wykresx.eps Acta Polytechnica Vol. 51 No. 1/2011 Current Exchanges for Reducible Higher Spin Modes on AdS A. Fotopoulos, M. Tsulaia Abstract 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) 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 have been constructed in the metric-like approach [8, 9, 10, 11]. Bearing in mind a possible application of the reducible Higher Spin modes (described by the so- called “triplet” [20]) for String Theory [21]–[22] and for AdS/CFT correspondence [23], we consider the problem of cubic interaction of a triplet on an AdS space. In particular we study the cubic interaction of 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 massless bosonic Higher Spin modes in a flat space time also works for an arbitrary dimensional AdS space. The second and more important result is that after this decomposition one can use the cubic vertex1 of [16], which describes an interaction of irreducible (Frons- dal) Higher Spin modes with two scalars, to obtain an interaction vertex for reducible Higher Spin modes with two scalars. Obviously this technique can be ap- plied not only for the particular vertex given in [16], but for constructing of more complicated interaction vertices 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 often technically complicated, due to repeated commuta- tors between covariant derivatives. The double trace- lessness condition for irreducible Higher Spin modes makes the problem at hand considerably simpler. Let us start from a free Lagrangian describing the propagation of reducible massless Higher Spin modes on a D dimensional AdS space. It contains a field ϕμ1,...,μs(x) of rank s, a field Cμ1,...,μs−1(x) of rank s− 1 and a field Dμ1,...,μs−2(x) of rank s− 2, and has the form [13] (see also [2] for details of the construc- tion), L = −1 2 (∇μϕ)2 + s∇ · ϕC + s(s− 1)∇ · C D + s(s− 1) 2 (∇μD)2 − s 2 C2 + s(s − 1) 2L2 (ϕ ′ ) 2 − s(s− 1)(s − 2)(s− 3) 2L2 (D ′ ) 2 − 4s(s− 1) L2 D ϕ′ − 1 2L2 [(s − 2)(D + s− 3)− s]ϕ2 + s(s− 1) 2L2 [s(D + s− 2) + 6] D2, (1) The symbol ∇· means divergence, while ∇ is the symmetrized 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 we have gA = g(μνAρ) = gμνAρ + gμρAν + gνρAμ. This Lagrangian is invariant under gauge transformations with parameter Λμ1,...,μs−1(x) δϕ = ∇Λ, δC = � Λ + (s− 1)(3− s −D) L2 Λ + 2 L2 g Λ′ δD = ∇ · Λ . (2) Let us note that the field C(x) has no kinetic term and can be eliminated via its own equations of motion to obtain L = −1 2 (∇μϕ)2 + s 2 (∇ · 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, 2010 1Let 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 ghost terms. 50 Acta Polytechnica Vol. 51 No. 1/2011 s(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) L2 D ϕ′ − 1 2L2 [(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 in terms of irreducible (Fronsdal) [6] modes, following the procedure given in [17] for a Minkowski space. Let us start with the simplest example of a s = 2 triplet which contains fields ϕμν(x), Cμ(x) and D(x). Let us make the ansatz ϕμν = Ψμν + 1 D − 2gμνΨ, ϕ′ − 2D = Ψ. (4) Inserting these expressions back to the Lagrangian (3), for s = 2 one obtains L = −1 2 (∇μΨρσ)2 + (∇νΨν μ) 2 +Ψ′∇μ∂νΨμν + 1 2 (∇μΨ′)2 − 1 2(D − 2)(∇μΨ)2 + (5) 1 L2 (Ψμν)2 + D − 3 L2(D − 2)(Ψ) 2 + D − 3 2L2 (Ψ′)2 Therefore, the initial Lagrangian (3) has been de- composed into a sum of two Fronsdal Lagrangians for s = 2 field Ψμν with the gauge transformation law δΨμν = ∇μΛν + ∇μΛν and a gauge invariant scalar Ψ. Let us describe this procedure for the spin 4 triplet, since in this case both a constraint on the parameter of gauge transformations and an off-shell constraint on the gauge field arise. Let us use the substitution [17] ϕ(4) = Ψ(4) + 1 D + 2gΨ (2) + 1 D(D − 2)(g) 2Ψ(0) D = 1 2 [ Ψ′(4) + 2 D + 2Ψ (2) + 1 D + 2gΨ ′(2) + (6) 2 D(D − 2)gΨ (0) ] . The field Ψ(4) is doubly traceless and transforms un- der the gauge transformations as δΨ(4) = ∇Λ̃, Λ̃ = Λ− 1 D + 2ηΛ ′ (7) Inserting these expressions into the Lagrangian (3), one can see again that it decomposes into the sum of the Fronsdal modes with spins 4, 2 and 0, described by the fields Ψ(4),Ψ(2) and Ψ(0). One can further generalize this procedure for an arbitrary spin. In particular, take ϕ = [ s2 ]∑ k=0 ρ̃k(D, s)(g)kΨ(s−2k) D = 1 2 [ s2 ]−1∑ k=0 ρ̃k(D, s)(g)kΨ ′(s−2k) + (8) [ s2 ]∑ k=1 ρ̃k(D, s)(g)k−1Ψ(s−2k). and Λ̃s−1−2k = [ s2 ]∑ q=0 ρq(D, s − 2k − 1)(g)qΛ[q+k](s−1) (9) with ρq(D − 2, s)= (−1) q(D + 2(s− q − 3))!! (D + 2(s− 3))!! , ρ̃k(D, s) = (D + 2(s− 2k − 2))!! (D + 2(s− k − 2))!! (10) and [q+k] denotes the number of traces. Finally, one can show that the normalization factor for the prop- agators for each of individual Fronsdal mode, i.e. the inverse of the prefactor of (∇μΨ (s−2k))2 terms mul- tiplied by 2, is Q(s, k,D) = 2 kk!(s− 2k)! s!ρ̃k(D, s) . (11) Now let us build a cubic interaction vertex of a Higher Spin triplet with two scalars on AdS. To this end, let us use the corresponding vertex for an individual Fronsdal mode [16] L00sint =Ψ (s) · Js + [ s − 1 6L2 [2s2 + (3D − 4)s− 6]− s− 2 L2 ] Ψ′(s) · Js−2 (12) where J1;2s−2q = s−2q∑ r=0 Cr s−2q(−1)r(∇μ1 . . .∇μrφ1) · (∇μr+1 . . .∇μs−2q φ2) (13) Therefore multiplying the interacting vertices (12) with the appropriate factor (11) and adding them to the free Lagrangian (3), one finds the expression for a cubic Lagrangian describing the interaction of re- ducible Higher Spin modes with two scalars on AdS. 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