n > , (1.2)where µ = 0 is a constant parameter which determines the curvature of AdS | , seesection 3.1 below. A complex tensor superfield G α ( m ) ˙ α ( n ) is said to be longitudinal linear if it satisfies the constraint¯ D ( ˙ α G α ( m ) ˙ α ... ˙ α n +1 ) = 0 ⇐⇒ ( ¯ D + 2 nµ ) G α ( m ) ˙ α ( n ) = 0 . (1.3)For n = 0 the first constraint in (1.2) is not defined, while the second condition( ¯ D − µ )Γ α ( m ) = 0 (1.4)defines a linear superfield. For n = 0 the constraint (1.3) defines a chiral superfield.The constraints (1.2), (1.3) and (1.4) are the the only differential constraints in AdS | which define off-shell supermultiplets with unconstrained component fields [15]. Certaintransverse linear and longitudinal linear supermultiplets play a fundamental role in themassless supersymmetric higher-spin gauge theories in AdS | [16] and their ancestors inMinkowski superspace [17, 18], see also [19] for a review.Ref. [15] described the projectors to the spaces of superfields which are constrainedby (1.2), (1.3) and (1.4), see eq. (3.10) below. However, these are not supersymmetricextensions of the Behrends-Fronsdal spin projection operators [13, 14], since all indepen-dent component fields of the resulting superfield are unconstrained. Supersymmetric spinprojection operators are required to turn an unconstrained superfield on AdS | into onewith the properties of a conserved current supermultiplet. It is pertinent to describethe structure of conformal supercurrents in AdS | following the more general analysis ofconserved current supermultiplets in a supergravity background [20].Let m and n be positive integers. A tensor superfield J α ( m ) ˙ α ( n ) on AdS | is called aconformal supercurrent of Lorentz type ( m/ , n/
2) if it obeys the two constraints D β J βα ( m −
1) ˙ α ( n ) = 0 ⇐⇒ (cid:0) D − m + 2)¯ µ (cid:1) J α ( m ) ˙ α ( n ) = 0 , (1.5a)¯ D ˙ β J α ( m ) ˙ β ˙ α ( n − = 0 ⇐⇒ (cid:0) ¯ D − n + 2) µ (cid:1) J α ( m ) ˙ α ( n ) = 0 . (1.5b)If m = n , it is consistent to restrict J α ( n ) ˙ α ( n ) to be real, ¯ J α ( n ) ˙ α ( n ) = J α ( n ) ˙ α ( n ) . The m = n = 1 case corresponds to the ordinary conformal supercurrent [21]. The case3 = n > | in [24]. The case m = n + 1 > | in [24].If m > n = 0, the constraints (1.5) should be replaced with D β J βα ( m − = 0 ⇐⇒ (cid:0) D − m + 2)¯ µ (cid:1) J α ( m ) = 0 , (1.6a)( ¯ D − µ ) J α ( m ) = 0 . (1.6b)The m = 1 case was first considered in [25], where it was shown that the spinor supercur-rent J α naturally originates from the reduction of the conformal N = 2 supercurrent [26]to N = 1 superspace.Finally, for m = n = 0 the constraints (1.6) should be replaced with( D − µ ) J = 0 , (1.7a)( ¯ D − µ ) J = 0 . (1.7b)These constraints describe a flavour current supermultiplet [27] in AdS . Irreduciblesupermultiplets of the types (1.5), (1.6) and (1.7) have been used in [28] for the covariantquantisation of the massless supersymmetric higher-spin gauge theories in AdS [16].The Behrends-Fronsdal projectors have been generalised to the case of AdS onlyrecently in [29]. One of the important outcomes of [29] was a new understanding of theso-called partially massless fields in AdS . Such fields in diverse dimensions were studiedearlier in [30–43]. Specifically, it was shown in [29] that the partially massless fields areassociated with the poles of the spin projection operators in AdS . In the present paperwe provide an N = 1 supersymmetric extension of the AdS spin projection operatorsand describe various applications of the resulting superprojectors. In particular, we willdemonstrate that the partially massless N = 1 supermultiplets in AdS , which haverecently been analysed in the component settings in [44, 45], are naturally associated withthe poles of the proposed superprojectors.This paper is organised as follows. Section 2 provides a brief review of the unitaryrepresentations of the AdS algebra so (3 ,
2) and its N = 1 supersymmetric extension osp (1 | . Section 3 is devotedto the construction of spin projection operators that map every unconstrained superfieldon AdS | into one with the properties of a conserved current supermultiplet. On-shellsupermultiplets in AdS are studied in section 4. Section 5 is devoted to the componentstructure of the on-shell supermultiplets. In section 6 we present an off-shell model for4he massive gravitino (superspin-1) multiplet. Factorisation of the superconformal higher-spin actions is described in section 7 and concluding comments are given in section 8. Themain body of the paper is accompanied by two appendices. Appendix A contains a list ofidentities that are indispensable for the derivation of many results in this paper. AppendixB is devoted to the derivation of partially massless gauge transformations, both in thenon-supersymmetric and supersymmetric cases.Throughout this work we will make use of various abbreviations to denote differenttypes of irreducible superfields. In particular, a superfield J α ( m ) ˙ α ( n ) satisfying (1.5) issimultaneously transverse linear and transverse anti-linear, and will be called TLAL. Asuperfield J α ( m ) satisfying (1.6) is simultaneously linear and transverse anti-linear, andwill be called LTAL. Finally, a scalar superfield J satisfying (1.7) is simultaneously linearand anti-linear, and will be called LAL. In this section we collate the well-known facts concerning (i) the unitary representa-tions of the AdS algebra so (3 ,
2) and its N = 1 supersymmetric extension osp (1 | . so (3 , and osp (1 | The irreducible unitary representations of the AdS algebra so (3 ,
2) were studied indetail in [46–54]. The results of these studies were used in [55] to classify the irreducibleunitary representations of the N = 1 AdS superalgebra osp (1 |
4) (see also [56, 57] for acomprehensive review).The irreducible unitary representations of the AdS algebra so (3 ,
2) are specified bythe lowest value E of the energy E and spin s , and are traditionally denoted by D ( E , s ). The allowed spin values are s = 0 , / , , · · · , the same as in Minkowski space. However,unlike in Minkowski space, unitarity imposes a bound on the allowed values of energy.According to the theorems proved in [52, 53], D ( E , s ) is unitary iff one of the followingconditions holds: (i) s = 0, E ≥ ; (ii) s = , E ≥
1, and (iii) s ≥ E ≥ s + 1. The The parameters E and s determine the values of the quadratic and quartic Casimir operators of so (3 , E is chosen to be dimensionless. To restore dimensionful energy,one has to rescale E → | µ | E , where µ ¯ µ determines the AdS curvature (2.7). D (cid:0) , (cid:1) = Rac and D (cid:0) , (cid:1) = Di are known as the Dirac singletons [47]. The representations D ( s + 1 , s ) for s > D (1 , ⊕ D (2 ,
0) are called massless sincethey contract to the massless discrete helicity representations of the Poincar´e group [54].These representations prove to be restrictions of certain unitary representations of theconformal algebra so (4 ,
2) to so (3 ,
2) [54, 60]. The remaining representations D ( E , s ) areusually referred to as the massive AdS representations.Irreducible unitary representations of the N = 1 AdS superalgebra osp (1 | so (3 ,
2) in analogy with the case of the super Poincar´e algebra in flatspace. Each term in the decomposition is identified with a particle (or field) carryingdefinite energy and spin. Unlike in flat space, particles in AdS belonging to the samesupermultiplet do not carry the same energy.There exist four types of unitary supermultiplets in AdS . The first are known asmassive representations and have the following decomposition S (cid:0) E , s (cid:1) := D (cid:0) E + 12 , s − (cid:1) ⊕ D (cid:0) E , s (cid:1) ⊕ D (cid:0) E + 1 , s (cid:1) ⊕ D (cid:0) E + 12 , s + 12 (cid:1) , (2.1a) s > , E > s + 1 , (2.1b)where the last inequality is the unitarity bound. This decomposition implies that amassive supermultiplet describes four particles in AdS . When s = 0 the first term inthe right-hand side of (2.1a) is absent in the decomposition and the unitarity bound isalso modified. The corresponding representations are referred to as the Wess-Zuminorepresentations: S (cid:0) E , (cid:1) := D (cid:0) E , (cid:1) ⊕ D (cid:0) E + 1 , (cid:1) ⊕ D (cid:0) E + 12 , (cid:1) , E > . (2.2)When the unitarity bound is saturated, E = s + 1 in eq. (2.1) or E = in eq. (2.2),the supermultiplets get shortened. This yields the other two types of representations.For s > s ≥ S (cid:0) s + 1 , s (cid:1) := D (cid:0) s + 1 , s (cid:1) ⊕ D (cid:0) s + 32 , s + 12 (cid:1) , s > . (2.3) It was found by Flato and Fronsdal [58, 59] that the singletons are the square roots of masslessparticles in the sense that all two-singleton states are massless. Our notation for the AdS supermultiplets follows the one introduced in [19]. Fronsdal [62] used adifferent notation for these representations, D S ( E , s ). consists of two physical componentfields. Finally, the fourth type of representation occurs for s = 0 , E = and is calledthe Dirac supermultiplet [59, 61] (or super-singleton) S (cid:0) , (cid:1) = D (cid:0) , (cid:1) ⊕ D (cid:0) , (cid:1) . (2.4)It unifies the bosonic and fermionic singletons.We will call the parameter s in (2.1) and (2.3) the superspin. The Wess-Zuminosupermultiplets (2.2) correspond to the superspin-0 representations. The geometry of AdS can be described in terms of torsion-free Lorentz-covariantderivatives of the form ∇ a = e a + ω a , e a = e am ∂ m , (2.5)where e am is the inverse vielbein, and ω a = 12 ω abc M bc = ω aβγ M βγ + ¯ ω a ˙ β ˙ γ ¯ M ˙ β ˙ γ , (2.6)is the Lorentz connection. The Lorentz generators M bc ⇔ ( M βγ , ¯ M ˙ β ˙ γ ) are defined inappendix A. The algebra of AdS covariant derivatives is (cid:2) ∇ α ˙ α , ∇ β ˙ β (cid:3) = − µ ¯ µ (cid:0) ε αβ ¯ M ˙ α ˙ β + ε ˙ α ˙ β M αβ (cid:1) ⇐⇒ (cid:2) ∇ a , ∇ b (cid:3) = − µ ¯ µM ab , (2.7)where the parameter µ ¯ µ > scalar curvature via R = − µ ¯ µ . Thecomplex parameter µ appears explicitly only in the algebra of AdS | covariant derivativesgiven by eq. (3.3).Below we make use of the quadratic Casimir of the AdS group, whose realisation ontensor fields is Q := − ∇ α ˙ α ∇ α ˙ α − µ ¯ µ (cid:0) M αβ M αβ + ¯ M ˙ α ˙ β ¯ M ˙ α ˙ β (cid:1) , (cid:2) Q , ∇ α ˙ α (cid:3) = 0 . (2.8)One should keep in mind that ∇ a ∇ a = − ∇ α ˙ α ∇ α ˙ α .Given two positive integers m and n , a tensor field h α ( m ) ˙ α ( n ) of Lorentz type ( m/ , n/ is said to be on-shell if it satisfies the equations0 = (cid:0) Q − ρ (cid:1) h α ( m ) ˙ α ( n ) , (2.9a)7 = ∇ β ˙ β h α ( m − β ˙ α ( n −
1) ˙ β . (2.9b)We say that such a field describes a spin s = ( m + n ) particle with pseudo-mass ρ . Thefollowing general result holds: ρ = (cid:2) E ( E −
3) + s ( s + 1) (cid:3) µ ¯ µ , (2.10)see [56, 57] for pedagogical derivations.As the name suggests, the pseudo-mass does not coincide with what is usually consid-ered to be the physical mass ρ phys . Rather, the two are related through ρ = ρ − τ (1 ,m,n ) µ ¯ µ , τ (1 ,m,n ) = 12 (cid:0) ( m + n ) − (cid:1) , (2.11)where τ (1 ,m,n ) is one of the partial massless parameters (2.17). The on-shell field h α ( m ) ˙ α ( n ) corresponds to the irreducible representation D (cid:0) E , ( m + n ) (cid:1) where the minimal energy E is related to the physical mass through ρ = (cid:2) E ( E − −
14 ( m + n + 2)( m + n − (cid:3) µ ¯ µ . (2.12)The unitarity bound for m + n > E ≥ ( m + n + 2), which in terms of the masses is ρ ≥ ⇒ ρ ≥ τ (1 ,m,n ) µ ¯ µ , m + n > . (2.13)With these relations in mind, we usually prefer to use the pseudo-mass as a representationlabel in place of E . As a caveat we note that there are two distinct values of E leadingto the same value of ρ , (cid:0) E (cid:1) ± = 32 ± s ρ µ ¯ µ + ( m + n − . (2.14)However, the solution ( E ) − always violates the unitarity bound for m + n >
1. Thus whenreferring to a unitary representation with s > and pseudo-mass ρ , we are implicitlyreferring to the representation corresponding to ( E ) + , E = 32 + 12 s ρ µ ¯ µ + ( m + n − , m + n > . (2.15)For s = 0 , (or, equivalently, m + n = 0 , E ≥ s + , andthe solution ( E ) − in (2.14) does not violate the unitarity bound for certain values of ρ . For s = 0 the values of ρ leading to ( E ) − ≥ are restricted by the condition − | µ | ≤ ρ ≤ | µ | , which is known as the Breitenlohner-Freedman bound [63].8 .3 Partially massless and massive fields Given two positive integers m and n , the tensor field h ( t ) α ( m ) ˙ α ( n ) is said to be partiallymassless with depth- t if it satisfies the on-shell conditions (2.9) such that its pseudo-masstakes the special value [29, 38, 39, 42] ρ = τ ( t,m,n ) µ ¯ µ , ≤ t ≤ min( m, n ) , (2.16)where the dimensionless constants τ ( t,m,n ) are defined by τ ( t,m,n ) := 12 (cid:16) ( m + n − t + 3)( m + n − t −
1) + ( t − t + 1) (cid:17) . (2.17)The specific feature of partially massless fields is that, for a fixed t , the system of equations(2.9) and (2.16) admits a depth- t gauge symmetry δ ζ h ( t ) α ( m ) ˙ α ( n ) = ∇ ( α ( ˙ α · · · ∇ α t ˙ α t ζ ( t ) α t +1 ...α m ) ˙ α t +1 ... ˙ α n ) . (2.18)This is true as long as the gauge parameter ζ ( t ) α ( m − t ) ˙ α ( n − t ) is also on-shell with the samepseudo-mass, 0 = (cid:0) Q − τ ( t,m,n ) µ ¯ µ (cid:1) ζ ( t ) α ( m − t ) ˙ α ( n − t ) , (2.19a)0 = ∇ β ˙ β ζ ( t ) α ( m − t − β ˙ α ( n − t −
1) ˙ β . (2.19b)It was demonstrated in [29] that the parameters τ ( t,m,n ) defined by (2.17) determine thepoles of the off-shell transverse projection operator for fields of Lorentz type ( m/ , n/ . This observation leads to a new understanding of the partially massless fields.Specifically, the gauge symmetry (2.18) of the field h ( t ) α ( m ) ˙ α ( n ) is associated with the pole ρ = τ ( t,m,n ) µ ¯ µ of the corresponding spin projection operator in AdS . In appendix B weprovide a systematic derivation of this claim using the spin projection operators of [29].Strictly massless fields carry depth t = 1 and therefore have mass given by ρ = τ (1 ,m,n ) µ ¯ µ = ⇒ ρ = 0 . (2.20)This saturates the bound (2.13) and hence defines a unitary representation of so (3 , τ (1 ,m,n ) > τ ( t,m,n ) , ≤ t ≤ min( m, n ) , (2.21)the true partially massless representations are non-unitary. In particular, this means thatthere are two minimal energy values,( E ) ± = 32 ±
12 ( m + n − t + 1) , (2.22)9hich are equally valid since they both violate the unitarity bound. In this work, wheneverthis ambiguity arises, we always implicitly choose the positive branch, E ≡ ( E ) + . Todistinguish the true partially-massless representations with depth t and Lorentz type( m/ , n/ P (cid:0) t, m, n (cid:1) , ≤ t ≤ min( m, n ) . (2.23)Such a representation carries minimal energy E = ( m + n ) − t + 2.The massive representation of so (3 ,
2) with spin s = ( m + n ) is realised on the spaceof fields h α ( m ) ˙ α ( n ) satisfying the equations (2.9) in which ρ is constrained by ρ > τ (1 ,m,n ) µ ¯ µ = ⇒ ρ > , (2.24)but is otherwise arbitrary. This restriction ensures the unitarity of the representation. In this section we construct spin projection operators that map every unconstrainedsuperfield on AdS | into one with the properties of a conserved current supermultiplet. Historically, the N = 1 AdS superspace, AdS | , was originally introduced [15, 64, 65]as the coset space AdS | := OSp (1 | / SO (3 , N = 1 AdS supergravity: (i) the well-known minimal theory (see, e.g., [6,66] for reviews);and (ii) the more recently discovered non-minimal theory [67]. Here we prefer to use thesecond definition, since it allows us to obtain all information about the geometry of AdS | from the well-known supergravity results.As usual, we denote by z M = ( x m , θ µ , ¯ θ ˙ µ ) the local coordinates of AdS | . The geometryof AdS | is described in terms of covariant derivatives of the form D A = ( D a , D α , ¯ D ˙ α ) = E A + Ω A , E A = E AM ∂ M , (3.1)where E AM is the inverse superspace vielbein, andΩ A = 12 Ω Abc M bc = Ω Aβγ M βγ + ¯Ω A ˙ β ˙ γ ¯ M ˙ β ˙ γ , (3.2)10s the Lorentz connection. The covariant derivatives obey the following graded commuta-tion relations: {D α , ¯ D ˙ α } = − D α ˙ α , (3.3a) {D α , D β } = − µM αβ , { ¯ D ˙ α , ¯ D ˙ β } = 4 µ ¯ M ˙ α ˙ β , (3.3b)[ D α , D β ˙ β ] = i¯ µε αβ ¯ D ˙ β , [ ¯ D ˙ α , D β ˙ β ] = − i µε ˙ α ˙ β D β , (3.3c)[ D α ˙ α , D β ˙ β ] = − µµ ( ε αβ ¯ M ˙ α ˙ β + ε ˙ α ˙ β M αβ ) . (3.3d)Here µ = 0 is a complex parameter. We recall that µ is related to the scalar curvature R of AdS by the rule R = − µ ¯ µ .In what follows, we make extensive use of the quadratic Casimir of the AdS superal-gebra, whose realisation on superfields takes the form [28] Q := ✷ + 14 (cid:16) µ D + ¯ µ ¯ D (cid:17) − µ ¯ µ (cid:16) M αβ M αβ + ¯ M ˙ α ˙ β ¯ M ˙ α ˙ β (cid:17) , ✷ = D a D a , (3.4a) (cid:2) Q , D A (cid:3) = 0 . (3.4b)It is an instructive exercise to check, using the relations (3.3), that (3.4b) holds.As discussed in section 1, a complex tensor superfield Γ α ( m ) ˙ α ( n ) is said to be transverselinear if it satisfies the constraint (1.2), whilst its conjugate obeys the constraint D β ¯Γ βα ( n −
1) ˙ α ( m ) = 0 ⇐⇒ ( D − n + 2)¯ µ )¯Γ α ( n ) ˙ α ( m ) = 0 (3.5)and is called transverse anti-linear. Similarly, a complex superfield G α ( m ) ˙ α ( n ) is said to belongitudinal linear if it satisfies the constraint (1.3), whilst its conjugate is constrained by D ( α ¯ G α ...α n +1 ) ˙ α ( m ) = 0 ⇐⇒ ( D + 2 n ¯ µ ) ¯ G α ( n ) ˙ α ( m ) = 0 , (3.6)and is called longitudinal anti-linear. We recall that for n = 0 the constraint (1.3) definesa chiral superfield, ¯ D ˙ α G α ( m ) = 0 ⇐⇒ ¯ D G α ( m ) = 0 . (3.7)We also recall that for n = 0 the first constraint in (1.2) is not defined, however one canconsistently define complex linear superfields constrained by (1.4).Given a complex tensor superfield V α ( m ) ˙ α ( n ) with n = 0, it can be uniquely representedas a sum of transverse linear and longitudinal linear multiplets, V α ( m ) ˙ α ( n ) = − µ ( n + 2) ¯ D ˙ γ ¯ D ( ˙ γ V α ( m ) ˙ α ... ˙ α n ) − µ ( n + 1) ¯ D ( ˙ α ¯ D | ˙ γ | V α ( m ) ˙ α ... ˙ α n ) ˙ γ . (3.8)11hoosing V α ( m ) ˙ α ( n ) to be transverse linear (Γ α ( m ) ˙ α ( n ) ) or longitudinal linear ( G α ( m ) ˙ α ( n ) ),the above relation gives Γ α ( m ) ˙ α ( n ) = ¯ D ˙ β ξ α ( m ) ˙ β ˙ α ( n ) , (3.9a) G α ( m ) ˙ α ( n ) = ¯ D ( ˙ α ζ α ( m ) ˙ α ... ˙ α n ) , (3.9b)for some prepotentials ξ α ( m ) ˙ α ( n +1) and ζ α ( m ) ˙ α ( n − . These relations provide general solu-tions to the constraints (1.2) and (1.3).Ref. [15] introduced projectors P ⊥ n and P k n which map the space of unconstrained su-perfields V α ( m ) ˙ α ( n ) to the space of transverse linear (1.2) and longitudinal linear superfields(1.3), respectively. These operators have the form P ⊥ n = 14( n + 1) µ ( ¯ D + 2 nµ ) , ¯ D ˙ β P ⊥ n V α ( m ) ˙ β ˙ α ( n − = 0 , (3.10a) P k n = − n + 1) µ ( ¯ D − n + 2) µ ) , ¯ D ( ˙ α P k n V α ( m ) ˙ α ... ˙ α n +1 ) = 0 , (3.10b)and satisfy the projector properties P ⊥ n P k n = 0 , P k n P ⊥ n = 0 , P ⊥ n + P k n = . (3.11)These properties imply that any superfield V α ( m ) ˙ α ( n ) can be uniquely represented as asum of transverse linear and longitudinal linear superfields, V α ( m ) ˙ α ( n ) = Γ α ( m ) ˙ α ( n ) + G α ( m ) ˙ α ( n ) , (3.12)which agrees with (3.8). It should be pointed out that the projectors (3.10) are non-analytic in µ .Finally, let us introduce the following operators: P (0) := − Q D β (cid:0) ¯ D − µ (cid:1) D β , (3.13a) P (+) := 116 Q (cid:0) ¯ D − µ (cid:1) D , (3.13b) P ( − ) := 116 Q (cid:0) D − µ (cid:1) ¯ D . (3.13c)By making use of the identities in appendix A, one may show that when restricted to thespace of scalar superfields they satisfy the projector properties = X i P ( i ) , P ( i ) P ( j ) = δ ij P ( j ) , (3.14)12or i = 0 , + , − . Here P (+) and P ( − ) are the chiral and antichiral projectors, respectively,while P (0) projects onto the space of LAL superfields. Using these projectors, it is alwayspossible to decompose an unconstrained complex scalar superfield V as V = L + σ + ¯ ρ , (3.15)where the complex superfield L is simultaneously linear and anti-linear, σ is chiral and ¯ ρ is anti-chiral (cid:0) D − µ (cid:1) L = (cid:0) ¯ D − µ (cid:1) L = 0 , ¯ D ˙ α σ = 0 , D α ¯ ρ = 0 . (3.16)If V is real, then L is also real and ¯ ρ = ¯ σ . In the flat superspace limit, the projectorsreduce to those constructed by Salam and Strathdee [1]. Our construction of AdS superprojectors will be based on certain properties of thefollowing theories: (i) the massless supersymmetric higher-spin gauge models in AdS [16];and (ii) the superconformal higher-spin (SCHS) gauge theories in AdS [20, 23, 68].Let s be a positive integer. We recall that there are two dually equivalent gaugeformulations for the massless superspin-( s + ) multiplet [16]. They both involve thesame superconformal prepotential, H α ( s ) ˙ α ( s ) , but different compensating multiplets. Theprepotential H α ( s ) ˙ α ( s ) is real and possesses the gauge freedom δ ζ H α ( s ) ˙ α ( s ) = ¯ D ( ˙ α ζ α ( s ) ˙ α ... ˙ α s ) − D ( α ¯ ζ α ...α s ) ˙ α ( s ) , (3.17)with the gauge parameter ζ α ( s ) ˙ α ( s − being unconstrained. Associated with H α ( s ) ˙ α ( s ) is thegauge-invariant chiral field strength W α (2 s +1) ( H ) = −
14 ( ¯ D − µ ) D ( α ˙ β · · · D α s ˙ β s D α s +1 H α s +2 ...α s +1 ) ˙ β ... ˙ β s . (3.18)The field strength W α (2 s +1) ( H ) and its conjugate are the only independent gauge-invariantfield strengths which survive on the mass shell.The massless superspin- s multiplet can be described using a superconformal prepo-tential Ψ α ( s ) ˙ α ( s − , its conjugate ¯Ψ α ( s −
1) ˙ α ( s ) and certain compensating multiplets [16, 24].The gauge transformation law of Ψ α ( s ) ˙ α ( s − is δ ζ,ξ Ψ α ( s ) ˙ α ( s − = ¯ D ( ˙ α ζ α ( s ) ˙ α ... ˙ α s − ) + D ( α ξ α ...α s ) ˙ α ( s − , (3.19)13ith the gauge parameters ζ α ( s ) ˙ α ( s − and ξ α ( s −
1) ˙ α ( s − being unconstrained. Associatedwith Ψ α ( s ) ˙ α ( s − is the gauge-invariant chiral field strength W α (2 s ) (Ψ) = −
14 ( ¯ D − µ ) D ( α ˙ β · · · D α s − ˙ β s − D α s Ψ α s +1 ...α s ) ˙ β ... ˙ β s − . (3.20)The field strength W α (2 s ) (Ψ) and its conjugate prove to be the only independent gauge-invariant field strengths which survive on the mass shell. The gauge prepotentials H α ( s ) ˙ α ( s ) and Ψ α ( s ) ˙ α ( s − and the corresponding gauge-invariantfield strengths (3.18) and (3.20) were used to construct superconformal higher-spin theo-ries [23, 68]. More general superconformal higher-spin models in AdS can be introducedby making use of gauge prepotentials Φ α ( m ) ˙ α ( n ) , with m and n positive integers. Thesuperfield Φ α ( m ) ˙ α ( n ) is defined modulo the gauge transformations [23, 68] δ ζ,ξ Φ α ( m ) ˙ α ( n ) = ¯ D ( ˙ α ζ α ( m ) ˙ α ... ˙ α n ) + D ( α ξ α ...α m ) ˙ α ( n ) , (3.21)with the gauge parameters being unconstrained. Associated with Φ α ( m ) ˙ α ( n ) and its com-plex conjugate ¯Φ α ( n ) ˙ α ( m ) are the chiral field strengths (also known as the linearised higher-spin super-Weyl tensors) W α ( m + n +1) (Φ) = − (cid:0) ¯ D − µ (cid:1) D ( α ˙ β · · · D α n ˙ β n D α n +1 Φ α n +2 ...α m + n +1 ) ˙ β ( n ) , (3.22a) W α ( m + n +1) ( ¯Φ) = − (cid:0) ¯ D − µ (cid:1) D ( α ˙ β · · · D α m ˙ β m D α m +1 ¯Φ α m +2 ...α m + n +1 ) ˙ β ( m ) , (3.22b)which are invariant under the gauge transformations (3.21) δ ζ,ξ W α ( m + n +1) (Φ) = 0 , δ ζ,ξ W α ( m + n +1) ( ¯Φ) = 0 . (3.23)The gauge-invariant action S ( m,n )SCHS [Φ , ¯Φ], which describes the dynamics of Φ α ( m ) ˙ α ( n ) andits conjugate ¯Φ α ( n ) ˙ α ( m ) , is typically written as a functional over the chiral subspace of thefull superspace [23, 68]. The specific feature of AdS | is the identity [69] Z d x d θ E L c = Z d x d θ d ¯ θ Eµ L c , ¯ D ˙ α L c = 0 , (3.24)which relates the integration over the chiral subspace to that over the full superspace;here E denotes the chiral integration measure and E − = Ber( E AM ). Keeping in mind(3.24), the gauge-invariant action S ( m,n )SCHS [Φ , ¯Φ] is given by S ( m,n )SCHS [Φ , ¯Φ] = 12 i m + n Z d | z Eµ W α ( m + n +1) (Φ) W α ( m + n +1) ( ¯Φ) + c . c . , (3.25) Off the mass shell, there exists one more gauge-invariant chiral field strength, W α (2 s ) ( ¯Ψ), definedaccording to the general rule (3.22b). However, it may be shown that W α (2 s ) ( ¯Ψ) ∝ W α (2 s ) (Ψ) on themass shell. In the flat superspace limit, W α (2 s ) ( ¯Ψ) = 0 on-shell. For the superconfomal gauge multiplets with either m = 0 or n = 0 see [20]. | z = d x d θ d ¯ θ . Upon integrating by parts, the action (3.25)may be written in the suggestive forms S ( m,n )SCHS [Φ , ¯Φ] = i m + n Z d | z E ¯Φ α ( n ) ˙ α ( m ) B α ( n ) ˙ α ( m ) (Φ) + c.c. (3.26a)= i m + n Z d | z E ¯Φ α ( n ) ˙ α ( m ) b B α ( n ) ˙ α ( m ) (Φ) + c . c . (3.26b)They are suggestive because, in addition to being gauge invariant, the linearised higher-spin super-Bach tensors [70] B α ( n ) ˙ α ( m ) (Φ) = 12 D ( ˙ α β · · · D ˙ α m ) β m D β m +1 W α ( n ) β ( m +1) (Φ) , (3.27a) b B α ( n ) ˙ α ( m ) (Φ) = 12 D ( α ˙ β · · · D α n ) ˙ β n ¯ D ˙ β n +1 W ˙ α ( m ) ˙ β ( n +1) (Φ) , (3.27b)are simultaneously transverse linear and transverse anti-linear (TLAL) D β B βα ( n −
1) ˙ α ( m ) (Φ) = 0 , ¯ D ˙ β B α ( n ) ˙ α ( m −
1) ˙ β (Φ) = 0 , (3.28a) D β b B βα ( n −
1) ˙ α ( m ) (Φ) = 0 , ¯ D ˙ β b B α ( n ) ˙ α ( m −
1) ˙ β (Φ) = 0 . (3.28b)Our ansatz for the TLAL projectors is based on the higher-derivative descendants(3.27a) and (3.27b), since the latter are each TLAL. However, it is clear that any projectormust preserve the rank of the tensor superfield on which it acts. This can be achievedby appropriately removing or inserting vector derivatives in (3.27) to convert the indices.With these remarks in mind, for positive integers m and n , we define the following twodifferential operators by their action on an unconstrained superfield Φ α ( m ) ˙ α ( n ) P α ( m ) ˙ α ( n ) (Φ) = − D ( ˙ α β . . . D ˙ α n ) β n D γ ( ¯ D − µ ) D ( β ˙ β . . . D β n ˙ β n D γ Φ α ...α m ) ˙ β ( n ) , (3.29a) b P α ( m ) ˙ α ( n ) (Φ) = 18 D ( α ˙ β . . . D α m ) ˙ β m ¯ D ˙ γ ( D − µ ) D ( ˙ β β . . . D ˙ β m β m ¯ D ˙ γ Φ β ( m ) ˙ α ... ˙ α n ) . (3.29b)From here one can define the operatorsΠ ( m,n ) Φ α ( m ) ˙ α ( n ) ≡ Π α ( m ) ˙ α ( n ) (Φ) := (cid:20) n +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1)(cid:21) − P α ( m ) ˙ α ( n ) (Φ) , (3.30a) b Π ( m,n ) Φ α ( m ) ˙ α ( n ) ≡ b Π α ( m ) ˙ α ( n ) (Φ) := (cid:20) m +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1)(cid:21) − b P α ( m ) ˙ α ( n ) (Φ) , (3.30b) Proving the equivalence of (3.27a) and (3.27b), and hence (3.26a) and (3.26b), is non-trivial and isclosely related to the coincidence relation (3.33). λ ( t,m,n ) are dimensionless constants defined by λ ( t,m,n ) = 12 (cid:16) ( m + n − t + 1)( m + n − t + 4) + t ( t − (cid:17) . (3.30c)The operators (3.30) project onto the space of TLAL superfields¯ D ˙ β Π α ( m ) ˙ α ( n −
1) ˙ β (Φ) = 0 , ¯ D ˙ β b Π α ( m ) ˙ α ( n −
1) ˙ β (Φ) = 0 , (3.31a) D β Π βα ( m −
1) ˙ α ( n ) (Φ) = 0 , D β b Π βα ( m −
1) ˙ α ( n ) (Φ) = 0 . (3.31b)The operators (3.29) also satisfy (3.31), but it is only the operators (3.30) which squareto themselves Π ( m,n ) Π ( m,n ) Φ α ( m ) ˙ α ( n ) = Π ( m,n ) Φ α ( m ) ˙ α ( n ) , (3.32a) b Π ( m,n ) b Π ( m,n ) Φ α ( m ) ˙ α ( n ) = b Π ( m,n ) Φ α ( m ) ˙ α ( n ) , (3.32b)and are hence TLAL projectors. Furthermore, one can consider (say) Π ( m,n ) to be theunique TLAL projector, since it can be shown that the two types of projectors (3.30a)and (3.30b) actually coincide Π α ( m ) ˙ α ( n ) (Φ) = b Π α ( m ) ˙ α ( n ) (Φ) . (3.33)In the case when m > n = 0, the operators analogous to (3.29) take the form P α ( m ) (Φ) = − D γ ( ¯ D − µ ) D ( γ Φ α ...α m ) , (3.34a) b P α ( m ) (Φ) = 18 D ( α ˙ β . . . D α m ) ˙ β m ¯ D ˙ γ ( D − µ ) D ( ˙ β β . . . D ˙ β m β m ¯ D ˙ γ ) Φ β ( m ) , (3.34b)whilst the projectors Π ( m ) := Π ( m, and b Π ( m ) := b Π ( m, areΠ ( m ) Φ α ( m ) ≡ Π α ( m ) (Φ) := (cid:16) Q − λ (1 ,m, µ ¯ µ (cid:17) − P α ( m ) (Φ) , (3.35a) b Π ( m ) Φ α ( m ) ≡ b Π α ( m ) (Φ) := (cid:20) m +1 Y t =1 (cid:0) Q − λ ( t,m, µ ¯ µ (cid:1)(cid:21) − b P α ( m ) (Φ) . (3.35b)These operators square to themselves but, in contrast to (3.31), they project onto thesubspace of simultaneously linear and transverse anti-linear (LTAL) superfields, D β Π α ( m − β (Φ) = 0 , D β b Π α ( m − β (Φ) = 0 , (3.36a) (cid:0) ¯ D − µ (cid:1) Π α ( m ) (Φ) = 0 , (cid:0) ¯ D − µ (cid:1) b Π α ( m ) (Φ) = 0 . (3.36b)16nce again one may show that the two types of projectors coincide, Π α ( m ) (Φ) = b Π α ( m ) (Φ).In the case when n > m = 0, the corresponding projectors Π ˙ α ( n ) ( ¯Φ) and b Π ˙ α ( n ) ( ¯Φ) can beobtained by complex conjugation and similar comments apply. This time however theyproject onto the subspace of simultaneously anti-linear and transverse linear superfields.When both m = 0 and n = 0, it may be shown that the orthogonal complement actingon Φ α ( m ) ˙ α ( n ) can be expressed as the following sumΠ ( m,n ) || Φ α ( m ) ˙ α ( n ) := (cid:0) − Π ( m,n ) (cid:1) Φ α ( m ) ˙ α ( n ) = ¯ D ( ˙ α Ψ α ( m ) ˙ α ... ˙ α n ) + D ( α Ω α ...α m ) ˙ α ( n ) (3.37)for some unconstrained superfields Ψ α ( m ) ˙ α ( n − and Ω α ( m −
1) ˙ α ( n ) . We see that Π ( m,n ) || projects Φ α ( m ) ˙ α ( n ) onto the union of spaces of longitudinal linear and longitudinal anti-linear superfields.If instead m > n = 0, the orthogonal complement may be written asΠ ( m ) || Φ α ( m ) := (cid:0) − Π ( m ) (cid:1) Φ α ( m ) = D ( α Ω α ...α m ) + Λ α ( m ) , ¯ D ˙ β Λ α ( m ) = 0 , (3.38)for some unconstrained Ω α ( m − and chiral Λ α ( m ) superfields. On the otherhand, if n > m = 0, then the orthogonal complement splits into the sum of a longitudinal linearsuperfield and an anti-chiral superfield,Π ( n ) || ¯Φ ˙ α ( n ) := (cid:0) − Π ( n ) (cid:1) ¯Φ ˙ α ( n ) = ¯ D ( ˙ α ¯Ψ ˙ α ... ˙ α n ) + ¯Λ ˙ α ( n ) , D β ¯Λ ˙ α ( n ) = 0 . (3.39)As the first application of the the superprojectors, we provide new expressions for thesuperconformal actions (3.25). For the higher-spin gauge supermultiplets H α ( s ) ˙ α ( s ) andΨ α ( s ) ˙ α ( s − they take the simple and elegant forms S ( s,s )SCHS [ H ] = 2( − s Z d | z E H α ( s ) ˙ α ( s ) s +1 Y t =1 (cid:0) Q − λ ( t,s,s ) µ ¯ µ (cid:1) Π ( s,s ) H α ( s ) ˙ α ( s ) , (3.40a) S ( s,s − [Ψ , ¯Ψ] = i( − s +1 Z d | z E ¯Ψ α ( s −
1) ˙ α ( s ) s Y t =1 (cid:0) Q − λ ( t,s,s − µ ¯ µ (cid:1) × D ˙ αβ Π ( s,s − Ψ βα ( s −
1) ˙ α ( s − + c.c. (3.40b)These expressions are analogous to those provided for the conformal higher-spin modelsin Minkowski space [71] and in AdS [29]. In section 3.2 it was shown that the operator Π ( m,n ) projects every unconstrained su-perfield Φ α ( m ) ˙ α ( n ) to the space of TLAL superfields, whilst its complement Π ( m,n ) || projects17nto the union of the spaces of longitudinal linear and anti-linear superfields. This allowsus to decompose Φ α ( m ) ˙ α ( n ) as followsΦ α ( m ) ˙ α ( n ) = φ α ( m ) ˙ α ( n ) + ¯ D ( ˙ α ζ α ( m ) ˙ α ... ˙ α n ) + D ( α ξ α ...α m ) ˙ α ( n ) , (3.41)where φ α ( m ) ˙ α ( n ) is TLAL and is hence irreducible, whilst ζ α ( m ) ˙ α ( n − and ξ α ( m −
1) ˙ α ( n ) areunconstrained and thus reducible. One can then repeat this decomposition on the lower-rank unconstrained superfields in (3.41). After reiterating this procedure a finite numberof times, one eventually arrives at a decomposition of Φ α ( m ) ˙ α ( n ) solely in terms of irre-ducible superfields. In presenting the resulting decompositions, we will use the notationalconvention introduced in [72] and adopted in [16]: V α ( n ) U α ( m ) = V ( α ...α n U α n +1 ...α n + m ) . (3.42)In the m > n case, the result of this process isΦ α ( m ) ˙ α ( n ) = n X t =0 ( D α ˙ α ) t φ α ( m − t ) ˙ α ( n − t ) + n − X t =0 (cid:2) D α , ¯ D ˙ α (cid:3) ( D α ˙ α ) t ψ α ( m − t −
1) ˙ α ( n − t − + n X t =0 ( D α ˙ α ) t D α χ α ( m − t −
1) ˙ α ( n − t ) + n − X t =0 ( D α ˙ α ) t ¯ D ˙ α ϕ α ( m − t ) ˙ α ( n − t − + ( D α ˙ α ) n (cid:16) σ α ( m − n ) + D α σ α ( m − n − (cid:17) , (3.43)for some irreducible complex superfields φ, ψ, χ and ϕ whose properties are summarisedin table 1. The superfields σ α ( m − n ) and σ α ( m − n − in (3.43) are chiral,¯ D ˙ α σ α ( m − n ) = 0 , ¯ D ˙ α σ α ( m − n − = 0 . (3.44)0 ≤ t ≤ n − t = n − t = nφ α ( m − t ) ˙ α ( n − t ) TLAL TLAL LTAL ψ α ( m − t −
1) ˙ α ( n − t − TLAL TLAL – χ α ( m − t −
1) ˙ α ( n − t ) TLAL TLAL LTAL ϕ α ( m − t ) ˙ α ( n − t − TLAL LTAL –Table 1: Properties of the superfields appearing in (3.43).If instead m = n = s , then we may further impose the reality condition H α ( s ) ˙ α ( s ) := Φ α ( s ) ˙ α ( s ) = ¯ H α ( s ) ˙ α ( s ) . (3.45)18n contrast to (3.43), H α ( s ) ˙ α ( s ) now decomposes as H α ( s ) ˙ α ( s ) = s X t =0 ( D α ˙ α ) t φ α ( s − t ) ˙ α ( s − t ) + s − X t =0 (cid:2) D α , ¯ D ˙ α (cid:3) ( D α ˙ α ) t ψ α ( s − t −
1) ˙ α ( s − t − + s − X t =0 ( D α ˙ α ) t (cid:16) ¯ D ˙ α χ α ( s − t ) ˙ α ( s − t − + c.c. (cid:17) + ( D α ˙ α ) s (cid:0) σ + ¯ σ (cid:1) , (3.46)for some irreducible complex superfield χ α ( s − t ) ˙ α ( s − t − and irreducible real superfields φ α ( s − t ) ˙ α ( s − t ) = ¯ φ α ( s − t ) ˙ α ( s − t ) , ψ α ( s − t −
1) ˙ α ( s − t − = ¯ ψ α ( s − t −
1) ˙ α ( s − t − , (3.47)whose properties are described in table 2. The scalar σ in (3.46) is chiral, ¯ D ˙ α σ = 0.0 ≤ t ≤ s − t = s − t = sφ α ( s − t ) ˙ α ( s − t ) TLAL TLAL LAL ψ α ( s − t −
1) ˙ α ( s − t − TLAL LAL – χ α ( s − t ) ˙ α ( s − t − TLAL LTAL –Table 2: Properties of the superfields appearing in (3.46).The above decompositions were used, albeit without derivation, in [28] for the covariantquantisation of the massless supersymmetric higher-spin models in AdS [16]. In this section we confine our attention to on-shell superfields φ α ( m ) ˙ α ( n ) . When m = 0and n = 0 we define these to satisfy the mass-shell equation (cid:0) Q − M (cid:1) φ α ( m ) ˙ α ( n ) = 0 , (4.1a)and the irreducibility conditions D β φ βα ( m −
1) ˙ α ( n ) = 0 , ¯ D ˙ β φ α ( m ) ˙ α ( n −
1) ˙ β = 0 . (4.1b)Hence the superfield φ α ( m ) ˙ α ( n ) is simultaneously transverse linear and transverse anti-linear (TLAL), and is said to have super-spin s = ( m + n + 1) and pseudo-mass M . Inthe case when m > n = 0 and n > m = 0, the condition (4.1b) should be modified to D β φ α ( m − β = 0 , (cid:0) ¯ D − µ (cid:1) φ α ( m ) = 0 , (4.2a)19 D ˙ β ¯ φ ˙ α ( n −
1) ˙ β = 0 , (cid:0) D − µ (cid:1) ¯ φ ˙ α ( n ) = 0 , (4.2b)respectively, whilst (4.1a) remains the same. In the case when m = n = 0, the on-shellsuperfields are discussed in sections 5.4 and 5.5.In analogy with (2.11), we define the physical mass M phys through M = M − λ (1 ,m,n ) µ ¯ µ . (4.3)As discussed in section 2, the pseudo-mass of an on-shell non-supersymmetric field is sub-ject to a unitarity bound. Since the on-shell supermultiplet φ α ( m ) ˙ α ( n ) contains a multitudeof such fields, this in turn induces a unitarity bound on the pseudo-mass of φ α ( m ) ˙ α ( n ) : M ≥ ⇒ M ≥ λ (1 ,m,n ) µ ¯ µ . (4.4)By going to components, we will see in section 5 that the on-shell supermultiplet φ α ( m ) ˙ α ( n ) furnishes the representation S (cid:0) E ( M ) , ( m + n + 1) (cid:1) of OSp (1 | E ( M ) = 1 + 12 s M µ ¯ µ − ( m + n )( m + n + 4) + 1 . (4.5) Massless supermultiplets correspond to those on-shell superfields carrying pseudo-mass M = λ (1 ,m,n ) µ ¯ µ . (4.6)There are at least two different angles from which this can be understood.The first is the fact that the system of equations (4.1) with m ≥ n > M satisfying (4.6) is compatible with gauge transformations of the form (3.21), δ ζ,ξ φ α ( m ) ˙ α ( n ) = ¯ D ( ˙ α ζ α ( m ) ˙ α ... ˙ α n ) + D ( α ξ α ...α m ) ˙ α ( n ) , (4.7a)where the gauge parameters ζ α ( m ) ˙ α ( n − and ξ α ( m −
1) ˙ α ( n ) are TLAL and obey the constraints D ( α ˙ β ξ α ...α m ) ˙ β ˙ α ( n − = i( n + 1) µζ α ( m ) ˙ α ( n − , (4.7b) D β ( ˙ α ζ βα ( m −
1) ˙ α ... ˙ α n ) = − i( m + 1)¯ µξ α ( m −
1) ˙ α ( n ) . (4.7c)These on-shell conditions imply that ζ α ( m ) ˙ α ( n − and ξ α ( m −
1) ˙ α ( n ) satisfy the equations (cid:0) Q − λ (1 ,m,n ) µ ¯ µ (cid:1) ζ α ( m ) ˙ α ( n − = 0 , (cid:0) Q − λ (1 ,m,n ) µ ¯ µ (cid:1) ξ α ( m −
1) ˙ α ( n ) = 0 . (4.8)20hus, from (4.8) and the gauge variation of (4.1a), we see that the gauge parameters areonly non-zero if M satisfies (4.6).When m = n = s one can consistently impose the reality condition H α ( s ) ˙ α ( s ) := φ α ( s ) ˙ α ( s ) = ¯ H α ( s ) ˙ α ( s ) , whereupon the gauge transformations (4.7a) take the form δ ζ H α ( s ) ˙ α ( s ) = ¯ D ( ˙ α ζ α ( s ) ˙ α ... ˙ α s ) − D ( α ¯ ζ α ...α s ) ˙ α ( s ) , (4.9a)whilst (4.7b) and (4.7c) become D β ( ˙ α ζ βα ( s −
1) ˙ α ... ˙ α s ) = i( s + 1)¯ µ ¯ ζ α ( s −
1) ˙ α ( s ) , (4.9b) D ( α ˙ β ¯ ζ α ...α s ) ˙ β ˙ α ( s − = − i( s + 1) µζ α ( s ) ˙ α ( s − . (4.9c)The presence of the gauge freedom (4.7) means that the OSp (1 |
4) representation (4.1)with M given by (4.6) is not irreducible. To obtain an irreducible representation, thespace of TLAL superfields (4.1) and (4.6) has to be factorised with respect to the gaugemodes. More specifically, two superfields φ α ( m ) ˙ α ( n ) and e φ α ( m ) ˙ α ( n ) are said to be equivalentif they differ by a gauge transformation (4.7). The genuine massless representation isrealised on the quotient space of the space of on-shell superfields (4.1) and (4.6) withrespect to this equivalence relation. The gauge degrees of freedom are automaticallyeliminated if one works with the gauge-invariant chiral field strength (3.22a), W α ( m + n +1) ( φ ) = − (cid:0) ¯ D − µ (cid:1) D ( α ˙ β · · · D α n ˙ β n D α n +1 φ α n +2 ...α m + n +1 ) ˙ β ( n ) , (4.10)instead of the prepotential φ α ( m ) ˙ α ( n ) . Making use of the constraint (4.1b) satisfied by φ α ( m ) ˙ α ( n ) , one may show that W α ( m + n +1) ( φ ) is constrained by0 = D β W βα ( m + n ) ( φ ) = ⇒ (cid:0) Q − λ (1 ,m,n ) µ ¯ µ (cid:1) W α ( m + n +1) ( φ ) , (4.11)where the second relation follows from the chirality of the field strength. Once again,the latter is consistent with (4.1a) only if M satisfies (4.6). The description in terms of W α ( m + n +1) ( φ ) provides the second way to formulate massless dynamics.Let us return to the gauge-invariant chiral field strengths (3.18) and (3.20), whichoriginate in the massless models for the superspin-( s + ) and superspin- s multiplets inAdS [16], respectively. As was demonstrated in [16], on the mass shell they satisfy theequations D β W βα (2 s ) ( H ) = 0 , (4.12a) D β W βα (2 s − (Ψ) = 0 , (4.12b)which are exactly the on-shell constraints (4.11).21 .2 Partially massless supermultiplets The transverse projection operators for non-supersymmetric tensor fields in AdS wereconstructed recently in [29]. There it was observed that the poles of the projectors wereintimately related to (partially) massless fields. The structure of on-shell N = 1 super-multiplets containing partially massless fields was recently discussed at the componentlevel in [44, 45]. However, their realisations in terms of on-shell superfields are not yetknown. In this section we address this gap.In the spirit of [29], it is natural to expect that the partially-massless supermultipletsare associated with the poles of the TLAL projectors (3.30). This motivates the definitionof a partially massless superfield to be one satisfying the on-shell constraints (4.1) suchthat the pseudo-mass takes one of the values (3.30c), M = λ ( t,m,n ) µ ¯ µ , ≤ t ≤ min( m + 1 , n + 1) . (4.13)We will say that φ α ( m ) ˙ α ( n ) carries ‘super-depth’ t . In section 5 we show that, with the abovedefinitions, the surviving component fields of an on-shell partially massless superfield withsuper-depth t > t or t −
1. We wouldlike to point out that λ ( t,m,n ) with 1 ≤ t ≤ max( m + 1 , n + 1) appear in the poles of theprojectors (3.30). The reason that we have chosen t = min( m + 1 , n + 1) as an upperbound on t in the definition (4.13) is because, as we will see, there is no gauge symmetrypresent for min( m + 1 , n + 1) < t ≤ max( m + 1 , n + 1).From (4.6), we see that partially massless supermultiplets with the minimal super-depth of t = 1 correspond to massless representations and are hence unitary. How-ever, partially massless supermultiplets whose super-depth lies within the range 2 ≤ t ≤ min( m + 1 , n + 1) have negative physical mass and describe non-unitary representationsof OSp (1 | λ (1 ,m,n ) > λ ( t,m,n ) , ≤ t ≤ min( m + 1 , n + 1) . (4.14)To distinguish the latter we may sometimes refer to them as true partially massless su-permultiplets. In analogy with (2.23), we will denote the true partially massless repre-sentation with super-depth t and Lorentz type ( m/ , n/
2) by P ( t, m, n ) , ≤ t ≤ min( m + 1 , n + 1) . (4.15)According to (4.5), such a representation carries minimal energy E = ( m + n + 1) − t + 2.22n the previous section it was observed that on-shell, (4.6) is the only pseudo-massvalue compatible with the massless gauge symmetry (4.7a). For true partially masslesssupermultiplets the story is considerably more complicated. This is because the correctgauge symmetry at the superspace level is not yet known, much less the gauge invariantactions and field strengths. However, by making use of the superprojectors proposedin section 3, it is possible to systematically derive the most general gauge symmetrycompatible with the on-shell conditions (4.1). In appendix B this procedure is carriedout in detail for the real supermultiplet H α ˙ α . The results of this analysis, and that of thesupermultiplet H α (2) ˙ α (2) obtained by analogy, are summarised below.Super-depth t = 1 t = 2 t = 3 δH α ˙ α ¯ D ˙ α ζ α − D α ¯ ζ ˙ α D α ˙ α ( σ + ¯ σ ) – δH α (2) ˙ α (2) ¯ D ˙ α ζ α (2) ˙ α − D α ¯ ζ α ˙ α (2) D α ˙ α (cid:0) ¯ D ˙ α ξ α − D α ¯ ξ ˙ α (cid:1) D α ˙ α D α ˙ α ( η + ¯ η )Table 3: Gauge symmetry for lower-rank PM supermultiplets.In table 3, the most general gauge symmetries for the aforementioned supermultipletsare given. In both cases it turns out that a gauge symmetry is present only when thepseudo-mass takes the values specified in (4.13). The gauge parameters are on-shell inthe sense that they each possess the same pseudo-mass as its parent gauge field and thatthey satisfy certain irreducibility constraints. Specifically, in the super-depth t = 1 case,the gauge parameters ζ α and ζ α (2) ˙ α in table 3 are LTAL and TLAL, respectively, and obeythe reality conditions (4.9b) and (4.9c). The gauge parameter ξ α , which corresponds to H α (2) ˙ α (2) with super-depth t = 2, is LTAL and obeys the reality condition D ˙ αβ ξ β = 3i¯ µ ¯ ξ ˙ α . (4.16)In the case of maximal super-depth, the gauge parameters σ and η are chiral and obeythe reality conditions − (cid:0) D − µ (cid:1) σ + 2¯ µ ¯ σ = 0 , (4.17a) − (cid:0) D − µ (cid:1) η + 3¯ µ ¯ η = 0 . (4.17b)These conditions are the equations of motion which follow from various Wess-Zuminomodels (c.f. (5.23)).The method used to derive the above results (see appendix B.2 for the details) is quitegeneral in that it deduces all types of gauge symmetry an on-shell supermultiplet can pos-sess, and at which mass values they appear. However, for higher-rank multiplets it quickly In table 3, and in the remainder of this subsection, we employ the notational convention (3.42). s + ) multiplets H ( t ) α ( s ) ˙ α ( s ) . For true partially-masslessmultiplets with super-depth 2 ≤ t ≤ s + 1, they take the form ≤ t ≤ s : δ ξ H ( t ) α ( s ) ˙ α ( s ) = (cid:0) D α ˙ α (cid:1) t − (cid:0) ¯ D ˙ α ξ α ( s − t +1) ˙ α ( s − t ) − D α ¯ ξ α ( s − t ) ˙ α ( s − t +1 (cid:1) , (4.18a) t = s + 1 : δ σ H ( s +1) α ( s ) ˙ α ( s ) = (cid:0) D α ˙ α (cid:1) s (cid:0) σ + ¯ σ (cid:1) . (4.18b)The system of equations (4.1) and (4.13) is invariant under the transformations (4.18a)as long as the gauge parameters are TLAL and satisfy the reality conditions D β ˙ α ξ βα ( s − t ) ˙ α ( s − t ) = i( s + 1)¯ µ ¯ ξ α ( s − t ) ˙ α ( s − t +1) , (4.19a) D α ˙ β ¯ ξ α ( s − t ) ˙ α ( s − t ) ˙ β = − i( s + 1) µξ α ( s − t +1) ˙ α ( s − t ) . (4.19b)The same is true for the transformations (4.18b) given that σ is chiral, ¯ D ˙ α σ = 0, and thatit satisfies the equations − (cid:0) D − µ (cid:1) σ + ( s + 1)¯ µ ¯ σ = 0 , (4.20a) − (cid:0) ¯ D − µ (cid:1) ¯ σ + ( s + 1) µσ = 0 . (4.20b)It may be shown that (4.19) and (4.20) imply that each gauge parameter satisfies thesame mass-shell equation as its corresponding gauge field, as required. We note that uponsubstituting t = 1 into (4.18a) and (4.19), one recovers the massless gauge transformations(4.9) with ξ α ( s ) ˙ α ( s − ≡ ζ α ( s ) ˙ α ( s − . We define a massive superfield φ α ( m ) ˙ α ( n ) to be one satisfying the on-shell conditions(4.1) and whose pseudo-mass satisfies M > λ (1 ,m,n ) µ ¯ µ , (4.21)but is otherwise arbitrary. By virtue of (4.4) this definition ensures that the physicalmass is positive and that the corresponding representation of OSp (1 |
4) is unitary. Interestingly, the higher-depth gauge transformations (4.18a) are different to those proposed for thegeneralised superconformal multiplets in [73]. The latter transformations possess the same functionalform as the second group of terms in (3.46). .4 Equivalent representations and reality conditions Given two positive integers m and n , with m + n ≡ s , we denote by L [ M ]( m,n ) the spaceof on-shell superfields φ α ( m ) ˙ α ( n ) satisfying the conditions (4.1). The following propositionholds: Provided the pseudo-mass satisfies M = λ ( t,m,n ) , ≤ t ≤ max( m + 1 , n + 1) , (4.22)the OSp (1 |
4) representations on the functional spaces L [ M ](2 s, , L [ M ](2 s − , , . . . L [ M ](1 , s − , L [ M ](0 , s ) are equivalent to S (cid:0) E ( M ) , s + (cid:1) , where E ( M ) is given by (4.5).To prove the above claim, we consider an arbitrary superfield φ α ( m ) ˙ α ( n ) ∈ L [ M ]( m,n ) andassociate with it the following descendants φ α ( m ) ˙ α ( n ) → ψ α ( m +1) ˙ α ( n − := D α m +1 ˙ β φ α ( m ) ˙ β ˙ α ( n − , (4.23a) φ α ( m ) ˙ α ( n ) → χ α ( m −
1) ˙ α ( n +1) := D β ˙ α n +1 φ βα ( m −
1) ˙ α ( n ) . (4.23b)It is obvious that ψ α ( m +1) ˙ α ( n − is completely symmetric in its undotted indices, while χ α ( m −
1) ˙ α ( n +1) is completely symmetric in its dotted indices. The descendant ψ α ( m +1) ˙ α ( n − proves to obey the conditions (4.1) if n = 1. In the n = 1 case, ψ α ( m +1) ˙ α ( n − is anon-shell superfield of the type (4.2a). The descendant χ α ( m −
1) ˙ α ( n +1) has analogous prop-erties. Therefore the relations (4.23a) and (4.23b) define linear mappings from L [ M ]( m,n ) to L [ M ]( m +1 ,n − and L [ M ]( m − ,n +1) , respectively.Making use of the identities (A.4) leads to the relations D β ˙ γ D β ˙ β φ α ( m ) ˙ β ˙ α ( n − = (cid:0) M − λ ( n +1 ,m,n ) µ ¯ µ (cid:1) φ α ( m ) ˙ γ ˙ α ( n − , (4.24a) D γ ˙ β D β ˙ β φ βα ( m −
1) ˙ α ( n ) = (cid:0) M − λ ( m +1 ,m,n ) µ ¯ µ (cid:1) φ γα ( m −
1) ˙ α ( n ) . (4.24b)In conjunction with the identities λ ( n +1 ,m,n ) = λ ( m +2 ,m +1 ,n − , λ ( m +1 ,m,n ) = λ ( n +2 ,m − ,n +1) , (4.25)the relations (4.24) tell us that the linear transformations (4.23a) and (4.23b) are one-to-one and onto, as long as M = λ ( n +1 ,m,n ) µ ¯ µ and M = λ ( m +1 ,m,n ) µ ¯ µ .Let us introduce the linear maps∆ ( m,n ) : L [ M ]( m,n ) → L [ M ]( m +1 ,n − , e ∆ ( m,n ) : L [ M ]( m,n ) → L [ M ]( m − ,n +1) , (4.26)defined by φ α ( m ) ˙ α ( n ) → φ α ( m +1) ˙ α ( n − := (cid:0) ∆ ( m,n ) (cid:1) α m +1 ˙ β φ α ( m ) ˙ β ˙ α ( n − , (4.27a)25 α ( m ) ˙ α ( n ) → φ α ( m −
1) ˙ α ( n +1) := (cid:0) e ∆ ( m,n ) (cid:1) β ˙ α n +1 φ βα ( m −
1) ˙ β ˙ α ( n ) , (4.27b)where we have introduced the dimensionless operators (cid:0) ∆ ( m,n ) (cid:1) α ˙ α = D α ˙ α p Q − λ ( n +1 ,m,n ) µ ¯ µ , (cid:0) e ∆ ( m,n ) (cid:1) α ˙ α = D α ˙ α p Q − λ ( m +1 ,m,n ) µ ¯ µ . (4.28)The above relations are equivalent to e ∆ ( m +1 ,n − ∆ ( m,n ) (cid:12)(cid:12)(cid:12) L [ M ]( m,n ) = , ∆ ( m − ,n +1) e ∆ ( m,n ) (cid:12)(cid:12)(cid:12) L [ M ]( m,n ) = . (4.29)Therefore, all the superfields φ α (2 s ) , φ α (2 s −
1) ˙ α , . . . , φ α ( m ) ˙ α ( n ) , . . . , φ α ˙ α (2 s − , φ ˙ α (2 s ) (4.30)also satisfy (4.1) and hence furnish the representation S (cid:0) E ( M ) , s + (cid:1) .In proving the above assertion it was assumed that the pseudo-mass of φ α ( m ) ˙ α ( n ) sat-isfies (4.22). This was to ensure that when φ α ( m ) ˙ α ( n ) is on the mass-shell, the operators(4.28) are well defined. When φ α ( m ) ˙ α ( n ) is partially-massless with super-depth t , at somestage the maps between tensor types become ill defined. Nevertheless, in this case it maybe shown that the corresponding representation may be equivalently realised on any ofthe functional spaces L [ M ]( m + n − t +1 ,t − , L [ M ]( m + n − t,t ) , . . . , L [ M ]( t,m + n − t ) , L [ M ]( t − ,m + n − t +1) .In general, φ α ( m ) ˙ α ( n ) and its conjugate ¯ φ α ( n ) ˙ α ( m ) describe two equivalent representationsof OSp (1 | φ α ( m ) ˙ α ( n ) . In the case when m + n ≡ s is even, φ α ( m ) ˙ α ( n ) can represented by φ α ( s ) ˙ α ( s ) which we require to be real,¯ φ α ( s ) ˙ α ( s ) = φ α ( s ) ˙ α ( s ) . (4.31)If m + n ≡ s − φ α ( m ) ˙ α ( n ) can be represented by φ α ( s ) ˙ α ( s − . In this case a realitycondition can be chosen in the form of a Dirac-type pair of equations D ( ˙ α β φ βα ( s −
1) ˙ α ... ˙ α s ) = e i γ M ( s +1) ¯ φ α ( s −
1) ˙ α ( s ) , (4.32a) D ( α ˙ β ¯ φ α ...α s ) ˙ α ( s −
1) ˙ β = e − i γ M ( s +1) φ α ( s ) ˙ α ( s − , (4.32b)where M s +1) := M − λ ( s +1 ,s,s − µ ¯ µ and γ is a constant real phase. The pair of equations(4.32) lead to the same mass-shell equation (4.1a).The above consideration offers a simple way to re-derive the superprojectors (3.30).Consider an off-shell superfield Φ α ( m ) ˙ α ( n ) and convert its dotted indices into undottedΦ α ( m ) ˙ α ( n ) → Φ α ( m + n ) := (cid:0) ∆ ( m + n − , (cid:1) ( α ˙ β . . . (cid:0) ∆ ( m,n ) (cid:1) α n ˙ β n Φ α n +1 ...α n + m ) ˙ β ( n ) . (4.33)26ext we apply the superprojector Π ( m + n ) , eq. (3.35a), to Φ α ( m + n ) . The resulting super-field ϕ α ( m + n ) := Π ( m + n ) Φ α ( m + n ) has the properties listed in (3.36). Finally we convert n undotted indices of ϕ α ( m + n ) back into dotted ones, ϕ α ( m + n ) → ϕ α ( m ) ˙ α ( n ) := (cid:0) e ∆ ( m + n, (cid:1) β ˙ α . . . (cid:0) e ∆ ( m +1 ,n − (cid:1) β n ˙ α n ϕ β ( n ) α ( m ) . (4.34)One observes that ϕ α ( m ) ˙ α ( n ) coincides with Π ( m,n ) Φ α ( m ) ˙ α ( n ) , eq. (3.30a). We now turn to the component analysis of the on-shell superfield φ α ( m ) ˙ α ( n ) satisfyingthe conditions (4.1) with arbitrary M . For this we make use of the bar-projection of atensor superfield V = V ( x, θ, ¯ θ ) (with suppressed indices) which is defined as usual: V | := V ( x, θ, ¯ θ ) (cid:12)(cid:12) θ =¯ θ =0 . (5.1)The supermultiplet of fields associated with V is defined to consist of all independent fieldscontained in (cid:8) V | , D α V | , ¯ D ˙ α V | , . . . (cid:9) . The covariant derivative of AdS , ∇ a , is related to D a according to the rule ∇ a V := ( D a V ) | , V := V | . (5.2)In practice, we assume a Wess-Zumino gauge condition to be imposed on the geometricobjects in (3.1) such that the background geometry is purely bosonic, D a | := E aM | ∂ M + 12 Ω abc | M bc = e am ∂ m + 12 ω abc M bc = ∇ a . (5.3)The supersymmetry transformation of the component fields of V is computed accordingto the rule (A.14)Below we make use of the non-supersymmetric AdS quadratic Casimir operator (2.8)which is related to the Casimir (3.4) via QV = h(cid:16) Q − (cid:0) µ D + ¯ µ ¯ D (cid:1)(cid:17) V i(cid:12)(cid:12)(cid:12) . In general there are four non-vanishing independent complex component fields, A α ( m ) ˙ α ( n ) := φ α ( m ) ˙ α ( n ) | , (5.4a)27 α ( m +1) ˙ α ( n ) := D ( α φ α ...α m +1 ) ˙ α ( n ) | , (5.4b) C α ( m ) ˙ α ( n +1) := ¯ D ( ˙ α φ α ( m ) ˙ α ... ˙ α n +1 ) | , (5.4c) E α ( m +1) ˙ α ( n +1) := (cid:18) (cid:2) D ( α , ¯ D ( ˙ α (cid:3) − i m − nm + n + 2 D ( α ( ˙ α (cid:19) φ α ...α m +1 ) ˙ α ... ˙ α n +1 ) | . (5.4d)They have each been defined in such a way that they are transverse0 = ∇ β ˙ β A βα ( m −
1) ˙ β ˙ α ( n − , (5.5a)0 = ∇ β ˙ β B βα ( m ) ˙ β ˙ α ( n − , (5.5b)0 = ∇ β ˙ β C βα ( m −
1) ˙ β ˙ α ( n ) , (5.5c)0 = ∇ β ˙ β E βα ( m ) ˙ β ˙ α ( n ) , (5.5d)which may be shown by making use of the on-shell conditions (4.1). Furthermore, onemay show that the bottom and top components satisfy the mass-shell equations0 = (cid:16) Q − (cid:2) M −
12 ( m + n + 4) µ ¯ µ (cid:3)(cid:17) A α ( m ) ˙ α ( n ) , (5.6a)0 = (cid:16) Q − (cid:2) M + 12 ( m + n ) µ ¯ µ (cid:3)(cid:17) E α ( m +1) ˙ α ( n +1) , (5.6b)whilst the two middle component fields satisfy the differential equations0 = i µ ∇ ( ˙ α β B βα ( m ) ˙ α ... ˙ α n +1 ) + (cid:16) Q − (cid:2) M −
12 ( m − n + 3) µ ¯ µ (cid:3)(cid:17) C α ( m ) ˙ α ( n +1) , (5.7a)0 = − i¯ µ ∇ ( α ˙ β C α ...α m +1 ) ˙ α ( n ) ˙ β + (cid:16) Q − (cid:2) M −
12 ( n − m + 3) µ ¯ µ (cid:3)(cid:17) B α ( m +1) ˙ α ( n ) . (5.7b)The latter imply that only one of the two fields B α ( m +1) ˙ α ( n ) and C α ( m ) ˙ α ( n +1) is independent,and lead to the higher derivative mass-shell equations0 = (cid:16) Q − M (cid:17)(cid:16) Q − M − (cid:17) B α ( m +1) ˙ α ( n ) , (5.8a)0 = (cid:16) Q − M (cid:17)(cid:16) Q − M − (cid:17) C α ( m ) ˙ α ( n +1) , (5.8b)where we have denoted M ± = M − µ ¯ µ ± µ ¯ µ s M µ ¯ µ − ( m + n )( m + n + 4) + 1 . (5.9)Using the relation (2.12), one may confirm that the above component results are inagreement with the decomposition (2.1), S (cid:0) E , s (cid:1) = D (cid:0) E + 12 , s − (cid:1) ⊕ D (cid:0) E , s (cid:1) ⊕ D (cid:0) E + 1 , s (cid:1) ⊕ D (cid:0) E + 12 , s + 12 (cid:1) , (5.10)as dictated by representation theory. Here s = ( m + n + 1) and E ≡ E ( M ) is definedaccording to (4.5). 28 .2 Partially massless supermultiplets In this section we restrict our attention to on-shell true partially massless supermulti-plets with super-depth 2 ≤ t ≤ min( m + 1 , n + 1). The massless case with t = 1 will beconsidered separately in the next section. Upon specifying the pseudo-mass to be givenby (4.13), one can show that the mass-shell equations (5.6) and (5.8) become0 = (cid:16) Q − τ ( t − ,m,n ) µ ¯ µ (cid:17) A α ( m ) ˙ α ( n ) , (5.11a)0 = (cid:16) Q − τ ( t − ,m +1 ,n ) µ ¯ µ (cid:17)(cid:16) Q − τ ( t,m +1 ,n ) µ ¯ µ (cid:17) B α ( m +1) ˙ α ( n ) , (5.11b)0 = (cid:16) Q − τ ( t − ,m,n +1) µ ¯ µ (cid:17)(cid:16) Q − τ ( t,m,n +1) µ ¯ µ (cid:17) C α ( m ) ˙ α ( n +1) , (5.11c)0 = (cid:16) Q − τ ( t,m +1 ,n +1) µ ¯ µ (cid:17) E α ( m +1) ˙ α ( n +1) . (5.11d)Here τ ( t,m,n ) are the non-supersymmetric partially massless values (2.17), which are relatedto the supersymmetric ones (3.30c) through λ ( t,m,n ) = τ ( t − ,m,n ) + 12 ( m + n + 4) . (5.12)In accordance with the discussion of on-shell partially massless fields given in section2.3, the following remarks hold: • The pair of equations (5.5a) and (5.11a) admits a depth t A = t − δ ζ A α ( m ) ˙ α ( n ) = ∇ ( α ( ˙ α · · · ∇ α tA ˙ α tA ζ α tA +1 ...α m ) ˙ α tA +1 ... ˙ α n ) (5.13)when 2 ≤ t ≤ min( m, n ). • The pair of equations (5.5d) and (5.11d) admits a depth t E = t gauge symmetry δ ξ E α ( m +1) ˙ α ( n +1) = ∇ ( α ( ˙ α · · · ∇ α tE ˙ α tE ξ α tE +1 ...α m +1 ) ˙ α tE +1 ... ˙ α n +1 ) (5.14)when 2 ≤ t ≤ min( m + 1 , n + 1). • Equation (5.5b) and the first branch of (5.11b) admit a depth t B = t − δ ρ B α ( m +1) ˙ α ( n ) = ∇ ( α ( ˙ α · · · ∇ α tB ˙ α tB ρ α tB +1 ...α m +1 ) ˙ α tB +1 ... ˙ α n ) (5.15)gauge symmetry when 2 ≤ t ≤ min( m + 1 , n + 1).29 Equation (5.5b) and the second branch of (5.11b) admit a depth t ′ B = tδ ρ B α ( m +1) ˙ α ( n ) = ∇ ( α ( ˙ α · · · ∇ α t ′ B ˙ α t ′ B ρ α t ′ B +1 ...α m +1 ) ˙ α t ′ B +1 ... ˙ α n ) (5.16)gauge symmetry when 2 ≤ t ≤ min( m + 1 , n ). • Comments similar to those above regarding B α ( m +1) ˙ α ( n ) , apply to C α ( m ) ˙ α ( n +1) . • The above gauge symmetries hold only for gauge parameters which are transverseand which satisfy the same mass-shell equation as the parent gauge field.One can reverse the logic and ask the question: for what values of M does there appeara gauge symmetry (of any depth) at the component level? The answer is precisely thosevalues which appear in the superprojectors (3.30c) (though, as above, special care mustbe taken when deducing the upper and lower bounds on the range of possible depths).From the above component analysis one can see that the true partially massless rep-resentation P (cid:0) t, m, n (cid:1) of OSp (1 |
4) decomposes into so (3 ,
2) subrepresentations as follows P (cid:0) t, m, n (cid:1) = P (cid:0) t − , m, n (cid:1) ⊕ P (cid:0) t − , m + 1 , n (cid:1) ⊕ P (cid:0) t, m + 1 , n (cid:1) ⊕ P (cid:0) t, m + 1 , n + 1 (cid:1) . (5.17)For integer ( m = n + 1 = s ) and half-integer ( m = n = s ) superspin, this decompositionis in agreement with the results of [44]. To study the component structure of an on-shell massless supermultiplet of superspin s >
0, it is advantageous to work with the gauge-invariant field strength W α (2 s ) definedby (4.10) instead of the prepotential φ α ( m ) ˙ α ( n ) , where 2 s = m + n + 1. The fundamentalproperties of W α (2 s ) ,¯ D ˙ β W α (2 s ) = 0 , D β W βα (2 s − = 0 ⇔ (cid:0) D − s + 1)¯ µ (cid:1) W α (2 s ) = 0 , (5.18)imply the following equations: D β ˙ β W βα (2 s − = 0 , (5.19a) (cid:0) Q − λ (1 , s − , µ ¯ µ (cid:1) W α (2 s ) = 0 . (5.19b)The chiral field strength W α (2 s ) has two independent component fields C α (2 s ) := W α (2 s ) | , (5.20a)30 α (2 s +1) := D ( α W α ...α s +1 ) | . (5.20b)Their properties follow from the equations (5.18) and (5.19) ∇ β ˙ β C βα (2 s − = 0 = ⇒ (cid:0) Q − τ (1 , s, µ ¯ µ (cid:1) C α (2 s ) = 0 , (5.21a) ∇ β ˙ β C βα (2 s ) = 0 = ⇒ (cid:0) Q − τ (1 , s +1 , µ ¯ µ (cid:1) C α (2 s +1) = 0 . (5.21b)These relations tell us that the component field strengths C α (2 s ) and C α (2 s +1) furnish themassless representation S (cid:0) s + 1 , s (cid:1) = D ( s + 1 , s ) ⊕ D (cid:0) s + , s + (cid:1) . Various aspects of the Wess-Zumino model in AdS were studied in the 1980s, see[15,63,74–77] and references therein. Here we will only discuss its group-theoretic aspectsfor completeness. In superspace, the Wess-Zumino model is formulated in terms of a chiralscalar superfield Φ and its conjugate ¯Φ. Without self-coupling, the model is described bythe action S WZ [Φ , ¯Φ] = Z d | z E n Φ ¯Φ + λ + ¯ λ o , ¯ D ˙ α Φ = 0 . (5.22)Here λ is a dimensionless complex parameter. The equations of motion correspondingto this model are − (cid:0) ¯ D − µ (cid:1) ¯Φ + λµ Φ = 0 , (5.23a) − (cid:0) D − µ (cid:1) Φ + ¯ λ ¯ µ ¯Φ = 0 . (5.23b)Using (5.23), it can be shown that the superfield Φ satisfies the mass-shell equation (cid:0) Q + µ ¯ µ − | λµ | (cid:1) Φ = 0 . (5.24)The on-shell chiral scalar Φ contains two independent component fields, which are ϕ : = Φ | , (5.25a) ψ α : = D α Φ | . (5.25b)By making use of the superfield mass-shell equation (5.24), it can be shown that thecomponent fields (5.25) satisfy the mass-shell equations (cid:0) Q − ρ (cid:1)(cid:0) Q − ρ (cid:1) ϕ = 0 , (5.26a) This parameter can be made real by applying a redefinition Φ → e i γ Φ, where γ = ¯ γ is constant. Q − ρ (cid:1) ψ α = 0 , (5.26b)where the pseudo-masses ρ i ) are ρ = | λµ | − µ ¯ µ (cid:0) | λ | + 2 (cid:1) , (5.27a) ρ = | λµ | + µ ¯ µ (cid:0) | λ | − (cid:1) , (5.27b) ρ = | λµ | − µ ¯ µ . (5.27c)In the massless limit λ →
0, the mass-shell conditions (5.27) take the form ρ = ρ = − µ ¯ µ = τ (1 , , µ ¯ µ , (5.28a) ρ = − µ ¯ µ = τ (1 , , µ ¯ µ , (5.28b)which is consistent with the on-shell massless field conditions (2.20). We wish to showthat the model (5.22) describes two Wess-Zumino representations (2.2). First, let us pointout that the equation for ϕ , (5.26a), is factorised into the product of two second-orderdifferential equations and, hence, describes two spin-0 modes with masses ρ (1) and ρ (2) .This means that the Wess-Zumino model describes three representations of so (3 , E ( i )0 associated with the pseudo-masses ρ ( i ) . Using the prescriptionadvocated in section 2.2, we find (cid:0) E (1)0 (cid:1) ± = 32 ± (cid:0) | λ | − (cid:1) , (5.29a) (cid:0) E (2)0 (cid:1) ± = 32 ± (cid:0) | λ | (cid:1) , (5.29b) (cid:0) E (3)0 (cid:1) ± = 32 ± | λ | . (5.29c)We see that there exist two branches of minimal energy solutions which furnish two Wess-Zumino representations S (cid:0) E , (cid:1) , eq. (2.2). We will call these branches the positivebranch and the negative branch. These solutions are given in table 4. These branchesComponent fields E (positive branch) E (negative branch)spin-0 1 + | λ | − | λ | spin-0 2 + | λ | − | λ | spin-
12 32 + | λ | − | λ | Table 4: Two branches of solutions.32urnish the Wess-Zumino representations S (cid:0) | λ | , (cid:1) and S (cid:0) − | λ | , (cid:1) , eq. (2.2). Inorder for these representations to be unitary, we require E > . Therefore, the positivebranch describes unitary representations for all values of λ . The negative branch describesunitary representations for | λ | < . In the massless case λ = 0 the two branches coincideand we have two massless scalars with energies E = 1 and E = 2 and a massless fermionwith energy E = . There are three special values of λ . The choice λ = 0 corresponds to the superconfor-mal or massless model. For λ = 1 the model has a dual formulation in terms of a tensorsupermultiplet [78]. In this case the equations of motion (5.23) can be recast in term ofa real superfield L = Φ + ¯Φ to take form (cid:0) ¯ D − µ (cid:1) L = (cid:0) D − µ (cid:1) L = 0 , (5.30a)while the chirality of Φ gives (cid:0) ¯ D − µ (cid:1) D α L = 0 . (5.30b)The equations (5.30) imply that L satisfies the mass-shell equation Q L = 0 , (5.31)which together define an on-shell scalar superfield. Finally for λ = 1 / The off-shell massive vector multiplet in a supergravity background was formulatedin [78, 79]. In the superspace setting, it is naturally described in terms of a real scalarprepotential V . Its action functional in AdS | is given by S [ V ] = 12 Z d | z E V n D α (cid:0) ¯ D − µ (cid:1) D α + M o V , ¯ V = V , (5.32)with M a non-zero real parameter. The corresponding equation of motion (cid:16) D α (cid:0) ¯ D − µ (cid:1) D α + M (cid:17) V = 0 (5.33) As pointed out in section 2.1, the massless spin-0 representations of osp (1 |
4) correspond to two pos-sible values of the minimal energy E = 1 or E = 2, and the massless spin- representation correspondsto E = . V is a linear superfield, (cid:0) ¯ D − µ (cid:1) V = (cid:0) D − µ (cid:1) V = 0 . (5.34a)Then making use of (A.4a) gives (cid:0) ✷ + 2 µ ¯ µ − M (cid:1) V = (cid:0) Q − M (cid:1) V = 0 . (5.34b)The equations (5.34a) and (5.34b) define an on-shell irreducible supermultiplet. It isworth pointing out that the above model has a dual formulation, which is the massivetensor multiplet model [78].On the mass shell, the independent component fields of V are: A = V | , (5.35a) ψ α = D α V | , (5.35b) h α ˙ α = 12 (cid:2) D α , ¯ D ˙ α (cid:3) V | , ∇ α ˙ α h α ˙ α = 0 . (5.35c)The corresponding equations of motion are:0 = (cid:0) Q + 2 µ ¯ µ − M (cid:1) A , (5.36a)0 = (cid:0) Q + 32 µ ¯ µ − M (cid:1) ψ α − i¯ µ ∇ α ˙ β ¯ ψ ˙ β , (5.36b)0 = (cid:0) Q + 32 µ ¯ µ − M (cid:1) ¯ ψ ˙ α + i µ ∇ β ˙ α ψ β , (5.36c)0 = (cid:0) Q − M (cid:1) h α ˙ α . (5.36d)The equations (5.36b) and (5.36c) lead to the quartic equation0 = (cid:0) Q + 32 µ ¯ µ − M (cid:1)(cid:0) Q + 32 µ ¯ µ − M − (cid:1) ψ α , (5.37)where M ± := M + 12 µ ¯ µ ± p µ ¯ µ (4 M + µ ¯ µ ) = 14 (cid:16) √ µ ¯ µ ± p M + µ ¯ µ (cid:17) . (5.38)The above results imply that the component fields furnish the massive superspin- rep-resentation of osp (1 , S (cid:0) E , (cid:1) = D (cid:0) E + 12 , (cid:1) ⊕ D (cid:0) E , (cid:1) ⊕ D (cid:0) E + 1 , (cid:1) ⊕ D (cid:0) E + 12 , (cid:1) , (5.39) It is instructive to compare the system of equations (5.30) with (5.33) and its corollaries (5.34). Theformer describes a non-conformal tensor multiplet in AdS , which reduces to the free massless tensormultiplet in the flat-superspace limit. E = 1 + q M µ ¯ µ + 1.The spinor equation (5.36b) and its conjugate (5.36c) are second-order partial differ-ential equations. First-order equations for the component spinor fields can be obtained ifone makes use of a St¨uckelberg reformulation of the above model. It is obtained throughthe replacement V → V + 1 M (cid:0) Φ + ¯Φ (cid:1) , ¯ D ˙ α Φ = 0 , (5.40)for some chiral superfield Φ. This leads to the action S [ V, Φ , ¯Φ] = 12 Z d | z E n V D α (cid:0) ¯ D − µ (cid:1) D α V + (cid:0) M V + Φ + ¯Φ (cid:1) o , (5.41)which is invariant under gauge transformations of the form δ Λ V = Λ + ¯Λ , δ Λ Φ = − M Λ , ¯ D ˙ α Λ = 0 . (5.42)The corresponding equations of motion are given by0 = 18 D α (cid:0) ¯ D − µ (cid:1) D α V + M V + M (cid:0) Φ + ¯Φ (cid:1) , (5.43a)0 = − (cid:0) ¯ D − µ (cid:1) V + µM Φ − M (cid:0) ¯ D − µ (cid:1) ¯Φ , (5.43b)0 = − (cid:0) D − µ (cid:1) V + ¯ µM ¯Φ − M (cid:0) D − µ (cid:1) Φ . (5.43c)Using the gauge freedom (5.42), one may impose the gauge condition Φ = 0 whereuponthe original model (5.32) is recovered. On the other hand, one can instead choose thefollowing Wess-Zumino gauge V | = 0 , D α V | = 0 , D V | = 0 , (cid:0) Φ − ¯Φ (cid:1) | = 0 , (5.44)which exhausts the gauge freedom. The remaining non-zero component fields are A = Φ | , (5.45a) ψ α = D α Φ | , (5.45b) h α ˙ α = 12 (cid:2) D α , ¯ D ˙ α (cid:3) V | , ∇ α ˙ α h α ˙ α = 0 , (5.45c) χ α = − (cid:0) ¯ D − µ (cid:1) D α V | . (5.45d)The spinor fields ψ α and χ α and their conjugates prove to satisfy the following first-orderdifferential equationsi ∇ α ˙ α χ α + M ¯ ψ ˙ α = 0 , − i ∇ α ˙ α ¯ χ ˙ α + M ψ α = 0 , (5.46a)35 ∇ α ˙ α ψ α + M ¯ χ ˙ α + ¯ µ ¯ ψ ˙ α = 0 , − i ∇ α ˙ α ¯ ψ ˙ α + M χ α + µψ α = 0 . (5.46b)Making use of the equations (5.46b) allows us to express the fields χ α and ¯ χ ˙ α in terms of ψ α and ¯ ψ ˙ α . Then the latter fields prove to satisfy the equations (5.36b) and (5.36c). Asregards the bosonic fields (5.45b) and (5.45c), they may be seen to obey the equations(5.36a) and (5.36d). As an illustration of our general discussion in section 4, here we present an off-shellmodel for the massive gravitino (superspin-1) multiplet in AdS , S (cid:0) E , (cid:1) := D (cid:16) E + 12 , (cid:17) ⊕ D ( E , ⊕ D ( E + 1 , ⊕ D (cid:16) E + , (cid:17) . (6.1)The unitarity bound for the superspin-1 case is E ≥
2, with the E = 2 value corre-sponding to the massless gravitino multiplet. We point out that on-shell models (i.e.without auxiliary fields) for the the massive gravitino multiplet in AdS have appeared inthe literature [80] (see also [81]).Two off-shell formulations for the massless gravitino in AdS were introduced in [16].One of them is described by the action S massless [ H, Ψ , ¯Ψ] = − Z d | z E n H D α (cid:0) ¯ D − µ (cid:1) D α H + µ ¯ µH + 14 H (cid:0) D α ¯ D ˙ α G α ˙ α − ¯ D ˙ α D α ¯ G α ˙ α (cid:1) + ¯ G α ˙ α G α ˙ α + 14 (cid:0) ¯ G α ˙ α ¯ G α ˙ α + G α ˙ α G α ˙ α (cid:1)o , (6.2)where H is a real scalar superfield, and G α ˙ α is a longitudinal linear superfield constructedin terms of an unconstrained prepotential Ψ α , G α ˙ α = ¯ D ˙ α Ψ α , ¯ G α ˙ α = −D α ¯Ψ ˙ α . (6.3)We propose a massive extension of the action (6.2) by adding a mass term S m [ H, Ψ , ¯Ψ] S massive [ H, Ψ , ¯Ψ] = S massless [ H, Ψ , ¯Ψ] + S m [ H, Ψ , ¯Ψ] . (6.4)We propose the following mass term: S m [ H, Ψ , ¯Ψ] = m Z d | z E n Ψ α Ψ α + ¯Ψ ˙ α ¯Ψ ˙ α + 12 H (cid:0) D α Ψ α + ¯ D ˙ α ¯Ψ ˙ α (cid:1) (6.5) This model has several dual versions given in [16,24,82]. In the flat-superspace limit, the action (6.2)reduces to that derived in [10]. (cid:0) m + 2 µ + 2¯ µ (cid:1) H o , where m is a real parameter of unit mass dimension.The equations of motion corresponding to S massive [ H, Ψ , ¯Ψ], eq. (6.4), are given by0 = 12 D α W α + 14 (cid:0) D α ¯ D Ψ α + ¯ D ˙ α D ¯Ψ ˙ α (cid:1) + m (cid:0) D α Ψ α + ¯ D ˙ α ¯Ψ ˙ α (cid:1) (6.6a) − (cid:0) mµ + m ¯ µ + 2 µ ¯ µ + 12 m (cid:1) H , − W α + (cid:0) m µ (cid:1) D α H + ¯ D ˙ α D α ¯Ψ ˙ α −
12 ¯ D Ψ α − m Ψ α , (6.6b)where we have introduced the field strength W α = − (cid:0) ¯ D − µ (cid:1) D α H . Acting on (6.6a)and (6.6b) with ¯ D gives (cid:0) ¯ D − µ (cid:1) D α Ψ α = (cid:0) m + 2¯ µ (cid:1)(cid:0) ¯ D − µ (cid:1) H , (6.7a) (cid:0) ¯ D − µ (cid:1) Ψ α = − W α . (6.7b)Next, acting on (6.6b) with D α yields¯ D ˙ α ¯Ψ ˙ α = (cid:0) m + 2 µ (cid:1) H − µ (cid:0) m + µ (cid:1) D α Ψ α . (6.8)Using relations (6.6a), (6.7) and (6.8), one can show that D α Ψ α = 0 . (6.9)Substituting (6.9) into (6.8), it immediately follows that H = 0 (6.10)on the mass shell. Now, the above relations imply that the spinor Ψ α obeys the followingmass-shell conditions: D α Ψ α = 0 , (cid:0) ¯ D − µ (cid:1) Ψ α = 0 , (6.11a)i D α ˙ α ¯Ψ ˙ α + (cid:0) m + µ (cid:1) Ψ α = 0 = ⇒ (cid:0) Q − ( m + µ )( m + ¯ µ ) − µ ¯ µ (cid:1) Ψ α = 0 . (6.11b)These are the on-shell irreducibility constraints for a massive superspin-1 multiplet, asdefined in section 4. The unitarity bound for the massive gravitino multiplet is | m + µ | > µ ¯ µ = ⇒ m + 2 µ = 0 . (6.12)These conditions have been used in the above derivation. One may think of the firstDirac-type equation in (6.11b) as the reality condition relating Ψ α and ¯Ψ ˙ α .37n conclusion, we give the explicit expression for the massive gravitino action (6.4) S massive [ H, Ψ , ¯Ψ] = Z d | z E n − H D α (cid:0) ¯ D − µ (cid:1) D α H − µ ¯ µH + 14 H (cid:0) D α ¯ D Ψ α + ¯ D ˙ α D ¯Ψ ˙ α (cid:1) + D α ¯Ψ ˙ α ¯ D ˙ α Ψ α + 14 (cid:0) ¯Ψ ˙ α D ¯Ψ ˙ α + Ψ α ¯ D Ψ α (cid:1) − m (cid:0) m + 2 µ + 2¯ µ (cid:1) H + m H (cid:0) D α Ψ α + ¯ D ˙ α ¯Ψ ˙ α (cid:1) + m (cid:0) Ψ α Ψ α + ¯Ψ ˙ α ¯Ψ ˙ α (cid:1)o . (6.13)Note that in the flat superspace limit, µ →
0, the action reduces to that describing themassive gravitino model derived in [83].In the massless limit, m →
0, the second mass-shell condition (6.11b) becomes (cid:0) Q − µ ¯ µ (cid:1) Ψ α = (cid:0) Q − λ (1 , , µ ¯ µ (cid:1) Ψ α = 0 . (6.14)This agrees with the definition of on-shell massless supermultiplets given in section 4.On the mass shell, the independent component fields of Ψ α are: χ α = Ψ α | , (6.15a) A α ˙ α = ¯ D ˙ α Ψ α | , ∇ α ˙ α A α ˙ α = 0 , (6.15b) B αβ = D ( α Ψ β ) | , (6.15c) ϕ αβ ˙ α = (cid:16)
12 [ D ( α , ¯ D ˙ α ] − i3 D ( α ˙ α (cid:17) Ψ β ) | , ∇ α ˙ α ϕ αβ ˙ α = 0 . (6.15d)Here A α ˙ α is a complex vector field. The component equations of motion, which followfrom the first equation in (6.11b), are:0 = i ∇ β ˙ α χ β + (cid:0) m + ¯ µ (cid:1) ¯ χ ˙ α , (6.16a)0 = i ∇ β ˙ α B αβ + ¯ µA α ˙ α − (cid:0) m + ¯ µ (cid:1) ¯ A α ˙ α , (6.16b)0 = i ∇ β ( ˙ α A β ˙ β ) − (cid:0) m + ¯ µ (cid:1) ¯ B ˙ α ˙ β , (6.16c)0 = i ∇ β ( ˙ α ϕ αβ ˙ β ) + (cid:0) m + ¯ µ (cid:1) ¯ ϕ α ˙ α ˙ β . (6.16d)These equations can be viewed as reality conditions expressing the component fields of ¯Ψ ˙ α in terms of those contained in Ψ α . Making use of the relations (6.16) and their conjugatesleads to the following equations:0 = (cid:0) Q + 32 µ ¯ µ − κ (cid:1) χ α , (6.17a)0 = (cid:0) Q − κ (cid:1) B αβ − i¯ µ ∇ ( α ˙ β A β ) ˙ β , (6.17b)0 = (cid:0) Q + µ ¯ µ − κ (cid:1) A α ˙ α + i µ ∇ β ˙ α B αβ , (6.17c)38 = (cid:0) Q − µ ¯ µ − κ (cid:1) ϕ αβ ˙ α , (6.17d)where κ := | m + µ | . (6.18)It can be shown that the equations (6.17b) and (6.17c) imply the quartic equation0 = (cid:0) Q − κ − √ µ ¯ µκ (cid:1)(cid:0) Q − κ + √ µ ¯ µκ (cid:1) A α ˙ α , (6.19a)0 = (cid:0) Q − κ − √ µ ¯ µκ (cid:1)(cid:0) Q − κ + √ µ ¯ µκ (cid:1) B αβ . (6.19b)The analysis above indicates that the component fields furnish the massive superspin-1representation (6.1), where E = 1 + q κ µ ¯ µ . Additionally, the mass-shell conditions areconsistent with the results in section 5.1, for a superfield with index structure m − n = 0, upon the redefinition κ = M − µ ¯ µ . In section 3.2 we introduced the gauge invariant actions for superconformal higher-spinmultiplets and used them to motivate our ansatz for the TLAL projectors. Let us comeback to this topic and demonstrate that these actions factorise into products of minimalsecond order differential operators.We begin with the familiar case when m = n = s and impose the reality condition(3.45). It is clear that for these values the higher-spin Bach tensor (3.27a) and thedescendant (3.29a) coincide, so that we have the relations B α ( s ) ˙ α ( s ) ( H ) = P α ( s ) ˙ α ( s ) ( H ) = s +1 Y t =1 (cid:0) Q − λ ( t,s,s ) µ ¯ µ (cid:1) Π α ( s ) ˙ α ( s ) ( H ) . (7.1)Now, one can employ the decomposition from section 3.3 (which is valid for all off-shellsuperfields) to split the unconstrained prepotential H α ( s ) ˙ α ( s ) in the action (3.26a) intoTLAL and longitudinal components, H α ( s ) ˙ α ( s ) = H α ( s ) ˙ α ( s ) + ¯ D ( ˙ α ζ α ( s ) ˙ α ... ˙ α s ) − D ( α ¯ ζ α ...α s ) ˙ α ( s ) . (7.2)Here H α ( s ) ˙ α ( s ) is TLAL, whilst ζ α ( s ) ˙ α ( s − is complex and unconstrained.By virtue of the gauge invariant and TLAL nature of the higher-spin Bach tensor, thelongitudinal parts give vanishing contribution to the action since they correspond to the39ure gauge sector. Therefore only the TLAL component of the prepotential remains and,since the projector Π ( s,s ) acts as the identity on this subspace, using (7.1) we obtain thefollowing factorisation S ( s,s )SCHS [ H ] = 2( − s Z d | z E H α ( s ) ˙ α ( s ) s +1 Y t =1 (cid:0) Q − λ ( t,s,s ) µ ¯ µ (cid:1) H α ( s ) ˙ α ( s ) . (7.3)This process is equivalent to fixing the gauge freedom (3.21) by imposing the gauge condi-tion H α ( s ) ˙ α ( s ) ≡ H α ( s ) ˙ α ( s ) , since the action (7.3) no longer possesses any gauge symmetry.Next we consider the complex supermultiplet Φ α ( m ) ˙ α ( n ) with n > m . In contrast to(7.1), one may show that in this case the following relation holds P α ( m ) ˙ α ( n ) (Φ) = D ( ˙ α β · · · D ˙ α n − m β n − m B α ( m ) β ( n − m ) ˙ α n − m +1 ... ˙ α n ) (Φ) , (7.4)which upon inverting yields B α ( n ) ˙ α ( m ) (Φ) = (cid:20) n +1 Y t = m +2 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1)(cid:21) − D ( α ˙ β · · · D α n − m ˙ β n − m × P α n − m +1 ...α n ) ˙ β ( n − m ) ˙ α ( m ) (Φ) . (7.5)In terms of the TLAL projector this reads B α ( n ) ˙ α ( m ) (Φ) = m +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1) D ( α ˙ β · · · D α n − m ˙ β n − m × Π α n − m +1 ...α n ) ˙ β ( n − m ) ˙ α ( m ) (Φ) . (7.6)Employing the same trick as in the previous case, one obtains the following factorisationof the SCHS action (3.26a) S ( m,n )SCHS [Φ , ¯Φ] = i m + n Z d | z E ¯ φ α ( n ) ˙ α ( m ) m +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1) × D ( α ˙ β · · · D α n − m ˙ β n − m φ α n − m +1 ...α n ) ˙ β ( n − m ) ˙ α ( m ) + c.c. , (7.7)where φ α ( m ) ˙ α ( n ) is the TLAL part of Φ α ( m ) ˙ α ( n ) . We see that, due to the mismatch of m and n , the SCHS action does not factorise wholly into products of second-order operators.However, upon taking appropriate derivatives of the equation of motion resulting fromvarying ¯ φ α ( m ) ˙ α ( n ) in (7.7), one arrives at the following fully factorised on-shell equation0 = n +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1) φ α ( m ) ˙ α ( n ) . (7.8)40ccording to our definitions in section 4, we see that some extra (non-unitary) massivemodes, corresponding to the values of λ ( t,m,n ) with m + 1 < t ≤ n + 1 enter the spectrumof the wave equation (7.8).Finally, if m > n , then one may show that the following relation holds b P α ( m ) ˙ α ( n ) (Φ) = D ( ˙ α β · · · D ˙ α m − n β m − n b B α ( n ) β ( m − n ) ˙ α m − n +1 ... ˙ α m ) (Φ) . (7.9)Upon inverting and expressing in terms of the TLAL projector, one may write B α ( n ) ˙ α ( m ) (Φ) = n +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1) D ( ˙ α β · · · D ˙ α m − n β m − n × Π α ( n ) β ( m − n ) ˙ α m − n +1 ... ˙ α m ) (Φ) , (7.10)where we have used the identities (3.33) and b B α ( n ) ˙ α ( m ) (Φ) = B α ( n ) ˙ α ( m ) (Φ). In this casethe factorised action takes the form S ( m,n )SCHS [Φ , ¯Φ] = i m + n Z d | z E ¯ φ α ( n ) ˙ α ( m ) n +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1) × D ( ˙ α β · · · D ˙ α m − n β m − n φ α ( n ) β ( m − n ) ˙ α m − n +1 ... ˙ α m ) + c.c. , (7.11)whilst the analogous on-shell equation is0 = m +1 Y t =1 (cid:0) Q − λ ( t,m,n ) µ ¯ µ (cid:1) φ α ( m ) ˙ α ( n ) . (7.12)Once again, (non-unitary) massive modes appear in the resulting wave equation.In the non-supersymmetric case, the factorisation of the conformal operators wasobserved long ago in [31, 84, 85] for the lower-spin values s = 3 / s = 2. Thefactorisation of the higher-spin conformal operators was conjectured in [86, 87] (see also[88]), and later proved by several groups [29, 89–91]. In this section we have provided thefirst derivation of the factorisation of the superconformal higher-spin actions in N = 1superspace. Just as it is for the non-supersymmetric case, the latter factor into productsof minimal second-order differential operators involving all partial mass values. In conclusion we briefly summarise the main results of this paper. We constructedthe superspin projection operators Π ( m,n ) in AdS | . They are given by eq. (3.30) for41 ≥ n >
0, and by eq. (3.35) for m > n = 0 (the other cases are obtained by complexconjugation). The operator Π ( m,n ) maps an unconstrained superfield Φ α ( m ) ˙ α ( n ) into onepossessing the properties of a conserved conformal supercurrent, see the equations (3.31)and (3.36). Making use of the superprojectors, we obtained a new representation for thesuperconformal higher-spin gauge actions in AdS given by eq. (3.40), which was used todemonstrate their factorisation, as discussed in section 7.We provided a systematic discussion of how to realise the unitary (both massive andmassless) and the partially massless representations of the N = 1 AdS superalgebra osp (1 |
4) in terms of on-shell superfields. In particular, we established a one-to-one cor-respondence between the on-shell partially massless supermultiplets φ α ( m ) ˙ α ( n ) with super-depth t and the poles of Π ( m,n ) determined by λ ( t,m,n ) µ ¯ µ belonging to the range (4.13).The corresponding gauge transformations were derived in the m = n case.Our results make it possible to address a number of interesting open problems in-cluding the computation of partition functions for the superconformal higher-spin gaugetheories in AdS . This will be discussed elsewhere. Acknowledgements:
SMK is grateful to I. L. Buchbinder and A. A. Tseytlin for email correspondence. Thework of EIB and SMK is supported in part by the Australian Research Council, projectNo. DP200101944. The work of DH is supported by the Jean Rogerson PostgraduateScholarship and an Australian Government Research Training Program Scholarship. Thework of MP is supported by the Hackett Postgraduate Scholarship UWA, under the Aus-tralian Government Research Training Program.
A AdS superspace toolkit
Our two-component spinor notation and conventions follow [66]. In particular, theLorentz generators M αβ and ¯ M ˙ α ˙ β act on two-component spinors as follows: M αβ ψ γ = 12 ( ε γα ψ β + ε γβ ψ α ) , M αβ ¯ ψ ˙ γ = 0 , (A.1a)¯ M ˙ α ˙ β ¯ ψ ˙ γ = 12 ( ε ˙ γ ˙ α ¯ ψ ˙ β + ε ˙ γ ˙ β ¯ ψ ˙ α ) , ¯ M ˙ α ˙ β ψ γ = 0 . (A.1b)In this paper we always work with tensor superfields that are symmetric in their undottedindices and separately in the dotted ones, Φ α ( m ) ˙ α ( n ) = Φ ( α ...α m )( ˙ α ... ˙ α n ) = Φ α ...α m ˙ α ... ˙ α n .42he following identities hold: M α β Φ βα ...α m ˙ α ( n ) = −
12 ( m + 2)Φ α ( m ) ˙ α ( n ) , (A.2a)¯ M ˙ α ˙ β Φ α ( m ) ˙ β ˙ α ... ˙ α n = −
12 ( n + 2)Φ α ( m ) ˙ α ( n ) , (A.2b) M βγ M βγ Φ α ( m ) ˙ α ( n ) = − m ( m + 2)Φ α ( m ) ˙ α ( n ) , (A.2c)¯ M ˙ β ˙ γ ¯ M ˙ β ˙ γ Φ α ( m ) ˙ α ( n ) = − n ( n + 2)Φ α ( m ) ˙ α ( n ) . (A.2d)We often make use of the following identities, which can be readily derived from thealgebra of covariant derivatives (2.7): D α D β = 12 ε αβ D − µ M αβ , ¯ D ˙ α ¯ D ˙ β = − ε ˙ α ˙ β ¯ D + 2 µ ¯ M ˙ α ˙ β , (A.3a) D α D = 4¯ µ D β M αβ + 4¯ µ D α , D D α = − µ D β M αβ − µ D α , (A.3b)¯ D ˙ α ¯ D = 4 µ ¯ D ˙ β ¯ M ˙ α ˙ β + 4 µ ¯ D ˙ α , ¯ D ¯ D ˙ α = − µ ¯ D ˙ β ¯ M ˙ α ˙ β − µ ¯ D ˙ α , (A.3c) (cid:2) ¯ D , D α (cid:3) = 4i D α ˙ β ¯ D ˙ β + 4 µ D α = 4i ¯ D ˙ β D α ˙ β − µ D α , (A.3d) (cid:2) D , ¯ D ˙ α (cid:3) = − D β ˙ α D β + 4¯ µ ¯ D ˙ α = − D β D β ˙ α − µ ¯ D ˙ α , (A.3e)where D = D α D α , and ¯ D = ¯ D ˙ α ¯ D ˙ α . Other useful identities are: D α ˙ β D ˙ ββ = δ αβ ✷ − µ ¯ µM αβ , (A.4a) D α ˙ α D α ˙ β = δ ˙ α ˙ β ✷ − µ ¯ µ ¯ M ˙ α ˙ β . (A.4b)Of special importance are the relations: ✷ + 2 µ ¯ µ = − D α (cid:0) ¯ D − µ (cid:1) D α + 116 (cid:8) D − µ, ¯ D − µ (cid:9) , (A.5a) D α (cid:0) ¯ D − µ (cid:1) D α = ¯ D ˙ α (cid:0) D − µ (cid:1) ¯ D ˙ α , (A.5b) (cid:2) D , ¯ D (cid:3) = − (cid:2) D β , ¯ D ˙ β (cid:3) D β ˙ β + 8 µ D − µ ¯ D = − D β ˙ β (cid:2) D β , ¯ D ˙ β (cid:3) − µ D + 8¯ µ ¯ D . (A.5c)The isometry group of AdS superspace, AdS | , is OSp (1 | | are generated by the Killing supervector fields ξ A E A which are definedto solve the Killing equation (cid:2) Ξ , D A (cid:3) = 0 , Ξ := − ξ β ˙ β D β ˙ β + ξ β D β + ¯ ξ ˙ β ¯ D ˙ β + ξ βγ M βγ + ¯ ξ ˙ β ˙ γ ¯ M ˙ β ˙ γ , (A.6)43or some Lorentz parameter ξ βγ = ξ γβ . Given a supersymmetric field theory in AdS formulated in terms of superfield dynamical variables V (with suppressed indices), itsaction is invariant under the isometry transformations δ V = Ξ V , (A.7)with ξ B being an arbitrary Killing supervector field.The Killing equation (A.6) implies the following [66]: D α ξ β ˙ β = 4i ε αβ ¯ ξ ˙ β , ¯ D ˙ α ξ β ˙ β = − ε ˙ α ˙ β ξ β , (A.8a) D α ξ β = ξ αβ , ¯ D ˙ α ξ β = − i2 µξ β ˙ α , (A.8b) D α ξ βγ = − µε α ( β ξ γ ) , ¯ D ˙ α ξ βγ = 0 . (A.8c)It is seen that the parameters ξ β , ξ βγ and their conjugates are expressed in terms of thevector parameter ξ β ˙ β , and the latter obeys the equation D ( α ξ β ) ˙ β = 0 = ⇒ D ( a ξ b ) = 0 . (A.9)It also follows from (A.8) that, for every element of the set of parameters Υ = { ξ B , ξ βγ , ¯ ξ ˙ β ˙ γ } ,its covariant derivative D A Υ is a linear combination of the elements of Υ. Therefore, allinformation about the superfield parameters Υ is encoded in their bar-projections, Υ | .Every Killing supervector superfield ξ B on AdS | , eq. (A.6), can be uniquely decom-posed as a sum of even and odd ones. The Killing supervector field ξ B is defined to beeven if v b := ξ b | 6 = 0 , ξ β | = 0 . (A.10)and odd if ξ b | = 0 , ǫ β := ξ β | 6 = 0 . (A.11)The fields v b ( x ) and ǫ β ( x ) encode complete information about the parent conformal Killingvector superfield. It follows from (A.9) that v b is a Killing vector field on AdS , ∇ ( a v b ) = 0 . (A.12)The relations (A.8) imply that ǫ β is a Killing spinor field satisfying the equation ∇ α ˙ α ǫ β = i4 µε αβ ¯ ǫ ˙ α . (A.13)44very Killing vector v b on AdS can be lifted to an even Killing supervector field ξ B onAdS | using the relations (A.8). A similar statement holds for Killing spinors.Given a tensor superfield V (with suppressed indices), its independent componentfields are contained in the set of fields ϕ = Φ | , where Φ := (cid:8) V , D α V , ¯ D ˙ α V , . . . (cid:9) . Inaccordance with (A.7), the supersymmetry transformation of ϕ is δ ǫ ϕ = ǫ β ( D β Φ) | + ¯ ǫ ˙ β ( ¯ D ˙ β Φ) | . (A.14) B Partially massless gauge symmetry
This appendix is devoted to the derivation of partially massless gauge transformationsboth in the non-supersymmetric and supersymmetric cases.
B.1 The non-supersymmetric case
Given two integers m and n such that m ≥ n >
0, let h α ( m ) ˙ α ( n ) be an on-shell field,0 = (cid:0) Q − ρ (cid:1) h α ( m ) ˙ α ( n ) , (B.1a)0 = ∇ β ˙ β h α ( m − β ˙ α ( n −
1) ˙ β . (B.1b)We would like to determine those values of ρ for which the above system of equations iscompatible with a gauge symmetry.We begin by positing a gauge transformation of the form δ ζ h α ( m ) ˙ α ( n ) = ∇ ( α ( ˙ α ζ α ...α m ) ˙ α ... ˙ α n ) (B.2)and look for a gauge parameter ζ α ( m −
1) ˙ α ( n − such that δ ζ h α ( m ) ˙ α ( n ) is a solution to theequations (B.1). Clearly, for gauge invariance of (B.1a) the gauge parameter must satisfy0 = (cid:0) Q − ρ (cid:1) ζ α ( m −
1) ˙ α ( n − . (B.3)Next we require (B.1b) to be gauge invariant, ∇ β ˙ β δ ζ h α ( m − β ˙ α ( n −
1) ˙ β = 0. To solve thisproblem, let us recall a technical result derived in [29]. Using the spin projection operators,it was shown in [29] that any unconstrained tensor field can be decomposed into irreducible(i.e. transverse) parts. For the gauge parameter ζ α ( m −
1) ˙ α ( n − , this decomposition takesthe form ζ α ( m −
1) ˙ α ( n − = ζ T α ( m −
1) ˙ α ( n − + n − X t =1 ∇ ( α ( ˙ α · · · ∇ α t ˙ α t ζ T α t +1 ...α m − ) ˙ α t +1 ... ˙ α n − ) , (B.4)45here ζ T α ( m − n ) is unconstrained, whilst the other fields are transverse,0 = ∇ β ˙ β ζ T α ( m − t − β ˙ α ( n − t −
1) ˙ β , ≤ t ≤ n − . (B.5)Inserting this into the gauge transformations (B.2) and requiring that the condition (B.1b)be gauge invariant, one arrives at the following equation n X k =1 k ( m + n − k + 1) (cid:16) ρ − τ ( k,m,n ) µ ¯ µ (cid:17) × ∇ ( α ( ˙ α · · · ∇ α k − ˙ α k − ζ T α k ...α m − ) ˙ α k ... ˙ α n − ) . (B.6)The right-hand side of (B.6) is the decomposition into irreducible parts. The wholeexpression may vanish only if there exists an integer t such that ρ = τ ( t,m,n ) µ ¯ µ , ≤ t ≤ n . (B.7)In addition, each ζ T α ( m − k ) ˙ α ( n − k ) in (B.6) except for ζ T α ( m − t ) ˙ α ( n − t ) must vanish identically.Hence the decomposition (B.4) reduces to ζ α ( m −
1) ˙ α ( n − = ∇ ( α ( ˙ α · · · ∇ α t − ˙ α t − ζ T α t ...α m − ) ˙ α t ... ˙ α n − ) , (B.8)and the gauge transformation (B.2) becomes the well-known one for a partially-masslessfield with depth t , δ ζ h α ( m ) ˙ α ( n ) = ∇ ( α ( ˙ α · · · ∇ α t ˙ α t ζ T α t +1 ...α m ) ˙ α t +1 ... ˙ α n ) , (B.9)with ζ T α ( m − t ) ˙ α ( n − t ) satisfying (B.3), (B.5) and (B.7). B.2 The supersymmetric case
We first consider the real supermultiplet H α ˙ α = ¯ H α ˙ α which is on-shell0 = (cid:0) Q − M (cid:1) H α ˙ α , (B.10a)0 = D β H β ˙ α = ¯ D ˙ β H α ˙ β . (B.10b)Once again we would like to determine those values of M for which the above systemof equations is compatible with a gauge symmetry. We will see that this occurs only atpartially massless values. For the supermultiplet H α ˙ α there are only two such values, M ≡ λ (1 , , µ ¯ µ = 5 µ ¯ µ , (B.11a) To derive this equation the identities in the appendix of [29] are indispensable. ≡ λ (2 , , µ ¯ µ = 3 µ ¯ µ , (B.11b)corresponding to the super-depths t = 1 (massless) and t = 2 respectively.We begin by positing a gauge transformation of the form δH α ˙ α = ¯ D ˙ α Λ α − D α ¯Λ ˙ α (B.12)for an unconstrained gauge parameter Λ α . From this we must deduce which constraintsmust be placed upon Λ α in order for δH α ˙ α to be a solution to the equations (B.10).Gauge invariance of (B.10a) requires Λ α to also have pseudo-mass M ,0 = (cid:0) Q − M (cid:1) Λ α . (B.13)Next we decompose the gauge parameter into irreducible parts. Performing this procedurein accordance with the discussion in section 3.3, we find that (B.12) takes the form δH α ˙ α = ¯ D ˙ α ζ α − D α ¯ ζ ˙ α + D α ˙ α ξ + (cid:2) D α , ¯ D ˙ α (cid:3) ζ + D α ˙ α (cid:0) σ + ¯ σ (cid:1) . (B.14)In (B.14), the gauge parameter ζ α is LTAL0 = (cid:0) ¯ D − µ (cid:1) ζ α , D α ζ α ⇔ (cid:0) D − µ (cid:1) ζ α . (B.15)The real parameters ξ = ¯ ξ and ζ = ¯ ζ are linear0 = (cid:0) ¯ D − µ (cid:1) ζ = (cid:0) D − µ (cid:1) ζ , (cid:0) ¯ D − µ (cid:1) ξ = (cid:0) D − µ (cid:1) ξ , (B.16)whilst σ is complex chiral, ¯ D ˙ α σ = 0.Before solving the equation 0 = D β δH β ˙ α , we first analyse the higher-order equations0 = D β ˙ β δH β ˙ β and 0 = (cid:2) D β , ¯ D ˙ β (cid:3) δH β ˙ β . Respectively, they yield the following equations (cid:0) M − µ D −
14 ¯ µ ¯ D (cid:1) ( σ + ¯ σ ) + (cid:0) M − µ ¯ µ (cid:1) ξ , (B.17a)0 = (cid:0) M + 34 µ D + 34 ¯ µ ¯ D (cid:1) ( σ − ¯ σ ) − (cid:0) M − µ ¯ µ (cid:1) ζ . (B.17b)One may show that both equations in (B.17) are consistent with (B.16) only if ζ = 0 , ξ = 0 , (B.18) The (non-unitary) masses M = 2 µ ¯ µ and M = 0 should be considered separately. The result of thisanalysis is that there is no gauge-symmetry present. M . In addition, we must also have σ = ¯ σ = 0 , (B.19)unless M = M , in which case σ must satisfy the equation0 = −
14 ( D − µ ) σ + 2¯ µ ¯ σ . (B.20)Finally, using the above information to solve the equation 0 = D β δH β ˙ α , one arrives atthe reality conditions (valid for all M ) D ˙ αβ ζ β = 2i¯ µ ¯ ζ ˙ α , D α ˙ β ¯ ζ ˙ β = − µζ α , (B.21)which imply the mass-shell condition0 = (cid:0) M − µ ¯ µ (cid:1) ζ α . (B.22)We see that unless H α ˙ α has pseudo-mass M = M , then ζ α = 0 . (B.23)In conclusion, the on-shell conditions (B.10) are only compatible with a gauge sym-metry when the pseudo-mass of H α ˙ α (and the corresponding gauge parameters) takes oneof the partially-massless values (B.11a) or (B.11b). For super-depth t = 1, the relevantgauge symmetry is δH α ˙ α = ¯ D ˙ α ζ α − D α ¯ ζ ˙ α , (B.24)where ζ α satisfies (B.15) and (B.21). On the otherhand, for super-depth t = 2 it is δH α ˙ α = D α ˙ α ( σ + ¯ σ ) , (B.25)where σ is chiral and satisfies (B.20).By conducting a similar analysis on the real supermultiplet H α (2) ˙ α (2) , one arrives atthe gauge transformations presented in table 3. In principle this procedure may be carriedout for any on-shell superfield φ α ( m ) ˙ α ( n ) with fixed m and n , but in practice this procedureis quite time consuming. In section 4.2 we use the above analysis to motivate an ansatzfor the case arbitrary half-integer superspin.48 eferences [1] A. Salam and J. A. Strathdee, “On superfields and Fermi-Bose symmetry,” Phys. Rev. D , 1521(1975).[2] E. Sokatchev, “Projection operators and supplementary conditions for superfields with an arbitraryspin,” Nucl. Phys. B , 96 (1975).[3] W. Siegel and S. J. Gates Jr., “Superprojectors,” Nucl. Phys. B , 295 (1981).[4] E. Sokatchev, “Irreducibility conditions for extended superfields,” Phys. Lett. B , 38-40 (1981).[5] V. Rittenberg and E. Sokatchev, “Decomposition of extended superfields into irreducible represen-tations of supersymmetry,” Nucl. Phys. B , 477-501 (1981).[6] S. J. Gates Jr., M. T. Grisaru, M. Roˇcek and W. Siegel, Superspace, or One Thousand and OneLessons in Supersymmetry , Benjamin/Cummings (Reading, MA), 1983, arXiv:hep-th/0108200.[7] E. I. Buchbinder, D. Hutchings, J. Hutomo and S. M. Kuzenko, “Linearised actions for N -extended(higher-spin) superconformal gravity,” JHEP (2019) 077 [arXiv:1905.12476 [hep-th]].[8] V. Ogievetsky and E. Sokatchev, “On vector superfield generated by supercurrent,” Nucl. Phys. B , 309 (1977).[9] V. I. Ogievetsky and E. Sokatchev, “Superfield equations of motion,” J. Phys. A , 2021 (1977).[10] S. J. Gates Jr. and W. Siegel, “(3/2, 1) superfield of O(2) supergravity,” Nucl. Phys. B , 484(1980).[11] S. J. Gates Jr., S. M. Kuzenko and J. Phillips, “The off-shell (3/2,2) supermultiplets revisited,”Phys. Lett. B , 97 (2003) [arXiv:hep-th/0306288].[12] E. I. Buchbinder, S. M. Kuzenko, J. La Fontaine and M. Ponds, “Spin projection opera-tors and higher-spin Cotton tensors in three dimensions,” Phys. Lett. B , 389-395 (2019)[arXiv:1812.05331 [hep-th]].[13] R. E. Behrends and C. Fronsdal, “Fermi decay of higher spin particles,” Phys. Rev. , 345 (1957).[14] C. Fronsdal, “On the theory of higher spin fields,” Nuovo Cim. , 416 (1958).[15] E. A. Ivanov and A. S. Sorin, “Superfield formulation of OSp(1,4) supersymmetry,” J. Phys. A (1980) 1159.[16] S. M. Kuzenko and A. G. Sibiryakov, “Free massless higher-superspin superfields on the anti-deSitter superspace,” Phys. Atom. Nucl. , 1257-1267 (1994) [arXiv:1112.4612 [hep-th]].[17] S. M. Kuzenko, A. G. Sibiryakov and V. V. Postnikov, “Massless gauge superfields of higher halfinteger superspins,” JETP Lett. , 534 (1993) [Pisma Zh. Eksp. Teor. Fiz. , 521 (1993)].[18] S. M. Kuzenko and A. G. Sibiryakov, “Massless gauge superfields of higher integer superspins,”JETP Lett. , 539 (1993) [Pisma Zh. Eksp. Teor. Fiz. , 526 (1993)].[19] A. G. Sibiryakov, “Superfield models for the massless higher-superspin multiplets,” Ph.D. thesis,Tomsk State University (1996).
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