Lagrangian description of massive higher spin supermultiplets in AdS 3 space

Lagrangian description of massive higher spin supermultiplets in AdS 3 space arxiv:1705.06163v1 [hep-th] 17 May 017 I.L. Buchbinder ab, T.V. Snegirev ac, Yu.M. Zinoviev d a Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk 634061, Russia b National Research Tomsk State University, Tomsk 634050, Russia c National Research Tomsk Polytechnic University, Tomsk 634050, Russia d Institute for High Energy Physics, of National Research Center Kurchatov Institute Protvino, Moscow Region, 1480, Russia Abstract We construct the Lagrangian formulation of massive higher spin (1,0) supermultiplets in three dimensional anti-de Sitter space. The construction is based on description of the massive three dimensional fields in terms of frame-like gauge invariant formalism and technique of gauge invariant curvatures. For the two possible massive supermultiplets (s,s+1/) and (s,s 1/) we derive explicit form of the supertransformations leaving the sum of bosonic and fermionic Lagrangians invariant. joseph@tspu.edu.ru snegirev@tspu.edu.ru Yurii.Zinoviev@ihep.ru

1 Introduction Three dimensional field models attract much attention due to their comparatively simple structure and remarkable properties of three-dimensional flat and curved spaces. One of the achievements in this area is a construction of three-dimensional higher spin field theories (see the pioneer works [1], [], [3], for modern development see e.g. the recent papers [4], [5] and references therein). The aim of this work is to construct the massive higher spin supermultiplets in the threedimensional anti-de Sitter space AdS 3. As is well known [6], the AdS 3 space possesses the special properties since all the AdS 3 superalgebras (as well as conventional AdS 3 algebra itself) factorize into left and right parts. For the case of simplest (1,0) superalgebra (the one we are working here with) such factorization has the form: OSp(1,) Sp() so that we have supersymmetry in the left sector only. It means that the minimal massive supermultiplet must contain just one bosonic and one fermionic degrees of freedom. Recently we constructed such supermultiplets using the unfolded formalism [7]. In this paper we develop the Lagrangian formulation for these supermultiplets. It generalizes the Lagrangian formulation for massive supermultiplets in three dimensional Minkowski space given in [8]. Note here that the off-shell superfield description for the supermultiplets containing topologically massive higher spin fields was constructed recently in [9] 1. We develop the component approach to Lagrangian construction for supersymmetric massive higher spin fields in AdS 3 on the base of gauge invariant formulation of massive higher spin fields. It was shown long enough [11, 1] that massive bosonic (fermionic) spin-s field can be treated as system of massless fields with spins s,s 1,s,...0(1/) coupled by the Stueckelberg symmetries. Later, a frame-like formulation of massive higher spin fields has been developed in the framework of the same approach [13]. It allows us to reformulate the massive higher spin theory in terms of gauge invariant objects (curvatures). In massless four-dimensional higher spin theory such curvatures are very convenient objects and allow, e.g, to built the cubic higher spin interactions [14 17]. However the construction of curvatures proposed in [14 17] is essentially adapted only for theories in four and higher dimensions. Nevertheless the formalism of gauge invariant objects can be successfully applied to massive higher spin fields in three dimensions as well. It was shown that in three dimensions the gauge invariant Lagrangians for massive higher spin fields [18, 19] can be rewritten in explicitly gauge invariant form [0]. In this work we elaborate this formalism for Lagrangian construction of massive higher spin supermultiplets. A general scheme for Lagrangian formulation of massive higher spin supermultiplets looks as follows. Let Ω A,Φ A is a set of fields and R A,F A is a set of curvatures corresponding to frame-like gauge invariant formulation of massive bosonic and fermionic fields respectively. The curvatures have the following structure R A = DΩ A +(eω) A, F A = DΦ A +(eφ) A 1 The off-shell N = massless higher spin superfields and their massive deformations has been elaborated in [10]. List of references on component and superfield formulations of higher spin supersymmetric theories is given in our paper [7]. 1

where e e µ αβ is a non-dynamical background AdS 3 frame. Lagrangian both for bosonic and for fermionic massive higher spins are presented as the quadratic forms in curvatures expression [0] L B = R A R A, L F = F A F A (1.1) where L B is a bosonic field Lagrangian and L F is a fermionic one. Note that for massless higher spin fields in three dimensions such form of the Lagrangians is impossible. Then, in order to realize supersymmetry between the bosonic and fermionic massive fields we deform the curvatures by gravitino field Ψ µ α with parameter of supertransformations ζ α. In the case of global supersymmetry we consider gravitino field as a non-dynamical background and parameter of supertransformations as global (i.e. covariantly constant). Such a construction can be interpreted as a supersymmetric theory in terms of background fields of supergravity. Schematically the curvature deformations are written as R A = (ΨΦ) A, F A = (ΨΩ) A and supertransformations have the form δω A (Φζ) A, δφ A (Ωζ) A The restrictions on the form of deformations and supertransformations are imposed by requirement of covariant transformations for deformed curvatures ˆR A = R A + R A δˆr A (Fζ) A, δ ˆF A (Rζ) A Finally the supersymmetric Lagrangian for given supermultiplets is the sum of Lagrangians (1.1) where initial curvatures are replaced by deformed ones R,F ˆR, ˆF. Possible arbitrariness is fixed by the condition that the Lagrangian must be invariant under the supertransformations: δ ˆL = [ˆR A δˆr A + ˆF A δ ˆF A ] = R A (Fζ) A = 0 Aim of the current paper is to demonstrate how such a general scheme can actually be realized for supermultiplets in AdS 3. The paper is organized as follows. First of all we fix the notations and conventions. In section, following the above scheme we consider as an example, the detailed Lagrangian construction of massive supermultiplet (, 3 ). Such an example illustrates all the key points of the construction. The next sections are devoted to generalizations of these results for arbitrary massive supermultiplets. In section 3 we consider the gauge invariant formulation of massive fields with spin s andspin s+ 1 which will be used for supersymmetric constructions. In particular, we write out all field variables and gauge-invariant curvatures and consider the Lagrangian in terms of these curvatures. Further, following the above general scheme, we study two massive higher spinsupermultiplets. Supermultiplets (s,s+ 1 ) isstudied insection 4 and (s,s+ 1 ) is studied in section 5. In conclusion we summaries and discuss the results obtained. For completeness, we include in the paper two appendices devoted to frame-like description of massless fields in three dimensions. Appendix A contains the bosonic fields, Appendix B contains the fermionic ones.

Notations and conventions. We use a frame-like multispinor formalism where all the objects (3,,1,0-forms) have totally symmetric local spinor indices. To simplify the expressions we will use the condensed notations for the spinor indices such that e.g. Ω α(k) = Ω (α 1α...α k ) Also we will always assume that spinor indices denoted by the same letters and placed on the same level are symmetrized, e.g. Ω α(k) ζ α = Ω (α 1...α k ζ α k+1) AdS 3 space will be described by the background frame (one-form) e α() and the covariant derivative D normalized so that D Dζ α = λ E α βζ β Basis elements of 1,,3-form spaces are respectively e α(), E α(), E where the last two are defined as double and triple wedge product of e α() : e αα e ββ = ε αβ E αβ, E αα e ββ = ε αβ ε αβ E. Also we write some useful relations for these basis elements E α γ e γβ = 3ε αβ E, e α γ e γβ = 4E αβ. Further on the sign of wedge product will be omitted. Massive supermultiplet (, 3 ) example In this section we consider in details the Lagrangian realization for massive supermultiplet (, 3/) example using the method mentioned above in the introduction. We start with the gauge-invariant formulations of free massive fields with spin and spin 3/ separately. Using such formulations, we present the full set of gauge-invariant curvatures for them. Then we consider the deformations of these curvatures by background gravitino field Ψ α and find suitable supertransformations. As a result we construct the supersymmetric Lagrangian..1 Free fields Spin In gauge invariant form the massive spin- field is described by system of massless fields with spins,1,0. In frame-like approach the corresponding set of fields consists of (Ω α(),f α() ), (B α(),a) and (π α(),ϕ) (see Appendix A for details). Lagrangian for free massive field in AdS 3 have the form [18] L = Ω αβ e β γω αγ +Ω α() Df α() +EB α() B α() e α() B α() DA Eπ α() π α() +E α() π α() Dϕ +me α() Ω α() A+mf αβ E β γb αγ +4 me α() π α() A + M 4 f αβe β γf αγ m me α() f α() ϕ+ 3 m Eϕϕ (.1) 3

It is invariant under the following gauge transformations δω α() = Dη α() + M 4 eα βξ αβ δf α() = Dξ α() +e α βη αβ me α() ξ δb α() = mη α() δa = Dξ m 4 e α()ξ α() δπ α() = m mξ α() δϕ = 4 mξ (.) where m is the mass parameter and m = M, M = m +λ The curvatures invariant under gauge transformations (.) have the form R α() = DΩ α() + M 4 eα βf αβ m E α βb αβ m me α() ϕ T α() = Df α() +e α βω αβ me α() A B α() = DB α() Ω α() + M 4 eα βπ αβ A = DA m 4 e α()f α() +me α() B α() Π α() = Dπ α() f α() +e α βb αβ + m M eα() ϕ Φ = Dϕ+4 ma+mme α() π α() (.3) Here we have changed a normalization for the two zero forms B α() mb α() π α() mmπ α() (.4) It is interesting to point out that, unlike massless theory in three dimensions, the Lagrangian for massive spin (.1) can be presented in manifestly gauge invariant form L = 1 R α()π α() 1 T α()b α() m 4 m e α()b α() Φ (.5) Spin- 3 In gauge invariant formulation, the massive spin-3/ field is described by system of massless fields with the spins 3/, 1/. In frame-like approach the corresponding set of fields consists of Φ α, φ α. (see Appendix B for details). The free Lagrangian for mass m 1 field in AdS 3 has the following form L = i Φ αdφ α + i φ αe α βdφ β i M 1 Φ αe α βφ β im 1 Φ α E α βφ β i 3M 1 Eφ αφ α (.6) 4

It is invariant under gauge transformations where δφ α = Dξ α +M 1 e α βξ β δφ α = m 1 ξ α (.7) M 1 = m 1 + λ 4 One can construct the gauge invariant curvatures with respect (.7). After change of normalization φ α m 1 φ α they look like F α = DΦ α +M 1 e α βφ β 4m 1 E α βφ β C α = Dφ α Φ α +M 1 e α βφ β (.8) As in the previous spin- case, one notes that Lagrangian (.6) can be rewritten in terms of curvatures (.8) L = i F αc α (.9) The above results onspin- andspin- 3 fields arebuilding blocks forsupersymmetrization. Full set of gauge invariant curvatures contains (.3) and (.8) respectively. Corresponding expressions for Lagrangians in terms of curvatures are (.5) and (.9).. Supersymmetric system Before we turn to Lagrangian formulation for massive supermultiplet (, 3/) let us consider this supermultiplet in massless flat limit. That is we consider the sum of Lagrangians (.3) and (.6) in the limit m,m 1,λ 0. It has the form ˆL = Ω αβ e β γω αγ +Ω α() Df α() +EB αβ B αβ B αβ e αβ DA Eπ αβ π αβ +π αβ E αβ Dϕ i Φ αdφ α + 1 φ αe α βdφ β (.10) Such a Lagrangiandescribes thesystem offree massless fields with spins,3/,1,1/,0. One can show there exist global supertransformations that leave the Lagrangian (.10) invariant. These supertransformations with the redefinition (.4) have form δf α() = iβ 1 Φ α ζ α δa = iα 0 Φ α ζ α im 1 β 0 e αβ φ α ζ β δϕ = im 1 δ0 φ γ ζ γ δφ α = β 1 Ω αβ ζ β mα 0 e β() B β() ζ α δφ α = 4mβ 0 m 1 B αβ ζ β + mm δ 0 m 1 π αβ ζ β (.11) The parameters β 1,β 0,α 0, δ 0 are arbitrary but, as it will be seen later, they are fixed in massive case. The choice of such notations for them will be clear from in next section for 5

arbitrary higher spin supermultiplets. To prove the invariance we need to use the Lagrangian equations of motion for auxiliary fields π α(),b α() corresponding to spins 0 and 1 E α βdb βγ = E γ βdb βα, E α γdπ βγ = 1 εαβ E γδ Dπ γδ (.1) Thus, we see that in the massless flat limit the Lagrangian formulation of pure massive supermultiplets (, 3 ) we must get the supertransformations (.11). This requirement will provide us the unique possibility for construction of the correct supersymmetric massive theory. Deformation of curvatures Now we are ready to realize the massive supermultiplets. We do it deforming curvatures by background gravitino field Ψ α with global transformation in AdS 3 δψ α = Dζ α + λ eα βζ β (.13) The main idea is to deform the curvatures so that they transform covariantly through themselves. It means that for all the deformed curvatures ˆR A = R A + R A the following relations should take place δˆr A = δ ζ R A +δ 0 R A = (Rζ) A (.14) where δ 0 is the transformation (.13) and δ ζ is the linear supertransformation. Let us consider the following ansatz for deformation of spin- curvatures R α() = iρ 1 Φ α Ψ α iˆρ 0 e α() φ γ Ψ γ B α() = iˆρ 1 φ α Ψ α T α() = iβ 1 Φ α Ψ α A = iα 0 Φ γ Ψ γ im 1 β 0 e αβ φ α Ψ β Π α() = iˆβ 1 φ α Ψ α Φ = im 1 δ0 φ γ Ψ γ (.15) Also one introduces the corresponding supertransformations δω α() = iρ 1 Φ α ζ α iˆρ 0 e α() φ γ ζ γ δb α() = iˆρ 1 φ α ζ α δf α() = iβ 1 Φ α ζ α δa = iα 0 Φ γ ζ γ im 1 β 0 e αβ φ α ζ β δπ α() = iˆβ 1 φ α ζ α δϕ = im 1 δ0 φ γ ζ γ (.16) The form of the supertransformations is completely determined by the form of deformations for curvatures. Besides β 1,β 0,α 0, δ 0 parameters which remain arbitrary from the massless case, we have new arbitrary parameters ρ 1, ˆρ 1, ˆρ 0, ˆβ 1. All the parameters will be fixed by requirement of covariant homogeneous curvature transformation (.14). In addition there are two mass parameters M,M 1 an we will see that one is related to another. 6

Let us check the requirement (.14) for curvatures R α(),t α(). One one hand δ ˆR α() = iρ 1 DΦ α ζ α +iˆρ 0 e α() Dφ β ζ β +i M β 1e α() Φ β ζ β +i( M β 1 1 λρ 1)e α γφ α ζ γ +i( m ˆρ 1 1 λˆρ 0 + 1 mmm 1 δ 0 )E α γφ α ζ γ +i( m ˆρ 1 1 λˆρ 0 1 mmm 1 δ 0 )E α γφ γ ζ α δˆt α() = iβ 1 DΦ α ζ α +i( mα 0 +ρ 1 )e α() Φ β ζ β +i(ρ 1 1 λβ 1)e α γφ α ζ γ and on the other hand +i( 4mm 1 β 0 4ˆρ 0 )E α γφ α ζ γ +i( 4mm 1 β 0 +4ˆρ 0 )E α γφ γ ζ α δ ˆR α() = iρ 1 F α ζ α iˆρ 0 e α() C β ζ β = iρ 1 DΦ α ζ α +iˆρ 0 e α() Dφ β ζ β +i(ˆρ 0 +M 1 ρ 1 )e α() Φ β ζ β +i(m 1 ρ 1 )e α γφ α ζ γ im 1ˆρ 0 E α γφ α ζ γ +i( M 1ˆρ 0 4m 1 ρ 1 )E α γφ γ ζ α δˆt α() = iβ 1 F α ζ α = iβ 1 DΦ α ζ α +i(m 1 β 1 )e α() Φ β ζ β +i(m 1 β 1 )e α γφ α ζ γ i4m 1 β 1 E α γφ γ ζ α (.17) Comparing the above relations, we obtain the solution M 1 = M λ, ρ 1 = M β 1, ˆρ 0 = δ 0 = 4β 0 = m 1 m β 1 For curvatures B α(),π α() we have on one hand δ ˆB α() = iˆρ 1 Dφ α ζ α iρ 1 Φ α ζ α M(M λ) (M λ) β 1, α 0 = m β 1 +i( ˆρ 0 + M ˆβ 1 )e α() φ β ζ β +i( M ˆβ 1 1 λˆρ 1)e α γφ α ζ γ δˆπ α() = iˆβ 1 Dφ α ζ α iβ 1 Φ α ζ α and on the other hand +i( mm 1 δ 0 M +ˆρ 1)e α() φ β ζ β +i(ˆρ 1 1 λˆβ 1 )e α γφ α ζ γ δ ˆB α() = iˆρ 1 C α ζ α = iˆρ 1 Dφ α ζ α iˆρ 1 Φ α ζ α +i(m 1ˆρ 1 )e α() φ β ζ β +i(m 1ˆρ 1 )e α γφ α ζ γ δˆπ α() = iˆβ 1 C α ζ α = iˆβ 1 Dφ α ζ α iˆβ 1 Φ α ζ α +i(m 1ˆβ1 )e α() φ β ζ β +i(m 1ˆβ1 )e α γφ α ζ γ (.18) Comparison of above relations yield ˆβ 1 = β 1, ˆρ 1 = ρ 1 7

The transformation laws for curvatures A and Φ look like δâ = iα 0F α ζ α +im 1 β 0 e αβ C α ζ β, δˆφ = im 1 δ0 C γ ζ γ (.19) Now we consider ansatz for deformation of spin-3/ curvatures F α = β 1 Ω αβ Ψ β mα 0 e γ() B γ() Ψ α +δ 0 f αβ Ψ β +γ 0 AΨ α γ 0 e αβ ϕψ β C α = 4mβ 0 m 1 B αβ Ψ β mm δ 0 m 1 π αβ Ψ β ρ 0 ϕψ α One introduces the corresponding supertransformations δφ α = β 1 Ω αβ ζ β mα 0 e γ() B γ() ζ α +δ 0 f αβ ζ β +γ 0 Aζ α γ 0 e αβ ϕζ β δφ α = 4mβ 0 m 1 B αβ ζ β + mm δ 0 m 1 π αβ ζ β +ρ 0 ϕζ α (.0) Let us require the conditions (.14). The we have on one hand δ ˆF α = β 1 DΩ αβ ζ β +mα 0 e β() DB β() ζ α +δ 0 Df αβ ζ β +γ 0 DAζ α + γ 0 Dϕe α βζ β +( λβ 1 M 1 β 1 )e γ() Ω αγ ζ γ +(M 1 β 1 )e γ() Ω γ() ζ α +( 1 λδ 0 M 1 δ 0 )e γ() f αγ ζ γ +(M 1 δ 0 )e γ() f γ() ζ α +(M 1 γ 0 1 λγ 0)e α γaζ γ +( 4mM 1 α 0 16mm 1 β 0 +mλα 0 )E α γb γβ ζ β +(4mM 1 α 0 mλα 0 )E γ() B αγ ζ γ mmm 1 δ0 E α γπ γβ ζ β +(4M 1 γ 0 4m 1 ρ 0 +λ γ 0 )E α γϕζ γ δĉα = 4mβ 0 m 1 DB αβ ζ β + mm δ 0 m 1 Dπ αβ ζ β +ρ 0 Dϕζ α β 1 Ω αβ ζ β δ 0 f αβ ζ β γ 0 Aζ α and on the other hand +( mβ 0 λ 4mM 1β 0 )e γ() B αγ ζ γ +(mα 0 + 4mM 1β 0 )e γ() B γ() ζ α m 1 m 1 +( mm δ 0 λ mmm 1 δ 0 )e γ() π αγ ζ γ 4m 1 m 1 m 1 +( mmm 1 δ 0 )e γ() π γ() ζ α +( γ 0 +M 1 ρ 0 1 m 1 λρ 0)e α γϕζ γ δ ˆF α = β 1 R αβ ζ β +mα 0 e β() B β() ζ α +δ 0 T αβ ζ β +γ 0 Aζ α + γ 0 Φe α βζ β = β 1 DΩ αβ ζ β +mα 0 e β() DB β() ζ α +δ 0 Df αβ ζ β +γ 0 DAζ α + γ 0 Dϕe α βζ β δ 0 e γ() Ω αγ ζ γ +( mα 0 +δ 0 )e γ() Ω γ() ζ α M β 1 e γ() f αγ ζ γ +( 1 4 mγ 0 + M β 1)e γ() f γ() ζ α +(mδ 0 4M γ 0 )e α γaζ γ +( m β 1 +mγ 0 )E α γb γβ ζ β +(m β 1 +mγ 0 )E γ() B αγ ζ γ +( mm γ 0 +4mM α 0 )E α γπ γβ ζ β +(mm γ 0 +4mM α 0 )E γ() π αγ ζ γ +mmβ 1 E α γϕζ γ (.1) 8

δĉα = 4mβ 0 m 1 B αβ ζ β + mm δ 0 m 1 Π αβ ζ β +ρ 0 Φζ α = 4mβ 0 m 1 DB αβ ζ β + mm δ 0 m 1 Dπ αβ ζ β +ρ 0 Dϕζ α 4mβ 0 m 1 Ω αβ ζ β mm δ 0 m 1 f αβ ζ β +4Mρ 0 Aζ α mm δ 0 m 1 e γ() B αγ ζ γ + mm δ 0 m 1 e γ() B γ() ζ α mm β 0 m 1 c 0 e γ() π αγ ζ γ +(mmρ 0 + mm β 0 m 1 )e γ() π γ() ζ α m M δ 0 4m 1 M eα γϕζ γ (.) Comparison of the above relations gives us at M 1 = M λ δ 0 = Mβ 1, γ 0 = γ 0 = M(M λ) (M λ) β 1, ρ 0 = m m β 1 Thus imposing the condition of the covariant curvature transformations (.14) we fix the supersymmetric deformations up to common parameter β 1 and relation between mass M 1 = M λ. The law of supersymmetry transformations is given by (.16),(.0). The obtained result is in agreement with our recent work [7] where we studied the analogous supermultiplets (, 3/) in unfolded formulation. Invariant Lagrangian Now we turn to construction of supersymmetric Lagrangian corresponding to supermultiplets (, ). Actually in terms of curvatures it presents a sum of Lagrangians for spin (.5) 3 and spin 3/ (.9), where the initial curvatures are replaced by deformed ones R ˆR ˆL = 1 ˆR α()ˆπα() 1 ˆT α() ˆBα() m 4 m e α() ˆB α()ˆφ+ i ˆF α Ĉ α Using the knowing supertransformations for curvatures (.17),(.18),(.19),(.1),(.) it is not very difficult to check the invariance of the Lagrangian. Indeed δ ˆL = i(ρ 1 mm 4m 1 δ0 )F α ζ α Π α() i(ˆβ 1 β 1 )R α() C α ζ α i(β 1 m m 1 β 0 )F α ζ α B α() i(ˆρ 1 1 δ 0)T α() C α ζ α i( mm 1 4 m δ 0 +mα 0 )e α() ˆBα() C γ ζ γ i (m mˆρ 1 + γ 0 )e α() C α ζ α Φ iγ 0 Aζα C α + iρ 0 ˆF α Φζ α + 1 iˆρ 0e α() C β Π α() ζ β Taking into account the equations of equation for fields B α(),π α() which are equivalent to the following relations we see that the variation δ ˆL vanishes. Φ = 0, A = 0 e γ() Π γ() = DΦ 4MA = 0 9

3 Free massive higher spin fields For completeness in this section we discuss the gauge invariant formulation of free massive bosonic and fermionic higher spin fields in AdS 3 [18,19]. Besides, we present the gauge invariant curvatures, the Lagrangians in explicit component form and the Lagrangians in terms of curvatures for bosonic spin s and fermionic spin s+1/ fields. 3.1 Bosonic spin-s field In gauge invariant form the massive spin s field is described by system of massless fields with spins s,(s 1),...,0. In frame-like approach the corresponding set of fields consists of (7) (Ω α(k),f α(k) ) 1 k (s 1), (B α(),a), (π α(),ϕ) Free Lagrangian for the fields with mass m in AdS 3 have the form L = s 1 ( 1) k+1 [kω α(k 1)β e β γω α(k 1)γ +Ω α(k) Df α(k) ] k=1 +EB αβ B αβ B αβ e αβ DA Eπ αβ π αβ +π αβ E αβ Dϕ s + ( 1) k+1 (k +) a k [ Ω α()β(k) e α() f β(k) +Ω α(k) e β() f α(k)β() ] k k=1 +a 0 Ω α() e α() A a 0 f αβ E β γb αγ +smπ αβ E αβ A s 1 + ( 1) k+1 b k f α(k 1)β e β γf α(k 1)γ +b 0 f α() E α() ϕ+ 3a 0 Eϕ (3.1) k=1 where a k = k(s+k +1)(s k 1) (k +1)(k +)(k+3) [M (k +1) λ ] a 0 = (s+1)(s 1) [M λ ] 3 b k = s M 4k(k +1), b 0 = sma 0, M = m +(s 1) λ 10

It is invariant under the following gauge transformations δω α(k) = Dη α(k) + (k +)a k e β() η α(k)β() k + a k 1 k(k 1) eα() η α(k ) + b k k eα βξ α(k 1)β δf α(k) = Dξ α(k) +e α βη α(k 1)β +a k e β() ξ α(k)β() + (k +1)a k 1 k(k 1)(k 1) eα() ξ α(k ) δω α() = Dη α() +3a 1 e β() η α()β() +b 1 e α γξ αγ (3.) δf α() = Dξ α() +e α γη αγ +a 1 e β() ξ α()β() +a 0 e α() ξ δb α() = a 0 η α(), δa = Dξ + a 0 4 e α()ξ α() δπ α() = Msa 0 ξ α(), δϕ = Msξ One can construct the curvatures invariant under these gauge transformation. After change of normalization the curvatures look like B α() a 0 B α(r) π α() b 0 π α() (3.3) R α(k) = DΩ α(k) + (k +)a k e β() Ω α(k)β() k + a k 1 k(k 1) eα() Ω α(k ) + b k k eα βf α(k 1)β T α(k) = Df α(k) +e α βω α(k 1)β +a k e β() f α(k)β() + (k +1)a k 1 k(k 1)(k 1) eα() f α(k ) R α() = DΩ α() +3a 1 e β() Ω α()β() +b 1 e α γf αγ a 0 E α βb αβ +b 0 E α() ϕ (3.4) T α() = Df α() +e α γω αγ +a 1 e β() f α()β() +a 0 e α() A B α() = DB α() Ω α() +b 1 e α βπ αβ +3a 1 e β() B α()β() A = DA+ a 0 4 e α()f α() a 0 E γ() B γ() Π α() = Dπ α() f α() +e α βb αβ a 0 sm eα() ϕ+a 1 e β() π α()β() Φ = Dϕ+MsA b 0 e α() π α() B α(k) = DB α(k) Ω α(k) + b k k eα βπ α(k 1)β + a k 1 k(k 1) eα() B α(k ) + (k +) a k e β() B α(k)β() k Π α(k) = Dπ α(k) f α(k) +e α βb α(k 1)β + (k +1)a k 1 k(k 1)(k 1) eα() π α(k ) +a k e β() π α(k)β() (3.5) 11

In comparison with massive spin case, the construction of curvatures for higher spins has some peculiarities. Namely, in order to achieve gauge invariance for all curvatures we should introduce the so called extra fields B α(k),π α(k), k s 1 with the following gauge transformations δb α(k) = η α(k) δπ α(k) = ξ α(k) As we already pointed out above, in three dimensions it is possible to write Lagrangian in terms of the curvatures only. In case of arbitrary integer spin field, the corresponding Lagrangian (3.1) can be rewritten in the form L = 1 s 1 ( 1) k+1 [R α(k) Π α(k) +T α(k) B α(k) ]+ a 0 sm e α()b α() Φ (3.6) k=1 3. Fermionic spin-(s+ 1 ) field In gauge invariant form the massive spin (s+1/) field is described by system of massless fields with spins (s+1/),(s 1/),...,1/. In frame-like approach the corresponding set of fields consists of (8) Φ α(k+1) 0 k (s 1), φ α Free Lagrangian for fields with mass m 1 in AdS 3 looks like s 1 1 i L = ( 1) k+1 [ 1 Φ α(k+1)dφ α(k+1) ]+ 1 φ αe α βdφ β k=0 s 1 + ( 1) k+1 c k Φ α(k 1)β() e β() Φ α(k 1) +c 0 Φ α E α βφ β k=1 s 1 + k=0 ( 1) k+1d k Φ α(k)βe β γφ α(k)γ 3d 0 Eφ αφ α (3.7) where c k = (s+k +1)(s k) (k +1)(k+1) [M 1 (k +1) λ 4 ] c 0 = s(s+1)[m 1 λ 4 ] d k = (s+1) (k +3) M 1, M 1 = m 1 +(s 1 ) λ The Lagrangian is invariant under the gauge transformations δφ α(k+1) = Dξ α(k+1) + c k d k (k +1) eα βξ α(k)β + k(k+1) eα() ξ α(k 1) +c k+1 e β() ξ α(k+1)β() (3.8) δφ α = c 0 ξ α 1

Let us construct gauge invariant curvatures with respect (3.8). After change of normalization φ α c 0 φ α the curvatures have form F α(k+1) = DΦ α(k+1) d k + (k +1) eα βφ α(k)β c k + k(k+1) eα() Φ α(k 1) +c k+1 e β() Φ α(k+1)β() F α = DΦ α +d 0 e α βφ β +c 1 e β() Φ αβ() c 0 E α βφ β C α = Dφ α Φ α +d 0 e α βφ β +c 1 e β() φ αβ() C α(k+1) = Dφ α(k+1) Φ α(k+1) + c k d k (k +1) eα βφ α(k)β + k(k+1) eα() φ α(k 1) +c k+1 e β() φ α(k+1)β() (3.9) As in the case of integer spins in order to achieve gauge invariance for all curvatures we should introduce the set of extra fields φ α(k+1), 1 k (s 1) with the following gauge transformation δφ α(k+1) = ξ α(k+1) Finally the Lagrangian (3.7) can be rewritten in terms of curvatures only as follows L = i s ( 1) k+1 F α(k+1) C α(k+1) (3.10) k=0 In section 5 we will need the description of massive spin-(s 1/) field. It can be obtained from the above description by replacement s (s 1). So we have considered the free massive fields with spins s and spins s+ 1. Also, we have formulated the gauge invariant curvatures (3.4), (3.5), (3.9) and gauge invariant Lagrangians (3.6), (3.10). In the next sections we will study a supersymmetrization of these results and construct the Lagrangian description of the supermultiplets (s,s+ 1) (s,s 1). 4 Massive supermultiplet (s,s+ 1 ) In this section we consider the massive higher spin supermultiplets when the highest spin is fermion. For these supermultiplets we construct the deformation of the curvatures, find the supertransformations and present the supersymmetric Lagrangian. Massless flat limit Before we turn to realization of given massive supermultiplets let us consider their structure at massless flat limit m,m 1,λ 0. In this case the Lagrangian will be described by the system of massless fields with spins (s+ 1),s,..., 1,0 in three dimensional flat space L = s 1 ( 1) k+1 [kω α(k 1)β e β γω α(k 1)γ +Ω α(k) Df α(k) ] k=1 +EB αβ B αβ B αβ e αβ DA Eπ αβ π αβ +π αβ E αβ Dϕ + i s 1 ( 1) k+1 Φ α(k+1) DΦ α(k+1) + 1 φ αe α βdφ β (4.1) k=0 13

One can show that this Lagrangian is supersymmetric. Indeed, if the equations of motion (.1) are fulfilled, the Lagrangian is invariant under the following supertransformations δf α(k) = iβ k Φ α(k 1) ζ α +iα k Φ α(k)β ζ β δf α() = iβ 1 Φ α ζ α +iα 1 Φ α()β ζ β δa = iα 0 Φ α ζ α +ic 0 β 0 e αβ φ α ζ β, δϕ = ic 0 δ 0 φγ ζ γ δφ α(k+1) α k = (k +1) Ωα(k) ζ α +(k +1)β k+1 Ω α(k+1)β ζ β δφ α = β 1 Ω αβ ζ β +a 0 α 0 e β() B β() ζ α δφ α = 8a 0β 0 c 0 B αβ ζ β + b 0 δ 0 c 0 π αβ ζ β Here we take into account the normalization (3.3). Thus, requiring that massive theory has a correct massless flat limit we partially fix an arbitrariness in the choice of the supertransformations. Parameters α k,β k,β 0,α 0, δ 0 at this step are still arbitrary. Deformation of curvatures Again we will realize supersymmetry deforming the curvatures by the background gravitino field Ψ α. We start with the construction of the deformations for bosonic fields R α(k) = iρ k Φ α(k 1) Ψ α +iσ k Φ α(k)β Ψ β T α(k) = iβ k Φ α(k 1) Ψ α +iα k Φ α(k)β Ψ β R α() = iρ 1 Φ α Ψ α +iσ 1 Φ α()β Ψ β +iˆρ 0 e α() φ β Ψ β T α() = iβ 1 Φ α Ψ α +iα 1 Φ α()β Ψ β A = iα 0 Φ α Ψ α +ic 0 β 0 e α() φ α Ψ β, Φ = ic 0 δ 0 φα Ψ α B α(k) = iˆρ k φ α(k 1) Ψ α iˆσ k φ α(k)β Ψ β Π α(k) = iˆβ k φ α(k 1) Ψ α iˆα k φ α(k)β Ψ β Corresponding ansatz for supertransformations has the form δω α(k) = iρ k Φ α(k 1) ζ α +iσ k Φ α(k)β ζ β δf α(k) = iβ k Φ α(k 1) ζ α +iα k Φ α(k)β ζ β δω α() = iρ 1 Φ α ζ α +iσ 1 Φ α()β ζ β +iˆρ 0 e α() φ β ζ β δf α() = iβ 1 Φ α ζ α +iα 1 Φ α()β ζ β δa = iα 0 Φ α ζ α +ic 0 β 0 e α() φ α ζ β, δϕ = ic 0 δ 0 φγ ζ γ δb α(k) = iˆρ k φ α(k 1) ζ α +iˆσ k φ α(k)β ζ β δπ α(k) = iˆβ k φ α(k 1) ζ α +iˆα k φ α(k)β ζ β All parameters will be fixed from requirement of covariant transformations of the curvatures (.14). First of all we consider δ ˆR α(k) = iρ k F α(k 1) ζ α +iσ k F α(k)β ζ β δˆt α(k) = iβ k F α(k 1) ζ α +iα k F α(k)β ζ β (4.) 14

It leads to relation M 1 = M + λ between mass parameters M 1 and M and defines the parameters α k β k σ k ρ k = k(s+k +1)[M +(k +1)λ]ˆα (k +1)(s k) = [M kλ]ˆβ k(k +1) (s+k +1) = [M +(k +1)λ]ˆσ k(k +1) (s k) = k 3 (k +1)(k+1) [M kλ]ˆρ (4.3) where From the requirement that ˆβ = ˆα, ˆρ = sm ˆα, ˆσ = sm ˆα δ ˆB α(k) = iˆρ k C α(k 1) ζ α +iˆσ k C α(k)β ζ β δˆπ α(k) = iˆβ k C α(k 1) ζ α +iˆα k C α(k)β ζ β (4.4) one gets ˆρ k = ρ k, ˆσ k = σ k, ˆβk = β k, ˆα k = α k Requirement of covariant transformations for the other curvatures δ ˆR α() = iρ 1 F α ζ α +iσ 1 F α()β ζ β iˆρ 0 e α() C β ζ β δˆt α() = iβ 1 F α ζ α +iα 1 F α()β ζ β gives the solution δâ = iα 0F α ζ α ic 0 β 0 e αβ C α ζ β, δˆφ = ic 0 δ 0 Cγ ζ γ (4.5) ˆρ 0 = 1 8 c 0 β 1, δ0 = 4β 0 = c 0 β 1, α 0 = c 0 β 1 a 0 4sMa 0 Now let us consider the deformation of curvatures for fermion. We choose ansatz in the form F α(k+1) = α k (k +1) Ωα(k) Ψ α +(k+1)β k+1 Ω α(k+1)β Ψ β +γ k f α(k) Ψ α +δ k f α(k+1)β Ψ β F α = β 1 Ω αβ Ψ β +a 0 α 0 e β() B β() Ψ α +δ 0 f αβ Ψ β +γ 0 AΨ α + γ 0 ϕe α βψ β C α = 8a 0β 0 c 0 B αβ Ψ β b 0 δ 0 c 0 π αβ Ψ β ρ 0 ϕψ α C α(k+1) = β k B α(k+1)β Ψ β α k B α(k) Ψ α δ k π α(k+1)β Ψ β γ k π α(k) Ψ α 15

and ansatz for supertransformations in the form δφ α(k+1) = α k (k +1) Ωα(k) ζ α +(k +1)β k+1 Ω α(k+1)β ζ β +γ k f α(k) ζ α +δ k f α(k+1)β ζ β δφ α = β 1 Ω αβ ζ β +a 0 α 0 e β() B β() ζ α +δ 0 f αβ ζ β +γ 0 Aζ α + γ 0 ϕe α βζ β δφ α = 8a 0β 0 c 0 B αβ ζ β + b 0 δ 0 c 0 π αβ ζ β +ρ 0 ϕζ α δφ α(k+1) = β k B α(k+1)β ζ β + α k B α(k) ζ α + δ k π α(k+1)β ζ β + γ k π α(k) ζ α From the requirement that δ ˆF α(k+1) = α k (k +1) Rα(k) ζ α +(k +1)β k+1 R α(k+1)β ζ β +γ k T α(k) ζ α +δ k T α(k+1)β ζ β (4.6) we have the same relation between masses M 1 = M + λ. Besides, it leads to γ k δ k = = (s+k +1) k(k +1) (k+1) [M +(k +1)λ]ˆγ (s k 1) [M (k +1)λ]ˆδ (k +1)(k +)(k +3) where Requirement ˆγ = sm ˆα, ˆδ = sm ˆα gives us δĉα(k+1) = β k B α(k+1)β ζ β + α k B α(k) ζ α + δ k Π α(k+1)β ζ β + γ k Π α(k) ζ α (4.7) γ k = γ k, δk = δ k, α k = At last, requirement for the other curvatures yields solution α k (k +1), βk = (k +1)β k+1 δ ˆF α = β 1 R αβ ζ β a 0 α 0 e β() B β() ζ α +δ 0 T αβ ζ β +γ 0 Aζ α + γ 0 Φe α βζ β δĉα = 8a 0β 0 c 0 B αβ ζ β + b 0 δ 0 c 0 Π αβ ζ β +ρ 0 Φζ α (4.8) Now, all the arbitrary parameters are fixed. γ 0 = γ 0 = c 0 β 1, ρ 0 = c 0 β 1 a 0 4sMa 0 16

Supersymmetric Lagrangian is the sum of free Lagrangians where the initial curvatures are replacement by deformed ones ˆL = 1 s 1 k=1 s 1 ( 1) k+1 [ ˆR α(k)ˆπα(k) + ˆT α(k) ˆBα(k) ]+ a 0 sm e α() ˆB α()ˆφ i ( 1) k+1 ˆFα(k+1) Ĉ α(k+1) (4.9) k=0 The Lagrangian is invariant under the supertransformations (4.),(4.4),(4.5),(4.6),(4.7),(4.8) up to equations of motion for the fields B α(),π α() Φ = 0, A = 0 e γ() Π γ() = DΦ sma = 0 (4.10) The Lagrangian (4.9) is a final solution for the massive supermultiplet (s,s+ 1 ). 5 Massive supermultiplet (s,s 1 ) In this section we consider another massive higher spin supermultiplet when the highest spin is boson. The massive spin-s field was described in section 3.1 in terms of massless fields. The massive spin-(s 1/) field can be obtained for the results in section 3. if one makes the replacement s (s 1). So the set of massless fields for the massive field with spin s 1/ is Φ α(k+1), 0 k s and φ α. The gauge invariant curvatures and the Lagrangian have the forms (3.8) and (3.10) where the parameters are c k = (s+k)(s k 1) (k +1)(k +1) [M 1 (k +1) λ 4 ] c 0 = s(s 1)[M 1 λ 4 ] (5.1) d k = (s 1) (k +3) M 1, M 1 = m 1 +(s 1 ) λ Following our procedure we should construct the supersymmetric deformations for curvatures. Actually the structure of deformed curvatures and supertransformations have the same form as in previous section for supermultiplets (s,s+1/). There is a difference only in parameters (5.1). Therefore we present here only the supertransformations for curvatures. Requirement (.14) for bosonic fields δ ˆR α(k) = iρ k F α(k 1) ζ α +iσ k F α(k)β ζ β δˆt α(k) = iβ k F α(k 1) ζ α +iα k F α(k)β ζ β δ ˆR α() = iρ 1 F α ζ α +iσ 1 F α()β ζ β iˆρ 0 e α() C β ζ β δˆt α() = iβ 1 F α ζ α +iα 1 F α()β ζ β δâ = iα 0F α ζ α ic 0 β 0 e αβ C α ζ β, δˆφ = ic 0 δ 0 Cγ ζ γ δ ˆB α(k) = iˆρ k C α(k 1) ζ α +iˆσ k C α(k)β ζ β δˆπ α(k) = iˆβ k C α(k 1) ζ α +iˆα k C α(k)β ζ β 17

gives us the relation M 1 = M λ between masses M 1 and M. Besides, it leads to σ k ρ k α k β k (s k 1) = [M (k +1)λ]ˆσ k(k +1) (s+k) = [M k 3 +kλ]ˆρ (k +1)(k +1) = k(s k 1)[M (k +1)λ]ˆα (k +1)(s+k) = [M +kλ]ˆβ k(k +1) where ˆβ = ˆα, ˆρ k = ρ k, ˆσ k = σ k, ˆβk = β k, ˆα k = α k ˆρ 0 = 1 8 c 0 β 1, δ0 = 4β 0 = c 0 β 1, α 0 = c 0 β 1 a 0 4sMa 0 ˆρ = sm ˆα, ˆσ = sm ˆα, ˆα = α s (s )[M (s 1)λ] From requirement of covariant supertransformations for fermionic curvatures δ ˆF α(k+1) = α k (k +1) Rα(k) ζ α +(k +1)β k+1 R α(k+1)β ζ β +γ k T α(k) ζ α +δ k T α(k+1)β ζ β δ ˆF α = β 1 R αβ ζ β a 0 α 0 e β() B β() ζ α +δ 0 T αβ ζ β +γ 0 Aζ α + γ 0 Φe α βζ β δĉα = 8a 0β 0 c 0 B αβ ζ β + b 0 δ 0 c 0 Π αβ ζ β +ρ 0 Φζ α δĉα(k+1) = β k B α(k+1)β ζ β + α k B α(k) ζ α + δ k Π α(k+1)β ζ β + γ k Π α(k) ζ α one gets γ k δ k = = (s k 1) k(k +1) (k +1) [M (k +1)λ]ˆγ (s+k +1) [M +(k +1)λ]ˆδ (k +1)(k +)(k +3) where γ k = γ k, δk = δ k, α k = α k (k +1), βk = (k +1)β k+1 γ 0 = γ 0 = c 0 β 1, ρ 0 = c 0 β 1 a 0 4sMa 0 ˆγ = sm ˆα, ˆδ = sm ˆα Supersymmetric Lagrangian have form (4.9) and it is invariant under the supertransformations up to equations of motion for auxiliary fields B α(),π α() (4.10). 18

6 Summary Let us summarize the results. In this paper we have constructed the Lagrangian formulation for massive higher spin supermultiplets in the AdS 3 space in the case of minimal (1,0) supersymmetry. For description of the massive higher spin fields we have adapted for massive fields in three dimensions the frame-like gauge invariant formalism and technique of gauge invariant curvatures. The supersymmetrization is achieved by deformation of the curvatures by background gravitino field and hence the supersymmetric Lagrangians are formulated with help of background fields of three-dimensional supergravity. In AdS 3 the space the minimal (1,0) supersymmetry combines the massive fields in supermultiplets with one bosonic degree of freedomand onefermionic one. As a result we have derived the supersymmetric andgauge invariant Lagrangians for massive higher spin (s, s + 1/) and (s, s 1/) supermultiplets. Acknowledgments I.L.B and T.V.S are grateful to the RFBR grant, project No. 15-0-03594-a for partial support. Their research was also supported in parts by Russian Ministry of Education and Science, project No. 3.1386.017. 7 Appendix A. Massless bosonic fields In this Appendix we consider the frame-like formulation for the massless bosonic fields in three dimensional flat space. For every spin we present field variables and write out the corresponding Lagrangian. All massless fields with spin s 1 are gauge ones so that we also present the gauge transformations for them. Spin 0 It is described by physical 0-form ϕ and auxiliary 0-form π α(). Lagrangian looks like L = Eπ αβ π αβ +π αβ E αβ Dϕ Spin 1 It is described by physical 1-form A and auxiliary 0-form B α(). Lagrangian looks like It is invariant under gauge transformations L = EB αβ B αβ B αβ e αβ DA δa = Dξ Spin It is described by physical 1-form f α() and auxiliary 1-form Ω α(). Lagrangian looks like L = Ω αβ e β γω αγ +Ω α(k) Df α(k) 19

The gauge transformations have the form δω α() = Dη α(), δf α() = Dξ α() +e α γη αγ Spin k It is described by physical 1-form f α(k ) and auxiliary 1-form Ω α(k ). Lagrangian looks like L = ( 1) k+1 [kω α(k 1)β e β γω α(k 1)γ +Ω α(k) Df α(k) ] The gauge transformations have the form δω α(k) = Dη α(k), δf α(k) = Dξ α(k) +e α βη α(k 1)β 8 Appendix B. Massless fermions fields In this Appendix we consider the frame-like formulation for the massless fermionic fields in three dimensional flat space. For every spin we present field variables and write out the corresponding Lagrangian. All massless fields with spin s 3/ are gauge ones so that we also present gauge transformations for them. Spin 1/ It is described by master 0-form φ α. Lagrangian looks like L = 1 φ αe α βdφ β Spin 3/ It is described by physical 1-form Φ α. Lagrangian and gauge transformations have the form L = i Φ αdφ α, δφ α = Dξ α Spin k +1/ It is described by physical 1-form Φ α(k 1). Lagrangian and gauge transformations have the form L = ( 1) k+1i Φ α(k+1)dφ α(k+1), δφ α(k+1) = Dξ α(k+1) References [1] M. P. Blencowe A Consistent Interacting Massless High er Spin Field Theory In D = (+1), Class. Quant. Grav. 6 (1989) 443. 0

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