Frame-like gauge invariant formulation for mixed symmetry fermionic fields

Transcription

1 Frme-like guge invrint formultion for mixed symmetry fermionic fields rxiv: v1 [hep-th] 3 Apr 2009 Yu. M. Zinoviev Institute for High Energy Physics Protvino, Moscow Region, , Russi Abstrct In this pper we consider frme-like formultion for mixed symmetry spin-tensors corresponding to rbitrry Young tbleu with two rows. First of ll, we extend Skvortsov formultion [24] for mssless mixed symmetry bosonic fields in flt Minkowski spce to the cse of mssless fermionic fields. Then, using such mssless fields s building blocks, we construct guge invrint formultion for mssive spin-tensors with the sme symmetry properties. We give generl mssive theories in (A)dS spces with rbitrry cosmologicl constnt nd investigte ll possible mssless nd prtilly mssless limits. E-mil ddress: Yurii.Zinoviev@ihep.ru

2 Contents 1 Introduction 1 2 Mssless cse Y (k + 3 2, 1 2 ) Y ( 5 2, 3 2 ) Y (k + 3 2, 3 2 ) Y ( 5 2, 5 2 ) Y (k + 3 2, k ) Y (k + 3 2, l ) Mssive cse Y ( 5 2, 3 2 ) Y (k + 3 2, 3 2 ) Y ( 5 2, 5 2 ) Y (k + 3 2, k ) Y (k + 3 2, l ) Conclusion 25 1 Introduction As is well known, in d = 4 dimensions for the description of rbitrry spin prticles it is enough to consider completely symmetric (spin-)tensor fields only. At the sme time, in dimensions greter thn four in mny cses like supergrvity theories, superstrings nd higher spin theories, one hs to del with mixed symmetry (spin-)tensor fields [1, 2, 3, 4]. There re different pproches to investigtion of such fields both light-cone [5, 6], s well s explicitly Lorentz covrint ones (e.g. [7, 8, 9, 10, 11, 12, 13]). For the investigtion of possible intercting theories for higher spin prticles s well s of guge symmetry lgebrs behind them it is very convenient to use so-clled frme-like formultion [14, 15, 16] (see lso [17, 18, 19]) which is nturl generliztion of well-known frme formultion of grvity in terms of veilbein e µ nd Lorentz connection ω b µ. There re two different frme-like formultions for mssless mixed symmetry bosonic fields. For simplicity, let us restrict ourselves with mixed symmetry tensors corresponding to Young tbleu with two rows. Let us denote Y (k, l) tensor Φ 1... k,b 1...b l which is symmetric both on first k s well s lst l indices, completely trceless on ll indices nd stisfies constrint Φ 1... k,b 1 )b 2...b l = 0, where round brckets men symmetriztion. In the first pproch [20, 21, 22, 23] for the description of Y (k, l) tensor (k l) one use one-form Y e (k1,l) µ s min physicl field. In this, only one of two guge symmetries is relized explicitly nd such pproch is very well dpted for the AdS spces. Another formultion Y [24] uses two-form e (k1,l1) µν s min physicl field in this, both guge symmetries re relized explicitly. Such formlism works in flt Minkowski spce while deformtions into AdS spce requires introduction of dditionl fields [25]. 1

3 In Section 2 of our pper we extend the formultion of [24] to the cse of mixed-symmetry spin-tensors Y (k+ 1, l+ 1 ) corresponding to rbitrry Young tbleu with two rows. Similrly to the bosonic cse both guge trnsformtions will be relized explicitly nd formultion will work in flt Minkowski spce only while deformtion into AdS spce turns out to be impossible (the only exception is spin-tensor Y (k + 1, k + 1 ) corresponding to rectngulr Young tbleu). Then in Section 3 we construct guge invrint frme-like formultion for mssive mixed symmetry spin-tensors corresponding to rbitrry Young tbleu with two rows (exmples for bosonic fields were considered in [26]). There re two generl pproches to guge invrint description of mssive fields. One of them uses powerful BRST pproch [27, 28, 29, 30, 11, 12, 31, 32]. Another one, which we will follow in this work, [33, 34, 35, 36, 19, 26, 37] is generliztion to higher spin fields of well-known mechnism of spontneous guge symmetry breking. In this, one strts with pproprite set of mssless fields with ll their guge symmetries nd obtin guge invrint description of mssive field s smooth deformtion. One of the nice feture of guge invrint formultion for mssive fields is tht it llows us effectively use ll known properties of mssless fields serving s building blocks. As we hve lredy seen in ll cses considered previously nd we will see gin in this pper, guge invrint description of mssive fields lwys llows smooth deformtion into (A)dS spce without introduction of ny dditionl fields besides those tht re necessry in flt Minkowski spce so tht restriction mentioned bove will not be essentil for us. As we will see in ll models constructed in Section 3, guge invrince completely fixes ll prmeters in the Lgrngin nd guge trnsformtions leving us only one free prmeter hving dimension of mss. It is hrdly possible to give meningful definition of wht is mss for mixed symmetry (spin)-tensor fields in (A)dS spces (see e.g. [38]) nd we will not insist on ny such definition. Insted, we will simply use this prmeter to nlyze ll possible specil limits tht exist in (A)dS spces. In this, only fields hving the sme number of degrees of freedom s mssless one in flt Minkowski spce we will cll mssless ones, while ll other specil limits tht pper in (A)dS spces will be clled prtilly mssless [39, 40, 41, 34, 17]. 2 Mssless cse In this Section we consider frme-like formultion for mssless mixed symmetry fermionic fields in flt Minkowski spce. We begin with some simple concrete exmples nd then consider their generliztion up to spin-tensors corresponding to rbitrry Young tbleu with two rows. In ll cses we lso consider possibility to deform such theories into AdS spce. As is well known, most of mixed symmetry (spin)-tensors do not dmit such deformtion without introduction of some dditionl fields [25], but the structure of possible mss terms nd corresponding corrections to guge trnsformtions will be hevily used in the next Section where we consider mssive theories. 2

4 2.1 Y (k + 3 2, 1 2 ) In wht follows we will need frme-like formultion for completely symmetric spin-tensors [15, 16, 26]. For completeness we reproduce here ll necessry formuls. Min object one-form Φ µ 1... k = Φµ k ) completely symmetric on locl indices nd stisfying constrint γ 1 Φ µ 1 k1 ) = (γφ) µ k1 ) = 0. To describe correct number of physicl degrees of freedom the free mssless theory hve to be invrint under the following guge trnsformtions: δ 0 Φ µ k ) = µ ζ k) + η µ k ) (1) where prmeters ζ nd η hve to stisfy: (γζ) k1) = 0, η, k) = 0, γ η, k) = (γη), k1) = 0 Here nd in wht follows round brckets denote symmetriztion. The free Lgrngin describing mssless prticle in flt Minkowski spce cn be written s follows: L 0 = i(1) k { µνα bc } [ Φ µ k ) Γ bc ν Φ α k ) 6k Φ µ k1 ) γ b ν Φ α c k1 ) ] (2) where reltive coefficients re fixed by the invrince under η shifts. Here nd further: { µνα bc } = e[µ e ν be α] c, Γ bc = 1 6 γ[ γ b γ c] nd so on. It is not hrd to construct deformtion into AdS spce. If we replce ordinry prtil derivtives in the Lgrngin nd guge trnsformtions by the AdS covrint ones, the Lgrngin cese to be guge invrint: δ 0 L 0 = i(1) k+1(d + 2k 1)(d + 2k 2) κ 2 Note tht the Lgrngin is completely ntisymmetric on world indices, so covrint derivtives effectively ct on locl indices (including implicit spinor one) only, e.g.: [D µ, D ν ]ζ k) = κ[e [µ 1 ζ ν] k1 ) Γ µνζ k) ], κ = 2Λ (d 1)(d 2) But guge invrince could be restored by dding pproprite mss-like terms to the Lgrngin s well s corresponding corrections to guge trnsformtions: provided: L 1 = (1) k b k { µν b }[ Φ µ k ) Γ b Φ ν k ) + 2k Φ µ k1 ) Φ ν b k1 ) ] (3) δ 1 Φ µ k ) = iβ k [γ µ ζ k) b k β k = 3(d 2), b k 2 = 9 4 (d + 2k 2)2 κ 2 (d + 2k 2) γ 1 ζ µ k1 ) ] (4) Note tht reltive coefficients in the mss-like terms re gin fixed by the invrince under η shifts, while the structure of vritions re chosen so tht they re γ-trnsverse. 3

5 2.2 Y ( 5 2, 3 2 ) Let us begin with the simplest exmple of mixed symmetry fermionic field. Frme-like description requires two-form Ψ µν which is γ-trnsverse γ Ψ µν = 0. Free mssless theory in flt Minkowski spce hs to be invrint under the following guge trnsformtions: δ 0 Ψ µν = [µ ξ ν] + η µν (5) where prmeter ξ µ is γ-trnsverse γ ξ µ = 0, while prmeter η bc is completely ntisymmetric nd γ-trnsverse γ η bc = 0. The Lgrngin cn be written in the following form: L 0 = i µναβγ bcde f [ Ψµν Γ bcde α Ψ f βγ 10 Ψ µν Γ bcd α Ψ e βγ ] (6) Being completely ntisymmetric on world indices, both terms re seprtely invrint under the ξ trnsformtions, while the reltive coefficients re fixed by the the invrince under the η shifts. As is well known it is impossible to deform such mssless theory into AdS spce without introduction of dditionl fields. Indeed, fter replcement of ordinry prtil derivtives by the AdS covrint ones, we could try to restore broken guge invrince by dding mss-like terms to the Lgrngin nd corresponding corrections to guge trnsformtions: L 1 = µναβ e 1 bcd [ Ψµν Γ bcd Ψ e αβ + 6 Ψ µν Γ bc Ψ d αβ ] (7) In this, vritions with one derivtive cncel provided: δψ µν = iα 1 [γ [µ ξ ν] + 2 d γ ξ [µ,ν] ] (8) α 1 = 1 5(d 4) but it is impossible to cncel vritions without derivtives by djusting the only free prmeter Y (k + 3 2, 3 2 ) It is pretty strightforwrd to generlize the exmple of previous Subsection to the cse corresponding to Young tbleu with k + 1 boxes in the first row nd only one box in the second row. Frme-like formultion requires two-form Ψ µν k ) completely symmetric on its k locl indices nd γ-trnsverse γ 1 Ψ µν 1 k1 ) = 0. Free mssless theory hs to be invrint under the following guge trnsformtions: δ 0 Ψ µν k ) = [µ ξ ν] k ) + η µν 1, k1 ) (9) where prmeter ξ µ k ) is γ-trnsverse, while prmeter η bc, k1) completely ntisymmetric on first three indices, completely symmetric on the lst k 1 ones nd stisfies: η [bc, 1] k2 ) = 0, γ η bc, k1) = (γη) bc, k2) = 0 4

6 Corresponding mssless Lgrngin hs the form: L 0 = i(1) k+1 µναβγ bcde [ Ψµν k ) Γ bcde α Ψ k ) βγ 10k Ψ k1 ) µν Γ bcd e α Ψ k1 ) βγ ] (10) Exctly s in the previous cse n ttempt to deform such theory into AdS spce without introduction of dditionl fields fils. Agin, fter replcement of ordinry derivtives by the covrint ones, we could try to restore broken guge invrince by dding mss-like terms to the Lgrngin s well s corresponding corrections to guge trnsformtions: L 1 = (1) k k { µναβ bcd } [ Ψµν k ) Γ bcd Ψ αβ k ) + 6k Ψ µν k1 ) Γ bc Ψ αβ d k1 ) ] (11) δ 1 Ψ k ) µν = iα k [γ [µ ξ k ) 2 ν] + (d + 2k 2) γ 1 ξ k1 ) [µ,ν] ] (12) In this, vritions with one derivtive cncel provided: k α k = 5(d 4) but it is impossible to chieve the cncelltion of vritions without derivtives. 2.4 Y ( 5 2, 5 2 ) Among ll mixed symmetry (spin)-tensors corresponding to Young tbleu with two rows whose with equl number of boxes in both rows turn out to be specil nd require seprte considertion. Let us begin with the simplest exmple Y ( 5, 5 ). Frme-like formultion requires two-form R b µν which is ntisymmetric on b nd γ-trnsverse γ R b µν = 0. Free mssless theory hs to be invrint under the following guge trnsformtions: δ 0 R µν b = [µ ξ ν] b + η [µ,ν] b where prmeters ξ b µ nd η bc µ re ntisymmetric on their locl indices nd γ-trnsverse. It is not hrd to construct guge invrint Lgrngin: L 0 = i µναβγ bcde [ Rµν fg Γ bcde α R fg βγ 20 R f µν Γ bcd α R ef βγ 60 R b µν γ c α R de βγ ] (14) where gin ech term is seprtely invrint under the ξ trnsformtions, while reltive coefficients re fixed by the invrince under the η shifts. One of the min specil fetures of such (spin)-tensors is the fct tht they dmit deformtion into AdS spce without introduction of ny dditionl fields. Indeed, let us replce ll derivtives in the Lgrngin nd guge trnsformtions by the AdS covrint ones. As usul, the initil Lgrngin cese to be invrint: δ 0 L 0 = 10i(d 1)(d 2)κ { µνα bc }[ R µν de Γ bc ξ α de 6 R µν d γ b ξ α cd ] In this, broken guge invrince cn be restored by dding mss-like terms to the Lgrngin nd corresponding corrections to guge trnsformtions: L 1 = µναβ 1,1 bcd [ Rµν ef Γ bcd R ef αβ + 12 R e µν Γ bc R de αβ 12 R b µν R cd αβ ] (15) provided: δ 1 R µν b = iα 1,1 [γ [µ ξ ν] b + α 1,1 = 1,1 5(d 4), 1,1 5(d 2)2 = κ 4 5 (13) 2 (d 2) γ[ ξ [µ,ν] b] ] (16)

7 2.5 Y (k + 3 2, k ) It is strightforwrd to construct generliztion of previous exmple for rbitrry k > 1. For this we need two-form R µν k ),(b k ) which is symmetric nd γ-trnsverse on both groups of locl indices nd stisfies R µν k,b 1 )(b k1 ) = 0. Moreover, R µν k ),(b k ) = R µν (b k ), k ). Free mssless theory hs to be invrint under the following guge trnsformtions: δ 0 R µν k ),(b k ) = D [µ ξ ν] k ),(b k ) + η [µ k ),(b k ) ν] (17) where prmeter ξ µ k ),(b k ) hs the sme properties on locl indices s R µν k ),(b k ), while prmeter η k),(b k ), stisfies: η µ k ),(b k,) = 0, (γη) µ k1 ),(b k ), = (γη) µ k ),(b k1 ), = γ η µ k ),(b k ), = 0 Mssless Lgrngin cn be constructed out of three terms seprtely invrint under ξ µ trnsformtions: L 0 = i µναβγ bcde [ Rµν k ),(b k ) Γ bcde α R k ),(b k ) βγ 20k R k ),(b k1 ) µν Γ bcd α R k ),e(b k1 ) βγ + 60k 2 Rµν k1 ),b(b k1 ) γ c α R βγ d k1 ),e(b k1 ) ] (18) where reltive coefficients re fixed by the invrince under η trnsformtions. As in the previous cse, such mssless theory could be deformed into AdS spce without introduction of ny dditionl fields. Guge invrince broken by the replcement of ordinry derivtives by the AdS covrint ones cn be restored if we dd to the Lgrngin mss-like terms of the form: L 1 = k,k { µναβ bcd } [ Rµν k ),(b k ) Γ bcd R αβ k ),(b k ) + 12k R µν k ),(b k1 ) Γ bc R αβ k ),d(b k1 ) + 12k 2 Rµν k1 ),b(b k1 ) R αβ c k1 ),d(b k1 ) ] (19) s well s corresponding corrections to guge trnsformtions: δ 1 R µν (k),(k) = iα k,k [γ [µ ξ ν] k ),(b k ) + provided: 2.6 Y (k + 3 2, l ) α k,k = 2 (d + 2k 4) (γ 1 ξ [µ,ν] k1 ),(b k ) + γ (b 1 ξ [µ k ),b k1 ) ν] )] (20) k,k 5(d 4), k,k 2 = 25 4 (d + 2k 4)2 κ Now we re redy to consider generl cse of Y (k + 3, l + 3 ) with k > l 1. This time we need two-form Ψ k ),(b l ) µν which is symmetric nd γ-trnsverse on both groups of locl indices nd stisfies Ψ k,b 1 )(b l1 ) µν = 0. Guge trnsformtions for free mssless theory hve the form: δψ k ),(b l ) µν = D [µ ξ k ),(b l ) ν] + η k ),(b l ) [µ ν] (21) where prmeter ξ µ k ),(b l ) hs the sme properties on locl indices s Ψ µν k ),(b l ), while prmeter η k),(b l ),c stisfies: η k,b 1 )(b l1 ),c = η µ k ),(b l,c) = 0, (γη) µ k1 ),(b l ),c = (γη) µ k ),(b l1 ),c = γ c η µ k ),(b l ),c = 0 6

8 This time we hve four terms seprtely invrint under ξ µ trnsformtions to construct mssless Lgrngin: i(1) k+l L 0 = µναβγ bcde [ Ψµν k ),(b l ) Γ bcde α Ψ k ),(b l ) βγ 10k Ψ µν k1 ),(b l ) Γ bcd α Ψ βγ e k1 ),(b l ) + 10l Ψ µν k ),(b l1 ) Γ bcd α Ψ βγ k ),e(b l1 ) 60kl Ψ µν k1 ),b(b l1 ) γ c α Ψ βγ d k1 ),e(b l1 ) ] (22) where s usul reltive coefficients re fixed by the invrince under η trnsformtions. It is not possible to deform this mssless theory into AdS spce without introduction of dditionl fields. Indeed, possible mss-like terms look s follows: (1) k+l L 1 = µναβ k,l bcd [ Ψµν k ),(b l ) Γ bcd Ψ k ),(b l ) αβ + +6l Ψ µν k ),(b l1 ) Γ bc Ψ αβ k ),d(b l1 ) + +6k Ψ µν k1 ),(b l ) Γ bc Ψ αβ d k1 ),(b l ) 12kl Ψ µν k1 ),b(b l1 ) Ψ αβ c k1 ),d(b l1 ) ] (23) In this, their non-invrince under the initil guge trnsformtions cn be compensted by corresponding corrections to guge trnsformtions: δ 1 Ψ µν k ),(b l ) = iα k,l [γ [µ ξ k ),(b l ) 2 ν] + (d + 2k 2) γ 1 ξ k1 ),(b l ) [µ,ν] + (24) 2 + (d + 2l 4) γ(b 1 ξ k ),b l1 ) 4 [µ ν] (d + 2k 2)(d + 2l 4) γ 1 ξ k1 )(b 1,b l1 ) [µ ν] ] provided: α k,l = k,l 5(d 4) but it is not possible to cncel vritions without derivtives by djusting the vlue of k,l. 3 Mssive cse In this Section we construct guge invrint frme-like formultion for mssive mixed symmetry fermionic fields. Once gin we begin with some simple concrete exmples nd then construct their generliztions. In ll cses our generl strtegy will be the sme. First of ll we determine set of mssless fields which re necessry for guge invrint description of mssive field. Then we construct the Lgrngin s sum of kinetic nd mss terms for ll fields involved s well s ll possible cross terms without derivtives nd look for the necessry corrections to guge trnsformtions. As we hve lredy mentioned in the Introduction, such guge invrint formlism works eqully well both in flt Minkowski spce s well s in (A)dS spce with rbitrry vlue of cosmologicl constnt. This, in turn, llows us to investigte ll possible mssless nd prtilly mssless limits tht exist in (A)dS spces. 7

9 3.1 Y ( 5 2, 3 2 ) Let us begin with the simplest cse Y ( 5, 3 ). To construct guge invrint description of mssive prticle we, first of ll, hve to determine set of mssless fields which re necessry for such description. In generl, for ech guge invrince of min guge field we hve to introduce corresponding primry Goldstone field. Usully, these fields turn out to be guge fields themselves with their own guge invrinces, so we hve to introduce secondry Goldstone fields nd so on. But in the mixed symmetry (spin)-tensor cse we hve to tke into ccount reducibility of their guge trnsformtions. Let us illustrte on this simplest cse. Our min guge field Y ( 5, 3) hs two guge trnsformtions (combined into one ξ µ trnsformtion in the frme-like pproch) with the prmeters Y ( 5, 1) nd Y (3, 3) nd reducibility corresponding to Y ( 3, 1 ). Thus we hve to introduce two primry Goldstone fields corresponding to Y ( 5, 1) nd Y (3, 3 ). Both hve its own guge trnsformtions with prmeters Y ( 3, 1 ) but due to reducibility of guge trnsformtions for the min guge field, it is enough to introduce one secondry Goldstone field Y ( 3, 1 ) only. This field lso hs its own guge trnsformtion with prmeter Y ( 1, 1 ) but due to reducibility of guge trnsformtions for the field Y ( 3, 3 ) the procedure stops here. Thus we need four fields: Y ( 5, 3), Y (5, 1), Y (3, 3) nd Y (3, 1 ). It is nturl to use frme-like formlism for ll fields in question so we will use Ψ µν, Φ µ, Ψ µν nd Φ µ respectively. In generl, guge invrint Lgrngin for mssive fermionic field contins kinetic nd mss terms for ll the components s well s number of cross terms without derivtives. Moreover, it is necessry to introduce such cross terms for the nerest neighbours only, i.e. min guge field with the primry ones, primry with secondry nd so on. Thus we will look for guge invrint Lgrngin in the form: L = i µναβγ f bcde [ Ψµν Γ bcde D α Ψ f βγ 10 Ψ µν Γ bcd D α Ψ e βγ Ψ µν Γ bcde D α Ψ βγ ] + +i { µνα bc } [ Φ d µ Γ bc D ν Φ d α 6 Φ µ γ b D ν Φ c α Φ µ Γ bc D ν Φ α ] + + µναβ bcd [0 Ψµν e Γ bcd Ψ e αβ Ψµν Γ bc Ψ d αβ + 2 Ψµν Γ bcd Ψ αβ ] + + { µν b }[ 1( Φ c µ Γ b Φ c ν + 2 Φ µ Φ b ν ) + 3 Φµ Γ b Φ ν ] + +ib 1 µνα bc } [ Ψ d µν Γ bc Φ d α 6 Ψ µν γ b Φ c α + Φ d µ Γ bc Ψ d να 6 Φ µ γ b Ψ c να ] + +ib µναβ 2 bcd [ Ψµν Γ bcd Ψ αβ Ψ µν Γ bc Ψ d αβ ] + +ib 3 { µν b }[ Φ µ γ b Φ ν Φ µ γ Φ ν b ] + ib 4 { µνα bc }[ Ψ µν Γ bc Φ α + Φ µ Γ bc Ψ να ] (25) where ll derivtives re AdS covrint ones. In order to compenste the non-invrince of these mss nd cross terms under the initil guge trnsformtions we hve to introduce corresponding corrections to guge trnsformtions. And indeed ll vritions with one derivtive cncel with the following form of guge trnsformtions: δψ µν δφ µ = D [µ ξ ν] i 0 5(d 4) [γ [µξ ν] + 2 b 2 d γ ξ [µ,ν] ] 10(d 2) [e [µ ξ ν] 1 d γ γ [µ ξ ν] ] b (d 3)(d 4) [Γ µνζ (d2 7d + 16) e (d + 1) [µ ζ ν] + 4(d 2) 4(d 2) γ γ [µ ζ ν] ] = D µ ζ 2b 1 ξ µ + i 1 3(d 2) [γ µζ 2 b 3 d γ ζ µ ] + 6(d 1) [e µ ζ 1 d γ γ µ ζ] 8

10 δψ µν = D [µ ξ ν] + b 2 20 ξ [µ,ν] + i 2 5(d 4) γ b 4 [µξ ν] 20(d 3)(d 4) Γ µνζ (26) δφ µ = D µ ζ + b 3 6 ζ µ + 2b 4 ξ µ i 3 3(d 2) γ µζ where ll coefficients re expressed in terms of Lgrngin prmeters 1,2,3,4 nd b 1,2,3,4. Now we clculte ll vritions without derivtives (including contribution of kinetic terms due to non-commuttivity of covrint derivtives) nd require their cncelltion. This gives us: 3(d 2) 1 = 5(d 4) 0, 2 = 0 2 = (d + 2) 0, 3 = 3(d2 4) d 5d(d 4) 0 5(d 4) 3(d 3) b (d 4)2 κ, 20(d + 1)(d 4)b 1 2 3d(d 3)b = 600(d + 1)(d 2)(d 3)κ b 3 2 = 9(d 1) 50(d 2) b, b (d 1) 4 = (d 2) b 1 2 Let us nlyze the results obtined. First of ll recll tht there is no strict definition of wht is mss in (A)dS spces. Working with guge invrint description of mssive prticles it is nturl to define mssless limit s the one where ll Goldstone fields decouple from the min guge one. For the cse t hnds, such limit requires tht both b 1 0 nd b 2 0 simultneously. As the third reltion bove clerly shows such limit is possible in flt Minkowski spce κ = 0 only. For the non-zero vlues of cosmologicl constnt we obtin one of the so clled prtilly mssless limits (depending on the sign of κ). To clrify subsequent discussion, let us give here Figure 1 illustrting the roles of cross terms b 1,2,3,4. In AdS b Ψ 2 µν Ψ µν b 1 b 4 b Φ 3 µ Φ µ Figure 1: Generl mssive theory for Y ( 5 2, 3 2 ) spin-tensor spce (κ < 0) one cn put b 2 = 0 nd thus b 3 = 0). In this, the whole system decomposes into two disconnected subsystems s Figure 2 shows. One of them, with the fields Ψ µν, Φ µ nd with the Lgrngin: L = i { µναβγ bcde } [ Ψµν f Γ bcde D α Ψ βγ f 10 Ψ µν Γ bcd D α Ψ βγ e ] + +i { µνα bc } [ Φ d µ Γ bc D ν Φ d α 6 Φ µ γ b D ν Φ c α ] + + µναβ e 0 bcd [ Ψµν Γ bcd Ψ e αβ + 6 Ψ µν Γ bc Ψ d αβ ] + 1 { µν b } [ Φ c µ Γ b Φ c ν + 2 Φ µ Φ b ν ] + +ib 1 { µνα bc } [ Ψ µν d Γ bc Φ α d 6 Ψ µν γ b Φ α c + Φ µ d Γ bc Ψ να d 6 Φ µ γ b Ψ να c ] (27) 9

11 Ψ µν Ψ µν b 1 b 4 Φ µ Φ µ Figure 2: Prtilly mssless limit for Y ( 5, 3 ) spin-tensor in AdS which is invrint under the following guge trnsformtions: δψ µν δφ µ = D [µ ξ ν] i 0 5(d 4) [γ [µξ ν] + 2 d γ ξ [µ,ν] ] b (d 3)(d 4) [Γ µνζ (d2 7d + 16) e (d + 1) [µ ζ ν] + 4(d 2) 4(d 2) γ γ [µ ζ ν] ] (28) = D µ ζ 2b 1 ξ µ + i 1 3(d 2) [γ µζ 2 d γ ζ µ ] describes unitry prtilly mssless theory corresponding to irreducible representtion of AdS group. At the sme time, two other fields Ψ µν nd Φ µ with the Lgrngin: L = i µναβγ bcde Ψµν Γ bcde D α Ψ βγ i { µνα + 2 { µναβ bcd bc } Φ µ Γ bc D ν Φ α + } Ψµν Γ bcd Ψ αβ + 3 { µν b } Φ µ Γ b Φ ν + +ib 4 { µνα bc }[ Ψ µν Γ bc Φ α + Φ µ Γ bc Ψ να ] (29) invrint under the following guge trnsformtions: δψ µν = D [µ ξ ν] + i 2 5(d 4) γ b 4 [µξ ν] 20(d 3)(d 4) Γ µνζ δφ µ = D µ ζ + 2b 4 ξ µ i 3 3(d 2) γ µζ (30) give guge invrint description of mssive ntisymmetric second rnk spin-tensor [42]. Let us turn to the ds spce (κ > 0). First of ll, from the eqution for the 2 0 bove we see tht there is unitry forbidden region b 2 1 < 15 (d 3)(d 4)κ. Inside this region 4 lives one more exmple of prtilly mssless theory corresponding to the limit b 1 0 nd hence b 4 0) s Figure 3 shows. In this, the min field Ψ µν together with Ψ µν describe this non-unitry prtilly mssless theory with the Lgrngin: L = i µναβγ bcde f [ Ψµν Γ bcde D α Ψ f βγ 10 Ψ µν Γ bcd D α Ψ e βγ Ψ µν Γ bcde D α Ψ βγ ] + + µναβ bcd [0 Ψµν e Γ bcd Ψ e αβ Ψµν Γ bc Ψ d αβ + 2 Ψµν Γ bcd Ψ αβ ] + +ib 2 { µναβ bcd } [ Ψµν Γ bcd Ψ αβ Ψ µν Γ bc Ψ αβ d ] (31) 10

12 b Ψ 2 µν Ψ µν b Φ 3 µ Φ µ Figure 3: Prtilly mssless limit for Y ( 5, 3 ) spin-tensor in ds spce which is invrint under the following guge trnsformtions: δψ µν = D [µ ξ ν] i 0 5(d 4) [γ [µξ ν] + 2 b 2 d γ ξ [µ,ν] ] 10(d 2) [e [µ ξ ν] 1 d γ γ [µ ξ ν] ] δψ µν = D [µ ξ ν] + b 2 20 ξ [µ,ν] + i 2 5(d 4) γ [µξ ν] (32) At the sme time, two other fields Φ µ, Φ µ provides guge invrint description for prtilly mssless spin 5/2 prticle [26] with the Lgrngin: L = i { µνα bc }[ Φ µ d Γ bc D ν Φ α d 6 Φ µ γ b D ν Φ α c Φ µ Γ bc D ν Φ α ] + + { µν b } [ 1( Φ µ c Γ b Φ ν c + 2 Φ µ Φ ν b ) + 3 Φµ Γ b Φ ν ] + +ib 3 { µν b }[ Φ µ γ b Φ ν Φ µ γ Φ ν b ] (33) which is invrint under the following guge trnsformtions: δφ µ 3.2 Y (k + 3 2, 3 2 ) = D µ ζ + i 1 3(d 2) [γ µζ 2 b 3 d γ ζ µ ] + 6(d 1) [e µ ζ 1 d γ γ µ ζ] δφ µ = D µ ζ + b 3 6 ζ µ i 3 3(d 2) γ µζ (34) We proceed with the construction of mssive theory for spin-tensor Ψ k ) µν with rbitrry k 1. Agin our first tsk to determine set of fields necessry for guge invrint description of such mssive field. Min guge field Y (k + 3, 1 ) hs two guge trnsformtions with prmeters Y (k + 3, 1 ) nd Y (k + 1, 3 ) so we need two corresponding primry fields. The first of them hs one guge trnsformtion with prmeter Y (k + 1, 1 ), while the second one hs two guge trnsformtions with prmeters Y (k 1, 3) nd Y (k + 1, 1 ). Tking into ccount reducibility of guge trnsformtions of the min guge field we hve to introduce two secondry fields Y (k 1, 3) nd Y (k + 1, 1 ) only. It is not hrd to check tht the procedure gin stops t the Y ( 3, 1) nd we need totlly Y (l + 3, 3) nd Y (l + 3, 1) with 0 l k. Thus we introduce the following fields: Ψ l ) µν nd Φ l ) µ, 0 l k. As we hve lredy noted guge invrint Lgrngin for mssive fermionic field contins kinetic nd mss terms for ll components s well s cross terms without derivtives for ll 11

13 nerest neighbours. Thus we will look for mssive Lgrngin in the form: where (1) l L(Ψ µν l ) ) = i { µναβγ bcde k k1 L = [L(Ψ l ) µν ) + L(Φ l ) µ )] + L cross (l) l=0 l=0 + l { µναβ bcd } [ Ψµν l ) Γ bcde D α Ψ βγ l ) 10l Ψ µν l1 ) Γ bcd D α Ψ βγ e l1 ) ] + } [ Ψµν l ) Γ bcd Ψ αβ l ) + 6l Ψ µν l1 ) Γ bc Ψ αβ d l1 ) ] (35) (1) l L(Φ µ l ) ) = i { µνα bc }[ Φ µ l ) Γ bc D ν Φ α l ) 6l Φ µ l1 ) γ b D ν Φ α c l1 ) ] + +b l { µν b } [ Φ µ l ) Γ b Φ ν l ) + 2l Φ µ l1 ) Φ ν b l1 ) ] (36) i(1) l L cross (l) = c µναβ l bcd [ Ψµν l ) Γ bcd Ψ l ) αβ Ψ l ) µν Γ bc d Ψ l ) αβ ] + +d l { µνα bc }[ Ψ µν l ) Γ bc Φ α l ) 6l Ψ µν l1 ) γ b Φ α c l1 ) ] + +d l { µνα bc }[ Φ µ l ) Γ bc Ψ να l ) 6lΦ µ l1 ) γ b Ψ να c l1 ) ] + +e l { µν b }[ Φ µ l ) γ b Φ ν l ) Φ µ l ) γ Φ ν b l ) ] (37) As usul, to compenste non-invrince of ll mss terms (both digonl s well s cross terms) under the initil guge trnsformtions, we hve to introduce corresponding corrections to guge trnsformtions. We hve lredy introduced such corrections for digonl mss terms with coefficients l nd b l in Subsection 2.3 nd Subsection 2.1 respectively. Let us consider three possible type of cross terms in turn. Ψ l+1 ) µν Ψ l ) µν. In this cse cross terms look s: L = i(1) l c l { µναβ bcd } [ Ψµν l ) Γ bcd Ψ αβ l ) Ψ µν l ) Γ bc Ψ αβ d l ) ] nd to compenste for their non-invrince we hve to introduce: δ Ψ µν l+1 ) δ Ψ µν l ) c l = 10(l + 1)(d + l 2) [e [µ 1 ξ l ) 1 ν] (d + 2l) γ 1 γ [µ ξ l ) ν] + = c l 10(l + 2) ξ [µ,ν] l) + 2 (d + 2l) g 1 2 ξ [µ,ν] l1 ) ] Ψ µν l ) Φ µ l ). Here the cross terms hve the following form: L = i(1) l d l { µνα bc } [ Ψ µν l ) Γ bc Φ α l ) 6l Ψ µν l1 ) γ b Φ α c l1 ) ] + h.c. nd to compenste for their non-invrince we hve to introduce the following corrections: δ Ψ µν l ) δ Φ µ l ) (l + 1)d l = 10(l + 2)(d 3)(d 4) [Γ µνζ l) (d 3)(d 4) + 2l + 2 e 1 [µ ζ l1 ) µ] + 2(l + 1)(d + l 3) (d + 2l 1) + 2(l + 1)(d + l 3) γ 1 γ [µ ζ l1 ) ν] ] = 2d l ξ µ l ) 12 (38) (39)

14 Φ µ l+1 ) Φ µ l ). The lst possible type of cross terms hve the form: L = i(1) l e l { µν b } [ Φ µ l ) γ b Φ ν l ) Φ µ l ) γ Φ ν b l ) ] while corrections to guge trnsformtions cn be written s follows: δ Φ µ l+1 ) δ Φ µ l ) = = e l 6(l + 1)(d + l 1) [e µ 1 ζ l) 1 (d + 2l) γ 1 γ µ ζ l) 2 (d + 2l) g 1 2 ζ l1 ) µ ] e l 6(l + 1) ζ l) (40) Collecting ll pieces together we obtin the following complete set of guge trnsformtions (for simplicity we omit here complicted terms which re necessry to ensure tht ll vritions re γ-trnsverse): δψ µν l ) δφ µ l ) = D [µ ξ l ) ν] + i l 5(d 4) [γ [µξ l ) c l ν] +...] + 10(l + 2) ξ [µ,ν] l) c l1 10l(d + l 3) [e [µ 1 ξ l1 ) (l + 1)d l ν] +...] 10(l + 2)(d 3)(d 5) [Γ µνζ l) +...] = D µ ζ l) ib l 3(d 2) [γ µζ l) +...] + 2d l ξ l ) e l µ + 6(l + 1) ζ l) + (41) e l1 + 6l(d + l 2) [e µ 1 ζ l1) +...] At this stge we hve complete Lgrngin nd guge trnsformtions, in this ll prmeters in guge trnsformtions re expressed in terms of the Lgrngin ones, b, c, d nd e so tht ll vritions with one derivtive cncel. Our next tsk clculte ll vritions without derivtives (including contribution of kinetic terms due to non-commuttivity of covrint derivtives) nd require their cncelltion. We will not give here these lengthy but strightforwrd clcultions presenting finl results only. First of ll we obtin number of recurrent reltions on digonl mss prmeters l nd b l which llows us to express ll of them in terms of the min one k = M: l = (d + 2k) (d + 2l) M, b 3(d 2) l = 5(d 4) l Then we obtin recurrent reltions on the prmeters d l which llows us to express ll of them in terms of min one d k = m (it is not mss, just nottion): d l 2 = (k + 1)(d + k 2) (l + 1)(d + l 2) m2 Further we get the following expressions for the prmeters c l nd e l : c l 2 = 10(k l)(l + 1)(d + k + l) (d + 2l) (k + 1)(d 4) [ (k + 2)(d 3) m2 + 10(l + 2)(d + l 2)κ] e l 2 = 9(l + 1)(d + l 1) 25(l + 2)(d + l 2) c l 2 13

15 At lst we obtin n importnt reltion on prmeters M nd m: M 2 = 5(k + 1)(d 4) 2(k + 2)(d 3) m (d 4)2 κ Now we re redy to nlyze the results obtined. To clrify the roles plyed by prmeters c, d nd e we give here Figure 4. First of ll note tht to obtin mssless limit we hve Ψ k µν c k1 Ψ k1 µν c k2 c 0 Ψ µν Ψ µν d k d k1 d 1 d 0 Φ k µ e k1 Φ k1 µ e k2 e 0 Φ µ Φ µ Figure 4: Generl mssive Y (k + 3 2, 3 2 ) theory to put m 0 nd c k1 0 simultneously. But s the expression on c k1 clerly shows such limit is possible in flt Minkowski spce (κ = 0) only. For non-zero vlues of κ we cn obtin number of prtilly mssless limits. Let us consider AdS spce (κ < 0) first. The most physiclly interesting limit ppers then c k1 0 nd hence e k1 0). In this the whole system decomposes into two disconnected subsystems s shown on the Figure 5. In Ψ k µν Ψ k1 µν c k2 c 0 Ψ µν Ψ µν d k d k1 d 1 d 0 Φ k µ Φ k1 µ e k2 e 0 Φ µ Φ µ Figure 5: Unitry prtilly mssless limit in AdS spce this, two fields Ψ µν k ) nd Φ µ k ) describe prtilly mssless theory corresponding to unitry irreducible representtion of AdS group [25]. The Lgrngin for such theory hs the form: (1) k L = i µναβγ bcde [ Ψµν k ) Γ bcde D α Ψ k ) βγ 10k Ψ k1 ) µν Γ bcd e D α Ψ k1 ) βγ ] + i { µνα bc }[ Φ k ) µ Γ bc D ν Φ k ) α 6k Φ k1 ) µ γ b c D ν Φ k1 ) α ] + + µναβ k bcd [ Ψµν k ) Γ bcd Ψ k ) αβ + 6k Ψ k1 ) µν Γ bc d Ψ k1 ) αβ ] + +b k { µν b }[ Φ µ k ) Γ b Φ ν k ) + 2k Φ µ k1 ) Φ ν b k1 ) ] + +d k { µνα bc } [ Ψ µν k ) Γ bc Φ α k ) 6k Ψ µν k1 ) γ b Φ α c k1 ) ] + +d k { µνα bc } [ Φ µ k Γ bc Ψ να k ) 6kΦ µ k1 ) γ b Ψ να c k1 ) ] (42) 14

16 while guge trnsformtions leving it invrint look s follows: δψ µν k ) δφ µ k ) = D [µ ξ k ) ν] + i k 5(d 4) [γ [µξ k ) ν] +...] (l + 1)d k 10(k + 2)(d 3)(d 5) [Γ µνζ k) +...] (43) = D µ ζ k) ib k 3(d 2) [γ µζ k) +...] + 2d k ξ µ k ) At the sme time ll other fields just give mssive theory for the Ψ k1 ) µν spin-tensor. Besides number of non-unitry prtilly mssless limits exists. Indeed, ech time when one of the c l 0 nd hence e l 0) the whole system lso decomposes into two disconnected subsystems. One of them with the fields Ψ m) µν nd Φ m) µ with l m k describes non-unitry prtilly mssless theory, while remining fields just give mssive theory for the Ψ l1 ) µν spin-tensor. Let us turn to the ds spce (κ > 0). From the lst reltion on prmeters M nd m we κ. Inside this region lives the only prtilly mssless limit possible. It ppers then we put m 0 nd this puts ll d l 0 simultneously). Once gin the whole system decomposes into two disconnected prts s shown on the Figure 6. One of them with the fields Ψ l ) µν 0 l k provides one see tht there is unitry forbidden region m 2 < 5(k+2)(d3)(d4) 2(k+1) Ψ k µν c k1 Ψ k1 µν c k2 c 0 Ψ µν Ψ µν Φ k µ e k1 Φ k1 µ e k2 e 0 Φ µ Φ µ Figure 6: Non-unitry prtilly mssless limit in ds spce more exmple of prtilly mssless theory in ds spce with the Lgrngin L = k (1) [ l i µναβγ bcde [ Ψµν l ) Γ bcde D α Ψ l ) βγ 10l Ψ l1 ) µν Γ bcd e D α Ψ l1 ) βγ ]+ l=0 + µναβ l bcd [ Ψµν l ) Γ bcd Ψ l ) αβ + 6l Ψ l1 ) µν Γ bc d Ψ l1 ) αβ ] ] + k1 +i l=0 (1) l c l { µναβ bcd } [ Ψµν l ) Γ bcd Ψ αβ l ) Ψ µν l ) Γ bc Ψ αβ d l ) ] (44) which is invrint under the following guge trnsformtions: δψ l ) µν = D [µ ξ l ) ν] + i l 5(d 4) [γ [µξ l ) c l ν] +...] + 10(l + 2) ξ [µ,ν] l) c l1 10l(d + l 3) [e [µ 1 ξ l1 ) ν] +...] (45) In this, remining fields Φ µ l ) 0 l k relize prtilly mssless theory constructed erlier [19]. 15

17 3.3 Y ( 5 2, 5 2 ) As we hve lredy noted, (spin)-tensors corresponding to Young tbleu with equl number of boxes in both rows re specil nd require seprte considertion. Here we consider simplest exmple mssive theory for R b µν spin-tensor. First of ll we hve to find set of fields necessry for guge invrint description of such mssive spin-tensor. Our min guge field Y ( 5, 5) hs one own guge trnsformtion with the prmeter Y (5, 3 ) only nd this is min feture mking this (spin)-tensors specil). Thus we need one primry Goldstone field Y ( 5, 3) only. This field hs two guge trnsformtions with prmeters Y (5, 1) nd Y (3, 3) but due to reducibility of our min field guge trnsformtions we need one secondry field Y ( 5, 1) only. This field lso hs one own guge trnsformtion with the prmeter Y (3, 1) but due to reducibility of primry field guge trnsformtions the procedure stops here. Thus we need three fields R b µν, Ψ µν nd Φ µ only. As in ll previous cses, we will construct mssive guge invrint Lgrngin s the sum of kinetic nd mss terms for ll three fields s well s cross terms without derivtives: L = i { µναβγ bcde +i { µναβγ bcde } [ Rµν fg Γ bcde D α R βγ fg 20 R µν f Γ bcd D α R βγ ef 60 R µν b γ c D α R βγ de ] + } [ Ψµν f Γ bcde D α Ψ βγ f 10 Ψ µν Γ bcd D α Ψ βγ e ] + +i { µνα bc } [ Φ d µ Γ bc D ν Φ d α 6 Φ µ γ b D ν Φ c α ] + + µναβ 0 bcd [ Rµν ef Γ bcd R ef αβ + 12 R e µν Γ bc R de αβ 12 R b µν R cd αβ ] + + µναβ 1 bcd e [ Ψµν Γ bcd Ψ e αβ + 6 Ψ µν Γ bc Ψ d αβ ] + 2 { µν b }[ Φ c µ Γ b Φ c ν + 2 Φ µ Φ b ν ] + +ib µναβ 1 bcd [ Rµν e Γ bcd Ψ e αβ 6 R b µν γ c Ψ d αβ Ψ e µν Γ bc R de αβ 6 Ψ µν γ b R cd αβ ] + +ib 2 { µνα bc } [ Ψ µν d Γ bc Φ α d 6 Ψ µν γ b Φ α c + Φ µ d Γ bc Ψ να d 6 Φ µ γ b Ψ να c ] (46) Now following our usul strtegy we clculte vritions with one derivtive to find pproprite corrections to guge trnsformtions. Relly most of them we re lredy fmilir with, the only new ones re relted with the cross terms R µν b Ψ µν. Clculting these new corrections nd collecting previously known results we obtin: δr µν b δψ µν δφ µ = D [µ ξ b ν] + i 0 5(d 4) [γ [µξ b 2 ν] + (d 2) γ[ ξ b] [µ,ν] ] + b (d 3) [e [µ [ ξ b] ν] 1 (d 2) γ[ γ [µ ξ ν] b] 2 (d 1)(d 2) Γb ξ [µ,ν] ] = D [µ ξ ν] b 1 10 ξ [µ,ν] i 1 5(d 4) [γ [µξ ν] + 2 d γ ξ [µ,ν] ] + (47) b (d 3)(d 4) [Γ µνζ (d2 7d + 16) e (d + 1) [µ ζ ν] + 4(d 2) 4(d 2) γ γ [µ ζ ν] ] = D µ ζ 2b 2 ξ 2 µ + 3(d 2) [γ µζ 2 d γ ζ µ ] Now we proceed with the vritions without derivtives (including contribution of kinetic terms due to non-commuttivity of covrint derivtives). First of ll, their cncelltion 16

18 leds to the reltions on the digonl mss terms: 1 = d (d 2) 3(d 2) 0, 2 = 5(d 4) 1 = Also we obtin two importnt reltions: b = 0 2 = (d 2) 8(d 1) b (d 2)2 κ 4 3d 5(d 4) 0 3(d 2) 20(d 1)(d 4) [(d 2)b (d 1)(d 3)κ] Simple liner structure of this theory R b µν Ψ µν Φ µ mkes n nlysis lso simple. First of ll we see tht mssless limit (i.e. decoupling of Ψ µν from R b µν ) corresponds to b 1 0. Such limit is possible in the AdS spce κ < 0 nd in the flt Minkowski spce, of course) in complete greement with the fct tht mssless theory for R b µν dmits deformtion into AdS spce without introduction of ny dditionl fields. In this, two other fields Ψ µν nd Φ µ describe prtilly mssless theory we lredy fmilir with. In the ds spce we once gin fce n unitry forbidden region b 2 1 < 25 (d 1)(d 2)κ. Inside this 2 region we find one more exmple of non-unitry prtilly mssless theory. It ppers then b 2 0, in this the field Φ µ decouples, while two other fields R b µν nd Ψ µν describe prtilly mssless theory. The Lgrngin nd guge trnsformtions for this theory cn be esily obtined from the generl formuls simply omitting the field Φ µ nd ll terms in the guge trnsformtions contining ζ. 3.4 Y (k + 3 2, k ) Let us consider now generl cse spin-tensor Y (k + 3, k + 3 ) with rbitrry k 1. Agin it is crucil tht the min field R k ),(b k ) µν hs one guge trnsformtion with prmeter Y (k + 3, k + 1 ) only so we need one primry field. This field hs two guge trnsformtions with prmeters Y (k + 3, k 1) nd Y (k + 1, k + 1 ) but due to reducibility of guge trnsformtions of min field we need one secondry field Y (k + 3, k + 1 ) only. It is not hrd to check tht complete set of fields necessry for guge invrint description contins Y (k + 3, l + 3) 0 l k nd Y (k + 3, 1). Following our generl procedure we will look for mssive guge invrint Lgrngin s the sum of kinetic nd mss terms for ll fields s well s cross terms for nerest neighbours: L = L(R µν k ),(b k ) ) + k1 l=0 L(Ψ µν k ),(b l ) ) + L(Φ µ k ) ) + L cross (48) where Lgrngin L(R µν )k),(b k ) ) is given by formuls (18) nd (19) of Subsection 2.5, while Lgrngin L(Ψ µν k ),(b l ) ) is given by formuls (22) nd (23) of Subsection 2.6. Here L cross = id µναβ k,k bcd [ Rµν k ),(b k1 ) Γ bcd Ψ k ),(b k1 ) αβ + 6k R k1 ),b(b k1 ) µν γ c d Ψ k1 ),(b k1 ) αβ Ψ µν k ), k1 ) Γ bc R αβ k ),d(b k1 ) + 6k Ψ µν k1 ),(b k1 ) γ b R αβ c k1 ),d(b k1 ) ] + 17

19 k1 + l=1 i(1) k+l d µναβ k,l bcd [ Ψµν k ),(b l1 ) Γ bcd Ψ k ),(b l1 ) αβ + +6k Ψ µν k1 ),b(b l1 ) γ c Ψ αβ d k1 ),(b l1 ) Ψ µν k ),(b l1 ) Γ bc Ψ αβ k ),d(b l1 ) + +6k Ψ µν k1 ),(b l1 ) γ b Ψ αβ c k1 ),d(b l1 ) ] + +i(1) k d k,0 { µνα bc }[ Ψ µν (k) Γ bc Φ α (k) 6k Ψ µν (k1) γ b Φ α c(k1) + + Φ µ k ) Γ bc Ψ να k ) 6k Φ µ k1 ) γ b Ψ να c k1 ) ] (49) As usul, to compenste for non-invrince of cross terms under the initil guge trnsformtions, we hve to introduce corresponding corrections to guge trnsformtions. Let us consider different cross terms in turn. R µν k ),(b k ) Ψ µν k ),(b k1 ). In this cse cross terms look like: L = id µναβ k,k bcd [ Rµν k ),(b k1 ) Γ bcd Ψ k ),(b k1 ) αβ + 6k R k1 ),b(b k1 ) µν γ c d Ψ k1 ),(b k1 ) αβ Ψ µν k ), k1 ) Γ bc R αβ k ),d(b k1 ) + 6k Ψ µν k1 ),(b k1 ) γ b R αβ c k1 ),d(b k1 ) ] nd to compenste for their non-invrince we hve to introduce: δ R µν k ),(b k ) δ Ψ µν k ),(b k1 ) d k,k = 20k(d + k 4) [ξ [µ k),(b k1 b e 1 ) ν] e 1 [ν ξ k1 )(b 1,b k1 ) µ] +...] = d k,k 10k ξ [µ k),(b k1 ) ν] (50) where gin dots stnd for the dditionl terms which re necessry for vritions to be γ-trnsverse. Ψ µν k ),(b l ) Ψ µν k ),(b l1 ). Corresponding cross terms hve the following form: (1) k+l L = id µναβ k,l bcd [ Ψµν k ),(b l1 ) Γ bcd Ψ k ),(b l1 ) αβ + +6k Ψ µν k1 ),b(b l1 ) γ c Ψ αβ d k1 ),(b l1 ) Ψ µν k ),(b l1 ) Γ bc Ψ αβ k ),d(b l1 ) + +6k Ψ µν k1 ),(b l1 ) γ b Ψ αβ c k1 ),d(b l1 ) ] nd to compenste for their non-invrince we hve to introduce the following corrections: δ Ψ µν k ),(b l ) d k,l = 10l(k l + 2)(d + l 4) [(k l + 1)ξ µ k),(b l1 b e 1 ) ν] e [ν 1 ξ µ] k1 )(b 1,b l1 ) +...] δ Ψ µν k ),(b l1 ) = d k,l 10l ξ [µ k),(b l1 ) ν] (51) Ψ µν k ) Φ µ k ). This cse we hve lredy considered in Subsection 3.2, so we will not repet corresponding formuls here. Collecting ll pieces together we obtin the following complete set of guge trnsformtions: δr µν k ),(b k ) = D [µ ξ ν] k ),(b k ) + i k,k 5(d 4) [γ [µξ ν] k ),(b l ) +...] 18

20 δψ µν k ),(b l ) δψ µν k ) δφ µ k ) d k,k 20k(d + k 4) [ξ [µ k),(b k1 b e 1 ) ν] e 1 [ν ξ k1 )(b 1,b k1 ) µ] +...] = D [µ ξ ν] k ),(b l ) + i k,l 5(d 4) [γ [µξ ν] k ),(b l ) +...] d k,l 10l(k l + 2)(d + l 4) [(k l + 1)ξ µ k),(b l1 b e 1 ) ν] (52) e [ν 1 ξ µ] k1 )(b 1,b l1 ) +...] d k,l+1 10(l + 1) ξ [µ k),(b l ) ν], 1 l k 1 = D [µ ξ k ) ν] + i k,0 5(d 4) [γ [µξ k ) ν] +...] d k,1 10 ξ [µ k), ν] (k + 1)d k,0 10(k + 2)(d 3)(d 4) [Γ µνζ k) +...] = D µ ζ k) ib k 3(d 2) [γ µζ k) +...] + 2d k,0 ξ µ k ) At this point we hve complete Lgrngin s well s complete set of guge trnsformtions, in this ll prmeters in guge trnsformtions re expressed in terms of Lgrngin prmeters k,l, d k,l nd b k so tht ll vritions with one derivtive cncel. Now we hve to clculte ll vritions without derivtives (including contributions of kinetic terms due to non-commuttivity of covrint derivtives) nd require their cncelltion. Once gin we omit these lengthy but strightforwrd clcultions nd give finl results only. First of ll we obtin number of recurrent reltions on digonl mss terms k,l which llow us to express ll of them in terms of the min one k,k = M: k,l = d + 2k 2) (d + 2l 2) M, b 3(d + 2k 2) k = M 5(d 4) Then we obtin recurrent reltions on prmeters d k,l which llow us to express ll of them in terms of the min one: d k,l 2 = l(k l + 2)(d + k + l 3) [m 2 100(k l)(d + k + l 4)κ], 1 l k (d + 2l 4) d k,0 2 = where we introduced nottion: (k + 2)(d + k 3) [m 2 100k(d + k 4)κ] 10(d 4) m 2 = (d + 2k 4) 2k(d + 2k 3) d k,k 2 At lst we obtin n importnt reltion on prmeters M nd m: 4M 2 = m 2 25(d + 2k 4) 2 κ Let us nlyze the results obtined. We hve lredy seen in Subsection 2.5 tht mssless spin-tensor R µν k ),(b k ) dmits deformtion into AdS spce without introduction of ny 19

21 dditionl fields. And indeed, s the lst reltion clerly shows, in AdS spce (κ < 0) nothing prevent us from considering limit m 0 when ll Goldstone fields decouple from the min one. From the other hnd, in the ds spce we gin obtin unitry forbidden region m 2 < 25(d + 2k 4) 2 κ. At the boundry of this region ll digonl mss terms become equl to zero so tht the theory gretly simplifies (though the number of physicl degrees of freedom remins to be the sme). Inside forbidden region we find number of (non-unitry) prtilly mssless theories. They pper ech time when one of the prmeters d k,l 0. In this, the whole system decomposes into two disconnected subsystems contining the fields R µν k ),(b l ), Ψ µν k ),(b n) l n k 1 nd Ψ µν k ),(b n) 0 n l 1, Φ µ k ), correspondingly. 3.5 Y (k + 3 2, l ) Now we re redy to consider generl cse mssive spin-tensor Y (k + 3, l + 3 ) with k > l 1. Our usul procedure (consider guge trnsformtions for ll fields nd tke into ccount their reducibility) leds to the following set of fields which re necessry for guge invrint description: Y (m + 3, n + 3 ) nd Y (m + 3, 1 ) where l m k nd 0 n l. These fields s well s prmeters determining pproprite cross terms (see below) re shown on Figure 7. Ψ k,l c k,l c k1,l c l+1,l Ψ k1,l Ψ l+1,l R l,l d k,l Ψ k,l1 d k1,l d l+1,l c k,l1 Ψ k1,l1 c k1,l1 Ψ l+1.l1 c l+1,l1 d l,l Ψ l,l1 d k,l1 d k1,l1 d l+1,l1 d l,l1 c k,0 c k1,0 c l+1,0 Ψ k,0 Ψ k1,0 Ψ l+1,0 Ψ l,0 d k,0 d k1,0 d l+1,0 e k e k1 e l+1 Φ k Φ k1 Φ l+1 Φ l d l,0 Figure 7: Generl mssive Y (k + 3 2, l ) theory As in ll previous cses, the totl Lgrngin contins kinetic nd digonl mss terms for ll fields s well s cross terms without derivtives: k l k L = L(Ψ m),(b n) µν ) + L(Φ m) µ ) + L cross (53) m=l n=0 m=l 20

22 Recll tht cross terms pper for nerest neighbours only, i.e. min guge field with primry fields, primry with secondry ones nd so on. Thus generl Ψ µν m),(b n) field hs cross terms with four other fields s shown on Figure 8. In the previous Subsection we hve lredy Ψ m,n+1 Ψ m+1,n d m,n+1 c m+1,n c m,n Ψ m,n Ψ m1,n d m,n Ψ m,n1 Figure 8: Illustrtion on possible cross terms considered cross terms for the pir Ψ µν m),(b n) Ψ µν m),(b n1 ), thus the only new terms we need re cross terms for the pir Ψ µν m),(b n) Ψ µν m1 ),(b n). They look s follows: (1) m+n L = ic µναβ m,n bcd [ Ψµν m1 ),(b n) Γ bcd Ψ m1 ),(b n) αβ + 6n Ψ µν m1 ),b(b n1 ) γ c Ψ αβ n1 ),d(b n1 ) Ψ µν m1 ),(b n) Γ bc Ψ αβ d m1 ),(b n) + 6n Ψ µν m1 ),(b n1 ) γ b Ψ αβ c m1 ),d(b n1 ) ] In this, to compenste for their non-invrince we hve to introduce the following corrections to guge trnsformtions: δ Ψ µν m),(b n) δ Ψ µν m1 ),(b n) = c m,n 10m(d + m 3) [e [µ 1 ξ ν] m1),(b n) +...] (54) c m,n = 10(m n + 1)(m + 1) [(m n + 1)ξ [µ,ν] m1),(b n) + ξ m1)(b 1,b n1) [µ ν] ] Note lso tht the lst two rows on Figure 7 re to those on the Figure 4 in Subsection 3.2, so ll necessry terms hve lredy been considered there. Collecting ll pieces together we obtin the following complete set of cross terms: L cross k l = i (1) { m+n µναβ bcd cm,n [ Ψ m1 ),(b n) µν Γ bcd Ψ m1 ),(b n) αβ + m=l n=1 6n Ψ m1 ),b(b n1 ) µν γ c Ψ n1 ),d(b n1 ) αβ Ψ m1 ),(b n) µν Γ bc d Ψ m1 ),(b n) αβ + 6n Ψ m1 ),(b n1 ) µν γ b c Ψ m1 ),d(b n1 ) αβ ] + +d m,n [ Ψ m),(b n1 ) µν Γ bcd Ψ m),(b n1 ) αβ + +6m Ψ m1 ),b(b n1 ) µν γ c d Ψ m1 ),(b n1 ) αβ 21

23 Ψ µν m),(b n1 ) Γ bc Ψ αβ m),d(b n1 ) + +6m Ψ µν ( m1 ),(b n1 ) γ b Ψ αβ c m1 ),d(b n1 ) ] } k +i (1) { m µναβ bcd cm,0 [ Ψ m) µν Γ bcd Ψ m) αβ Ψ m) µν Γ bc d Ψ m) αβ ]+ m=l +d m,0 [ Ψ m) µν Γ bc Φ m) α 6m Ψ m1 ) µν γ b c Φ m1 ) α + + Ψ µν m) Γ bc Φ α m) 6m Ψ µν m1 ) γ b Φ α c m1 ) ] + +e m [ Φ µ m) γ b Φ ν m) Φ µ m) γ Φ ν b m) ] } (55) Similrly, combining results of this nd previous Subsections, we obtin complete set of guge trnsformtions for ll fields involved: δψ µν m),(b n) δψ µν m) δφ µ m) = D [µ ξ m),(b n) ν] + i m,n 5(d 4) [γ [µξ m),(b n) ν] +...] c m+1,n 10(m n + 2)(m + 2) [(m n + 2)ξ [µ,ν] m),(bn) + ξ m)(b 1,b n1) [µ ν] ] c m,n 10m(d + m 3) [e [µ 1 ξ m1),(b n) ν] +...] d m,n+1 10(n + 1) ξ [µ m),(b n1) ν] d m,n 10n(m n + 2)(d + n 4) [(m n + 1)ξ [µ m),(b n1 b e 1 ) ν] e 1 [ν ξ m1 )(b 1,b n1 ) µ] +...], 1 n l = D [µ ξ m) ν] + i m,0 5(d 4) [γ c [µξ m) m,0 ν] +...] + 10(m + 2) ξ [µ,ν] m) (56) c m1,0 10l(d + m 3) [e [µ 1 ξ m1) ν] +...] (m + 1)d m,0 10(m + 2)(d 3)(d 5) [Γ µνζ m) +...] = D µ ζ m) ib m 3(d 2) [γ µζ m) e +...] + 2d m,0 ξ m) m µ + 6(m + 1) ζm) + e m1 + 6m(d + m 2) [e µ 1 ζ m1) +...] Hving in our disposl totl Lgrngin nd complete set of guge trnsformtions where ll vritions with one derivtive cncel, we proceed with vritions without derivtives. After lengthy but strightforwrd clcultions we obtin the following results. First of ll we obtin number of reltions on digonl mss terms m,n nd b m which llow us to express ll them in terms of min one k,l = M: m,n = (d + 2k)(d + 2l 2) (d + 2m)(d + 2n 2) M Similrly, we get number of reltions on the prmeters d m,n, c m,n nd e m (determining cross terms) so tht ll of them cn be expressed in terms of one min prmeter. We choose d k,l s such min prmeter nd introduce nottion: m 2 = (k l + 1)(d + 2l 4) l(k l + 2)(d + 2l 3) d k,l 2 22

24 Then we obtin the following importnt expressions for the prmeters d k,n corresponding to leftmost column on Figure 7: d k,n 2 = n(l n + 1)(k n + 2)(d + l + n 3) [m 2 100(l n)(d + l + n 4)κ], n 1 (k n + 1)(d + 2n 4) d k,0 2 = (k + 2)(l + 1)(d + l 3) [m 2 100l(d + l 4)κ] 10(k + 1)(d 4) s well s for prmeters c m,l corresponding to topmost row on Figure 7: c m,l 2 = m(k m + 1)(d + k + m 1) [m (m l + 1)(d + m + l 3)κ] (d + 2m 2) It is very importnt nd this gives nice check for ll clcultions) tht ll prmeters d m,n corresponding to the sme row on Figure 7 turn out to be proportionl to the leftmost one d k,n : d 2 (k n + 1)(d + k + n 2) m,n = (m n + 1)(d + m + n 2) d k,n 2 Similrly, ll prmeters c m,n nd e m corresponding to the sme column turn out to be proportionl to the topmost one c m,l : c m,n 2 = e m 2 = (m l)(d + m + l 2) (m n)(d + m + n 2) c m,l 2 9(m l)(d + m + l 2) 25(m + 1)(d + m 3) c m,l 2 At lst but not lest, we obtin n importnt reltion on two min prmeters M nd m: 4M 2 = m 2 25(d + 2l 4) 2 κ We hve lredy mentioned in Subsection 2.6 tht mssless spin-tensor Ψ µν k ),(b l ) does not dmit deformtion into AdS spce without introduction of dditionl fields. In the guge invrint formultion for mssive spin-tensor such limit would require tht both d k,l 0 nd c k,l 0 simultneously nd such possibility exists in flt Minkowski spce (κ = 0) only. For non-zero vlues of cosmologicl constnt we obtin number of prtilly mssless limits insted. Let us consider AdS spce (κ < 0) first. The most physiclly interesting limit rises when m 2 = 100(k l + 1)(d + k + l 3)κ. In this, prmeter c k,l nd hence ll prmeters c k,n nd e k ) becomes equls to zero nd fields Ψ µν k ),(b n) 0 n l nd Φ µ k ) (corresponding to leftmost column in Figure 7) decouple nd describe (the only) unitry prtilly mssless theory. The Lgrngin for this theory hs the form: l L = L(Ψ k ),(b n) µν ) + L(Φ k ) µ ) + L cross (57) n=0 23

25 L cross l = i (1) k+n d µναβ k,n bcd [ Ψµν k ),(b n1 ) Γ bcd Ψ k ),(b n1 ) αβ + n=1 +6k Ψ k1 ),b(b n1 ) µν γ c d Ψ k1 ),(b n1 ) αβ Ψ µν k ),(b n1 ) Γ bc Ψ αβ k ),d(b n1 ) + +6k Ψ ( k1 ),(b n1 ) µν γ b c Ψ k1 ),d(b n1 ) αβ ] +id µναβ k,0 bcd [ Ψµν k ) Γ bc Φ k ) α 6k Ψ k1 ) µν γ b c Φ k1 ) α + + Ψ µν k ) Γ bc Φ α k ) 6k Ψ µν k1 ) γ b Φ α c k1 ) ] (58) nd is invrint under the following guge trnsformtions: δψ µν k ),(b n) δψ µν k ) δφ µ k ) = D [µ ξ k ),(b n) ν] + i k,n 5(d 4) [γ [µξ k ),(b n) ν] +...] d k,n+1 10(n + 1) ξ [µ k),(b n1) ν] d k,n 10n(k n + 2)(d + n 4) [(k n + 1)ξ [µ k),(b n1 b e 1 ) ν] e [ν 1 ξ µ] k1 )(b 1,b n1 ) +...], 1 n l = D [µ ξ k ) ν] + i k,0 5(d 4) [γ [µξ k ) ν] +...] (59) (k + 1)d k,0 10(k + 2)(d 3)(d 5) [Γ µνζ k) +...] = D µ ζ k) ib k 3(d 2) [γ µζ k) +...] + 2d k,0 ξ µ k ) All other fields just give mssive theory for the spin-tensor Ψ µν k1 ),(b l ). Besides, number of non-unitry prtilly mssless limits exist. It hppens ech time then one of the c m,l nd hence ll c m,n with 0 n l nd e m ) goes to zero. In this, the whole system decomposes into two disconnected subsystems nd digrm on Figure 7 splits horizontlly into two blocks s shown on Figure 9). The left block describes non-unitry prtilly mssless theory, while Ψ k,l Ψ m,l Ψ m1,l R l,l Φ k Φ m Φ m1 Φ l Figure 9: Exmple of non-unitry prtilly mssless limit in AdS the right one gives mssive theory for spin-tensor Ψ µν m1 ),(b l ). Recll tht our definition 24

26 of msslessness is bounded to flt Minkowski spce. From the nti de Sitter group point of view ech verticl column on Figure 7 corresponds to unitry irreducible representtion which cn be clled mssless [25]. In this, ll other representtions (mssive or prtilly mssless) cn be constructed out of pproprite set of mssless ones s it should be. Let us turn to the ds spce (κ > 0). Here we once gin fce n unitry forbidden region m 2 < 25(d + 2l 4) 2 κ (which follows from the reltion between M nd m). Inside this forbidden region we obtin number of prtilly mssless limits (but ll of them led to the non-unitry theories). The first one rises then prmeter d k,l nd hence ll prmeters d k,n ) becomes zero. In this, the fields Ψ µν m),(b l ) with l m k (corresponding to upper row on Figure 7, see Figure 10) decouple nd describe prtilly mssless theory which corresponds to irreducible representtion of the de Sitter group nd from the de Sitter group point of view cn be clled mssless). Contrry to wht we hve seen in AdS cse, ll other fields Ψ k,l R l,l Ψ k,l1 Ψ l,l1 R l1,l1 Φ k Φ l Φ l1 Figure 10: Prtilly mssless limit in ds spce lso gives prtilly mssless theory. The reson is tht to describe complete mssive theory for spin-tensor Ψ µν k ),(b l1 ) we need one more column of fields s lso shown on Figure 10. Similrly, prtilly mssless limits hppens ech time when one of the prmeters d k,n nd hence ll prmeters d m,n with l m k) becomes zero. Once gin the whole system decomposes into two disconnected subsystems nd digrm on Figure 7 splits verticlly into two blocks). In this, both upper nd bottom blocks describe non-unitry prtilly mssless theories. The reson gin is tht bottom block does not hve enough fields for description of mssive spin-tensor Ψ µν k ),(b n1 ). 4 Conclusion Once gin we hve seen tht frme-like formlism gives simple nd elegnt wy for description of (spin)-tensors with different symmetry properties. The formultion for mssless mixed symmetry spin-tensors constructed here turns out to be nturl nd strightforwrd 25