We show how to decompose a Lagrangian for reducible massless bosonic Higher Spin modes into the ones describing irreducible (Fronsdal) Higher Spin modes on a D dimensional AdS space. Using this decomposition we construct a new nonabelian cubic interaction vertex for reducible higher spin modes and two scalars on AdS from the already known vertex which involves irreducible (Fronsdal) mode

Fotopoulos, A.

Tsulaia, M.

Acta Polytechnica

English

arXiv

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 51 No. 1/2011Current Exchanges for Reducible Higher Spin Modes on AdSA. Fotopoulos, M. TsulaiaAbstractWe show how to decompose a Lagrangian for reducible massless bosonic Higher Spin modes into the ones describingirreducible (Fronsdal) Higher Spin modes on a D dimensional AdS space. Using this decomposition we construct a newnonabelian cubic interaction vertex for reducible higher spin modes and two scalars on AdS from the already knownvertex which involves irreducible (Fronsdal) modes.Keywords: gauge symmetry, AdS-CFT correspondence, string field theory.Higher Spin gauge theories (see [1]–[2] for recent re-views) are usually formulated either in frame-like[3]–[4] or metric-like [5]–[20] approaches.Recently several interesting cubic vertices havebeen constructed in the metric-like approach [8, 9,10, 11]. Bearing in mind a possible application of thereducible Higher Spin modes (described by the so-called “triplet” [20]) for String Theory [21]–[22] andfor AdS/CFT correspondence [23], we consider theproblem of cubic interaction of a triplet on an AdSspace. In particular we study the cubic interactionof a triplet with two scalar fields.The main result of this paper is twofold. Firstly,we show that the procedure derived in [17] for de-composing the free Lagrangian for reducible masslessbosonic Higher Spin modes in a flat space time alsoworks for an arbitrary dimensional AdS space. Thesecond and more important result is that after thisdecomposition one can use the cubic vertex1 of [16],which describes an interaction of irreducible (Frons-dal) Higher Spin modes with two scalars, to obtainan interaction vertex for reducible Higher Spin modeswith two scalars. Obviously this technique can be ap-plied not only for the particular vertex given in [16],but for constructing of more complicated interactionvertices in AdS following the method given in [14].The advantage of this approach is that the construc-tion of interaction vertexes for triplets in AdS is oftentechnically complicated, due to repeated commuta-tors between covariant derivatives. The double trace-lessness condition for irreducible Higher Spin modesmakes the problem at hand considerably simpler.Let us start from a free Lagrangian describing thepropagation of reducible massless Higher Spin modeson a D dimensional AdS space. It contains a fieldϕμ1,...,μs(x) of rank s, a field Cμ1,...,μs−1(x) of ranks− 1 and a field Dμ1,...,μs−2(x) of rank s− 2, and hasthe form [13] (see also [2] for details of the construc-tion),L = −12(∇μϕ)2 + s∇ · ϕC + s(s− 1)∇ · C D +s(s− 1)2(∇μD)2 − s2C2 +s(s− 1)2L2(ϕ′)2 −s(s− 1)(s− 2)(s− 3)2L2(D′)2 −4s(s− 1)L2Dϕ′ −12L2[(s− 2)(D + s− 3)− s]ϕ2 +s(s− 1)2L2[s(D + s− 2) + 6] D2, (1)The symbol ∇· means divergence, while ∇ is thesymmetrized action of ∇μ on a tensor. The sym-bol ′ means that we take the trace of a field. Mul-tiplication of a tensor by the metric g implies sym-metrized multiplication, i.e., if A is a vector Aμ wehave gA = g(μνAρ) = gμνAρ + gμρAν + gνρAμ. ThisLagrangian is invariant under gauge transformationswith parameter Λμ1,...,μs−1(x)δϕ = ∇Λ,δC =  Λ +(s− 1)(3− s−D)L2Λ +2L2g Λ′δD = ∇ · Λ . (2)Let us note that the field C(x) has no kinetic termand can be eliminated via its own equations of motionto obtainL = −12(∇μϕ)2 + s2 (∇ · ϕ)2 +s(s− 1)∇ · ∇ · ϕD + s(s− 1) (∇μD)2 +Talk given at the XIXth International Colloquium on Integrable Systems and Quantum Symmetries, Prague, Czech Republic,June 17–19, 20101Let us point out that the method given in [14] describes the construction of nonabelian cubic interaction vertices, see also [16]for some explicit nonabelian examples. A particular example of an abelian vertex given in [15] is in some sense a “degenerate”solution of the method, where however the abelian property is maintained in a nontrivial way, due to the structure of the ghostterms.50Acta Polytechnica Vol. 51 No. 1/2011s(s− 1)(s− 2)2(∇ ·D)2 +s(s− 1)2L2(ϕ′)2 −s(s− 1)(s− 2)(s− 3)2L2(D′)2 − (3)4s(s− 1)L2Dϕ′ −12L2[(s− 2)(D + s− 3)− s]ϕ2 +s(s− 1)2L2[s(D + s− 2) + 6] D2 .Now we would like to decompose this Lagrangian interms of irreducible (Fronsdal) [6] modes, followingthe procedure given in [17] for a Minkowski space.Let us start with the simplest example of a s = 2triplet which contains fields ϕμν(x), Cμ(x) and D(x).Let us make the ansatzϕμν = Ψμν +1D − 2gμνΨ, ϕ′ − 2D = Ψ. (4)Inserting these expressions back to the Lagrangian(3), for s = 2 one obtainsL = −12(∇μΨρσ)2 + (∇νΨνμ)2 +Ψ′∇μ∂νΨμν +12(∇μΨ′)2 − 12(D − 2)(∇μΨ)2 + (5)1L2(Ψμν)2 +D − 3L2(D − 2)(Ψ)2 +D − 32L2(Ψ′)2Therefore, the initial Lagrangian (3) has been de-composed into a sum of two Fronsdal Lagrangiansfor s = 2 field Ψμν with the gauge transformationlaw δΨμν = ∇μΛν + ∇μΛν and a gauge invariantscalar Ψ.Let us describe this procedure for the spin 4triplet, since in this case both a constraint on theparameter of gauge transformations and an off-shellconstraint on the gauge field arise. Let us use thesubstitution [17]ϕ(4) = Ψ(4) +1D + 2gΨ(2) +1D(D − 2)(g)2Ψ(0)D =12[Ψ′(4) +2D + 2Ψ(2) +1D + 2gΨ′(2) + (6)2D(D − 2)gΨ(0)].The field Ψ(4) is doubly traceless and transforms un-der the gauge transformations asδΨ(4) = ∇Λ˜, Λ˜ = Λ− 1D + 2ηΛ′ (7)Inserting these expressions into the Lagrangian (3),one can see again that it decomposes into the sum ofthe Fronsdal modes with spins 4, 2 and 0, describedby the fields Ψ(4),Ψ(2) and Ψ(0).One can further generalize this procedure for anarbitrary spin. In particular, takeϕ =[ s2 ]∑k=0ρ˜k(D, s)(g)kΨ(s−2k)D =12[ s2 ]−1∑k=0ρ˜k(D, s)(g)kΨ′(s−2k) + (8)[ s2 ]∑k=1ρ˜k(D, s)(g)k−1Ψ(s−2k).andΛ˜s−1−2k =[ s2 ]∑q=0ρq(D, s− 2k − 1)(g)qΛ[q+k](s−1) (9)withρq(D − 2, s)= (−1)q(D + 2(s− q − 3))!!(D + 2(s− 3))!! ,ρ˜k(D, s) = (D + 2(s− 2k − 2))!!(D + 2(s− k − 2))!!(10)and [q+k] denotes the number of traces. Finally, onecan show that the normalization factor for the prop-agators for each of individual Fronsdal mode, i.e. theinverse of the prefactor of (∇μΨ(s−2k))2 terms mul-tiplied by 2, isQ(s, k,D) = 2kk!(s− 2k)!s!ρ˜k(D, s) . (11)Now let us build a cubic interaction vertex of a HigherSpin triplet with two scalars on AdS. To this end,let us use the corresponding vertex for an individualFronsdal mode [16]L00sint =Ψ(s) · Js +[s− 16L2[2s2 + (3D − 4)s− 6]−s− 2L2]Ψ′(s) · Js−2(12)whereJ1;2s−2q =s−2q∑r=0Crs−2q(−1)r(∇μ1 . . .∇μrφ1) ·(∇μr+1 . . .∇μs−2qφ2)(13)Therefore multiplying the interacting vertices (12)with the appropriate factor (11) and adding them tothe free Lagrangian (3), one finds the expression fora cubic Lagrangian describing the interaction of re-ducible Higher Spin modes with two scalars on AdS.Then one can perform a current–current exchangeprocedure following the lines of [18, 19, 17].51Acta Polytechnica Vol. 51 No. 1/2011AcknowledgementIt is a pleasure to thank A. P. Isaev, S. O. 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Current Exchanges for Reducible Higher Spin Modes on AdS

A. Fotopoulos

M. Tsulaia

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Current Exchanges for Reducible Higher Spin Modes on AdS

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