Simple Unfolded Equations for Massive Higher Spins in AdS 3
PPrepared for submission to JHEP
Simple Unfolded Equations for Massive Higher Spinsin AdS Pan Kessel a and Joris Raeymaekers b a Machine Learning Group, Technische Universit¨at, Berlin Marchstrasse 23, 10587 Berlin, Ger-many. b CEICO, Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 182 21 Prague 8,Czech Republic.
E-mail: [email protected] , [email protected] Abstract:
We propose a simple unfolded description of free massive higher spin particlesin anti-de-Sitter spacetime. While our unfolded equation of motion has the standard formof a covariant constancy condition, our formulation differs from the standard one in that ourfield takes values in a different internal space, which for us is simply a unitary irreduciblerepresentation of the symmetry group. Our main result is the explicit construction, for thecase of AdS , of a map from our formulation to the standard wave equations for massivehigher spin particles, as well as to the unfolded description prevalent in the literature.It is hoped that our formulation may be used to clarify the group-theoretic content ofinteractions in higher spin theories. a r X i v : . [ h e p - t h ] A ug ontents d +1 d +1 Symmetry 6 sl (2 , R ) ⊕ sl (2 , R ) Basis 83.2 Particle Representations 103.3 Topologically Massive Equations 123.4 Unfolded Massive Equations 133.5 Projecting on Lorentz Tensors 153.6 Recovering the Topologically Massive Equations 183.7 Mapping to the Unfolded System of [15] 183.8 Explicit Mode Solutions 19 Conventions 21B Representations of the Lorentz Subalgebra 23
The study of relativistic wave equations, whose solution spaces carry unitary irreduciblerepresentations of the spacetime symmetry group, lay at the birth of quantum field theoryand was undertaken by several towering figures in the field. For example, in 1939 Fierzand Pauli [1] wrote down equations describing free, massive particles transforming as asymmetric rank s tensor:( (cid:3) − M ) φ µ ...µ s = 0 , ∇ µ φ µµ ...µ s = 0 , (1.1)where φ µ ...µ s is a totally symmetric traceless tensor. In the anti-de Sitter space AdS d +1 the set of solutions of these equations, upon imposing suitable boundary conditions, forma unitary, irreducible representation D (∆ , s ) of the symmetry algebra so (2 , d ) with lowestenergy ∆, where M = ∆(∆ − d ) − s . (1.2)– 1 –e refer to [2] for a modern review of relativistic wave equations and references to theoriginal literature.In recent years a different yet equivalent formulation of relativistic wave equations hasproven useful, namely the unfolded formulation due to Vasiliev and collaborators (see [3]and [4] for reviews). In this formulation, covariant wave equations like (1.1) are replacedby a system of coupled first-order equations typically containing an infinite number ofauxiliary fields. A beautiful feature of unfolded equations is that they geometrize covariantwave equations like (1.1), since they can be interpreted as a covariant constancy conditionon a section of a certain vector bundle over AdS d +1 . Unfolded equations were first proposedfor massless higher spins in (A)dS [5], and subsequently generalized to other dimensions,massive fields and flat backgrounds [6]-[17]. One advantage of the unfolded formulationis that it formally facilitates the coupling of massive fields to massless higher spin gaugefields, and therefore it lies at the core of Vasiliev’s construction of interacting higher spintheories [18],[19]. This has in turn played an important role in recently uncovered examplesof holographic duality, where bulk Vasiliev theories were argued to be dual to holographicboundary CFTs possessing conserved currents of spin greater then two.One example in the context of AdS /CFT holography is the minimal model hologra-phy proposed by Gaberdiel and Gopakumar [20]. Here, the bulk theory contains a massivescalar field with mass m = λ −
1, which is coupled to massless higher spin fields with hs [ λ ]gauge symmetry through unfolded equations [9],[10]. A second example, which inspiredthe current work, is provided by the tensionless limit of string theory on the AdS × S × T background with Ramond-Ramond flux. In this case, the bulk theory contains masslesshigher spin fields with a gauge symmetry which goes under the name of the ’higher spinsquare’ (HSS) [21],[22],[23]. Furthermore, the symmetric orbifold CFT contains manyspinning primaries with spins s = | h − ¯ h | ∈ N / and the standard one. In both formulations, the basic field C ( x ) isa zero-form section of a vector bundle over AdS , taking values in an infinite-dimensionalrepresentation R of the symmetry algebra so (2 , d ). The field equations simply state that C ( x ) is covariantly constant: ( d + A R ) C = 0 . (1.3)In this equation, A stands for the flat AdS d +1 connection made out of the vielbein andspin connection A = e a P a + 12 ω ab M ab . (1.4) Recently, it was shown [24],[25] that the symmetric orbifold CFT also describes a subsector of thetensionless limit of string theory on the S-dual background with NS-NS flux. – 2 –he subscript R in (1.3) means that the generators are taken in the representation R . Theequation (1.3) states that the general solution is obtained by picking an arbitrary vector C at the origin and parallel transporting it. In terms of the group element G in writing A = G − dG , the general solution is C ( x ) = G − R ( x ) C .In the standard unfolded formulation [9],[10],[15] for a massive spin- s field on AdS ,the representation R acts on basis vectors V ( t ) a , with | a | ≤ t and t = s, s + 1 , . . . , which forfixed t transform as a spin- t representation under the Lorentz subalgebra so (1 , V ( t ) a are generators of the higher spin algebra hs [ λ ]and the action of the AdS translation generators comes from the ‘lone-star’ product. Theresulting system of equations are equivalent to the Fierz-Pauli description (1.1), since onecan prove a ‘central on mass-shell’ theorem which states that the lowest spin- s componentof C is precisely the Fierz-Pauli field φ µ ...µ s above.In this work, we will explore a different unfolded formulation, where the representation R in (1.3) is instead simply taken to be the unitary irreducible representation D (∆ , s ) itself.One advantage of this formulation is that the space of solutions to (1.3) forms a Hilbertspace with inner product inherited from D (∆ , s ). While for this choice for R the factthat (1.3) gives a field theory realization of D (∆ , s ) is almost tautological, it is not apriori clear if there is an analogue of the central on mass-shell theorem allowing one toreconstruct the Fierz-Pauli field from C , nor how this unfolded formulation is related tothe standard one. The main goal of this work is to address these questions. A key propertyis that the generators V ( t ) a of the standard unfolded formulation can be constructed asnon-normalizeable state in the D (∆ , s ) Hilbert space . This allows us to construct a linearmap or ‘intertwiner’ between the two representations, and construct from our field theunfolded field of the standard formulation. The restriction to the spin- s component of thelatter then leads to the desired on mass-shell theorem. As a corollary, our results allow fora completely algebraic construction of the mode solutions of the Fierz-Pauli equations, seeequation (3.90) below.Since in the present unfolded formulation, the group theoretic meaning is completelytransparent and involves only the representation D (∆ , s ), it may be hoped that it mayshed light on the group-theoretic content of the interaction vertices in Vasiliev theory.This may be of use in constructing as yet unknown interactions in the theory based on thehigher spin square. As a first step towards such a construction, we will show in a separatepublication how the equation proposed in [26] combines an infinite set of our massive higherspin equations of the form (1.3) into a single multiplet of the higher spin square. d +1 In this section, we review some aspects of the geometry of AdS d +1 and propose and analyzeour unfolded equations. In [11], a similar construction was performed for the case of massless representations in AdS D with D ≥ – 3 – .1 Coset Description We start out by recalling the coset description of anti-de Sitter space. The d +1-dimensionalanti-de Sitter spacetime AdS d +1 can (up to global issues which are not relevant at present)be described as the homogenous symmetric space SO (2 , d ) /SO (1 , d ), where the isotropysubgroup SO (1 , d ) is the Lorentz group in d +1 dimensions. We will denote by M AB , A, B =0 (cid:48) , , . . . d the generators of the Lie algebra so (2 , d ) with commutation relations[ M AB , M CD ] = i ( η BC M AD − η AC M BD − η BD M AC + η AD M BC ) , (2.1)where η AB = diag( − − + . . . +). In a unitary representation, the M AB are represented byHermitian operators. We will split them up in ‘AdS translations’, i.e. the coset generators P a = M (cid:48) a , a = 0 , . . . d , and Lorentz generators M ab ∈ SO (1 , d ). They satisfy [ P a , P b ] = iM ab [ M ab , P c ] = − iη c [ a P b ] [ M ab , M cd ] = − iη c [ a M b ] d + 2 iη d [ a M b ] d . (2.2)Due to homogeneity, points in AdS d +1 can be viewed as coset representatives G ( x ) ∈ SO (2 , d ), for example we could use a ‘canonical’ parametrization where G ( x ) = exp x a P a .The symmetry group SO (2 , d ) acts on the coset element as gG ( x ) = G ( x (cid:48) ) h, g ∈ SO (2 , d ) , h ∈ SO (1 , d ) . (2.3)The infinitesimal version of this relation, setting g = 1 + (cid:15) AB M AB , defines the Killingvectors l µAB ∂ µ through x (cid:48) µ = x µ − (cid:15) AB l µAB and what we will call the ‘Lorentz-compensator’fields W abAB through h = 1 − (cid:15) AB W abAB M ab . It follows from (2.3) that these satisfy G − M AB G = − l µAB G ( x ) − ∂ µ G ( x ) − W abAB M ab . (2.4)It can be shown that the Killing vectors l µAB obey the same commutation relations (2.1) asthe generators M AB . We refer to [27] for a proof and a review of the differential geometryof coset spaces.From the coset representative we construct the flat so (2 , d )-valued connection A = G − dG . (2.5)It can be decomposed into vielbein and spin connection parts as follows A = e a P a + 12 ω ab M ab ≡ e + ω. (2.6)Using this relation in (2.4), one finds an expression for the Killing vectors and Lorentz com-pensators in terms of the vielbein, spin connection and adjoint representation componentsof G : l µAB = − e µa ( G − M AB G ) a (2.7) W abAB = − ( G − M AB G ) ab − l µAB ω abµ . (2.8) Note that we set the AdS radius to one. – 4 – .2 Unfolded equations
Following Wigner’s definition, a quantum mechanical particle can be identified with aunitary, irreducible representation of the spacetime symmetry group in which the energyis bounded from below. In the case of AdS d +1 , particle representations are built on a setof primary states | ∆ , s (cid:105) which form a unitary irreducible representation of the maximalcompact subalgebra so (2) ⊕ so ( d ). Here, ∆ is the eigenvalue of the energy operator P while s denotes quantum numbers specifying a unitary irreducible representation of so ( d ).The states | ∆ , s (cid:105) are annihilated by the energy lowering operators J − a = M a + iP a . (2.9)The representation is built up by acting on the states | ∆ , s (cid:105) with the energy raising oper-ators J + a = M a − iP a and will be denoted by D (∆ , s ). If s is a totally symmetric rank-stensor, the quadratic Casimir takes the value12 M AB M AB = ∆(∆ − d ) + s ( s + d − . (2.10)By a field theory realization of the particle representation D (∆ , s ), we mean a set ofspacetime-dependent fields which satisfy a set of equations (and possibly boundary con-ditions) which are invariant under the spacetime isometry algebra, such that the solutionspace transforms as the representation D (∆ , s ). For example, the Fierz-Pauli equations(1.1) in AdS d +1 give, upon imposing suitable boundary conditions, a field theory real-ization of the representation D (∆ , s ), where s stands for the symmetric rank s tensorrepresentation.As anticipated in the Section 1, we will now show that an alternative field theoryrealization of a particle representation R = D (∆ , s ) is provided by the system of equations( d + A R ) C ( x ) = 0 , (2.11)where C ( x ) is a zero-form which takes values in an internal space which is precisely therepresentation space R . The connection A is the AdS d +1 connection (2.6), and the subscript R means that the generators in (2.6) are taken in the representation R . For notationalsimplicity, we will drop this subscript in what follows. Note that the equation (2.11) isintegrable due to the fact that A is a flat connection. Let us first show the covariance of the equations (2.11) under diffeomorphisms and localLorentz transformations. For this, we observe that the equations are gauge-invariant un-der local SO (2 , d ) transformations under which both the background A and the field C transform, in the following way: A → Λ( A + d )Λ − (2.12) C → Λ C (2.13) This statement holds only for d >
2. For AdS , where the subgroup of spatial rotations reduces to so (2),we will review in Section 3.3 below that the Fierz-Pauli equations describe two irreducible representationswith opposite signs of the spatial so (2) helicity. – 5 –hen Λ = exp( λ ab ( x ) M ab ) belongs to the Lorentz subgroup SO (1 , d ), these transforma-tions encode the covariance of the equation (2.11) under local Lorentz transformations. In-deed, the first equation (2.12) implies, using the commutation relations (2.2), the standardtransformation of the vielbein and spin connection under local Lorentz transformations.The second equation (2.13) elucidates the Lorentz tranformation character of our masterfield C : it transforms as the representation R , decomposed under the Lorentz subalgebra so (1 , d ). Therefore, C doesn’t transform irreducibly under Lorentz tranformations in gen-eral, in contrast to e.g. symmetric tensor field of the Fierz-Pauli system. We note that theequation (2.11) can be rewritten as ( ∇ + e a P a ) C = 0 (2.14)where ∇ ≡ d + 12 ω ab M ab (2.15)is the Lorentz covariant derivative.Similarly, it can be shown that taking Λ = exp( λ a ( x ) P a ) encodes covariance underlocal diffeomorphisms, albeit mixed with local Lorentz tranformations (see [28] for details). The equations of motion (2.11) imply that C is a covariantly constant section, and thereforethe general solution can be obtained by picking an arbitrary value C ∈ R to be the valueof C in the origin x = 0 (which we take to correspond to the identity, G (0) ≡
1) andparallel transporting it: C ( x ) = G ( x ) − C . (2.16)Since the representation R is unitary, the space of solutions to (2.11) has the structure ofa Hilbert space, where the inner product is defined as (cid:0) C, C (cid:48) (cid:1) ≡ (cid:0) C ( x ) , C (cid:48) ( x ) (cid:1) R = (cid:0) C , C (cid:48) (cid:1) R . (2.17)Here, ( · , · ) R is the inner product on R . The result is independent of x due to (2.16) andunitarity.If { e p } p forms an orthonormal basis of R , a complete orthonormal basis of solutions isgiven by { C p } p with C p ( x ) = G − ( x ) e p . (2.18) d +1 Symmetry
For a fixed background A , i.e. a specific choice for the AdS d +1 vielbein and spin connection,the global symmetries of eq. (2.11) are the subset of transformations (2.12, 2.13) whichleave A invariant. From (2.16), it is easy to see that these are generated by infinitesimalgauge parameters of the form λ AB = G − M AB G , which obviously generate the anti-de-Sitter algebra so (2 , d ). Their action on the field C is δ AB C = G − M AB G C . (2.19)– 6 –he basis of solutions (2.18) transforms precisely as the representation R of the symmetryalgebra: δ AB C p = ( M AB ) qp C q , (2.20)where the indices p, q refer to components in the representation R . It is therefore clearthat (2.11) provides a field theory realization for the particle representation R .Using the equation of motion (2.11) and the identity (2.4), we can reexpress the right-hand side of (2.19) as the action of the scalar Lie derivative plus an ‘internal’ part deter-mined by the Lorentz compensator given in (2.8): δ AB C = l aAB ∂ a C − W abAB M ab C = l µAB ∇ µ C + 12 ( G − M AB G ) ab M ab C, (2.21)where in the second equality we have used (2.8). We note that the second order Casimirdifferential operator constructed from δ AB is constant, for example for the symmetric tensorrepresentation D (∆ , s ) it evaluates to12 δ AB δ AB C = 12 M AB M AB C = (∆(∆ − d ) + s ( s + d − C . (2.22)We end this section with some comments: • The unfolded equations (2.11) are consistent for general representations R , for ex-ample the representation s in D (∆ , s ) is not restricted to be a symmetric tensorbut can have mixed symmetry. Though we will focus on the massive case, where D (∆ , s ) is a ‘long’ multiplet, in what follows, the above unfolded description also ap-plies to the massless or partially massless cases, when D (∆ , s ) saturates a unitaritybound and becomes ‘short’. The representation R in (2.11) could in principle evenbe non-unitary, though of course in this case the solutions would not form a Hilbertspace. • The unfolded description in this section generalizes in a straightforward manner toMinkowski space (the coset Poincar´e d +1 /SO (1 , d )) and de Sitter space (the coset SO (1 , d + 1) /SO (1 , d )). • While our unfolded equations carry by construction a representation of the AdS d +1 symmetry algebra, and therefore also of the simply connected part of the symmetrygroup, they are not guaranteed to be invariant under additional discrete symmetries(such as parity in d = 2, as we shall illustrate below). To construct a system invariantunder an additional discrete Z symmetry may require considering a doublet of fields C, ˜ C which are exchanged by the discrete symmetry. Our proposed unfolded equations (2.11) give a simple field theory realization of an arbi-trary particle representation of the symmetry group. However, they do so at the cost ofintroducing an infinite number of fields: since unitary representations are infinite dimen-sional, the field C has an infinite number of components. Most of these components are– 7 –xpected to be in some sense auxiliary, and it will be the goal of this section to understandhow to extract the physical components of C , focusing on the case of AdS and on massiveparticle representations case for simplicity. In this case, we will find the explicit linearcombinations of components of our master field C which satisfy the topologically massiveequations (3.50). This provides a version of the ‘central on mass-shell’ theorem for ourunfolded equations. In deriving this result, we will also find a map from our master field C to the field obeying the unfolded equations of [15]. sl (2 , R ) ⊕ sl (2 , R ) Basis
Let us first specialize the general equations of the previous section to the case of AdS . Thethree-dimensional case is somewhat special in that symmetry algebra is not semisimple, so (2 , (cid:39) sl (2 , R ) ⊕ sl (2 , R ), and it will be convenient to work in a basis L m , ¯ L m , m = − , , L m , L n ] = (cid:15) mnp η pq L q = ( m − n ) L m + n , (3.1)[ ¯ L m , ¯ L n ] = (cid:15) mnp η pq ¯ L q = ( m − n ) ¯ L m + n , (3.2)[ L m , ¯ L n ] = 0 . (3.3)Here, the (cid:15) -tensor is defined to have (cid:15) − = 2 and η mn is the inverse of η mn = −
20 1 0 − . (3.4)The latter is proportional to the Cartan-Killing form which we normalize as K ( L m , L n ) = K ( ¯ L m , ¯ L n ) = 12 η mn , K ( L m , ¯ L n ) = 0 . (3.5)The generators can be combined into AdS -translation generators P m and Lorentz genera-tors M m , which generate the diagonal sl (2) subalgebra, as follows P m = L m − ¯ L m , M m = L m + ¯ L m . (3.6)In terms of the original so (2 ,
2) generators M AB , A, B = 0 (cid:48) , , , P = M (cid:48) , M = M , (3.7) P ± = M (cid:48) ± iM (cid:48) , M ± = M ∓ iM . (3.8)We note that in unitary representations of sl (2 , R ), the generators must satisfy L † = L , L †± = L ∓ , and similarly for the barred generators. The generators of the maximalcompact subalgebra so (2) ⊕ so (2) are the energy operator P = L − ¯ L , which generatesglobal time translations, and the helicity operator M = L + ¯ L which generates spatial U (1) rotations. – 8 –he AdS connection splits into sl (2 , R ) and sl (2 , R ) parts: A = e m P m + ω m M m = A m L m + ¯ A m ¯ L m , (3.9)where A m = ω m + e m , ¯ A = ω m − e m . (3.10)Noting that the coset element G splits as G ( x ) = g ( x )¯ g ( x ) , g ∈ SL (2 , R ) , ¯ g ∈ SL (2 , R ) , (3.11)we can work out the equations (2.8) for the Killing vectors and Lorentz compensator tofind l µm = −
12 ( g − L m g ) n e µn , ¯ l µm = 12 (¯ g − ¯ L m ¯ g ) n e µn (3.12) W nm = − l µm ¯ A nµ , ¯ W nm = − ¯ l µm A nµ . (3.13)From (3.12), we can derive the following useful identities involving the Killing vectors: η mn l µm l νn = 14 g µν , η mn ¯ l µm ¯ l νn = 14 g µν , (3.14) ∇ [ n ( l m ) p ] = l µm e qµ (cid:15) qnp , ∇ [ n (¯ l m ) p ] = − ¯ l µm e qµ (cid:15) qnp . (3.15)To derive the identities in the second line we have used the flatness of A .It is a simple exercise to find explicit expressions of the above quantities in the Poincar´ecoordinate system. We take the group elements g, ¯ g to be g = e x + L e ρL , ¯ g = e x − ¯ L − e − ρ ¯ L . (3.16)This leads to A = L dρ + e ρ L dx + , ¯ A = − ¯ L dρ + e ρ ¯ L − dx − , (3.17) e = P dρ + 12 e ρ P dx + − e ρ P − dx − , ω = 12 e ρ M dx + + 12 e ρ M − dx − . (3.18)Computing the metric one indeed finds the AdS metric in Poincar´e coordinates: ds = K ( e, e ) = dρ + e ρ dx + dx − . (3.19)For the Killing vectors one finds, from (3.12), l − = e − ρ ∂ − + x + ∂ ρ − x ∂ + , ¯ l − = − ∂ − , (3.20) l = − ∂ ρ + x + ∂ + , ¯ l = − x − ∂ − + 12 ∂ ρ , (3.21) l = − ∂ + , ¯ l = − x − ∂ − + x − ∂ ρ + e − ρ ∂ + . (3.22)– 9 – .2 Particle Representations In this section, we will give explicit matrix elements for the unitary representations D (∆ , s )in the sl (2 , R ) ⊕ sl (2 , R ) basis. We start by reviewing and introducing some notation for thehighest- and lowest-weight representations of the sl (2 , R ) Lie algebra, which are the onlyones relevant for our purposes. We refer to [29], [30] for reviews of sl (2 , R ) representationtheory.The lowest weight infinite-dimensional representations D + ( h ), for 2 h (cid:54) = Z − , are builton a lowest weight or primary state | (cid:105) h satisfying L | (cid:105) h = 0 , L | (cid:105) h = h | (cid:105) h . (3.23)If the primary state is normalized, h (cid:104) | (cid:105) h = 1, the normalized states in the representationare labelled as | m (cid:105) h , m ∈ N and given by | m (cid:105) h = ( m !(2 h )(2 h + 1) . . . (2 h + m − − ( L − ) m | (cid:105) h . (3.24)The generators are represented in this basis as L − | m (cid:105) h = (( m + 1)(2 h + m )) | m + 1 (cid:105) h , (3.25) L | m (cid:105) h = ( h + m ) | m (cid:105) h , (3.26) L | m (cid:105) h = ( m (2 h + m − | m − (cid:105) h . (3.27)For later convenience, we note that the generators can be written in ket-bra notation as L − = (cid:88) m ∈ N (( m + 1)(2 h + m )) | m + 1 (cid:105) h h (cid:104) m | (3.28) L = (cid:88) m ∈ N ( h + m ) | m (cid:105) h h (cid:104) m | (3.29) L = (cid:88) m ∈ N (( m + 1)(2 h + m )) | m (cid:105) h h (cid:104) m + 1 | . (3.30)Using these expressions one checks that, on D + ( h ), the quadratic Casimir C ≡ η mn L m L n = L −
12 ( L L − + L − L ) (3.31)takes the value C = h ( h − . (3.32)The representations D + ( h ) are unitary for h > D − ( h ) for 2 h (cid:54) = Z − , whose weightsare sign-reversed compared to those of D + ( h ). These are infinite-dimensional highest weightrepresentations built on a highest weight or anti-primary state with L -eigenvalue − h , andare unitary for h >
0. The quadratic Casimir takes again the value (3.32). The statesin these representations can be conveniently denoted as kets h (cid:104) m | , on which the sl (2 , R )generators act from the right with an extra minus sign to get the right commutationrelations. In particular, h (cid:104) | is indeed an anti-primary state satisfying h (cid:104) | ( − L − ) = 0 , h (cid:104) | ( − L ) = ( − h ) h (cid:104) | . (3.33)– 10 –et us also comment on the cases which were excluded above, built on a lowest weightstate with negative half-integer weight or a highest weight state with positive half-integerweight. In this case we obtain a finite-dimensional irreducible representation of dimension2 | h | + 1, which contains both a highest weight | h | and a lowest weight −| h | state. Thequadratic Casimir takes the value C = | h | ( | h | + 1). These representations, with the excep-tion of the singlet h = 0, are non-unitary, and will be denoted by D ( | h | ). They are analyticcontinuations of the unitary finite-dimensional spin | h | representations of su (2) and we willtherefore also refer to them as ‘spin | h | ’.We are now ready to work out the particle representations of AdS in the sl (2 , R ) ⊕ sl (2 , R ) basis. Recall from the previous section that particle representations of so (2 ,
2) arelabelled as D (∆ , η ) where ∆ is the energy (eigenvalue of P = L − ¯ L ) and η the helicity(eigenvalue of M = L + ¯ L ) of the lowest energy state in the multiplet. From the aboveconsiderations, we see that in the sl (2 , R ) ⊕ sl (2 , R ) basis these are identified as D (∆ , η ) = (cid:0) D + ( h ) , D − (¯ h ) (cid:1) (3.34)where ∆ = h + ¯ h, η = h − ¯ h. (3.35)The case where either h or ¯ h vanishes describes a short multiplet and corresponds toa massless higher spin particle. We leave the more challenging problem of relating ourdescription of the massless case to the standard Fronsdal equations for future work, andfocus here instead on the case where both h, ¯ h >
0, which corresponds to massive higherspin fields. The periodicity of the global angular coordinate furthermore restricts thehelicity η to be integer (for bosons) or half-integer (for fermions), i.e. h − ¯ h ∈ Z / . (3.36)Adopting the notation where the vectors in D − (¯ h ) are bra states as discussed above, or-thonormal basis states of (cid:0) D + ( h ) , D − (¯ h ) (cid:1) can be represented as ket-bra states of the form | m (cid:105) ¯ h h (cid:104) n | , m, n ∈ N . (3.37)These are orthogonal with respect to the inner product (cid:0) ψ, ψ (cid:48) (cid:1) = tr ψ † ψ (cid:48) (3.38)where the trace is taken in the D − ¯ h Hilbert space. Note that the states in particle rep-resentations can be interpreted as linear maps (or intertwiners) of sl (2 , R ) representationspaces D +¯ h → D + h . (3.39) Note that in our representations D + ( h ) and D − ( h ), the generators are manifestly unitarily represented,i.e. L † m = L − m . This has the advantage that, when taking the limit where h becomes a negative half-integer, no null states appear. This fact will simplify some parts of the subsequent analysis, in particularthe results derived in Appendix B. A different convention, which often appears in the literature, is related to ours by the redefinition¯ L m → − ¯ L − m , which preserves the algebra. In this convention, the particle representations are of the(primary, primary) type (cid:0) D + ( h ) , D + (¯ h ) (cid:1) , though sl (2 , R ) ⊕ sl (2 , R ) is embedded differently into so (2 , P m = L m + ¯ L − m , M m = L m − ¯ L − m . – 11 – .3 Topologically Massive Equations Before studying our unfolded equations in more detail, it will be useful to recall a peculiarityof massive higher spin equations in
AdS which was anticipated in Footnote 4. In spacetimedimension three, the subgroup of spatial rotations reduces to SO (2), and the correspondingquantum number is the helicity η = h − ¯ h in (3.35). Since parity changes the sign of η , theparticle representation (cid:0) D + ( h ) , D − (¯ h ) (cid:1) , while furnishing a representation of the componentof SO (2 ,
2) connected to identity, is therefore not invariant under parity. The Fierz-Pauliequations (1.1) in AdS , which don’t depend on the sign of η and are parity-invariant,actually describe the direct sum (cid:0) D + ( h ) , D − (¯ h ) (cid:1) ⊕ (cid:0) D + (¯ h ) , D − ( h ) (cid:1) . (3.40)The free equations which instead describe only a single helicity (cid:0) D + ( h ) , D − (¯ h ) (cid:1) (and arenecessarily parity non-invariant for η (cid:54) = 0) are generalizations of the topologically massiveequations for spin one and two [31]. It will facilitate our discussion in Section 3.6 below torederive these equations here from a purely group-theoretic point of view.We start from a field transforming in the spin- s representation of the Lorentz group,where s = | η | = | h − ¯ h | . We can describe this field as a completely symmetric multi-spinor φ ( s ) α ...α s . The Killing vectors of AdS act on it through a generalization of the standardLie derivative, the so-called Lie-Lorentz derivative (see [36] and Appendix A) L l m φ ( s ) α ...α s = l µm (cid:16) ∇ µ φ ( s ) α ...α s − s e nµ ( γ n ) β ( α φ ( s ) | β | α ...α s ) (cid:17) , L ¯ l m φ ( s ) α ...α s = ¯ l µm (cid:16) ∇ µ φ ( s ) α ...α s + s e nµ ( γ n ) β ( α φ ( s ) | β | α ...α s ) (cid:17) . (3.41)In the (cid:0) D + ( h ) , D − (¯ h ) (cid:1) representation, the sl (2 , R ) and sl (2 , R ) Casimirs are equal to h ( h −
1) and ¯ h (¯ h −
1) respectively. Therefore we impose the following field equations:( η mn L l m L l n − h ( h − φ ( s ) α ...α s = 0 , (cid:0) η mn L ¯ l m L ¯ l n − ¯ h (¯ h − (cid:1) φ ( s ) α ...α s = 0 . (3.42)Using the identities (3.14) for the Killing vectors l µm , ¯ l µm , these can be rewritten as (cid:0) ∇ µ ∇ µ − M (cid:1) φ ( s ) α ...α s = 0 , (3.43) ∇ β ( α φ ( s ) | β | α ...α s ) + µφ ( s ) α ...α s = 0 , (3.44)where M = ∆(∆ − − s, µ = sgn η (∆ − . (3.45)We note that for integer spin the first equation (3.43) is the first equation in the Fierz-Paulisystem (1.1) in the spinor basis.For the spin-0 case, the second equation (3.44) is actually absent. For s (cid:54) = 0, the(3.43,3.44) equations can be significantly simplified as follows. It is convenient to introducean operator D which acts on a general multispinor τ α ...α s as( D τ ) α ...α s ≡ ∇ βα τ β...α s . (3.46)– 12 –e should note that, when acting with D on a symmetric multispinor the result is ingeneral no longer symmetric, although the square D does map symmetric tensors intoeach other. Indeed, one can show using (A.9) that( D φ ( s ) ) α ...α s = ( (cid:3) + s + 1) φ α ...α s . (3.47)Equation (3.44) can be rewritten as2 sµφ ( s ) α ...α s = ( D φ ( s ) ) α α ...α s + ( D φ ( s ) ) α α ...α s + . . . + ( D φ ( s ) ) α s α ...α . (3.48)Acting with D on both sides of this equation, the right-hand side is symmetric due to(3.47), which allows us to derive the integrability condition ∇ β [ α φ ( s ) | β | α ] ...α s = 0 . (3.49)This means that the symmetrization in equation (3.44) can be dropped and we can replaceit with ( D φ ( s ) ) α ...α s + µφ ( s ) α ...α s = 0 . (3.50)These equations replace the full system (3.43, 3.44) for s (cid:54) = 0, since they also imply theKlein-Gordon equation (3.43): using (3.47) we can write( ∇ µ ∇ µ − M ) φ ( s ) α ...α s = ( D − µ )( D + µ ) φ ( s ) α ...α s . (3.51)Furthermore, by contracting two indices in (3.50), we see that they imply the divergence-free condition ∇ β β φ ( s ) β β α ...α s = 0 (3.52)which for integer spin is precisely the second Fierz-Pauli constraint in (1.1). The equa-tions (3.50) therefore imply the Fierz-Pauli equations (1.1) (and their generalization forhalf-integer spin), while it follows from (3.51) that the parity-invariant Fierz-Pauli systemdescribes a pair of topologically massive fields φ ( s ) , ˜ φ ( s ) which satisfy (3.50) with oppositesigns of µ , and are exchanged by parity. The equations (3.50) are arbitrary spin general-izations [32],[33] of the linearized topologically massive spin-1 and spin-2 equations [31],and are sometimes referred to as self-dual equations. It can be shown [34] that they indeedcontain the representation D ( h + ¯ h, h − ¯ h ). After these preliminaries, let us describe in more detail our unfolded equations (2.11) inAdS . Our unfolded master field C is a zero-form taking values in the internal space( D + ( h ) , D − ( h )), and can be expanded in components in the ket-bra basis (3.37) as follows C = (cid:88) m,n ∈ N C mn ( x ) | m (cid:105) h ¯ h (cid:104) n | . (3.53)The inner product (2.17) on the space of solutions becomes (cid:0) C, C (cid:48) (cid:1) = tr C † ( x ) C (cid:48) ( x ) . (3.54)– 13 –n terms of the coefficients (3.53), the inner product equals (cid:80) mn ¯ C mn ( x ) C (cid:48) mn ( x ), and it isactually independent of x as argued below (2.17).We recall that in the basis (3.53), the generators of sl (2 , R ) act on C as the operators L m in the h -primary representation (see (3.30)) from the left, while the generators of sl (2 , R ) act as the operators − L m in the ¯ h -primary representation from the right. Inother words, the AdS translations and Lorentz generators act as anticommutators andcommutators respectively P m C = L m C + CL m , M m C = L m C − CL m . (3.55)The unfolded equations (2.14) read ∇ C + e m P m C = 0 , (3.56)where the Lorentz covariant derivative acts as ∇ C = ( d + ω m M m ) C. (3.57)It is sometime useful to write (3.56) in tangent space indices as( ∇ m + P m ) C = 0 . (3.58)We also note that, in terms of the gauge potentials A = A m L m and ¯ A = ¯ A m L m (see (3.10),the equations take a form similar to Vasiliev’s unfolded equation for the zero form [9] dC + AC − C ¯ A = 0 , (3.59)although, as we already stressed in the Introduction and will explain in detail below, theirgroup-theoretic content is rather different.The equations of motion are invariant under the sl (2 , R ) ⊕ sl (2 , R ) symmetries of theAdS background which act on the fields as, using (2.21,3.13), δ l m C = l µm (cid:0) ∇ µ − e mµ M m (cid:1) C, δ ¯ l m C = ¯ l µm (cid:0) ∇ µ + e mµ M m (cid:1) C. (3.60)We note that, just like the topologically massive equation (3.50), our unfolded equation isnot parity-invariant. Indeed, the natural action of parity on the background gauge fields A is, in Poincar´e coordinates ( t, x, ρ ), P : A mt ( t, x, ρ ) → ¯ A mt ( t, − x, ρ ) A mx ( t, x, ρ ) → − ¯ A mx ( t, − x, ρ ) A mρ ( t, x, ρ ) → ¯ A mρ ( t, − x, ρ ) (3.61)and similarly for ¯ A , so that P = 1. One can check that this leaves the gravitationalaction S CS [ A ] − S CS [ ¯ A ] invariant, where S CS [ A ] is the Chern-Simons action. There is nonatural transformation law on C which makes the equation (3.59) invariant under parity.Instead, we can introduce a second field ˜ C , taking values in ( D + (¯ h ) , D − ( h )), with equationof motion d ˜ C + ¯ A ˜ C − ˜ CA = 0 . (3.62)– 14 –he combined system (3.59,3.62) is then invariant under the parity transformation C ( t, x, ρ ) → ˜ C ( t, − x, ρ ) (3.63)and similarly for ˜ C . The combined system can also be shown to be time-reversal invariant.To make matters a little more concrete, we can explicitly work out some of the equa-tions in Poincar´e coordinates. The equations of motion (3.59) read: ∂ ρ C + L C + CL = 0 , ∂ + C + e ρ L C = 0 , ∂ − C − e ρ CL − = 0 . (3.64)Using (2.16), we can also find the general solution to (3.59). A basis of solutions { C [ pq ] } p,q is labelled by two natural numbers p, q and obtained by applying the gauge transformation(2.16) on constant basis vectors | p (cid:105) h ¯ h (cid:104) q | of ( D + ( h ) , D − (¯ h )). One finds C [ pq ] ( x ) = g − ( x ) | p (cid:105) h ¯ h (cid:104) q | ¯ g ( x )= e − ρ ( h +¯ h + p + q ) p (cid:88) j =0 q (cid:88) k =0 N p,qj,k e ρ ( j + k ) x j + x k − | p − j (cid:105) h ¯ h (cid:104) q − k | , (3.65)where N p,qj,k = ( − j (cid:18)(cid:18) pj (cid:19)(cid:18) h + p − j (cid:19)(cid:18) qk (cid:19)(cid:18) h + q − k (cid:19)(cid:19) . (3.66)For later reference, let us stress that the solutions (3.65) only have a finite number of non-vanishing components. These solutions are by construction orthonormal with respect tothe inner product (3.54): ( C [ pq ] , C [ p (cid:48) q (cid:48) ] ) = δ pp (cid:48) δ qq (cid:48) . (3.67) In this and the following subsection, we show how our unfolded equations (2.11) are relatedto other field theory realizations describing the same massive higher spin particle, namelythe topologically massive equations (3.50) and the alternative unfolded description of [15].Concretely, we will show that both the topologically massive fields and the unfolded fieldsof [15] can be constructed as linear combinations of our components fields C mn ( x ). Theconstruction relies on interesting group-theoretic properties which allow us to project ourfield C on an irreducible spin- s Lorentz tensor, yielding the topologically massive equations,or on the so (2 ,
2) representation which underlies the unfolded formulation of [15].To illustrate a crucial difference between our unfolded equations and the standard waveequations, it is instructive to compute the result of acting with the covariant Laplacian onsolutions of our unfolded equations. Using the fact that, on (cid:0) D + ( h ) , D − (¯ h ) (cid:1) , we have theCasimir identity η mn P m P n + η mn M m M n = η mn L m L n + η mn ¯ L m ¯ L n = 2 h ( h −
1) + 2¯ h (¯ h − . (3.68)we compute, using the equation of motion (3.56) for C , ∇ µ ∇ µ C = η mn P m P n C = (cid:0) h ( h −
1) + 2¯ h (¯ h − − η mn M m M n (cid:1) C. (3.69)– 15 –he last term in the brackets is the Casimir operator of the Lorentz subalgebra, whichdoes not evaluate to a constant since our field C transforms in the reducible representation D + ( h ) ⊗ D − (¯ h ) under the Lorentz subalgebra; therefore no component of C itself satisfiesa covariant wave equation, in contrast to the standard unfolded formulation [9],[10],[15].To make contact with covariant wave equations such as the topologically massive equa-tion (3.50), we should therefore project our master field C onto a field φ ( s ) transformingin the finite-dimensional representation D ( s ), with s = | h − ¯ h | . In other words, we shouldconstruct a covariant linear map or intertwiner between these two Lorentz representations.The existence of such an intertwiner is somewhat nontrivial, as it maps a unitary infinitedimensional representation to a nonunitary finite-dimensional one. It can therefore notbe constructed from the known [35] Clebsch-Gordan decomposition of D + ( h ) ⊗ D − (¯ h ) interms of unitary representations (which involves members of the continuous series withunbounded energies).To construct the desired projections we proceed as follows. In Appendix B, we con-struct vectors in the (cid:0) D + ( h ) , D − (¯ h ) (cid:1) state space transforming in finite-dimensional rep-resentations under the Lorentz subalgebra. We find vectors spanning precisely the spin s + k, k ∈ N , representations which we denote as V ( t ) a with t ≥ s, | a | ≤ t . For ¯ h ≥ h , theyare of the form V ( t ) a = (cid:88) n ∈ N v n ( t, a ) | n (cid:105) h ¯ h (cid:104) n − a − s | , (3.70)where the coefficients are given in Appendix B, see (B.10). They transform under theLorentz subalgebra as M m V ( t ) a = ( mt + a ) V ( t ) a − m . (3.71)We should stress at this point that, as shown in Appendix B, the vectors V ( t ) a arenot normalizeable and are therefore not states in (cid:0) D + ( h ) , D − (¯ h ) (cid:1) considered as a Hilbertspace. However, it is sufficient for our purposes that they have finite overlap with the fields C which solve the equations of motion (3.56). This can be seen by inspecting the explicitsolutions (3.65): each basis solution for C has only a finite number of nonzero coefficients.Therefore it makes sense to consider the spin- t projections of C defined as the overlap φ ( t ) a ( x ) ≡ (cid:16) V ( t ) a , C ( x ) (cid:17) . (3.72)These indeed transform in a spin- t representation under Lorentz transformations, sincefrom (3.71) we find the intertwining relation (cid:16) V ( t ) a , M m C ( x ) (cid:17) = ( − mt + a ) φ ( t ) a + m ( x ) ≡ R ( t ) ( M m ) ba φ ( t ) b ( x ) . (3.73)One checks that R ( t ) ( M m ) ba ≡ ( − mt + a ) δ ba + m define basis matrices for the spin- t repre-sentation D ( t ) of the Lorentz subalgebra.From (3.73), we can derive a number of useful properties. First of all, the spin- t projection of the Lorentz-covariant derivative ∇ C is precisely the covariant derivative of φ ( t ) a ( x ): (cid:16) V ( t ) a , ∇ C (cid:17) = dφ ( t ) a + ω m R ( t ) ( M m ) ba φ ( t ) b ≡ ∇ φ ( t ) a . (3.74)– 16 –urthermore, using (3.60), we find that the spin- t projection of an infinitesimal symmetrytransformation acting on C gives precisely the Lie-Lorentz derivative (see (A.19, A.20))with respect to the corresponding Killing vector: (cid:16) V ( t ) a , δ l m C (cid:17) = l µm (cid:16) ∇ µ φ ( t ) a − e pµ R ( t ) ( M p ) ba φ ( t ) b (cid:17) ≡ L l m φ ( t ) a (3.75) (cid:16) V ( t ) a , δ ¯ l m C (cid:17) = ¯ l µm (cid:16) ∇ µ φ ( t ) a + e pµ R ( t ) ( M p ) ba φ ( t ) b (cid:17) ≡ L ¯ l m φ ( t ) a . (3.76)The full set of states V ( t ) a for t ≥ s, | a | ≤ t constructed above have the remarkableproperty that they form an irreducible representation of the full AdS symmetry. To showthis one needs to check that they transform among themselves under AdS translations. Asshown in Appendix B, this is indeed the case, with P m V ( t ) a = 2 V ( t +1) a − m − µst ( t + 1) ( mt + a ) V ( t ) a − m + ( s − t )( µ − t )2 t (2 t + 1) d m ( t, a ) V ( t − a − m . (3.77)where the coefficients d m ( t, a ) are given in (B.13). This means that the full set of projections φ ( t ) a ( x ) for t ≥ s, | a | ≤ t also form an irreducible multiplet of sl (2 , R ) ⊕ sl (2 , R ), withtranslations acting as( V ( t ) a , P m C ) = 2 φ ( t +1) a + m − µst ( t + 1) ( − mt + a ) φ ( t ) a + m + ( s − t )( µ − t )2 t (2 t + 1) d − m ( t, a ) φ ( t − a − m ≡ P m φ ( t ) a . (3.78)The spin s = 0 case, h = ¯ h , deserves a further comment, since the vectors V ( t ) a constructed above then possess extra structure related to higher spin algebras. This extrastructure arises because in this case, the particle representation ( D + ( h ) , D − ( h )) can beviewed as a map from D + ( h ) to itself, D + h → D + h , (3.79)and it makes sense to consider the product or commutator of the V ( t ) a .The state of Lorentz spin 0 is simply the identity operator V (0)0 = (cid:88) n | n (cid:105) h h (cid:104) n | = (3.80)so that the projection on the Klein-Gordon field is simply φ (0) = tr C . The spin-1 vectorsare simply the sl (2 , R ) operators V (1) m = L − m given in (3.30). The lowest weight vector ofLorentz spin t is V ( t ) − t = ( L ) t (3.81)and the other V ( t ) − t are constructed from these using (B.7). By construction, the V ( t ) a for t (cid:54) = form a hs [ λ ] algebra under taking commutators, with λ = 4 h ( h −
1) + 1, while underoperator multiplication we expect recover the ‘lone-star’ product [37]. This structure plays– 17 – key role in the standard unfolded description of the free massive spin-0 field [9],[10] .Note that, by extending A to be an arbitrary flat gauge potential with values in hs [ λ ],the unfolded equations consistently describe the massive spin-0 field in a background ofmassless higher spin fields. This can be used to efficiently compute holographic three-pointfunctions of the scalar-scalar-current type [38].To recapitulate, we constructed through eqs. (3.70) and (3.73) an intertwiner betweenparticle representations and tensor representations of the Lorentz algebra for the case ofmassive higher spin particles in AdS . We would like to point out that for the case ofmassless representations in AdS D with D ≥
4, a similar intertwiner was constructed in[11].
With these results in hand, it is now straightforward to show that our unfolded equationsimply the topologically massive equations (3.50). We start by converting the index a ofthe fields φ ( s ) a into a rank-2 s symmetric spinor index. In our conventions, this amounts toa simple relabeling of indices, since our spin- t representation matrices (3.73) are preciselythe 2 t -th symmetric tensor product of our spin- matrices. Concretely, we define the fields φ ( t ) α ...α t , with α j ∈ {− , + } as φ ( t ) a ↔ φ ( t ) α ...α t (3.82)where the indices are related as a = 12 t (cid:88) i =1 α i , α . . . α t = + · · · + (cid:124) (cid:123)(cid:122) (cid:125) t + a − · · · − (cid:124) (cid:123)(cid:122) (cid:125) t − a (3.83)and we should keep in mind that φ ( t ) α ...α t is defined to be totally symmetric.Under this relabelling, the symmetry under Lie-Lorentz derivatives (3.76) gets con-verted to their equivalent spinorial expressions (3.41). By construction, the field φ ( s ) α ...α s satisfies the Casimir relations (3.42) and, as shown in Section 3.3, it follows that they alsosatisfy the topologically massive equations (3.50). We can also show that the full set of fields φ ( t ) a ( x ) for t ≥ s, | a | ≤ t satisfies the unfoldedequations of [15]. To this end, we combine (3.78) with the equation of motion (3.56) toobtain ∇ m φ ( t ) a = 2 φ ( t +1) a + m − µst ( t + 1) ( − mt + a ) φ ( t ) a + m + ( s − t )( µ − t )2 t (2 t + 1) d − m ( t, a ) φ ( t − a + m . (3.84) Those works make use of an oscillator realization, which describes the direct sum of two irreduciblerepresentations of the symmetry algebra, see [26] for details, and therefore correspond to a pair of unfoldedequations in our approach. For example, the case λ = can be described by a single harmonic oscillator,and gives rise to the direct sum (cid:0) D + ( ) , D − ( ) (cid:1) ⊕ (cid:0) D + ( ) , D − ( ) (cid:1) . We have checked that the oscillatorrealization of the V ( t ) s is indeed a special case of our expressions (B.10) for general h, ¯ h . – 18 –e convert this to spinor form using the identities φ ( t +1) a + m = 12 ( γ m ) βγ φ ( t +1) βγα ...α t (3.85)( − mt + a ) φ ( t ) a + m = t ( γ m ) β ( α φ ( t ) | β | α ...α t ) (3.86) d − m ( t, a ) φ ( t − a + m = ( γ m ) ( α α φ ( t − α ...α t ) . (3.87)The relation (3.84) therefore becomes ∇ m φ ( t ) α ...α t =( γ m ) βγ φ ( t +1) βγα ...α t − µs ( t + 1) ( γ m ) β ( α φ ( t ) | β | α ...α t ) + ( s − t )( µ − t )2 t (2 t + 1) ( γ m ) ( α α φ ( t − α ...α t ) . (3.88)This is, up to convention-dependent normalization factors, precisely the unfolded systemof Boulanger et. al. in spinor variables, see eq. (2.28) in [15]. As was shown there, wecan also use (3.88) to give an alternative and more direct derivation of the topologicallymassive equations (3.50). Taking (3.88) for t = s and converting the index m into a pairof spinor indices using (A.13) we obtain ∇ αβ φ ( s ) α ...α s = − φ ( s +1) αβα ...α s + µss + 1 (cid:16) (cid:15) α ( α φ ( s ) | β | α ...α s ) + (cid:15) β ( α φ ( s ) | α | α ...α s ) (cid:17) . (3.89)Upon contracting the β and α indices in this expression, we obtain the linearized topo-logically massive higher spin equations (3.50).The fully symmetrized part of equation (3.89) shows that the spin- s + 1 field φ ( s +1) α ...α s +2 can be obtained by acting with covariant derivatives on the spin- s field φ ( s ) α ...α s . By asimilar argument holds the same statement holds for components φ ( t ) α ...α t with t > s : onecan convert the spacetime index m in (3.88) to spinorial indices. The last two terms inthis equation will then be proportional to at least one epsilon tensor and therefore dropout upon considering the fully symmetric component of the equation. All the informationis therefore contained in the spin- s field, while the spin t > s fields are auxiliary. Using the projections defined above, we can also give a purely algebraic construction ofthe mode solutions of the topologically massive gravity equations (3.50). In Poincar´ecoordinates, we start from the basis of solutions { C [ pq ] } p,q ∈ N of (3.65) and work out theprojection on a spin- s tensor (assuming ¯ h ≥ h ) using (3.72, 3.70) to find φ ( s )[ pq ] a ≡ (cid:16) V ( s ) a , C [ pq ] (cid:17) = e ( a + s − ∆) ρ ∞ (cid:88) n =0 v n ( s, a ) N pqp − n,q + a + s − n e − nρ x p − n + x q + a + s − n − , (3.90)where the normalization factor N was defined in (3.66) and the explicit expression for v n is given in Appendix B, see (B.10). – 19 –hese solutions form a basis of the topologically massive equations transforming as (cid:0) D + ( h ) , D − (¯ h ) (cid:1) under the AdS symmetries. A potential caveat in our reasoning so farwas that the spin- s projections of our solutions might actually vanish; the above expressionshows this not to be the case. Indeed, they yield the full multiplet of solutions to thetopologically massive equations. For example, for the lowest component φ ( s )[ pq ] − s = φ ( s )[ pq ] −− ... − (3.90) reduces to, up to a normalization factor, φ ( s )[ pq ] − s ∼ e − ( h +¯ h ) ρ x p + x q − F (cid:18) − p, − q, h, − e − ρ ( x − x + ) (cid:19) . (3.91)This expression is nonvanishing and finite since the hypergeometric function truncates toa polynomial with a finite number of terms. One checks that for h = ¯ h the physical fieldsatisfies the Klein-Gordon equation with the correct mass term (1.2):( ∇ µ ∇ µ − h ( h − φ (0)[ pq ]0 = 0 . (3.92) In this work, we have proposed a simple unfolded description of particles in an arbitraryrepresentation of the spacetime symmetry. It is somewhat nontrivial to connect this for-mulation with the standard covariant wave equations, which requires one to construct anintertwiner between this representation and the appropriate tensor under the Lorentz al-gebra, which we worked explicitly for massive fields of arbitrary spin in AdS (see [11] forthe case of massless fields in AdS D ≥ ). One could say that in this approach, the problem ofunfolding a given relativistic wave equation reduces to the representation theoretic problemof constructing the appropriate intertwiner.We end by pointing out some open problems and possible generalizations. • Our construction of the projection on spin- t tensors, using the non-normalizeablevectors V ( t ) a , was somewhat pedestrian and deserves a more rigorous treatment. Thiscould also elucidate whether our unfolded equation is truly equivalent to the alterna-tive unfolded formulation of [15]: though we found a map from our master field to theone in the formulation of [15], it is not clear if this map is invertible (see AppendixA of [39] for a discussion of this issue in the spin-0 case). • As we show in Appendix B, it is possible to construct highest weight (for h > ¯ h )or lowest weight (for h < ¯ h ) vectors with Lorentz spin lower than s in the represen-tation space ( D + ( h ) , D − (¯ h )). These are however not part of an irreducible Lorentzrepresentation, rather they form an indecomposable structure. The meaning of theprojection of our unfolded field C on this indecomposeable structure is unclear to us,though it is somewhat suggestive of a dual formulation involving gauge fields. • Our explicit construction for AdS could be generalized in various ways. For example,in higher dimensions one might expect to be able to construct an intertwiner betweenthe particle representation D (∆ , s ) and the multiplet that underlies the unfoldedmassive equations of [14]. It would also be interesting to study, using the results– 20 –f [11], the relation between our formulation for massless particles and the Fronsdalequations and their standard unfolded form [6], and to study the partially masslesscase. • One of our motivations for studying the current unfolded formulation is that it arisesnaturally in the AdS theory with higher spin square gauge symmetry. In a separatepublication [40], we will show that the natural equation describing matter coupled tothe higher spin square [26] describes an infinite set of unfolded massive higher spinequations of the type studied in this work. • Since the present unfolded formulation has a clear group theoretic meaning whichinvolves only the particle representation D (∆ , s ), it may be hoped that it providesa natural framework to describe higher spin interactions. It would be interesting togive a more group-theoretic characterization of the interaction vertices in Vasilievtheory, especially in their recently developed local form [41], in our framework. Itmay be also be hoped that the current setup is the natural one for addressing theopen problem of constructing the fully interacting theory with higher spin squaregauge symmetry. Acknowledgments
We thank Stefan Fredenhagen, Carlo Iazeolla, Tom´aˇs Proch´azka, Evgeny Skvortsov andMisha Vasiliev for useful discussions. We are indebted to Mitya Ponomarev and NicolasBoulanger for carefully reading our manuscript. P.K. would like to thank the Albert Ein-stein Institute for generous support by which his contribution to the present work becamepossible. The research of J.R. was supported by the Grant Agency of the Czech Republicunder the grant 17-22899S.
A AdS Conventions
In this appendix, we spell out some of our conventions for AdS . The AdS symmetryalgebra is [ M m , M n ] = (cid:15) pmn M p , [ M m , P n ] = (cid:15) pmn P p , [ P m , P n ] = (cid:15) pmn P p , (A.1)where η mn = −
20 1 0 − , (cid:15) − ≡ . (A.2)In terms of the sl (2 , R ) ⊕ sl (2 , R ) basis we have M m = L m + ¯ L m , P m = L m − ¯ L m . Fromthe dreibein and spin connection, we can form the AdS connection A = e m P m + ω m M m (A.3)– 21 –hose flatness, d A + A ∧ A = 0, is equivalent to the structure equations de m + (cid:15) mnp e n ∧ ω p = 0 , dω m + 12 (cid:15) mnp ( ω n ∧ ω p + e n ∧ e p ) = 0 . (A.4)Let Φ R be a field transforming in a representation R under local Lorentz tranforma-tions. The Lorentz covariant derivative is ∇ µ Φ R = ( ∂ µ + ω mµ R ( M m ))Φ R . (A.5)where R ( M m ) are the representation matrices. For example, on a tangent vector theappropriate representation is R ( M m ) v n = − (cid:15) nm p v p . (A.6)The covariant derivative can be extended to tensors with curved indices in the usual wayusing the Christoffel symbols. The covariant derivative has the following properties ∇ µ e mν = 0 , (A.7) ∇ µ ( R ( M m )Φ R ) = R ( M m ) ∇ µ Φ R , (A.8)[ ∇ m , ∇ n ]Φ R = − (cid:15) pmn R ( M p )Φ R , (A.9)where the last identity is derived from (A.4).We often use spinor notation, with indices α, β, . . . ∈ {− , + } , which are raised andlowered with (cid:15) αβ and (cid:15) αβ , where (cid:15) − + = (cid:15) − + = 1 . (A.10)We use ‘northwest-southeast’ conventions: (cid:15) αβ v β = v α , v β (cid:15) βα = v α . (A.11)The gamma matrices are denoted as γ m , m ∈ {− , , + } are given by( γ − ) βα = (cid:32) (cid:33) , ( γ ) βα = (cid:32) − (cid:33) , ( γ + ) βα = (cid:32) −
20 0 (cid:33) , (A.12)and they satisfy γ m γ n = η mn + (cid:15) pmn γ p , ( γ m ) α α ( γ m ) β β = − δ ( β α δ β ) α . (A.13)The spin- s representation of the Lorentz algebra acts on a rank-2 s symmetric multi-spinor as R s ( M m ) φ α ...α s = s ( γ m ) β ( α φ | β | α ...α s ) . (A.14)In the main text we make use of an operator D defined as( D φ ) α ...α s ≡ ∇ βα φ β...α s (A.15)one can show, using (A.9), that D φ α ...α s = ( (cid:3) + s + 1) φ α ...α s . (A.16)– 22 –ote that D does not map symmetric spinors into symmetric spinors in general, while D does.We also make use of the Lorentz-covariant definition of the Lie derivative with respectto a Killing vector k , acting on fields in arbitrary representations of the Lorentz algebra,see [36]. This definition extends the standard definition of the Lie derivative of tensor fieldsand is called the Lie-Lorentz derivative. For a field φ a transforming in a representation R under the Lorentz algebra, it is defined as L k φ a = k µ ∇ µ φ a + 12 ∇ [ m k n ] (cid:15) mnp R ( M p ) ba φ b . (A.17)From this definition, using (A.6) and the fact that k is a Killing vector, one shows that L k e mµ = 0 , L k R ( M m ) ba = 0 . (A.18)On AdS , using the identities (3.15), the Lie-Lorentz derivative simplifies to L l m φ a = l µm (cid:16) ∇ µ φ a − e pµ R ( M p ) ba φ b (cid:17) , (A.19) L ¯ l m φ a = ¯ l µm (cid:16) ∇ µ φ a + e pµ R ( M p ) ba φ b (cid:17) . (A.20)In spinor notation, this leads to (3.41). B Representations of the Lorentz Subalgebra
In this appendix, we will construct vectors in the representation space ( D + ( h ) , D − (¯ h )) with h, ¯ h > M m act on this space as M m ψ = L m ψ − ψL m , with the L m given explicitly in (3.30). Westart by constructing all lowest Lorentz weights λ w and highest weights ν w defined by theproperties M λ w = wλ w , M λ w = 0 (B.1) M ν w = wν w , M − ν w = 0 . (B.2)It follows that if λ w is a lowest weight w state, then λ w − k ≡ L k λ w = (cid:18) P (cid:19) k λ w , k ∈ N , (B.3)if nonvanishing, is another lowest weight state of weight w − k . Similarly if ν w is a highestweight w state, then ν w +˜ k ≡ L ˜ k − ν w = (cid:18) P − (cid:19) ˜ k ν w , ˜ k ∈ N , (B.4) Viewed here as the vector space
Span {| m (cid:105) h ¯ h (cid:104) n | , m, n ∈ N } , in particular we will not insist on normal-izeability with respect to the norm (3.54) on ( D + ( h ) , D − (¯ h )). – 23 –f nonvanishing, is another highest weight state of weight w + ˜ k . Starting from an ansatzdescribing the most general vector with fixed Lorentz weight under M , one finds that thefull set of lowest (highest) weight vectors can be obtained in this way, through repeatedaction of of L ( L − ) on a single starting vector.One finds lowest weight vectors at weights − s − k and highest weight vectors at weights − s + ˜ k , with k, ˜ k ∈ N and s ≡ | h − ¯ h | . Assuming for the moment that h ≤ ¯ h (we will returnto the case h > ¯ h below), these vectors are given by λ − s − k = L k λ − s , λ − s = (cid:88) n ∈ N (cid:18) (2¯ h + n − h + n − (cid:19) | n (cid:105) h ¯ h (cid:104) n | , (B.5) ν − s +˜ k = L ˜ k − ν − s , ν − s = (cid:88) n ∈ N (cid:18) (2 h + n − h + n − (cid:19) | n (cid:105) h ¯ h (cid:104) n | . (B.6)Let us now proceed to arrange these highest and lowest weight vectors and their Lorentzdescendants into multiplets, see Figure 1 for the resulting weight diagram in the case ofspin 2.Setting ˜ k = 2 s + k in (B.6), we find that ν s + k ∼ M s + k ) − λ − s − k and hence these vectorsare the lowest and highest weights of a 2( s + k ) + 1-dimensional spin-( s + k ) representation,denoted as D ( s + k ) in Section 3.2. We will denote the basis vectors in the D ( t ) multipletas V ( t ) a , with a running from − t to t and normalized as follows: V ( t ) a = ( − t + a ( t − a )!(2 t )! M t + a − (cid:0) L t − s λ − s (cid:1) , t ≥ s, | a | ≤ t (B.7)Our normalization constants are chosen such that V ( t ) − t = λ − t , V ( t ) t = ν t and such that theLorentz generators act as M m V ( t ) a = ( mt + a ) V ( t ) a − m . (B.8)For the explicit component expression of the V ( t ) a one finds V ( t ) a = (cid:88) n ∈ N v n ( h, ¯ h ; t, a ) | n (cid:105) h ¯ h (cid:104) n − a − s | , for h ≤ ¯ h (B.9)with (recall ∆ ≡ h + ¯ h ) v n ( h, ¯ h ; t, a ) = ( t − a )!(2 t )! t + a (cid:88) l =0 ( − ) l (cid:18) t + al (cid:19)(cid:16) (1 − h − n ) l ( − n ) l (1 − l + n ) t − s (2 h − l + n ) t + s (2 − a − s + n ) t + a − l − (1 + ∆ − a + n ) t + a − l − (1 − a − s + n )(∆ − a + n ) (cid:17) / , (B.10)where ( x ) n = x ( x + 1) . . . ( x + n −
1) denotes the Pochhammer symbol. To find similarexpressions for the case h > ¯ h , one notes that the Hermitean conjugate ( V ( t ) a ) † is a stateof weight − a in the space ( D + (¯ h ) , D − ( h )), leading to V ( t ) a = (cid:88) n ∈ N v n (¯ h, h ; t, − a ) | n + a − s (cid:105) h ¯ h (cid:104) n | , for h ≥ ¯ h (B.11)– 24 – - ν - ν ν ν ν ν λ - λ - λ - M Figure 1 . Weight diagram showing the highest (in red) and lowest (in green) weight states underthe Lorentz algebra and their descendants (in blue), for the spin 2 case with ¯ h − h = 2. An arrowpointing up (down) means that the states are linked by the action of M − ( M ). We note thatthe states ν − and ν − are null primaries. The states in the blue shaded region are the V ( t ) a whichform an irreducible representation of the full AdS algebra sl (2 , R ) ⊕ sl (2 , R ). Only the states in thegreen shaded region are actually normalizeable. As we saw in (B.8), the vectors V ( t ) a span the representation ⊕ ∞ t = s D ( t ) under theLorentz subalgebra generated by M m . What is more, they also transform among themselvesunder the full sl (2 , R ) ⊕ sl (2 , R ) symmetry, under which they form a single irreduciblerepresentation. To see this, we have to work out how the AdS translation generators acton them; one can derive the following relation: P m V ( t ) a = 2 V ( t +1) a − m − µst ( t + 1) ( mt + a ) V ( t ) a − m + ( s − t )( µ − t )2 t (2 t + 1) d m ( t, a ) V ( t − a − m . (B.12)where µ was defined in (3.45) and we defined the coefficients d ± ( t, a ) = ( a ± t )( a ± ( t − t (2 t − , d ( t, a ) = a − t t (2 t − . (B.13)The coefficients in (B.12) are completely fixed by the properties (B.3,B.4), consistency withthe AdS algebra (A.1) and the Casimir relation η mn P m P n V ( t ) a = (cid:0) h ( h −
1) + 2¯ h (¯ h − − t ( t + 1) (cid:1) V ( t ) a . (B.14)– 25 –e also verified (B.12) using the explicit expressions (B.9).We also mention a further useful relation involving the action of the translation gen-erators P m on the vectors V ( t ) a . Recalling that these act as P m ψ = L m ψ + ψL m one canderive the following identity M n +1 − P = n ( n + 1) P − M n − − − n + 1) P M n − + P M n +1 − . (B.15)Combining this with (B.7), we find a relation expressing the vectors in the spin- t + 1representation in terms of the action of the translation generators on vectors in the spin- t representation: V ( t +1) a = (cid:88) m = − c m ( t, a ) P m V ( t ) a + m (B.16)where c − = ( t + a )( t + a + 1)2(2 t + 2)(2 t + 1) , c = ( t + a + 1)( t − a + 1)(2 t + 2)(2 t + 1) , c = ( t − a + 1)( t − a )2(2 t + 2)(2 t + 1) . (B.17)In (B.6) we also found, for ¯ h > h , a number of highest weight vectors, namely ν − s +˜ k for 0 ≤ ˜ k < s , for which there is no corresponding lowest weight vector and which hencedo not fit in finite-dimensional representations. Though these do not play a role in thepresent work (see however the Outlook section), we now briefly comment on the Lorentzrepresentations carried by these states. One checks that ν − s +˜ k for 0 ≤ ˜ k < s − is actuallya Lorentz descendant of ν s − ˜ k − : ν − s +˜ k ∼ M s − ˜ k ) − ν s − ˜ k − for 0 ≤ k < s − . (B.18)The ν − s +˜ k for 0 ≤ ˜ k < s − and their descendants therefore form an invariant subspacewhose complement is not invariant; in that case ν − s +˜ k and ν s − ˜ k − belong to an infinite-dimensional reducible but indecomposable representation of the Lorentz subalgebra.Let us also discuss the normalizablility of the vectors constructed above with respectto the inner product (3.54). 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