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 inmind 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 reduciblemassless 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 interactionvertex for reducibleHigherSpinmodes with two scalars. Obviously this technique canbe 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 interactionvertexes for triplets inAdS is often technically complicated, due to repeated commuta- tors betweencovariantderivatives. Thedouble trace- lessness condition for irreducible Higher Spin modes makes the problem at hand considerably simpler. Let us start froma free Lagrangiandescribing the propagation of reduciblemasslessHigher Spinmodes 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 ∇ · ϕ C + 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(μν Aρ) = gμν Aρ + 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 andcanbe eliminatedvia its ownequationsofmotion to obtain L = − 1 2 (∇μϕ)2 + s 2 (∇ · ϕ)2 + s(s −1)∇ · ∇ · ϕ D + 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 tripletwhich contains fields ϕμν(x), Cμ(x) and D(x). Let us make the ansatz ϕμν =Ψμν + 1 D −2 gμνΨ, ϕ ′ −2D =Ψ. (4) Inserting these expressions back to the Lagrangian (3), for s =2 one obtains L = − 1 2 (∇μΨρσ)2 +(∇νΨνμ) 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 +2 gΨ(2) + 1 D(D −2) (g) 2 Ψ(0) D = 1 2 [ Ψ′ (4) + 2 D +2 Ψ(2) + 1 D +2 gΨ′ (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 +2 ηΛ′ (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) k Ψ(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−1 Ψ(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 Fronsdalmode, i.e. the inverse of the prefactor of (∇μΨ(s−2k))2 terms mul- tiplied by 2, is Q(s, k, D)= 2kk!(s −2k)! s!ρ̃k(D, s) . (11) Nowletusbuilda cubic interactionvertexofaHigher 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 J 1;2 s−2q = s−2q∑ r=0 Crs−2q(−1) r(∇μ1 . . . ∇μr φ1) · (∇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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Giuria 1, I-10125 Torino, Italy Mirian Tsulaia E-mail: tsulaia@liv.ac.uk Department of Mathematical Sciences University of Liverpool Liverpool, L69 7ZL, United Kingdom 53