JHEP07(2009)014 July 3, 2009 May 28, 2009 : June 10, 2009 : — referred to : nd INFN, Sezione di Published Received Accepted , edded into a single master sor fields of arbitrary shape on of Skvortsov’s equations ly project the latter onto the Vasiliev frame-like potentials 10.1088/1126-6708/2009/07/014 tentials that leads to a smooth The remainder of the reduced MV) conjecture. In the unitary arXiv:0812.3615 tly via either bosonic (symmetric doi: tors. At critical masses the reduced NRS associate researcher on leave from the ma, Italy. Hainaut, Belgium. Published by IOP Publishing for SISSA [email protected] , and Per Sundell 2 Carlo Iazeolla 1 Dipartimento di Fisica, Universit`adi Roma “Tor Vergata” a 0812.4438 Gauge Symmetry, Field Theories in Higher Dimensions Following the general formalism presented in [email protected] Work supported by a “Progetto Italia” fellowship. F.R.S.-F Previous address: 1 2 SISSA 2009 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126E-mail: Pisa, Italy [email protected] c ArXiv ePrint: as Paper I — we deriveand the mass unfolded equations in of constantly motionin for curved one ten backgrounds higher by dimension. radialfield, The valued reducti complete in a unfolded tensorial system Schurbasis) module is or realized emb fermionic equivalen (anti-symmetric basis)Weyl vector zero oscilla -form modules become indecomposable.submodules We carrying explicit Metsaev’s masslesssystem contains representations. a set of fieldsflat St¨uckelberg and limit dynamical in po accordance withmassless the cases Brink-Metsaev-Vasiliev (B in AdS,and we explicitly identify disentangle the their Alkalaev-Shaynkman- unfolded field equations. Keywords: Service de et M´ecanique Gravitation, Universit´ede Mons-
Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Ro Abstract: Nicolas Boulanger, Unfolding mixed-symmetry fields inBMV AdS conjecture: and the II. Oscillator realization JHEP07(2009)014 3 6 6 7 1 3 16 18 19 19 21 22 22 24 25 25 27 28 29 29 31 35 35 35 37 10 11 12 13 14 15 16 38 37 15 hadow sor ules ses x 1 − ,D 2 R 1) 1) , , D -cohomology for unitarizable ASV gauge potential AdS − σ 3.7.3 Smooth flat limit 3.6.1 Occurrence3.6.2 of non-trivial potential module On 3.7.1 The3.7.2 example of Θ = The (2 example of Θ = (3 3.5.1 Proof3.5.2 of indecomposability for massless Harmonic3.5.3 cases expansion via most electric Harmonic primary expansion Weyl via ten most magnetic Weyl tensor and s 3.3.1 Twisted-adjoint3.3.2 module and mass formula Proposition for indecomposability in the3.4.1 critical ca Harmonic3.4.2 expansion Characteristic equation 3.2.1 Radial3.2.2 Lie derivatives and unfolded Initial mass comments terms on criticality/reducibility 2.2.4 Equivalent bosonic and fermionic universal Schur2.3.1 mod Bosonic2.3.2 oscillators (symmetric basis) Fermionic oscillators (anti-symmetric basis) 2.2.1 Howe2.2.2 duality and Schur states Cell2.2.3 operators: general definitions Cell and operators: properties explicit oscillator realization 3.7 fields St¨uckelberg and flat limit 3.5 The critically massless case 3.6 Unitarizable ASV potential 3.4 The generically massive case 3.1 Transverse and3.2 parallel components in Radial reduction 3.3 Radially reduced Weyl zero-form 2.3 Master-field reformulation of Skvortsov’s equations 2.1 Skvortsov’s unfolded2.2 equations Interlude: oscillator realization of the Young tableau 4 Conclusion A Notation and conventions 3 Tensor fields in Contents 1 Introduction 2 Tensor gauge fields in flat spacetime JHEP07(2009)014 ], D 41 43 44 45 47 3 41 48 50 51 no- ] and AdS space- 9 ]. The -forms. – p D 6 ; (ii) the 16 – dS ) 13 A , see [ 3 , AdS = 2 ime by Skvortsov [ g + sing algebraic techniques evel, peculiarities that are Weyl zero -forms q n broken spontaneously by ntial algebras [ the present Paper II, we use ities as well as definitions of by consistency conditions on lete sets of generalized curva- aynkman and Vasiliev (ASV), led ds freely propagating in or fields and ordinary mmetry fields in ( field content and symmetries of ropagating in constantly curved ds. r modules in order to effectively s have previously been given by ] whereby the concepts of space- nlinear field equations for totally gebras) and their representations. 12 – = 1 and 10 1 h ) C ; D ( – 1 – sp ) and -chains with ) C − C ; ; σ ]. It is therefore likely to be helpful also in addressing the D D ( ( 18 gl so , 17 ] referred to as Paper I, we introduced some general tools and 1 ]. For some recent works on mixed-symmetry fields in 5 , 4 ], and was recently performed in the case of Minkowski spacet 2 As in Paper I, we use the unfolded formalism [ Tensor fields of mixed symmetry exhibit, already at the free l B.1 Howe-dual LieB.2 algebras Generalized SchurB.3 modules Fock-space realizations B.4 Schur modulesB.5 for Schur modules for 1 Introduction In a companion paper [ C Radial reduction of the backgroundD connection Tensorial content of the E Mixed-symmetry gauge fields and singleton composites B Oscillator realizations of classical Lie algebras times [ tation adapted to the unfolded descriptionbackgrounds. of Such tensor an fields analysis p waswho initiated proposed an by action Alkalaev, in Sh frame-like formalism for mixed-sy who provided the correspondingthese unfolded tools field together equations. with In anwrite oscillator down formulation unfolded of field Schu equations for arbitrary tensor fiel references therein. key features are thatauxiliary (i) fields equations are encoded oftures, into motion, flatness including Bianchi conditions in ident on general comp an infinite set of zero -forms cal challenging issue of interacting mixed-symmetry gauge fiel time, dynamics and observables are derived from free differe absent in the “rectangular” case, including symmetric tens spacetime. Metric-like and partiallyMetsaev gauge-fixed in equation [ diffeomorphism invariance is manifestgiven solutions); (this symmetry andthe is (iii) coupling the the gauge constantsfor in invariance deformations is the of ensured curvatures associative algebras thatThis (including powerful can Lie framework is al be instrumentalhigher-spin in solved gauge controlling u the theory and underliessymmetric Vasiliev’s gauge fully fields no [ JHEP07(2009)014 2) 1)- ntz- − = 4 , − D 0) that D ( , D > so (1 p = 0 — the iso -form module p p en as the gener- ] and henceforth. 4 are concerned, formulation here. ]. That the corre- ], unitary massless -forms ( 24 > 20 ] proposed in p 19 ] directly generalizes onal Weyl zero -form D n-Wigner equations to ] to arbitrarily-shaped 25 27 n . More precisely, the set bastida’s formalism was 29 1 tida [ smooth flat limit — in the 6 and in constantly curved − analysis of the pioneering on [ must be the uppermost one. ,D ssless degrees of freedom was ith the leftover gauge invari- > d — given by the direct sum kvortsov’s unfolded equations 1 ]. The local wave equation pro- ermined by the Young diagram R uations” in [ Labastida’s formulation has ap- chur modules realized explicitly quation [ D y relies on the properties of the n in order to extend Skvortsov’s constitute on-shell folding language which facilitates 23 p 1)-tensor of shape Θ under − D ( decompose in the flat limit into topological 4 0 and infinite-dimensional for so (Λ; Θ) , in terms of an “unbroken” gauge field ϕ ]. – 2 – -form modules. that break the gauge symmetries associated with AdS 32 p p > } ] anticipates a field-theoretic realization of an AdS ) -forms with fixed ′ p 19 ). 3 allows us to unfold a conjecture due to Brink, Metsaev , 1 (Λ; Θ ] and results in a system consisting of D R χ ] for a review and references. The proof of the propagation ] for Fronsdal fields in flat space [ 27 { ], see also [ -form variables arise that generalize the vielbein and Lore , 22 ) of irreducible gauge fields in 28 AdS p 3 ′ 31 , 4 (in accordance with the analysis done in [ and its traceless part, defined in [ 30 > , K 2 , thereby making contact with the equations and the D ] — see [ (Λ=0; Θ D χ 21 ′ Θ AdS ] for an arbitrary tensor gauge field in flat spacetime can be se L ], though we shall not make direct contact with this off-shell 24 , ⊕ 26 23 ], are given here, comprising the complete infinite-dimensi 2 should be given by the reduction of the As we mentioned previously, the unfolded presentation of La As far as free tensor gauge fields in flat spacetime of dimensio In Paper I, we reviewed and extended the generalized Bargman The present analysis in } ′ (Λ=0; Θ) Θ module as well as the finite-dimensional alization to arbitrary dimensions of the Bargmann-Wigner e given recently by Skvortsov [ are traceless Lorentz tensors ofof various the symmetry massless types metric-like det field. The modules that are finite-dimensional for a Lagrangian formulation was proposed some time ago by Labas spacetime as soon as { all blocks but one, in suchsense a that way the that the number combined ofϕ system local has degrees a of freedom is conserve Such fields must be considered in flat spacetime as soon as plus a set of fields St¨uckelberg proposed by ASV in [ constantly curved spacetimes, translating them into thetheir un integration whereupon gauge fields. In the present paperin we terms review of and master reformulate fields S using taking their oscillators values and in Fock generalized spaces. S formulation We to use this reformulatio mixed-symmetry gauge field with shape Θ , mixed-symmetry two-row tensor fields in sponding equations of motion indeedunderstood propagate the later proper [ ma generalized curvature Vasiliev’s first-order action [ of the proper massless physical degrees of freedom cruciall posed in [ Let us finally mention that apeared trace-unconstrained in version of [ subject to the conditionance, that must one remain block, untouched. the In one the associated unitary w case, that block connection of spin-2work theory. [ The results, that complete the and were therefore called “generalized Bargmann-Wigner eq dittos plus one massless field in aforementioned Weyl zero -form module. The first-order acti and Vasiliev. The BMV conjecture [ JHEP07(2009)014 , , ) 2 D D (Θ) (2.1) 3.27 ϕ )). In AdS AdS -massless 3.10 ). Finally, D ] and the di- ge field 5 .) In section , 3.74 AdS ave at most six 4 A [ + 1)-dimensional bosonic singletons D D , in the presentation of P , +1 ’s critical cases is given AdS B ] h ). Our conclusions and signatures, and that the in erlying the analysis in this +1 esulting smoothness of the ’s massless fields in B e stress that our treatment and in appendix ameter, introduced by con- h 0 3.100 ; work. ensional reductions as “world” h sm for free tensor gauge fields alizations of Young diagrams. with a one-form ASV potential. , +1 uge field in ( shows that some products of B ses” (see items (i)-(iv) in section +1 s [ B E details the radial reduction of the contains a review of Howe duality s , and then cast them into a master- ] ]. In particular, see equations ( , C B 1 0 B s = 1 . (We note that the general sigma- 19 h − ; 1 ,D B h 1 s [ R 1) . In this paper, we carry out this procedure – 3 – lists shapes occurring in the computation of the ,..., ].) Appendix ] − 1 D -forms in the unfolded system (see eq. ( 33 h . Appendix p ; 4 1 , D suggest that fields the can St¨uckelberg be incorporated s [ } , ′ ] 0 Θ h { ; 0 s . Appendix [ 1 − ]. We then cast them into a compact master-field form using an ,D 2 27 we derive the unfolded equations for general tensor fields in R 1 Θ = , 3 -dimensional Minkowski spacetime of an on-shell tensor gau 3 D -type m ), and the corresponding value of the mass parameter in eq. ( = 2 . Besides, the Metsaev’s mixed-symmetry that may appear h ) for the zero-forms. The appropriate projection to Metsaev 3.72 P 3.28 The partially massive nature of mixed-symmetry gauge fields The paper is organized as follows: The general formalism und In the following, we shall frequently suppress the labels -cohomology groups for ASV potentials with 1 − and “fiber” indices areaccommodates treated any separately combinations from of theradial ambient outset. reduction and W allows tangentstraining for space the arbitrary radial values derivatives of of the all mass par using the unfolded language, which is readily adapted to dim sitting in the flat ambient space with signature (2 explicitly via a suitable radial reduction of an unbroken ga minus construction was introduced in [ we review Skvortsov’s unfolded equations in an outlook arein presented the in context section of classical Lie algebras. Appendix lowest-weight unitary representations may arise inonly tensor if blocks, the first of height one, and are therefore2 associated Tensor gauge fieldsIn in this flat section spacetime wein first flat review spacetime Skvortsov’s unfolded [ formali oscillator realization of Young tableaux. 2.1 Skvortsov’s unfoldedThe equations unfolding in mensional reduction leading to particular, the reduction allowsI.4.3.4), for though general we “critical shall mas focus mainly on the case of Metsaev and ( in eq. ( the unfolded equations for the unitary ASV potential are ( background fields in σ leaving a number of details in other special casespaper for is future contained in Paper I. (We recall some of our notation field form suitableFinally, for in radial section reduction using oscillator re Young diagrams associated to dynamical fields. analyze critical limits forflat the limit mass in accordance parameter with and the show BMV the conjecture r [ JHEP07(2009)014 . 1 s := 0 (2.2) (2.9) (2.3) (2.8) (2.4) (2.6) (2.5) (2.7) + . (2.13) (2.10) (2.11) (2.12) , , +1 ] α , ) -module B ] B 0 } 0 s ] h p ] g 1 − [ B ; )= B . The global h − α B h ; s the primary (Θ 1 ]; s ; [ +1 I s B − p B s (1) [ I,I [ s T [ . | (Θ ,..., β g ] , Θ ,..., | ted-adjoint ] ∞ +1 q q ,..., ,...,s ] I 0 R +2 h > Θ) := I +1 ; | , I := h ∈{ E α h ; +1 I ; I q Θ . =0; | s +1 +2 + [ 2 I := 0 +1 1 defined by B , I s sits in the smallest Lorentz type ] h q } } [ , k s M [ , B N 1 R B and +1 , , } h } + 1] B p ; β +1 E 7→ I I ; 1] p B + 1] I s h s [ R {z k , ..., h (Λ=0; 6 · · · > s (Θ) with indecomposable structure + is obtained from Θ by first deleting one +1 : 1; q I 1 I T α {z + {z g E h − add one row k o B p R ; I α q ,..., > +1 Z 0 the submodules are finite-dimensional and s , ] ]; add one row I := | [ insert one row + +1 ∈ I 2 s , I q 1 2 p [ | ] > s h [ s − Θ) , 1 [ | ; – 4 – ] I > } B Θ L , , 2 − I := 0 . The submodules are given by ) then ] I q p } s , α n α I [ β , h · · · 0 h , h ]; J , R , =0; h ) p ] α 1 s 2 1; ) 1; (Θ I p ] 1; [ > {z E p ), I − − [ − − M p (Θ) = + 1] I 0 . For Θ > + [ q 1 − 1 I s > g T 1 + − (Θ s I blocks of Θ and then inserting one extra row of variable given by Θ plus one extra first row of length ⊗ 1 I cut one column [ ) h (Θ B s , the set s := ; [ ) α + 1)th blocks in compliance with row order (with p q I q < [ } 1 +1) > s U I I s +1) [0];0 (Λ=0; R ,...,B {z + I,J [ ( p (Θ − I ] , h ( {z I p ,..., + g q ] ( ∞ = p ,..., ,..., 1 ] + = 1 T ] -module ⊗T cut one column 1 0 α h , s ⊗T 1 = 0 g cut one column I ) ,...,B p h := 0 := | g ) h [ 0 ∼ = 1; q ] Θ=:Θ U ∞ 0 0 , ( 0 th and ( Ω 1; U ( 1; p s R ( h I ) − − [ 1 J q ] ∈ { q s := − − I 1 h 0 1 p + − s 1 I is given by the one-to-one map − [ ∞ [ | I s s M = | [ =1 | [ p 0 given by Θ minus its first column, and the smallest zero -form i α := Ω I J (Θ 1 R , s ). In particular, the form of highest degree s q q = = = Θ=Θ P ] − +1) = = Ω I R ]; I α ∞ m p ] = p | − + [ ]; q B :=0 I ( I I p p q + T Θ [ − [ p := [ I p +1 = 0 the submodule is infinite-dimensional and defines the twis R q p Θ 0 B = 0 : Θ s I s 1 : R I > -grading of I e It has the property that if N that is, for fixed column from each of the first results in a triangular For length between the Upon defining where which vanishes trivially if Weyl tensor sitting in Θ:=Θ and one has one has JHEP07(2009)014 = I k (2.18) (2.15) (2.19) (2.14) (2.20) (2.17) (2.16) , . , 1 , , . , 1 1 ed by the 0 +1 1 1 I − − s . − − ≡ . +1 +1 . + +1 +1 ) = 0 − q I,I I,I I s σ ω I,I I,I . k ······ s matrix the and operator setting th JHEP07(2009)014 ], 22 , as de- (2.21) (2.22) (2.23) (Θ) it 21 ϕ ± m of the S . = 0 yields 0 0 α ) bases leads is the (trace- p R − 0 our analysis in suffice for han- ∩ R . The restriction ) , 1 ∩ 0 ) R 2.2.2 Weyl ra of canonical trans- T | ∩ T ≡ | x ) − . − σ T | ( σ ction +1 ∗ ( − . m module and may be put +1 a generalized Schur module α ∗ σ α H ) s well as the generic massive ( R ǫ of freedom carried by H ∗ ∗ he maximal finite-dimensional representations of the classical rs is needed as was realized by are by definition the subspaces ) oscillator realizations, the Lie ions take the form e Θ +1 +1 H s and more generally to describe ( − ± α α α α nt of B i S p 1 − 1 ). − − σ X = 0; Θ) through harmonic expansion. ) of the reducible Weyl zero -form) it σ determines the on-shell content of the consisting of states with 2 H h 3.74 a + T θ M + i 3.72 ( α α Weyl R D ǫ · · · T ∇ ∇ 1 a – 6 – ] and to give these an explicit oscillator realiza- θ i = = ) 40 -irrep ] for reviews) and also pointed out later in [ and associated generalized Schur modules – 0 D α α ] ( ] g 36 ± ǫ Θ sl 37 R , ) e m ) P , 1 . The corresponding − − 1 − 35 23 σ σ m . However, in order to examine the critically massless cases + + D (Θ) = Θ) as the non-trivial content of to its submodule ∇ ∇ ( ] (see [ ϕ T C AdS 34 in the triangular module := [( := [( . Using bosonic (+) and fermionic ( Θ) and it is not necessary to actually extract the precise for − α α ( B σ say, using manifestly symmetric (+) and anti-symmetric ( Z C X ǫ δ m ]. Such an expression is rederived here and will be crucial to 5 arises as a subalgebra of the infinite-dimensional Lie algeb , (namely in analyzing the projection ( 4 ± — more precisely, for our derivation of ( ] e 5 m 3 D , 4 As realized early in [ The general properties of the cell operators presented in se AdS algebra formations of the oscillatorsubalgebra algebra that and commutes is with identified with t scribed in appendix suffices to analyze Thus, for the purpose of counting the local on-shell degrees to the notion of Howe dual algebras The first levels of Bianchi identities and gauge transformat tion [ the local degrees ofin freedom correspondence are with encoded the in massless the Weyl zero-for Labastida operator. 2.2 Interlude: oscillatorIn realization order of to the study Young thetensor tableau integrability fields of of Skvortsov’s arbitrary equation shapes,and related one hyperform may adopt complex the [ notion of dling the unfolded master-fieldmaster-field equations equations in in flat spacetime a appears that a moreMetsaev explicit [ expression for the cellsection operato 2.2.1 Howe duality andThe Schur decomposition states of tensor productsmatrix algebras, of finite-dimensional The cohomology of in Skvortsov equations. In particular, the non-trivial conte of the triangular module constrained) Labastida gauge field The Labastida field equation is the non-trivial content of the primary Weyl tensor JHEP07(2009)014 , ] , n = ∼ = 2 1) D 38 and − ± ( m ∓ , e (2.27) (2.28) (2.25) (2.29) (2.24) (2.26) m )] ± ν ± i 37 − ± = 0and ) . e λ S ν . ν , efined for + D ) 5 (2 = i 6 ( ,b j ) , so ∆ ± i j so [ | 4 ( e ) 6 w σ that induces a i ij ( α [12] 6 ± T T ∓ ) for obeying: ors [ 1 , ) tensors, such that − M ,a ± ν D ) , i ( and S ( (2 0 where σ so s denoted here by 1) so =0 α ) imply > (¯ − i ( , f ± 1), and use the notation of + − ∆ )] ± ν | , ± 2.25 odules ) for − e F F + w ν ,a = , ) ν ) and i ∞ ( > + 6 ,a (2 . +1 ± (2 ν , leads to finite-dimensional Howe ) j , ¯ β sp ∓ +1) ) and ( (2 so · · · [ ± i j,b =1 -irrep with highest weight given by 6 S ν i ¯ − = 0in α sp ∈ = 0 in i } ∼ = > m , ± i a that are annihilated by a Borel subalge- ν 2.24 +1)) ∼ = − ± 1 ( ¯ + 6 α i,a − ) = (0 i ± i e , ± e i ± (11) w m α 1 i a i e λ m ± β ( ∆ T ( ∆ ( | α | i f j S ∆ ǫ by | { δ arise the universal Howe-dual algebras , π )) : g (11) [12] – 7 – σ − ) and i = 1) denoting the reverse permutation in T T j = 0 ∞ B.16 N ( ,a and ( ) → ∞ ± i ) tensors, and (2 ( → ∞ i f ,..., ) : ± D ± , π sp , ∆ ± ( ¯ ν | β ) ± D ν ) g ( sl ∼ = ν ( =0 ± i ) then the Schur states also obey the tracelessness conditio σ i so + e λ i j π D e ∆ m ) ( δ = | ) for f with ( ,a so ( − ± ) m i σ ν i consisting of states j 1) ( ( = π ± N ± sl oscillator flavors, say ,..., ( β m ± F as a set of operators on the oscillator module ± ν ) = ( )) and parameterized by 1)) ν ) being the linear automorphisms of the oscillator algebra d ; and π D ± − ). If ± fg ± =1 ( ν ν i contains exactly one copy of the ( ) then the Schur states can be chosen to obey ± ± i σ S
:= ν ) tensors, and e λ sp D π ( ± ( ,a , ( i ) i π D j ∈ otherwise. ) i sl i ∈ S ( δ sl ( ∆ . Using D | σ ± sl B ∆ ∈ − ( = 0. In both cases one can show that | ( = ¯ . Moreover, in the limit i ± β i j j , so − e σ m N m N i ± , =1 ,a where arbitrary composite operators ( π for ν i ) ) ) for i ∆ } If The oscillator formalism can be used to define the cell operat ( | D ] ± i ± ( ∞ (i) the amputation and generation properties ij ( e (ii) the conjugation rule β [ w sl ( appendix taking the leading traces of Schur states such that ( dual algebras, namely where in a three-graded splitting (see ( of the Fock modules bra of non-trivial and regular action on the corresponding Schur m sl where { 2.2.2 Cell operators:From general now definitions on and properties we consider the general classical matrix algebra and that T JHEP07(2009)014 2 . The (2.31) (2.30) (2.35) (2.36) (2.37) (2.32) (2.33) (2.34) ,a 1, i.e. ) i ( ± ± β th row (+) i )= m -traceless (see ) for ( J . ǫ 2.60 . , = 0 − + . . i ) and ( ,i , ∆ ℓ +1 +1 | i ± i i +1 ℓ j pose into Young tableaux, i δ . e 2.59 ,a w w = +1. +1 h ℓ +1 ) ℓ ǫ n m s ( i .] Thus, the Schur modules = = m ( = i − ) is well-defined since i − = 0 i ± i ± h ± ± i ∆ i
s , B [ c ) s + M (2 + leaves with radius i ) [ ity theorem, which forbids inde- (1) f [0] e algebra (see also the comments ( ξ ; 1] B 2 b ξ 1 1 b a, ξ D ∆ α L i iso P b b β N λ λ ( ,..., a ] + ρ ,..., e 2 ≈ ] 2 1 AdS = 1 h s L ; [ = h 0 2 ) = ; , ) act canonically on − s b a f [0] 1 i X [ ( ] b 0 s Θ) : P , = [ b b e ∆ ∇ ( ab m ξ ∗ L [ α ≈ i i 2 c M =0; + 1] 0 C [0]; 2 – 19 – and reducible with an indecomposable structure 1 = b b X Θ b Θ= h and descend from the smallest M f and ; 0 ≡ , , ρ 1 , b s X (1) i -module +1 0 0 [ b e d λ α AB D ξ i g b Θ) i S ≈ ≈ c M [0]; ; (Λ=0; − = f b = Θ b 0 0 -form module and the infinite-dimensional Weyl zero-form T 1 f [0] 0 b X X =0 h ∈ ∞ (Λ; ) ∆ b α M )] X b [0];0 one has S ξ a b | C (1) b Θ λ = P λ L b , where ξ ) yields a λN n ( g i -type with shape to an irreducible twisted-adjoint ρ b h m ab + , determined by the value of | a b 2 + m 0 c 3.25 f M b ∇ b C b i e Θ := b Θ) X f [0] ab 0 can be computed using b Θ) := ω ∆ ; i ≈ 2 =0; λ := f ∇− 2 ( 1 1 − b M -types are realized in R b (Λ; R d b b := [ m S := 1 sitting in the (Λ=0; R 1 ∇ + 1 dimensions. The pull-back of the latter to b − T Let us restrict The constraint ( The fact that in AdS, differently from the flat-space case, the D ,D 2 for critical values of that is irreducible for generic values of with corresponding to the primary Weyl tensor of a tensor gauge fie and that of subtypes. Thus, at fixed can be glued together are a be obtained using R where the module is a direct consequence ofcomposable Weyl’s finite-dimensional complete modules reducibil for a semi-simplein Li section I.3.4). 3.3 Radially reduced Weyl3.3.1 zero-form Twisted-adjoint moduleThe and radially mass reduced formula Weyl zero -form obeys in JHEP07(2009)014 - − − λ g D (1) b ξ (3.38) (3.34) (3.36) (3.33) (3.41) (3.39) (3.40) (3.37) (3.35) (3.42) + ) . 1 1 2 D ) . b N , + (1) )( b , 1 ξ b f [0] 1 Θ) ; ( ) [0] b f N f . − (2 (Λ; + ,... b (1) S 3 1 3 1
+ 1)∆ i λ A, b b N N b β ) = Θ ( f [0] , . 2 2 [0] ]) + A (1) 0 − b ) f (∆ b N ] β ] — which is a nontrivial m α Θ 2 ) it can be seen that ( [ trivial ideal arises) leaves (1) m − b λ − 2 m ± [ µ [ a, 2 ,... 1 o C f 2 -type is given by b ∀ β are invariants for the action 0 3.29 C ≈ on this state generates a C m , (1) − α , − +1 corresponds to two mass oper- a (1) } , ] ≡ ab B − λ A, b 2 D β ] + D b ) b i i m g β c ] M | [ ] b , m + b 2 Θ b [ + m (1) B ab | b Θ +1 A Θ) [ 2 b | {z ξ C 1 . Further simplifications follow from o entries h 2 1 ; D by the state ( ( B C +1 c ; ,...,s M i b f 2 C N B − (1) B B α c = +1 1 2 M λ J,J = h s | +1 )( a, n b − s [ D Θ) . From ( | b [0]; ) (Λ; β b D ; S , r (Θ) is given by [0] B µ b b , b Θ) b S S f ; Θ b α f )= ξ a ± | | µ f ] := 2 f [0] +1) + λ f Θ ,...,s B λ J + | ,..., m g P 2 1 } ] p [ (Λ; (Λ; h ( − c 1 ( )∆ (Θ) 1 1 M b 2 b S b S ξ ρ – 20 – 2
b h ± n ( µ C N D ) ; − C ( . The action of f ab 2 + 1) 1 B {z =1 , B + entries b s ξ the resulting smallest 0 Y P c − [ J ,...,s M of ≡ r +1 ǫ 1 ( λ 1 1 1 ] and α ρ ab D m b | s µ h -subtypes in b = N + m 2 b + Θ ( +1 S [ c m 1 M L 2 1 ± D (2 µ AB r i 2 b +1 Ca f C 1 ⊂ 2 α N α − c M λ M ( D Θ = c i M − -module with smallest type Θ, viz. D [0]; a q λ Θ AB S = | b B g Θ AB ≈ − | b ± Θ) under i c M c ; M ,µ c M (Θ) := 1 2 ), it follows that f C (1) b ;Θ) := ± Θ) [0] ± µ b b ξ ξ 2 + 1 ; f AB ) f B f 3.2 0 b Θ). Hence (Λ; b ξ M ǫ D c M ; ] := ) takes the simplified form b 2 S f 1 2 b λ m + (Λ; -types and their [ -covariant (Λ; b := 1 S 2 b 1 m 2 | 3.32 m T (Λ; b C − λ λ N b ,µ S g -independent massive parameter on h (2 given by = ± [0] α 2 f m ) = iλ ,µ C 2 ± ), and ( where [0] P f ( 2 [0] and ρ It follows that any given value ) Decomposing f b m − (1) − b ξ where the Using the relations ( as discussed in section I.4.3.4 and below. The operator module. Removing the idealsan (as irreducible we shall see, at most one non- ( ators belonging to the subspace This shape is represented in the Schur module 1 where the last identity can be seen by expanding the Casimir of that hold in and non-constant operator in the module JHEP07(2009)014 , I ϕ ) is a (3.44) (3.45) (3.47) (3.48) (3.46) (3.43) (3.49) . of Θ is − µ . ] 2 f − + µ µ B ] f f M h , B ) for a fixed 1) . This field ; h = = tion matrices B ; − ≡ s ,...,B f f B ; Θ) [ , 3.36 s ,t + [ µ ) 1 +1 2 f I = 2 − I,I Θ M ) for ) for t ,..., I ; ] ( f f 2 I 2 I,S ǫ Θ Θ +2 ses , in accordance with our +1 , I ; ; M M t ) 2 2 h ′ f ′ f ′ I − ,...,..., 4 ; s ] ) (Λ; (Λ; ; Θ) M M +2 ;Θ = 1 . +1 2 I I (1) 2 I ted-adjoint representations of T T t ′ I s h ∇ [ any block ), namely that in the critically M (Λ; (Λ; ; , M -module (see section I.4.3.1); and T T ] iff ) corresponding to a partially massless 1] ) = +1 λ I ; Θ) (that is, (Λ; 3.44 41 − I g D E s (Λ; I − − µ 1) = ( f [ ;Θ) = I 2 f ϕ th block of Θ, viz. , s,s,t − Θ , χ I − I +1 . ; 2 f I M ; Θ) ; Θ) or of [ − h I + 2 2 f 2 f µ 1; 1] s,t ′ f M ; ( ; Θ) f associated with massive gauge symmetries b Θ=( (Λ;
2 f − M M
+1 M δχ I I T (Λ; I instead gives cut twisted-adjoint modules, as integrate integrate χ s M s [ [ T (Λ; (Λ; I , with (Λ; ∼ = , ) – 21 – of Metsaev’s critically massless gauge fields I T T (Λ; T 1] − µ ) are partially massless, I Θ ; Θ) f T ( + 1] ′ I χ ; − ; Θ) 2 I +1 I : C I + µ h ;Θ =1= h I,I − 2 f M ; 2 I ; ′ I I 1 f I ) in general. s 2 I,S M h s and [ b b M Θ) = Θ) = [ (Λ; = , ; , ; ] I M I ] 1 f f f ϕ 3.43 (Λ; 1 ϕ , one obtains shapes (Λ; − − I C I C T I B (Λ; h χ (Λ; (Λ; h I , ; ; b b χ S S 1 1 ± µ − C : : − I f 1) having gauge invariance I denotes the two roots of the characteristic equation ( s s [ ± ± =2= µ µ − [ , two dual indecomposable structures arise: The representa ± f f we then prove a part of the claim ( µ we prove ( I f ± µ s,t = = ( ] , then f ,..., 3.5 χ 3.4 λ ,..., ] f f ] 1 g 1 [ h 2 h ; ; critically massless C 1 is obtained by deleting one cell from the 1 s s [ ′ I [ = th block i.e. critical I µ are transposed upon exchanging generically massive twisted-adjoint (irreducible) := = In section In section Actually, setting I ′ I 4 non-critical (i) for non-critical (ii) at critical Θ Θ where We claim that, if 3.3.2 Proposition for indecomposability in the critical ca of height one — while the case and of the corresponding fields St¨uckelberg reduces to a non-generic partially massless symmetric tens general definition in item (iv) of section I.4.3.4, whenever value massless cases (see item (iii) I.4.3.4) it follows that We identify the above twothe modules, primary respectively, as Weyl the tensors twis where Θ tuklegfield St¨uckelberg where in the These fields St¨uckelberg JHEP07(2009)014 to 2 I ) . + 1)- ζ (3.51) (3.50) (3.52) (3.53) (3.54) ′ I consist- D (Θ 1 I − is obtained χ } o D , ′′ J,I ⊗ S Θ ∪··· ; Θ) are lowest-weight ≡ , 0 I ; Θ) e 0 6= ′′ I,J th block which means ± e ( ) and then send ( I 2 ± I J,K + ∪··· ; Θ) carries the massive ζ 2 D D same shape as =1; =1 imit though the number of , M ; Θ) . To examine the critical , and so on. For generic mass B I,J 2 B J,K ese cut modules actually arise ′ I
(Λ; ; Θ) ) M ) ) where . 2 0 generated by 1 C . − ′′ I,J } M o , ′′ J,K − D (Λ; diag ϕ 5 (Θ D rs ) ] (Θ th block of Θ , Θ (Λ; I − χ T s b (Θ) c b χ I 0 ϕ M S 42 ∪··· th block: The radially reduced ( {z ( f C ⊕ from the equations of motion/action by =: I + ∪ ⊕ ∪ − + =1 I ) s is the subspace of B J th block of Θ rs I 1 b 1 χ → =1
− 6= J B I ) M J − + D
= ( (Θ) ) I diag – 22 – D ] ; Θ) b ′′ I,J ′ I = χ f D =1; − 1 0 T {z B J e =: S − diag ( (Θ (Θ (
s that becomes an independent — generically partially D 1 ) χ χ diag rs I ′ J = − ) one may arrange the reduced field content as follows: b χ + D M I 0 ⊗ S -module (see item (i) of section I.4.3.4) from the Weyl ∪ ∪ (Θ e ) can be factored out is a manifestation of the fact that λ D ) 6= 0 (which involves division by χ ′ I g ′ I 2 : I (Θ) ; Θ) ζ ;Θ 0 (Θ ∪ ϕ T 2 e ′ χ I b ( S Θ) (with constrained radial derivatives) decomposes into 0 (fixed n | ( + M (Θ) → b ϕ D → ϕ ) spaces. ) ∪ n | (Λ; ! − ;Θ) with 2 = 0 for b I Θ) I 0 ( χ I e b M ϕ ( 0 there is enhancement of gauge symmetry in the b χ → − D ; Θ) := [ → 2 0 2 e I ( b M ζ Θ) system decouples from 1 ( I is obtained by deleting one cell from the − b ϕ b ϕ + D ′ I := ( 2 D I ζ The fact that The map extends to real Weyl tensors in ( 5 dimensional gauge field (+) and highest-weight ( In the limit by deleting one cell from the last row of the that the 3.4.1 Harmonic expansion To this end we first construct the harmonic map [ zero. The equations ofdegrees motion/action of freedom remain change. smooth in this l 3.4 The genericallyLet massive case us show that the generic Weyl zero -form module representations defined in item (ii) of section I.4.3.4. In the latter case, th from factoring out a tensorial massless — field system. One may remove where Θ all fields St¨uckelberg are “eaten” by the massive field fixing the gauge where zero-form module generated from a primary Weyl tensor of the there is enhancement of gauge symmetry in the limit ing of states that are invariant under JHEP07(2009)014 s b ) w −→ K h otally (3.59) (3.55) (3.62) (3.58) (3.56) (3.61) (3.60) (3.57) =1 J K (Θ) + , w P −→ + s θ , L , = + r 1 )= J ⊗ L θ , − p ) ( . + D Θ rs U | i block of Θ in order 1 w δ ( −→ ′ 0 s θ 0 ; th ;Θ . n Ω + := 0 J ψ e 0 | . The diagonal ground with n e ) ∈ x n ) and decomposes under + r = 0 k +1 = , x + ( l L =0 ′ 0 k ∞ ollows that ( Θ) ) L 1 X | n j 3.41 1 r ( − θ ( − β 6 + D · · · i , e + D ) j . ; Θ) . This set contains a unique l i 1 1 0 ∀ := k k θ is defined to be the state of minimal ) := ;Θ e + ( − ) ( 0 , C (Θ), i.e. its most electric component, , x ); j 1 L + D e + ( ( + ) i θ due to ( | − C ( ) Θ L m . The diagonal states are +1 | D + D Θ) , e l 0 j | ′ | 0 D i ;Θ ( 1 j e i , θ ′ θ n 0 0 ). By its definition this state obeys S ; Θ) to the primary (massive) Weyl tensor − ( + e θ Θ = 0 x 0 2 − | | D ; > e L C ′ 0 1 ′ 0 ( S +1 θ E e l 1 − , ψ | ⊗ | Θ) ( j | -type of · · · − i – 23 – + D + J,J θ s b ′ ) 0 i C ( s , + D 1 L ′ 0 ) θ J | l , j ) ; θ ( ) n D , =0 J ∞ ′ 0 act in ; ) j,k n p M ′ 0 + e e,θ + ( δ | ∗ ( e = 0 rs ) L | L +1) ) C ;Θ) x ( J 1 =1 j c h 0 ( M n l + ( p e − := ( -types arising in 0 + =1 Θ ( B L | ¯ ]) Y M ) + D β P J + 1 ′ 0 i ( s θ ∈S − 42 e,θ θ − yields X = ( ψ =1 B + D ;Θ Y l i J 1 0 + = θ s b e s modulo traces terms and − | j,j := | ; ;Θ) and δ + Θ + D 0 n i − r s + r i e ∈
X ′ 0 ( L θ Θ) = Θ 1 L =1 + | | θ m l ) + − ′ ′ . The lowest-spin state 0 0 j ; ( = ) it suffices to map θ θ ′ 0 ′ 0 + D m ; Θ) . This tensor belongs to P D | ( r, e Θ) . Let us seek a harmonic expansion given by the Ansatz (cf. t θ 2 s b | | ¯ | β + C ′ 0 = there are no singular vectors. θ M ; Θ) 3.53 0 ( 0 e := 0 (Θ) | j e e C Θ) is the number of boxes which are removed from the ) . It follows that the smallest ( act in ;Θ) is the set of -type | (Λ; w C j 0 ( θ θ 0 + D , e ( rs s C ( + J S M L n ∈ To show ( Decomposing under -type that minimizes the energy (see figure (Θ) + where where C as follows: where: such that state obeys Under the assumption that there are no singular vectors, it f For generic where minimal where s to obtain is given by symmetric massless tensors [ JHEP07(2009)014 -type (Θ)) (3.67) (3.69) (3.66) (3.64) (3.70) (3.65) (3.68) s f − +2 0 ǫ ), one finds to be an ) . (3.63) , . i − r +2 L Θ 3.29 α | ′ , 0 , . = 0 + + = 0 θ 1 | 1 1) , and the right-hand + r ) s ) 1)) B L − B − s + 2) ( ): s ( − , 0 ( 2 1 0 B ǫ D 0 ǫ ven by the rescaled Bessel B h ( ǫ h (Θ)) ( . = − 3.68 f so − r ; Θ)] ;Θ); and 0 + 2 ,...,t 0 2 0 ,...,t ) e + n ) n ( = e α + 1 B α ( M B s M s s ( + -types given in ( + = + ( 1 + d t 1 dx b 1 ; t 1 m E D ; ) (Λ; s s ( ... 0 ( ... ); ǫ + r )( ); |T 1 − 1 L s α λ s − ( are determined by the characteristic equa- ( g Θ] with 1 [ | 0 + 1 + , ν h 0 2 s h e +2) e ) [ 1 are states with fixed energy and spin; rs C 0 2 s x acting in ǫ ∗ ,...,r C ,...,r } M √ – 24 – } ) + 4( ( D ) + s 1 +2 ν 1 s 2 s ( 2 ; Θ)] J α ( iL 1 0 2) + Θ]+( ) which yield the useful relation d 1 ( r ) is determined by demanding ν | dx e traces = 0 where r + AdS i ;Θ)] = 1 s x ( i − − 1 0 [ x 0 ( − ′ ) 0 2 s θ + e 0 A.8 n θ Θ x ( ; ǫ | , i− C ; D ′ 0 ) ( 2 ′ 0 [ ′ 0 2 √ D e θ Θ nx | α | e | := 4 − | ψ ) λ ′ 1) 0 ) + ] = 0 g θ x |∈ 2 i [ | x 1 ( e − θ r ( α s ( 2 ] = 4 ; D Θ 0 0 0 | Θ b ) = ( n e Θ | C ǫ ′ e 0 | h ′ 0 x +( M θ ) of the lowest energy θ ( b ,x 2 + m − ψ [ ) ψ Θ (1) − r 2 | 2 ). Using the parametrization of 1 r, ′ 0 r L e,θ ). D C θ 3.52 + [ ¯ ( { +1) β (Θ) 1 0 ψ 0 r C ;Θ)] = f ǫ + { 3.36 0 M 3.52 e L := ( = ( = ( | denotes symmetric and traceless projection, and D ) 2 | λ 1) e,θ ∗ g ( is a coset representative of [ − 2 C + h 0 {· · · } C ǫ + in Θ, i.e. L − i) 0 ii) iii) the embedding function e ( The latter condition amounts to that 3.4.2 Characteristic equation Finally, the values ( tion leading to the following form of the characteristic equatio It follows thatfunction there exists a regular embedding function gi Using the commutation relations ( the embedding condition can be rewritten as with the roots ( where where side is given by ( JHEP07(2009)014 for ξ ξ 0 ξ ξ 0 f (3.71) . } ξ ξ 0 of ) becomes b Θ) ξ ξ 0 I ; f f ξ ξ 0 3.31 0 (Λ; b S 0 ∈ ; (3) the corresponding 0 b D C 0 assless primary Weyl tensor for AdS 0 n b C ξ ξ 0 b Θ) defined in ( ; ) +1) I ξ ξ 0 ∗ I f p Θ ) ( ξ ξ 0 | ∗ b ξ ∗ 0 (Λ; { Θ θ ξ ξ 0 b ( ( S C C ξ ξ 0 b Θ)= ; f -module (Λ; – 25 – λ b S g ; and (4) the most electric component of (2). ξ ∩ D ) ξ +1 AdS I ξ p b ξ ) in the ξ a P ξ ( ρ ∇ ). To this end let us seek the critical values ∇ ; Θ) := Im( 3.45 ∇ +1 ∇ I,I 2 I,S ∇ M ξ ξ . The four shapes associated with (1) the original strictly m ; (2) the reduced, critically massless primary Weyl tensor i (Λ; 1 χ ) ) ξ − ∗ T ∗ b Θ ,D ξ 2 ( (Θ b R C ϕ ξ in Figure 1 indecomposable with ideal which the representation critically massless gauge potential in 3.5 The critically massless3.5.1 case Proof ofLet indecomposability for us massless first cases show ( JHEP07(2009)014 = a and 1 can- upon b . X a (3.75) (3.74) (3.76) (3.72) (3.73) ) B b e , +1 by λ -type Θ 1 D ) for the , − i = m +1 I − ξ 0 D I L I I Θ 2.55 b | S p p ) ≈ a 6 > anchi identity P 0 ( p p b that simplifies the ρ 6 X , A ν ). According to the for for . α e), this reduces to the case (1) 0 A b ξ = 0 b +1 ξ +1 i 3.40 I I p p ≈ + = +1 b N ero -form module as follows: 0 D I f [0] I , ) represented in i p p b − +1) sitting in the same I X ∆ . b I valent to the ideal property of N p +1 λ p Θ ( | D 3.47 D b +1) ξ − − i I b Θ) contains the proper submodule p p b +1) Θ ; ( ]. | I b I ξ AdS . p + 1 − p 45 ; Θ) is massless except if ( ) = 0 : +1 f J b 2 , ξ I I I I h p p ; Θ), given by ( J,J Θ 44 2 M s ; (Λ; ) ( I =1 2 I given by ( b S X M J 7 + , I ;Θ) in (Λ; M +1) have the same scaling dimensions, viz. ( 2 I ) ϕ Θ J +1 0 modulo – 26 – +1 a can consistently deform the background vielbein I (Λ; p := p p be . This embedding implies that ( M a (Λ; ϕ b ξ b ≡ 1 6= N ( I 2 (in which case C ]. See also [ +1 b p I X (Λ; and J 0 + 2 = 3 using the explicit expression ( = D 41 =1 6= B > b ϕ b a +1] Y I S J J X th block. It follows that the generalized Verma module I ] 1 + 1)st row p p I p +1 b f [ X ⊂ I I ] = = 0 ( s + 1 p = [ ∆ D (1) I ∇ b e ν p +1 +1 ⇒ , are the scaling dimensions appearing in the radial velocity D D ∈ S b Θ it follows that = i i I I +1 ) is that +1 +1) + 1)-row projection ν I [0] I I +1 Θ Θ p I p p | | f ( D ) p b 3.74 b i ξ N in the primary Weyl tensor) (freezing ( ( J [ I + p i ( − ], later revisited in [ Θ b ξ 1 1 | + 43 b ). We claim that this equation has the unique solution N +1 , namely, only one block of height one (totally symmetric cas I ; Θ) , amounts to +1) B p ) contains a singular vector corresponding to the primary Bi I h +1 ) with primary type of shape 3.10 := I p ( I I,I := b e Θ Θ ; = 0 , so that the “graviton field” f [0] λ I ; 2 I 2 I,S a p 2 I =1= b e f ) M M B M λ = = 1 in which case it is partially massless. One consequence of ( If − 6 7 (Λ; (Λ; f ξ (Λ; ∗ B χ L which means that h where not be anti-symmetrized into the nomenclature of section I.4.3.4 the field cell operators. ForT fixed switching on interactions. Setting this ideal to zero amounts to constraining the Weyl z first investigated in [ V We have shown this for ( Integration yields theas gauge the field generically massive gauge field Cartan integrability of the aboveT constraint, which is equi calculations somewhat) and where ∆ constraints ( JHEP07(2009)014 + I b Θ) . s = 1, ; I (3.79) (3.81) (3.77) (3.80) (3.78) (3.82) k − p := f in other I 0 . , e + (Λ; +1 , b f I S s + s , b Θ). It appears = L 0 ; + I s as sor + r f I − p s L , ≈ f ) implies that . , using the notation sless fields if ) ) becomes rs − I . In other words, the 0 I 0 δ 1 (Λ; ) , i.e. that there exists e 1 Θ ) for b s 3.79 X I diag b ; S ; Θ) with singular vector − ] 3.52 k := 2 I I 0 1 Θ ) + D + ≡ e ave this for future work. ; i − t the following generaliza- 3.44 2, since the projection then ( I 0 M 2 I D 2 e + +1) the primary Weyl tensor. haracterized by means of any ;Θ > I M , x I 0 D p hat of − = (Λ; ( k e ⊗ S 0 ′ b | I ξ f 1 I 0 p C (Λ; ] assumes the same value in ergies in ( − λ , C − + D ; Θ) g ; Θ) , is smaller than in the massive i [ I 0 → 0 , e I 0 θ 2 D e e 1 : ( ( ( C ); ≈ − + + + θ 1 + D ( diag I I D i ) as an ideal. We claim that the indecom- D e p − p | I annihilates 1 + D n 1+ − ;Θ i ) 1+ Θ x I I 0 contractions. ; b D p N e 2 I + ( | ;Θ – 27 – A 1) I 0 ⊗ L b ξ M +1 e ⊗ S = + 1)st row | > C ) I J,J I I k p s p (Λ; p − ) =0 + ( ∞ ; Θ) ) f n M T I 0 L J e p − ( + ( it follows that ∈S θ + M 1 L b Θ) takes the form given in eq. ( ( -types arising in ; I I is now given by D − − + = p I 6= s + 1 p f : 1 cells in ( D + ,J f − B s Y
+ D − =1 i 1 := ≡ J I (Λ; 0 k − b θ S I electric + D ; + = p ′ C I f I 0 1 p S e | ; Θ) f − I 0 b + 1)-row projection ( Θ) must consist of the same twisted-adjoint representation + D e ; i I ( b Θ) does not contain I I 0 p . Let us show that this irrep is carried by + ; p ): θ ( (reducing in the primary Weyl tensor) D I f ; ;Θ), the set of + ′ p corresponding to the massless lowest-weight irrep I 0 3.4 I 0 f b e Θ) where e I 3.72 ( | ; (Λ; p I b S S + p (Λ; − b f S Identifying Referring to item (iv) in section I.4.3.4 it is plausible tha To this end we note that the existence of the singular vector ( 2 − (Λ; b presented here as a state in the doubled space [ of section a harmonic map D where to us that the reversedalgebraic indecomposable subsidiary structure condition cannot involving be c But Hence tion of ( case presented above since the operator with reference state given by the most electric component of posable structure of words, the decomposition order is reversed with respect to t which — as we already have shown — leads to mixed-symmetry mas S lowest-spin state will give rise to mixed-symmetry partially massless fields i creates a block of height one in3.5.2 the primary Weyl tensor. Harmonic We expansion le In via the critical most limit electric the primary larger of Weyl the ten two characteristic en JHEP07(2009)014 . . | I 0 e Θ due | − (3.84) (3.89) (3.83) (3.87) (3.88) (3.86) (3.90) (3.85) D 1 S − θ. 2 , D ⊗ i ). I I ) 0 := θ Θ U ; hadow I 0 ( . ′ ; e e I 0 2 I 0 e . | Ω ) , M I x ) ∈ p = 0 ) belongs to ( I ) I I , (Λ; − ) I Θ Θ 0 Θ , | ; B ;Θ). This module has a Θ ; I s 0 C 2 I | I 0 +1 ( θ 2 I nent, viz. I e e θ ; Θ) B s nce state given by the most ( ψ ( 2 I M h + 1 M ) is given by D as in the generically massive = 0 with + = 1 spelled out for composite M d 0 I (Λ; ′ ,...,t := 0 dx 1 tor with rank smaller than ) (Λ; ǫ 0 θ Θ , C ) h | 0 3.52 B i I (Λ; C I 0 s ) C I ( p θ ϕ = I 1 . t ψ Θ ; ∈ | − Θ 2 → I | I 0 = 1, -spin ... ) → θ +1 θ D ); I + +1 ( I | 1 I B s s 1)- module s Θ C s ( diag ( J 1 − − diag h n C 1 , ν I ) , ) − 1 ) p as follows: , D J i − x ,...,r D p I s +1+ } ( 0 – 28 – D (2 ) √ ¯ 0 Θ 1 β ( | ǫ s ( so ⊗ S I ( I 0 I ν 1 ) ⊗ S θ = 1+ r 6= | J I i I 0 I + ,J I + 4( 0 B has the same Θ ν ˇ e Y θ L ; ;Θ). ; Θ) 2 | − =1 ), and hence by all elements of i ; I 0 2 I 0 ) ′ I 0 ) ′ I 0 J e e e d I 0 e e θ x I dx (ˇ ( e ( Θ e,θ ; n | ∗ ( ( + x + √ ) ′ 0 + ) e e C C x D D | D h . It follows that there exists a harmonic map ( +1) I + I = 4 : : ) Θ p 3.4 | ( 0 I 0 ) = ( 2 ϕ ¯ e,θ θ β X ( x ( S D ψ ( magn s ]. | 1 I C I r Θ { | 42 Θ S ) = 0 I 0 X ∈ I θ ; Θ) carries θ M 2 I Θ ψ | = I 0 M s θ
( ) ; Θ) is carried by I I 0 C (Λ; e Θ ). Therefore, it decomposes under ϕ ( ( D C The critical limit of the smaller energy eigenvalue in ( 3.75 where the embedding function obeys This is a direct generalization of the special case We then seek a harmonic expansion for the most electric compo Thus massless fields in [ where the lowest-energy is given by leading to the regular embedding function On the other hand, the primary Weyl tensor different pattern of singular vectors. It has no singular vec to ( case analyzed in section so that This energy corresponds to the shadow Hence its lowest-spin state 3.5.3 Harmonic expansionOne via can most also show magnetic that Weylmagnetic there tensor exists component a and of harmonic s the map primary with Weyl refere tensor as follows: This implies the second-order differential equation JHEP07(2009)014 = 1 (3.91) (3.92) k , 0 belongs ) 1 k ( α p > ]; 1 can be replaced h
. [ and the lowest-spin Rank b i
Θ [0] (Λ; Θ) . For generic f - f
;Θ ned in section I.4.4.4) 0 e := 0 is non-empty for some R
e | ) ).
⊥ I 1 ly by the Weyl zero -form. p f, -forms with k
( p R α 3.23
]; 1 ∩ ]) so that
h [ ) B
h b Θ , i.e. ; ) +1) )
B I 1 s p k [
( ( , α 1 iξ 1
]; h h 1
b + h N f [ ,..., ] ] obtained by constrained radial reduction of
I 1 ∆ − p f f [ h
; − R ∆ 1 +1 | ] – 29 – +1
1 1 λ s 1 Θ h h ⊂ h p p p p p p •
f [ b N , as already stated in ( @ (∆ ; Θ) with its lowest-energy state ∅ 0 @ , respectively. = e b 6= ( Θ = ([ @ ⋆ (Λ; Θ) ] D 1 ⊥ f 1 − | | 1 @ , h ′ h f, 0 f [ R θ and @ | p p p p p p p p p p p p @ R ⋆ • = 0 ∩ ) b Θ) with 1 ) k (Θ) , that in its turn implies that all ( =1:∆ f α I +1) ]; S is nilpotent, the kernel is spanned by the 1 1 (Θ) and let us denote its potential module by h h f [ ------6 ( f (Λ= 0; ∆ b iξ ϕ 0 ′ 0 +1) e e 1 + h ] ( 1, that obey 1 indicated by the ξ h f [ i − (Λ; Θ) contains a non-trivial potential (in the sense explai ′ 0 Energy belongs to . A lowest-weight module ∆ 2 θ f , ; λ 1 ′ 0 ⊥ R 1 (Θ) . This is so even in case Ker( e h f, | f 1 , because higher-degree potentials are not sourced direct Since R S , ,...,s iff Ker( Thus, I > to Figure 2 by its eigenvalue the gauge field 3.6 Unitarizable ASV3.6.1 potential Occurrence ofLet non-trivial us potential consider module the subsector f 0 state Restricting our analysis to the critically massless values JHEP07(2009)014 0 b X ) . The (1) λ (3.98) (3.93) (3.99) (3.94) (3.95) (3.96) (3.97) -types 1 iξ h (3.100) g . b 1 − X , ) . − ] 1 2 0 , h [ 1 . , b Θ ro-form module ≈ ( 0 traint on 1 1 h 0 ≈ = 1 ,...,s b b U X I I > , 0 , 1) +1) = 0 , − 1 X 1 λ , 0 implies that exp( 0 f ]) +1 8 h 1 ) , possibly together with +1 ( 1 I 1) I C b k B ξ ≈ p i + 1 p , V potential in form-degree h 0 − e b y theorem only allows gluing isely, taking different combi- 1 igher form-degree, the latter E ; N 3.20 acting canonically on b 2 X h , B + 1 ) 1 ( − 1 1 − − s I = -type h f [ s 1 b P 0 f (1) 1 g he form AB 0 for h e R C h ( = iξ c +1) M b 1 P 1 1 + − > ) h ASV ,..., h k ] − ( 1 b f [0] 2 X b e I = +1) h s ∆ + 1 with 1 ; ∋ that yields a potential module, i.e. λ h I ]. λ · · · 2 reads ( − g p 2 s b e − ⊥
1 [ 1 , AB L , (1) f − h 1 I ) sits in the − + 1 b ] e ′ f c is given by h f M · · · 1 i − h R 1 +1 R [ – 30 – +1 + 1] 1 (1) h AB 1 − b Θ h b e ∩ 1 h b ( Ω 1 ( X + 1 + 1 = 0 iff b h ) i 1 2 h N ∗ L -connection. I 1 h b 1; , g U s p − b +1) ) U i i = 1 − I − d − + h ′ f ( ] − ) into 1 +1) 1 1 ] 1 R s = 1 − I 1 k k 1 iξ h h f [ h h ( [ ⊕ leaf with radius b ∇ b + + + ξ U b Θ ] ∗ L 2 2 − I ( D i 1 := ([ b f s s − ∇ 1 1 iN h ] f [ 1:∆ ∗ L b h 1 S i ) “untwists” the “twisted” covariant derivative in the cons h ∆ b = = U [ AdS and λ = b b ∇− Θ ) ) := ) I > 3.98 ) being the flat i 1 1 λ ( a -form potential k k g 1 1 ( (
a, + α α h b := ( β − I 1 ]; ]; , λe a f h ASV 1 1 1 − − e 1 I that will appear directly glued to the corresponding Weyl ze h h ab R f [ f [ + R := Ker( 1 ω 1 = h h ASV 1 ) − 1 b i R h f ( = ( b e 1:∆ e R ∗ L i =1:∆ AB I I > We stress again that, although Weyl’s complete reducibilit The embedding of The exponential in ( 8 the infinite-dimensional zero-form module toneed not one necessarily module be in the h nations unitary of ASV the potential. fields More occurring prec on the right-hand side of eq. ( and Ω and the resulting generalized curvature constraint takes t is logarithmically divergent. some zero-forms, it shouldhigher be that possible to find non-unitary AS in the reduced equation. zero -form constraint cannot be untwisted, however, since ( it follows that Its pullback to a fixed where Thus it is only the unitarizable critical value The unitarizable which we identify as the ASV gauge potential [ JHEP07(2009)014 := where is non- ′ ASV (3.109) (3.106) (3.108) (3.103) (3.105) (3.110) (3.114) (3.112) (3.104) (3.113) (3.111) (3.107) (3.102) (3.101) ′ a R k,q )-tensors. e T cells below ∈ D N ( with respect g , ∈ , , X . If sl k 2 + 1 q
, , 1 σ 1 , L s . ]) 1 , h ′ ASV + − h B 1) = + with ′ k,q 1 , R h − q ′ q T s α , α ] σ 0 where 1; = 0 1 R +1] := ⊕ + 1 + + 6 h 1 , [ = , ≈ | | − h ′ 0 1 ) 2 [ 0 0 ′ by inserting q , ′ e k,q h Θ Θ 1 B σ R | | σ ( X This amounts to T s s ) [ P , (0) + 0 + α , σ , p
:= := ]; +1) ) 1 1 α +1] 1 + m h | | ) = 1 ′ [ ,..., where 0 h
i i width(Ξ , ] (1) − ( h α , α [ α 0 ∇ 2 R ′ q a, . 1 ) ]; σ h b g 1 s +1 β (Θ g , R [0]; ( 1 h e ) := q ⊗ [ α + 1; h is the direct product of Z Θ + 6 is defined by + ]; a + α , | takes into the account degeneracies (due ∈ Θ 0 p ( | , | ) α β q − ′ [ e ′ ⊗ = 1 0 g N α p Ξ Ba ′ 2 q Ba R | σ decompose into L s · · · + c R M c M ⊗ 0 = = , := 1 k = B , B ) σ | (1) , 2 b 0 ξ ′ ⇒ ⋄ , ) b , ξ )) )= s 1 U 0 σ ,...,n – 31 – λ a Ξ g α 2 m a ( ) σ | ( , ( m (
i ∈ M 1 q β ie a
′ ASV , viz. ] α ⋄ s ) ] + Ξ := ([ 1 = 1 T ]; α,g − α ie (1+ e 0 have the form ( 1 Θ) +1 and p i h | q ) : i ( Ξ 1 [ h ) : width(Ξ 6 α [ − a h ) is obtained from , g Ω -tensors and g p ≈ b ] 1) , Θ a [ ( = ab } b g P ( q Θ 1 e m ( 1 1 + − ] e c h ⊗ p M ⊗ ∈ T ∈ ) = ′ q -types of fixed rank). The resulting 1 Ξ) (Θ + . 0 a , a , h 1 i i q θ · · · ab [ 0 1; ) = m ] i R Ξ Ξ i h ASV α α P = ( + a 1 ρ α 1 ω ( ]; i ∩ ] a a ]; − ∈ ∈ q R h 1 P i α 2 1 [0]; p ,...,s defined by e e ( ′ ′ ] ) h [ σ · · · 1 q p h [ 0 1 1; 0 [ ′ k s -type in , , Ξ Ξ 1 Θ h − N ( 0 0 [ 0 a R m 1 − θ + 1 + 1 i i ρ ρ d α i ∈ { − a a 1 =1 1 1 → n 0 and i M g = ([ = s g = g h h ie ie ′ q ( ≈ ) ) := − − − − : Θ R ) ∇ = = g := g 1 ( g : ( 2 , ( α = = = = α R s 0 α p ]; g ⋄ ′ ′ ]; -cohomology for unitarizable ASV gauge potential k,q k,q ]; 1 e C Θ = ([ 0 p 1 ] , h ′ [ 6 : Θ T T − Ξ ′ [ 1 h 1 0 +1] +1] ′ [ E σ R h 1 1 σ α α R 0 [ , , R h h 1 [ [ − 0 0 1 is the direct product of 6 6 h f σ σ 1 e 0 R , e ⊗ Θ is the smallest 2 s = − 1 ′ f to the ordering R where degenerate, then the maps 3.6.2 On The constraints The corresponding triangular module The space to that there are many internal the first block while adhering to the rules of Young diagrams. It follows that where the primary type-setting index where and the secondary type-setting index JHEP07(2009)014 n = 1) 2 . , , ′ 1 (2 R ϕ = 1 e Weyl (3.117) (3.116) (3.115) (3.118) (3.119) (3.120) (3.121) (3.122) (3.123) 1 , h 0 -chains are , with vari- − → , ) σ ) anti-symmetric (0) 2 . The chain with , ′ α , k ⊕ ··· ,... # 0 R 3 ) ′ 2 ] . ( g 1 1 = 0, two fields 4 3 ] ( R p → 2 [ g Z − e ⊗ p ) [ e ⊕ ⊗ 1, constraints in s , → Ξ e ⊗ (0) # · · · ′ 1 ) 4 the lowest ′ α ; ] " ) R 1 } 3 [2] at R (1) > 2) k ǫ ∈ ( ] k = 0 ′ α , 2 ⊕ 1 3 ǫ,X,R,Z,Z , ) − . s R ( ) k − (3 g 0 Z ′ 0 ( g = 0 and one Labastida-like field 1 0 ( g 3 − 2; h R ( ′ α g → → 1 Z . e − =2 R ⊗ , X 1; s {z ) ) ⊕ ( g 1); Ξ] 1 → with 2] 0 =0 Ξ − s 1 , ) ] ǫ (0) (1) [ − ,g 2 1 ′ − 0, and first level of Bianchi identities in ′ ′ α α o [3 k ⊕··· ∋ 1 s [ (2) In this generic case the triangular module [ 0 =1 R R 1 R i s ′ 3 ′ ′ α > q ( ( c [3] " 0 . For [( 2 2 X | T , 0 R R ǫ g ; ( ǫ = 1 is a parameter 0 R R 4. } 0 > 2 0 ⊕ n 1 2] ∋ ′ , – 32 – > , R > ′ 2 6 α > → → → ) 3 T 1 =1 [2 k {z q = 0 = R - cohomology in more detail in the cases 2 p ) ) ) with − g 1 → 0 3 − s − h σ − ) ⊕ 1 1 | C (0) (1) (2) ( o σ k 6 , k k ; 0 0 for ′ ′ ′ α α α ′ 1 ′ − − , (3) M 1 1 − } 1 − 1 p ′ h α , R R R R 1 p R · · · c [3] ≡ k ( ( ( − s 6 R 1 {z = 1 1 1 =0 ǫ = ( ⊕ , 1 H g 2 2); | U 2 k 1 The irreducible module carrying the unitary representatio p ′ 0 0 X X X k , 0 . X R (3 → → → → > p =0 1 via a constraint in = ⊕ → k 0 and ) ) ) ) M 2 and g 1). ) 1 , , R g (0) (1) (2) (3) =1 − > - cohomology for (4) > ′ ′ ′ ′ α α α α 2] ,g 2 ′ ASV − ′ α R ∼ = α< , , R R R R 1 B with σ R R =0 s ( ( 0 ( ( [3 ′ ( q = 0 0 0 0 1 o 0 ǫ ǫ ǫ ǫ T ǫ , parameters in R q,g ) for ′ ]; = 0, two Proca-like field equations at · · · → → → → 1 = 1, = 1 . The degree 1 module is “glued” to the degree 0 module via th ֒ ֒ ֒ ֒ U → ′ ASV R ֒ 2]; h ( ′ g [ 1)) is given by 1 , T g -content of the parameters is given by ) 0 1) in , T c [3]; 3 h [2 , = m Ω Ξ denotes the direct sum of shapes given by the contraction of 1 contains the differential gauge parameter given by 1 ] [3] 1 : 0 2 (2 ∈ ′ 0 − R k 1; (2 Z [ − C =0:0 =1:0 =2:0 =3:0 ǫ i R (1) at n = q q q q − = S c [3]; In what follows we examine the The non-trivial q q + + + + = 2 D g g g g ( ′ 2 + + R g where The corresponding triangular module where such that cells from the shape Ξ . D tensor The example Θ = (2 The case equation at R ables (see figure and n g where the JHEP07(2009)014 0, and ≈ ǫ S Ξ (3.124) (3.130) (3.131) (3.126) (3.127) (3.128) /Bianchi (1) ∇ 0 implies 9 , leaving a metric-like ∇ ; Proca-like ≈ . A + , S . A ϕ # Ξ] Ξ # P , 0 containing the # and Ξ] Ξ [1] 1 ∇ 2 i 1 (1) s # and [1] [1] (1) S ≈ Ξ] (3.125) i i ); − 1 ∇ − := e ⊗ 1 Ξ " [1] Ξ A 1 A 2; s 1 i − s A F Ξ s , [( [1] ⊕ 1 − P i ϕ " " = 1, where s , leaves: i) Labastida-like ϕ 1); # 1 , leaves Noether P 0, s q " D only in case one has an action 1 ⊕ ⊕ [ D − Ξ ⊕ A ≈ # # 1 + ⊕ − 0 and s [2] , so that N g 1 1 2); Ξ] ; (3.129) # S 2 i Ξ] [( denotes all possible divergencies [2] s (1) ϕ ≈ 2 . ⊕ − [2] S − − e (1) ; fields ⊗ Ξ " e i ⊗ 1 ϕ 1 ǫ (1) 1 − Ξ ∇ ⊕ Ξ] ∇ Ξ] s , s Ξ s ⊕ F 1 Ξ ∇ 1); + [( s [1] [2] " # # " i i φ A " − 1 2 ⊕ Ξ ); F ); ⊕ Ξ 1 2); Ξ] 1 1 ⊕ − − # s ∇ Noether identity Ξ can be used to remove s # s ⊕ [1] 1 1 # [( i − [( [( 1 ǫ 1 s s S 2 := 1 ϕ (2) [2] 1 B " " s P − − (1) e e ϕ N ⊗ ⊗ ); Ξ] [( 1 − 1 – 33 – ∋ 1 e P ⊕ ⊕ ⊕ ⊗ s s 1 s ⊕ Ξ Ξ A ) # # s [( Ξ " " Ξ] − ⊕ ϕ " σ [1] ⊕ ⊕ F (1) (1) ( 1 1 . i 0 ⊕ 1); Ξ] e e , # # ⊗ ⊗ φ ∋ − − 2 ); Ξ] # 2); 1 − 1 1 ) Ξ Ξ H 1 s (1) (1) s s 1 Ξ 1 1 − s − [1] [1] [( s e e s s ⊗ ⊗ i i σ 1 " . Since all field are now on-shell, the divergencies ϕ ( [( s " " 1 1 Ξ Ξ ϕ , S , respectively. In the above ∈ [( ∋ 1 " ⊕ " ⊕ N A ) A ) H P ∈ ∈ ∋ − (0) = 2, whose content is listed in appendix σ = 0, where = 3, whose content is listed in appendix ) ) ′ α ) ∋ ( and q q − g 0 R ) , (1) (0) ( denote the three Lorentz-irreps that occur in the dynamical σ 0 φ − ′ ′ α α + 1 ( + + σ 2 H S R R , g g q ( X ( ( 0 ; and Labastida-like field equations 2 0 , 2 1 1 H ≈ R X and H S A Ξ , ∇ ϕ , respectively, of the residual parameter ǫ := Strictly speaking, one should use the terminology 9 (1) S P The chain with principle. field equations in identities in and ii) a Bianchi identity for the Proca-like equations, in transverse on-shell Lorentz tensor leaves dynamical tensor gauge fields in in Ξ . The parameter can be used to gauge away d’Alembertians of field. We shall use similar notation below. The chain with where The chain with Thus the dynamical system consists of a parameter ∇ equations of motion of the schematic form a mass-shell condition for leaves Proca-like field equations in the Lorentz-irreps The chain with JHEP07(2009)014 ) + 1 s A ; Ξ), 2; Ξ); ϕ Ξ) := Ξ) := 0 and ) Ξ := (3.132) − 3 s,s [3] [2] ( i i , ∇ ≈ s ,s ) ϕ 2 1; 3 2; − R ...... − − s ,... ( 2; Ξ) 3 ,s Ξ) := r field equations; ,s A 1 1 − [1] i − − s 2; ( s s,s ( S ( − 2; Ξ); fields A Ξ) and P A S ϕ 2 − [2] P i F s,s R ...... ( . - cohomology restricted to the ,s 0, 1; A 0, 1 − η,ǫ,X,R,Z,Z 0 and ϕ NH• • • ϕ − ( σ ≈ e to the Weyl zero -form module. are higher Bianchi identities. The P − ≈ ≈ ,s is the differential gauge parameter; S H 1 s 0, A ( ⋆ − (1) S A = 1 are the Proca-like first-order field s , respectively, can be used to gauge ; Ξ) ≈ ǫ ( ∇ and q A (1) S ǫ S N 1 ⊕··· ∋ + s,s ∇ ϕ at ( ′ 3 S R ...... ϕ + Ξ), • (2) ϕ R and F A Ξ • • • [1] ∇ i Ξ) and Ξ ⊕ S =1: i) The ∇ ǫ ′ 2 Here the triangular module (see figure + [1] ∇ 2; 1 , i R – 34 – h ǫ − ϕ 1; Ξ ⊕ 4. ,s 4 then the dynamical system contains ( 1 Ξ) := ∇ ′ 1 − Ξ) := > 0 − > [2] R i s [1] ,s 4 R ...... i ( 2 1 ⊕ s , whereafter all fields are on-shell. The on-shell gauge 1; cohomology, it is part of the A s ′ 0 2; A Ξ) := S − − − − ϕ R − − [1] s σ 1 i ( 1 s ⊕ (2) Ξ), ; ,s s S s,s 2 1 ǫ ( ∇ [1] ′ − 2, i S s,s − ϕ ( 1 R s 1; > = 1 are the Einstein-Fronsdal-Labastida-like second-orde ϕ P ( − 0 . The parameters and − q P ⊕ A R ...... 1; Ξ), 0, B 2 P S ≈ at ′ − s,s − cohomology in the case of ϕ ⋆ ≈ ( 0, R S − A (1) ≈ ϕ σ = 2, S s,s 2; Ξ) = ) ∇ = 0 are the dynamical fields; iii) the ( S 2 1 ǫ , (1) = 2 are Noether/Bianchi identities; vi) the , − q 1 h s (2) ϕ q ∇ (0) (1) (2) (3) (4) ( ′ ASV ,s 2; Ξ), . The at α α α α α α 0; and Labastida-like field equations ∇ (2) 2 at T + R R R R R R − + • ∇ ≈ S − A ϕ s,s s S S ( ( is the primary Weyl tensor which “glues” the potential modul (2) Ξ Ξ A A The case potential module. While it is not part of the total v) the equations; iv) the ∇ Proca-like equations ϕ ∇ Figure 3 parameters ∇ away ii) the F parameters can then be used to gauge away all fields except and one can show that if JHEP07(2009)014 ′ ) g he 3.6.1 3.96 (3.133) . associated D I 0) associated equation for χ . = 1 is depicted p> defined in ( 3 1)) is given by I = 3 +1 to [3] , 1 − I [3] 0 ′ le f D g κ X . ; (3 R -2 objects that could with potential whereas the D -forms ( ( p 1)-irreducible Skvortsov − [2] I 1 f D X − · · · odule R n [1; 1]) . All the fields take value ; ASV , , } 2) tuklegfield St¨uckelberg ss limit. We interpret the field sts of the ASV module plus the [1] , D 2 , gauge parameter and the zero- forms in flat space coming from [2] 0 X (2 th the various fields. St¨uckelberg — that is projected away by the (3 (see caption) to all the radially κ , 0 ′ , 0. l reduction from has a smooth flat limit in the sense iso = 2 g [3] o different gradings used in section 0 [1] ′ 0 =3 [1] 1 , X {z − ≈ I b g Θ = ([2; 1] g η ǫ Y 3.3.2 f 2] , R [3 0 [2] 0 obtained by constrained radial reduction of traces Y X | b f Θ) with ; , − [1; 1]) . } R , • 2] 2 – 35 – , is associated with the [2] • 1 X ⊂ 1 =2 [2 ǫ {z g , g (Λ=0; χ 0 , b = 1 ϕ [1] | C 1 (1) • ′ 0 [1] 0 ; Y g η ǫ 1) 1) } ∇ 2 2] , , , (Λ; Θ) , 1 b Θ = ([3; 1] f , d [3 {z R 1 =0 g = 0 1) := • | U 1 ′ , • 0 b Θ = (3 Θ = (2 ). We then argue that Θ) . [2] that remains upon imposing the subsidiary condition on t Y g — ǫ ( 1 (2 b ϕ 1 χ 2 . The grading b Θ) with = χ 3.72 ′ − C -form fields obtained upon radial reduction of the ′ + 2 R g p g (Λ=0; 2 1 = = b 0 − − . These quantities do not form a trivial pair because the field ϕ g ′ 0 R g R R R form sector thus obtained consists of the fields listed in tab th block — see the discussion in section , where we have assigned a new grading − in I 2 p . The set of -cohomology of the triangular module associated with [2] 0 − Y σ as a zero-mode for loses its source precisely in the unitary critically massle The irreducible module carrying the unitary representatio The We note that the cohomology contains two antisymmetric rank [2] [2] 0 0 3.7 fields St¨uckelberg and flatIn this limit section we first look at some examples of the unfolded m grading is associated with all the fields obtained upon radia 3.7.2 The example of of the BMV conjecture albeitthe with sector additional St¨uckelberg in topological AdS. p- 3.7.1 The example of and in figure is Table 1 the gauge field with the Skvortsov module starting from in table Let us consider the subsector which we, based on the analysisunfolded fields performed minus St¨uckelberg so the far, Weyl claim zero consi -form of the subsidiary constraint ( The with the module associated with reduced unfolded variables, including thoseAll associated these wi fields are thus various components of the in Lorentz-irreducible shapes. The relation between the tw form a trivial pair,form namely the cohomologically nontrivial Y Y primary Weyl tensor JHEP07(2009)014 2] , 2] , b Θ = [3 1 [3 0 =5 . X — κ g 1 2] , [3 1] 0 , b Θ) with e Y [3 0 spacetime. , κ ⊕ , 1] D , 1] [3 1 , (Λ= 0; e [3 0 X b ϕ AdS η , ⊕ , ◦ 2] 1] , , 2] • , [2 [3 1 1 =4 [2 0 2 . The solid shapes represent e X X g — κ , see caption of table R — — — — ′ D , g 2 1) field in [3] 1 R — — , AdS pes. X [3] , 0 ⊕ ⊕ η • 1] 2] , , , [1 2 [2 1] 1 0 , 1 e Y X [2 0 R — R — — — , , κ , 1] 1] , , 1] [3 [3 , 0 0 e [2 Y Y 0 0 η ⊕ 1) gauge field in , , , , R — — 1] 1] – 36 – , , 1] 1). The dashed shapes represent the cohomology for 1] , , [2 [2 , 1 1 =3 [1 0 [1 1 1 e X g ζ ǫ X (2 − ⊕ ϕ , — R — — 1] , [2] 0 1 [2 0 2 e X — R — — — Y [2] 0 − , , η — — R — — , 1] [1] 1 2 , c [3] . For the definition of the grading − [2 [1] 1 X 0 0 ǫ , — R — — — Y Y , , 1] = 2 = 3 = 0 = 1 , 1] , ′ ′ ′ ′ 2 [1 [3] =2 -form fields obtained upon dimensional reduction of 1 0 [1] 0 [1 0 g g grade g g − p g ζ ǫ Y Y — R — — — — • 2 -cohomologies for the unitary, massless, spin-(3 -cohomology of the unitary (2 • 1 X − − ǫ , σ σ , = 4 = 0 = 1 = 2 = 3 . [1] [2] =1 1 0 • 0 [1] 0 g g g g g g ζ ǫ Y Y . The set of . The • 1 =0 [1; 1]) . All the fields take value in Lorentz-irreducible sha • 0 — ǫ Y , g Table 4 2 1 − − 0 R R R ([3; 1] Table 3 Table 2 the closed Weyl zero-form the cohomology for the dynamical field JHEP07(2009)014 ) , one leave 3.10 + 1)- nd the (3.134) (3.138) (3.136) (3.135) (3.137) D 1)-module: background. − D b Θ) in ( +1)-dimensional ( he reformulation ) has the limit , , D b ϕ 6= 0 containing the 1 D AdS (2 Θ , ; so ) 2 ′ 1 e aforementioned frozen Θ M ry massless lowest-weight he radial reduction ( ] to the , 4) for Λ , ge field (Λ=0; Θ) 3 , ) ′ (Λ; =0; ) 2 1 .2) fields St¨uckeberg associated Θ BMV quations. M v in [ ⊗T ; (Λ; Θ) R 2 1 ) ] at the level of the field equations which were found to be very useful − 1 ′ f U M ∪ 19 ( (Λ=0; Θ R 0 (Λ=0; (Λ; Skv ⊕ T 0 R C ] by using an oscillator formalism. Certain )=Ω (Θ) 3 1 E (Θ) Θ (Λ; Θ) 1) and radially reduced the ( (Λ=0; Θ) ; 1 BMV 1 − M 1 2 1 – 37 – − 1 BMV Σ smooth limit h ASV f M ) do not contain any elements of form degree less ∈ Σ ′ ′ e M R S ∈ extra , D ′ Θ still form a massively contractible cycle, so that the Θ R 0 1 (Λ; 0 ) lead to the following reducible 1) contracts to the direct sum of massless irreps of h → 0 −→ λ → C X − −→ λ ). ) 3.72 with first block smaller than that of Θ . 1 (Λ = 0; Θ +1) (Λ; Θ) = (Λ; Θ) = 1 Θ 1 e , D R ; − − h h 1 1 3.3.2 ( 2 1 ′ f f (2 b ξ (Λ; Θ) R R so M − 1 f (Λ=0; Θ) := and (Λ; R ) are the Skvortsov modules predicted by the BMV conjecture a 1 ′ T contains a finite set of topological fields. For a fixed Θ and Λ = 0 6= 0 without affecting the smoothness of the flat limit, which we − BMV 1 h R Y extra R (Λ=0; Θ 1) in accordance with the BMV conjecture. Skv (Λ; Θ) is a massively contractible cycle (see section I.4.4. − , nor of form degree 1 R S th block (see section h I , D The above reducible module has the We then proceeded with the following steps: We started from t Finally, we note that it should be possible to project away th The Weyl zero -form module (1 3.7.3 Smooth flatTo limit repeat, the analysis so far shows that in the unitary case t operators were constructed, the so-called cell operators, followed subsidiary constraint ( by extending the unfolding analysis carried out by Skvortso for an alternative proof of the consistency of Skvortsov’sof e Skvortsov’s unfolded equations for a mixed-symmetry gau To this end, we reformulated the equations of [ which together with harmonicspace expansion representation shows of that the unita for future studies. 4 Conclusion In the present paper we studied the BMV conjecture [ where iso field content for Λ dimensional flat space with signature (2 than BMV fields St¨uckelberg as well as thewith frozen (see section I.5 where complement can show that flat-space potential modules JHEP07(2009)014 2 / ra- + 1)th ] for the I 7 p ;Θ). al-reduction 0 e ise expression ( mulation of ar- D , an ideal, is said I 0) of the unfolded under a subalgebra ing comments and -2004-512194 and by R )] and verified that the orted in part by Scuola p > case of mixed-symmetry 8 construction and it would ting to make a precise link rt. This work is supported ni, P.P. Cook, J. Demeyer, 3.72 f this paper. We also wish ost four rows. The nontrivial ints in the general case. For ozen fields. St¨uckelberg ith one cell in the second row. M.A. Vasiliev for discussions. sor products of two spin-1 is a Lie algebra (or more gen- . ( eyl tensors is a good sign that d-symmetry fields. cq, T. Poznansky, A. Sagnotti, -module l tion in terms of subsets of un- l act enveloping-algebra approach y gauge fields. In a future work operators and their commutation . If B -form sector ( ⊎ p )]. Next, we constrained the ( A 3.17 – 38 – mixed-symmetry fields cannot be seen as single- along the lines proposed by Zinoviev, see [ ) and ( containing an invariant subspace D R 3.12 AdS generic ), ( , (Θ) carries Metsaev’s unitary representation E ϕ 3.10 . Then we constrained the Lie derivatives of the fields along a ]. The oscillator realization of the constraints in our radi . A module D k | 46 R AdS that we displayed in the present work, in particular the prec -forms and the gauge fields needed for a first-order action for D p AdS is denoted by This non-factorization property maybe is an artefact of our As shown in appendix In relation with the previous issue, it would be very interes In particular, we were able to prove the BMV conjecture in the l ⊆ unfolded fields to construction does not appearconstrained to oscillators. allow for a strict factoriza row of the internalgeneralized indices Weyl carried tensor by of the zero -forms [see eq cases where the shape associated with the field is a long hook w be very important, we believe, toto investigate about the an lowest-weight abstr modules corresponding to generic mixe Acknowledgments We are thankfulE. Skvortsov to for several K.to useful thank Alkalaev remarks G. on and Barnich,F.A. an F. Dolan, E. early Bastianelli, J. Engquist, version X. Skvortsov D. o Bekaert,E. Francia, M. A. for S. Sezgin, G¨unaydin, Lecler D. Campoleo Sorokin, enlighten Ph.A. Spindel, F. Sagnotti Strocchi, is E. also Tonniin and thanked part for by his the encouragementsthe EU and NATO contracts suppo grant MRTN-CT-2004-503369 PST.CLG.978785. and MRTN-CT TheNormale research Superiore of and P.S. by was the supp MIUR-PRIN contract 2007-5ATT7 A Notation and conventions The direct sum of two vector spaces is written as dial vector field [see eqs. ( system in of the constraints that would enable one to project out the fr between these gauge fields whose corresponding Young diagramsconsistency possess at of m the constraints imposedthese on constraints the are generalized correct W we for would arbitrary like mixed-symmetr to furtherthis, it study is the crucial to consistency haverelations. of a better our understanding Also constra of of the cell interest is to study further the bitrary mixed-symmetry fields in ton composites, though certainfermionic singletons long-hook [ fields arise in ten erally an associative algebra) thenk the decomposition of an JHEP07(2009)014 ) 1 i is 1, ( ± a, p and ¯ β > (A.2) (A.3) (A.1) = +1 σ i I h and m where , th column. ) i i ) ( p > a, i β D . . . ) where ) acting in the th row/column, i 2 } m -irreps with non- ± ab h m k . ν , + ( ) ) 1 -types then we refer σ, η sl h n n := k are commuting or not. )-types in symmetric are invariant in which ) , with formal limits etc. labeled by indices } and heights λ . D a . − g ( 2 i I epending on how they are P s ν ] C ] where α / ] sl {z { ( a +1 = ( I A R P +1 ,...,m cells in the ,...,m ,...,m so = diag( b , Θ n [ 1 c + ite-dimensional target spaces i M c α +1 ν -labeled box in the B 1 + 1 − m [ > s AB iη a h and i i η D his paper). In unfolded dynamics t of symplectic algebras, the only point 6 . We let is not invariant in which case one consisting of | I I I ,m i η ) and s rs k ,...,m I 1 } | 2 h + 0 h M he generators ν ] = 2 R − m c =1 ,...,m ,...,m (2 B I by using “cell operators” 1 1 1 D ] , P s sp {z and P A m m obeying the commutation rules = ab by removing or adding, respectively, a cell ,...,m c 1) denote the “canonical” Lorentz and spin M ab 1 M forms a module, the Schur module, for the . In particular, we refer to finite-dimensional := [ ) and [ AB B S [ − | m M R ) = ( ) = ( – 39 – C c B M S +1 one refers to finite-dimensional n n +1) with metric , p n explicitly D l i η ( D ab denote Young projections on shape Θ . We also use ( ⊂ so th row. Schematically, M i ]) := ( k Θ so 2 ) , obtained as a formal limit of B P ] = 2 ,...,m ,...,m := ,...,m h i i iλ )-types in anti-symmetric bases). Extension to traceless ∞ 0 ; obeying s ( CD ) . D B m ( -types, which we denote by Θ sl I s in the c 10 k M [ / sl , ] = a b R -typesetting of (( ,...,m ,...,m ( k can be treated AB 1 1 , P , respectively, remove and add an − 1) and ,..., ⊕ a ] ) ; or (ii) decomposable if both ] c ν S m m M i rectangular blocks of lengths [ [ 2 ( I ( P I − [ ) ) / h a, i i := 0 . We let 6 ( ( ; ¯ )-index = β B R ), respectively. 2 a, a, 1 l ( ) , and with generators , D s | D ¯ s +1 β [ β ( ∞ n , rs . The space of shapes (1 R ] D and sl (2 1 m , δ ] so are labeled by ( I h i 1 denote the real form of [ so ; of shapes with total height a, 1 − g b = := s β and ± l ,...,B ± | ([ ν = ( 2 m S R ∞ , ) and ,...,n etc.. Correspondingly, if there exists a slicing ab η i Young diagrams, or row/column-ordered shapes, with Infinite-dimensional modules can be presented in many ways d The Schur module We let ∞ := We are here abusing a standard terminology used in the contex α = 1 (2 0 = 1 10 , Lorentz-irreps as Lorentz types (that willone be may tensorial view in t typesettingfor as unfolded local sigma models. coordinatizations of We set infin aside issues of topology. i We let degenerate bilinear forms as sliced under various subalgebras. If m case one writes defined (see Paper II) to act faithfully in sp and α the block-notation to such expansions as an subalgebras, respectively, with generators Lorentz tensors leads to Howe-dual algebras containing the bases) or widths spanned by the transvections writes being to make clear the distinction between the cases where t to be either (i) indecomposable if the complement of spaces for a shape with Similarly, I universal Howe-dual algebra JHEP07(2009)014 , + , is a (Θ) iL ) := ρ e φ 0 ) (A.4) (A.5) (A.7) (A.9) (A.6) (A.8) ). − ξ -types (A.11) ( h ǫ ab . ;Θ δ 0 0 := . 3) (A.10) ] M e ± r , ( = − ab is invertible west-weight dS 0 with Killing L a ) modules of D s D ω P ) L [ a e , the isometry ( t ξ 1 − 1) into ξ i e | 2 ( 2 1 ǫ 2 − iδ L − δ λ λ − | and , := / D d = 0 ( 2 . The Lie derivative L ] = 2 . λ 2 so . We let := ab ± t . − λ 0 es related to Θ , ǫ ⊕ := Θ] e M ∇ | ,L 2) = ) and purely imaginary for ab ± , 1) m b rs − (2) [ isometries σ b e 2 D − AdS c D from a unique lowest-energy M a M and so 2 [ C e L ω D ab 6= 0) with representation 1)( 2 ) = 0 (one has ( c ∼ = − ab − λ with a ω M , AdS ] , E D so h ω 1 2 ω ( 0 + ) ξ g b + i Θ − r := − e + − | ab ∼ = ab ( are defined as follows s L s = -eigenvalue ξ [ R ab λ 2 M ( E − g AB . -irrep (Λ ab R of an unfolded Lorentz tensor field 0 the connection Ω can be frozen to a , 1 2 C ǫ λ ] dω ab , + r ≈L ab 0 2 g M , ω + L η ) ± r ) are the irreps obtained by factoring out a Θ ( | 2 1 M 0 a e ω with L )] + := AB s = ξ 0 2 [ P R ≈ iλ i ± ǫ + + b 2 a ± M ;Θ = 1) with radius and we use conventions where the exterior total – 40 – i e ab 0 e a 2 1 C T 6= 0 then = 0 = a e ( 1) algebra reads d σ P θ ) act from the left. If the frame field ]= 2 a r ( i ξ ± i a ǫ ( ξ ρ − ;Θ λ ± r ξ ( ǫ i e i 2 0 2(1 + − , R D δ ± e by λ b + i − , D +1)]+ D e a dS = E,L ξ 0 0 with − [ θ b ≡ (2 ǫ |± e a [ to denote equations that hold on the constraints surface. b 2 . The generalized Verma module is irreducible for generic d i is assumed to be real for 0 2( so e ) ω := + e 1) and if a i ≈ 1 ab − ω + P θ := ǫ 0 − − , P E , = ; a e ) obeying 1) and a ξ λ ) [ e + − rs Ω+Ω ) state 2 | P ω 0 − L − r , D δ ) to denote lowest-weight (+) and highest-weight ( ( d e de a e − 0 ǫ ρ L + M := (1 is := = 2 e ∇ + 2 := := − + ;Θ ( i L L ξ σ iso )] = ξ 0 θ a ( i = rs 0 + r e Ω := 0 one sometimes deals with harmonic expansions involving lo ; R T ∼ = e L a ± | D ( e < ;Θ ( iM λ ) 0 2 1 )= ± ξ g e ξ ( -valued connection Ω and curvature ( ǫ ( and the inner derivative D λ = AdS δ ǫ , i.e. singular vectors arise only for certain critical valu D g ] = 2 d 0 r | ) and 0 e + s λ 0 g [ M ,L . In compact basis, the 2 ;Θ 0 − r where = 0 then C ± . The We use We use weak equalities e In unfolded field theory the mass-square i L 2 a ( [ that are sliced under its maximal compact subalgebra θ D λ + T ; λ ξ e along a vector field respectively, where derivative we define the inverse frame field If algebras of leading to the mass formula g fixed background value, breaking the diffeomorphisms down to In the maximally symmetric backgrounds | In the case of Λ the eigenvalue of spaces where parameters dS i and are associated with a cosmological constant Λ = By their definition, the modules (+) or highest-energy ( all proper ideals in the generalized Verma module generated D values of (dynamical field, Weyl tensor, . . . ) carrying a JHEP07(2009)014 ) ± is ∼ = i − N ( ) and a, − , D (B.3) (B.2) (B.4) (B.1) γ AB + where ). The (2 i c (Θ ) ,...,ν M ) iff the i and non- 2 ( ) = ¯ so ∓ − , (1) a a, T ab ¯ γ P and 0 such that (Θ . The linear ng a strictly ( ) ) for some κ c i ) ⋆ = 1 ∓ M ] λ i ( -types Θ ′ − ( i > ′∓ T , + m ρ , ε,ε κ ). Factoring out [ (Θ ∼ = δ ) := N + 1) Ba ), viz. ∗ , ab − ,N , and , and ¯ c ) ] α (Θ M , l i ∓ b )) a ) ]= ( M i ′ B δ + . ± i,a ( ( (1) ε α, T b j i ξ +1 λ ( ¯ α a, ) T δ ; , ε , δ λ , (Θ β ∼ = C ,κ ε ) ) ng projector, respectively. ) i ; ± δ i ¯ ∗ ( = α ± ( − als T ρ r [ )= ) ) are contractions of α, ensional subalgebras ) := a ( i,a + ± a Dν ) of “depth” P ε ) = (0 1 . As a special case for all values of α ( l P variant under the canonical trans- ± (2 ), the oscillators obey ) ( (Θ 2 and in (Θ ( . i ) j,b + ( > λ ab i C ± ; + so = [ ( ¯ − α (1) ρ (Θ ; > i ,κ T M T ) oscillators, corresponding to tensors ) 1) i D i ,N 2 1) representation that decomposes into − ( ) ( ± i,a − := i ± ρ ( , ǫ iλ ¯ (Θ) := Θ, the irrep consisting of a single (1 α − so T nor Θ 2 1 ε − ± )) l (0) 2 and ) for + 1) C with canonical T ]= , D 1 . Their finite-dimensional representations and b ± ; b Θ with its canonical representation ] where the unfolding of mixed-symmetry fields in flat − . In particular, ( > λ − (1 ; ) D 47 i i , P – 41 – (Θ ± ( ( ,κ , δ , ) a ) + ( iso i = ± a, i ] ) ( ± ( sp P 2 the duals T , ǫ C b i,a j,b Θ ) κ β ; ¯ 1)-irreps with (a) largest and smallest α > = 0 . 1)-type C + ) with , α i ; + − ) i,a κ ± − ¯ We also let α D α Dν ( , D Ba (Θ α, ) (2 11 i so ± , D c ( (1 M ( , . For ε sp ] := . T ) B (2 b iso ξ a N C j,b -plets if so = [ ; P λ := ¯ α + 6 , (0) D + i,a ( yields a reducible T l n transform in the fundamental representation of i,a α gl ) , the dual of the twisted-adjoint representation containi κ ε α ) := [ δ − 1)-irreps a ) denote P ± 0 iff − := ( ( λ l / ; (Θ ≡ ,...,D , D 6= 0 and ) ,κ i ± ) n ( i 2= 0; Θ κ (1 + ( T = 1 0 at fixed ρ ) is assumed to be even for M determined from the shape of Θ a iso P → 1 (these are representations for [ D ( a, b ′∓ λ − ,N We let The translations are nilpotent in The ) In a similar context, see also the recent work [ -plets if , respectively; and (b) translations represented by i (Λ= 0; = ± ( ) i 11 ∗ ± − -type Θ annihilated by ρ ( 2 th row does not form a block of its own in Θ b massless primary Weyl tensor. Θ space was reformulated using BRST-cohomological methods. forming an infinite-dimensional Lie algebra with commutato are auxiliary flavor indices. Theformations oscillator generated algebras by are arbitrary in Grassmann even polynomi where homogeneous canonical transformations form the finite-dim (the trace-corrected cell creation operator) for fixed T ideals yields “cut” finite-dimensional modules m and Θ B Oscillator realizations ofB.1 classical Lie algebras Howe-dual LieWe algebras denote the classical algebras by T i 1)-types as follows: the limit ξ where isomorphic to twisted representations canonical can be realized using bosonic (+) and fermionic ( in manifestly symmetric or anti-symmetric bases for the You Omitting the tensor-spinorial representations of JHEP07(2009)014 = a (B.5) (B.6) (B.7) (B.8) (B.9) X (B.16) (B.14) (B.12) (B.10) (B.15) (B.13) (B.11) 1). The . − and ab in the case , . J b = ) ) D j,b X ǫ C C , , ¯ , ; ; α b c ab = k i j δ − − ) ( J d i,a q ν ν N i| C α i l a , = ; (2 (2 = δ + i , k M := ¯ j p D so sp a − b ( δ i ac l ij b i |h l | h ), X J = = c i sp T i, h N ab T α − − k j | l l iJ p,q e e on δ a , , h c ab J i,b a Lie algebra which are their mutual ] = 2 ]= J α k i,c l kl , J ] = 4 . . ¯ j,b α i,a T i cd ba α , N , ji α i i j +1 j which is defined to be the maximal ǫJ i,a T N , M ǫ T N , = ¯ = 2 , [ α [ in the case of ± ) ) = l ab e ⊕ a ± b ab C C := 0 reads ; M ; ab , 0 M [ 1 12 ij M l + ) + N := ν ν 1 ± reads 0 of some signature ( ⊕ ν ) : i − (2 ) : (2 ( – 42 – ± 1 ij , T C l , e C h ab sp so gl ; − ) , J ; c η T b . One has T D = i,a = = = ba l D ( M = ( = ¯ α a = d + + ± gl ab , l l l δ e e e sp ǫM ab ij i j,a , ± i k J T l − i α e − i δ l j l = Ω a h δ d + j k i h := h T ) and M ab j,a c b 2 J N α C i δ ; − ab i,a h D α 1, the above bases exhibit explicitly the three-grading ( (¯ M ) : ) : ] = 4 ]= ) : ± ] = 1 2 so c C C d kl kl C ; ; = ; T , T , )= D D }≡ i D , M together with its Howe dual j l that commutes with ( ( ( ij a ( ab b l N j,a ǫ T = +1), and ± [ so sp gl [ l M M ǫ [ , α = = = l l l i,a ) ( ¯ α C { ; 2 1 . For definiteness, we take D ( ba := A Howe dual pair of Lie algebras is a pair of Lie subalgebras in J so i j b 12 N centralizers. These contain The oscillator realization of the generators of where subalgebra of Their commutation rules take the form where In the cases of with the commutation rules and indices are raised and lowered according to the conventi X of oscillator realization of the generators of JHEP07(2009)014 ± M , can l (B.20) (B.21) (B.18) (B.19) (B.24) (B.17) (B.25) (B.22) (B.23) in ] for all ± ab ± l ) below) e e λ | . For given has a non- M l [ ± ab ± C , B.34 M M ) M i 1 2 ]= λ , then the spectrum | l l , and hence only the [ ) and ( )]:= ± C C e λ , , , and . labels the degeneracy of ; i ) ± l µ e B.33 µ ± D )) ; ( i e λ , | (no sum on 2 ± into o ( ) sp λ e λ [ ) of l ( useful Howe-duality relation is | ± − 2 ± j λ ± e λ ) obeying e λ ( ( D r-algebra modules 6 ], the enveloping algebra of ± ± does not yield any singular vectors. + 1 ) l ⊗ | ± , [ i e λ , are the ground states of , C ], and hence assumes a fixed value, i ± D | i ± i C D ) . ± e λ λ ± e l ) ab ± | w e ( ± [ e λ ( ± ∈ U e λ λ | e 2 ± λ M D ( e λ ] | l ± D ± λ ab [ ⊗ | ( ∈ U ∓ ǫ =1 . By making a canonical choice of the Borel ± C µ M ] uniquely in terms of C M ± i − mult( ± 2 1 are multiplicities. Consequently, ± e λ =1 l mult λ e ± . ν i [ ± ) mult D e λ Schur states } } – 43 – e M Λ ( ) C ± -irreps action on := ± i l ∈ e λ ± ± i = 0 =0 for M ) decomposes under )]:= e λ ± e λ ± e := ± w Λ( i e λ i l i { ,... ± M e C Λ( ∈ e 1 M ; ± ± e w λ ± ± λ ∈ , ( = = =1 e e λ λ ν λ of all i 0 D | | X S ± ( ) ± = ± ij l λ ± e D ± i e λ
l T so e λ ∈ {
M [ e λ ± ] = ) i 2 j ) δ ± ± = ± M e λ e λ e λ ) are the Schur states and the index l | − ( | l
[ ± i j ) decomposes into finite-dimensional irreps of , C ± λ ± 2 e generalized Schur module λ ( b ± a | N D S C ± generate lowest-weight spaces e λ λ are multiplicities. For simplicity, we assume that ) . Hence, ( M ( i ± a b ± D N ± e λ ( e λ is sufficiently large to fix M | ∈ , the Schur states can be chosen to obey ± } ± ) D l : ( e i ∈ D C ± ǫ { )] := µ 1 : e λ , and mult ; ∀ ) contains the labels C ± , which we shall refer to as the ± C ; ± i e λ e λ = | ± D ( e λ λ ǫ | | ] say, in gl [ ± By construction an invariant polynomial , the corresponding 2 e λ | ± C l [ The Schur states where mult( B.2 Generalized SchurThe modules oscillator algebra can be realized in various oscillato We also define the shifted Howe-dual highest weights (see als that assume the values subalgebra for degenerate inner product and that the M with Howe-dual highest weights where multiplicity remains a free parameter.that In of what the follows, quadratic one Casimir operators be rewritten as an invariant polynomial C invariants of invariants where Λ( the construction. If where JHEP07(2009)014 . 0 − > . In D 2 a ]). (B.31) (B.30) (B.29) i|| (B.26) (B.27) (B.32) (B.33) (B.28) (B.34) th and 42 i = 2 ∆ | ) . For the i j − D . b ν N ( ; , i.e. u || − ± D − l e T F ) Dν , , C ; + − D , := ( th row and column ing the trivial dia- between the i for some values of sp onal tensorial representa- . The various Fock-space ± l h and dim i,a ,...,ν ,...,ν (∆)) , )= α − ± cd D ght e λ ( | act real form of = 1 = 1 M bd , D usp . J (∆) = 0. One also notes that ) R > D ) λ ac i 1)) − ( − nite inner product. From J ) ν ν = 0 yields the standard Fock space ∆ , , and ± | − − ≡ j (2 (2 i ν i ab R D ( nnihilated by ¯ ; are subspaces of states of fixed rank 0 − δ , N R | sp so u l e ± ) =0 if ( ab ij ν =
e oscillator algebra (see, for example, [ ; = ± generate inner product matrix with alternating i,a T R M ± ± ν i ompact real forms of ; presentations of the maximally split non-compact D α ab as follows: ; ± − η ∆ F ± | ) ν = , i , i = 0 ; i D j ± + | = l D e − D ) † =0 + − N i i ( ∞ i ,...,D, ) ,ν + † M R e e F × ) w w (∆ ) ν + ab ) with X – 44 – l + ν | ± ǫ ij · · · (2 ( ν = = 1 = = M D obeying T ( ( ( + ( i i sp so u l 1) i e ± R so 0 ν ; | mult ± ), − ± D , i , ± i D F b ∆ a ( j ) i M D N u ) D N M ( ) i ( (∆) being the number of cells in the D with equality iff ± = i ( D = = ( P h , i.e. usp so u l D l ± † † , h l ! e i ) ) , w i 1 j = a × b h i < j R i l
N : i M w ( + ± ( h decomposes under = ν 0 if i ; + , i 2 ± D ± D R i > X ; ν | ; 2 D F w − ± ± j ± D ν e λ ; F . The Fock space thus decomposes into unitary finite-dimensi ± D = − ab F (∆) and ± i R i and unitary finite-dimensional representations of the comp J ; e λ (∆) = (∆) = (∆) = w ± i + = l + i ν − i 13 e ; dim λ ∗ e e λ λ = ± ) D i ab F J w The standard Fock space can be equipped with the positive defi . These spaces have dimensions 13 Generalized Fock spaces can be built on anti-vacua that are a realizations are subsumed into the Moyal quantization of th the case of bosonic oscillators, these modules have a non-de signature that yields unitary representations of the non-c real form of Howe-dual algebra one finds unitary infinite-dimensional re where tions of the compact real form of using ( B.3 Fock-space realizations Acting with the oscillators on a state We note that for fermionic oscillators, dim The Fock space where the sumgram) and runs over all possible Young diagrams ∆ (includ R with of ∆, respectively. We shall say that ∆ contains a block of hei it follows that JHEP07(2009)014 (B.41) (B.42) (B.38) (B.40) (B.39) (B.35) (B.37) (B.36) (B.43) , . i +1) of ∆ to be the j , (∆) − T i ± e λ − | j , (∆)) . ) h ± , ± The total dimension of , (∆) + e λ ) ν i | λ ; − | ± w , ) arises exactly once. Cor- racy in the sense that the ( e (∆)). D ) ). ∆ ± | i (∆) | ∆ y use dimension formulae or = >ν ) e λ h λ ∈ | ) ( , D 1 ± (∆ stics of the oscillators. To show i λ Y , ν ± + B.35 w . ± i,j ( ∆ ( ν )= d | D T invariant, and } gl T = or ) 1 ) | R , ; ± ± ) and the multiplicities 0 (∆ ν min( ∆ ± ν i | ν ; = ∆ ; ; , w > D > ) D D ∈ { D | i | B.34 − 1 1 F ) ∆) dim( h h ∆ e ∆ | , (∆ ± (∆ ) )–( ν ± 0 ± 0 ) ⊗ | ; D j 0 if 1 else 0 if 1 else ( ) . – 45 – leave D C | − , and we define the transpose ∆ C B.32 gl , ( ( i 1 ; mult ± mult ) ( l e − (∆ + D R h ± ( ∆ | ± ǫ ν + ) = ) = X N ) = ∆ ± i ; gl ( ∆ − + and e | C (∆) (and hence w ν ν ∆ D + i ; l | ; ; ∈ mult e ) = dim( ) = w ) ∆ = M D D D ( ± ± (∆ | | i,j ν ν ( under gl ± ǫ ) is given by ; ; · · · )= ∆: rank(∆)= (∆ (∆ Q R T D D ; − 0 + 0 = ), both | ( is one-to-one and each dual pair ( ± = mult C ν B.37 ± R (∆ ± ; i ; i (∆ ± d R e ± ; D w h e λ ± ∆ mult mult D ± ∆) = F M ( d | ν ; ) ↔ gl D = N λ F ( = ± l ν gl ; ± D S dim( 1)th rows if − h As we shall demonstrate below, the multiplicities Let us examine more carefully the determination of ( + i where the dual Young diagrams and Young diagram with ( In the case of B.4 Schur modules for The vanishing conditions follow immediatelythat from the the stati non-vanishing multiplicitiesdirectly are decompose equal to 1, one ma Calculation of multiplicitiesthe using right-hand dimension side of formulae. ( The Fock space realizationcorrespondence is thus completely free of degene where the highest-weights are given by ( respondingly, the decomposition of the Schur module reads JHEP07(2009)014 (B.49) (B.45) (B.44) (B.46) (B.47) (B.48) , n ) , ) ± R , ν ± ( ν ) must hold in = m ) is a polynomial i D ± m and ν B.39 ; , + =1 (∆) ν i D i , D X ( ∆ + ); and | ± R ± m,n d ij d p ) and ( ) , on of the regular representa- j B.33 =0 i ) into i R ,...,ν 0 C N X | B.38 ; ( ) m,n ). + D = 1 + ν = 0 . We note that the sub-leading ( ν = m gl 6 ( α (and it is invariant under insertions , + ) B.45 ν Y j given by ( ! i