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Nonlinear Equations for Symmetric Massless Higher Spin Fields in $(A)dS_d$

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FIAN/TD/07–03Nonlinear Equations for Symmetric Massless Higher Spin Fields in (A )dS d M.A.Vasiliev I.E.Tamm Department of Theoretical Physics,Lebedev Physical Institute,Leninsky prospect 53,119991,Moscow,Russia Abstract Nonlinear ?eld equations for totally symmetric bosonic massless ?elds of all spins in any dimension are presented.1Introduction Higher spin (HS)gauge theories are theories of gauge ?elds of all spins (see e.g.[1]for a review).Because HS gauge symmetries are in?nite-dimensional,HS gauge theories may correspond to most symmetric vacua of a theory of funda-

mental interactions presently identi?ed with superstring theory.The problem is to introduce interactions of HS ?elds in a way compatible with nonabelian HS gauge symmetries containing di?eomorphisms and Yang-Mills symmetries.Full nonlinear dynamics of HS gauge ?elds has been elaborated so far at the level of equations of motion for d =4[2,3]which is the simplest nontrivial case since HS gauge ?elds do not propagate if d <4.Some lower order inter-actions of HS ?elds in the framework of gravity were worked out at the action level for d =4[4]and for d =5[5].As a result,it was found out that (i)consistent HS theories contain in?nite sets of in?nitely increasing spins;(ii)HS gauge interactions contain higher derivatives;(iii)in the framework of gravity,unbroken HS gauge symmetries require a non-zero cosmological constant;(iv)HS symmetry algebras [6]are certain star product algebras [7].The properties (i)and (ii)were deduced in the earlier work [8]on HS interactions in ?at space.

The feature that unbroken HS gauge symmetries require a non-zero cosmo-logical constant is crucial in several respects.It explains why the analysis of HS–gravitational interactions in the framework of the expansion near the?at background led to negative conclusions[9].The S-matrix Coleman-Mandula-type no-go arguments[10]become irrelevant because there is no S-matrix in the AdS space.Also it explains why the HS gauge theory phase is not di-rectly observed in the superstring theory prior its full formulation in the AdS background is found,and?ts the idea of the correspondence between HS gauge theories in the bulk and the boundary conformal theories[11,12,13,14,15,16].

In the context of applications of the HS gauge theory to the superstring theory(d=10)and M theory(d=11),it is important to extend the4d results on the HS–gravitational interactions to higher dimensions.The aim of this paper is to present the full nonlinear formulation of the?eld equations for totally symmetric bosonic HS?elds in any dimension.The form of the proposed equations is analogous to that of the4d equations of[3].

2Free higher spin gauge?elds

There are two equivalent approaches to description of totally symmetric bosonic massless?elds of all spins at the free?eld level.The approach developed by Fronsdal[17]and de Wit and Freedman[18]is parallel to the metric formula-tion of gravity.Here an integer spin s massless?eld is described by a totally

symmetric tensor?a

1...a s (a,b,c=0...d?1)subject to the double traceless-

ness condition[17]?b b c ca

5...a s =0which is nontrivial for s≥4.The free?eld

Abelian HS gauge transformation is

δ?a

1...a s =?{a

εa

2...a s}

,(2.1)

where the parameterεa

1...a s?1is a totally symmetric rank(s?1)traceless

tensor,εb ba

3...a s?1=0.A nonlocal version of the same theory with unrestricted

gauge parameters was developed in[19].

An alternative approach operates in terms of a1-form frame like HS gauge

?eld e a

1...a s?1=dx n a

1...a s?1

[20,21,22]which is traceless in the?ber indices,

e b ba3...a s?1=0.(2.2)

(We use underlined letters for indices of vector?elds and forms.)The Abelian HS gauge transformation law in the?at space-time is

δe a1...a s?1=dεa1...a s?1+h bεa1...a s?1,b.(2.3)

Here h a is the?at space frame?eld(i.e.,h a=dx a in the Cartesian coordi-nates).εa1...a s?1is a totally symmetric traceless gauge parameter equivalent to that of the Fronsdal’s formulation.The gauge parameterεa1...a s?1,b is also traceless and satis?es the condition that the symmetrization over all its indices is zeroε{a1...a s?1,a s}=0.It is a HS generalization of the parameter of the local Lorentz transformations in gravity(s=2)which isεa,b.The Lorentz type ambiguity due toεa1...a s?1,b can be gauge?xed by requiring the frame type HS gauge?eld to be totally symmetric

e n a

2...a s =?na

2...a s

,(2.4)

thus establishing equivalence with the Fronsdal’s formulation.Note that such

de?ned?a

1...a s is automatically double traceless as a consequence of(

2.2).

The Lorentz type HS symmetry with the parameterεa1...a s?1,b assumes a HS1-form connectionωa1...a s?1,b.From the analysis of its transformation law it follows[21,22]that,generically,some new gauge connections and symmetry parameters have to be introduced.As a result,the full set of HS connections associated with a spin s massless?eld consists of the1-forms dx n a1...a s?1,b1...b t which take values in all irreducible representations of the d-dimensional mass-less Lorentz group o(d?1,1)described by the traceless Young tableaux with at most two rows such that the upper row has length s?1

s-1

q q

(2.5) In other words,the1-forms dx n a1...a s?1,b1...b t are symmetric in the Lorentz vector indices a i and b j separately and satisfy the relationsωn

a1...a s?3c

,b1...b t=0.(From here it follows that all other traces of the?ber indices are also zero.)For the case t=0,the?eldωa1...a s?1identi?es with the dynamical spin s frame type?eld e a1...a s?1.

The formalism of[21,22]works both in(A)dS d and in?at space.In (A)dS d it allows to build the free HS actions in terms of manifestly gauge invariant linearized(Abelian)?eld strengths.Explicit form of these linearized curvatures provides the starting point towards determination of nonabelian HS symmetries and HS curvatures.For the case of d=4this program was realized in[21,6].Here we extend these results to the bosonic HS theory in any dimension.To this end it is convenient to use the observation of[5] that the collection of the HS1-formsωa1...a s?1,b1...b t with all0≤t≤s?1can be interpreted as a result of the“dimensional reduction”of a1-form

ωA1...A s?1,B1...B s?1carrying the irreducible representation of o(d?1,2)or o(d,1) (A,B=0,...,d)described by the traceless two-row rectangular Young tableau of length s?1

ω{A1...A s?1,A s}B2...B s?1=0,ωA1...A s?3C C,B1...B s?1=0.(2.6) Let us?rst recall how this approach works in the gravity case.d dimen-sional gravity can be described by a1-form connectionωAB=?ωBA of the (A)dS Lie algebra(o(d,1))o(d?1,2).The Lorentz subalgebra o(d?1,1) is identi?ed with the stability subalgebra of some vector V A.Since we are discussing local Lorentz symmetry,this vector can be chosen di?erently in dif-ferent points of space-time,thus becoming a?eld V A=V A(x).The norm of this vector is convenient to relate to the cosmological constantΛso that V A has dimension of length

V A V A=?Λ?1.(2.7)Λis supposed to be negative and positive in the AdS and dS cases,respectively (within the mostly minus signature).This allows for a covariant de?nition of the frame?eld and Lorentz connection[23,24]

E A=D(V A)≡dV A+ωAB V B,ωL AB=ωAB+Λ(E A V B?E B V A).(2.8) According to these de?nitions E A V A=0,D L V A=dV A+ωL AB V B≡0.When the frame E A n

E B m

It is manifestly invariant under the linearized HS gauge transformations

δωA1...A s?1,B1...B s?1(x)=D0εA1...A s?1,B1...B s?1(x)(2.11) because D20≡R(ω0)=0.The(A)dS covariant form of the free HS action of [22]is[5]

S s2=

(s?p?2)!

(2.13) are?xed by the condition that the action is independent of all those compo-

nents ofωA1...A s?1,B1...B s?1

1for which V B

V B

ωA1...A s?1,B1...B s?1

1=0.As a result

of this condition,the free action(2.12)depends only on the frame type dynam-ical HS?eld e a1...a s?1and the Lorentz connection type auxiliary?eldωa1...a s?1,b expressed in terms of the?rst derivatives of e a1...a s?1by virtue of its equation of motion equivalent to the“zero torsion condition”

0=T1A

1...A s?1≡R1A1...A s?1,B1...B s?1V B1...V B s?1.(

2.14) Plugging the expression forωa1...a s?1,b back into(2.12)gives rise to the free HS action expressed entirely(modulo total derivatives)in terms of e n

a1...a s?1equivalent to?m

1...m s contributes to the action.Although

this action is de?ned to be independent of the“extra?elds”ωa1...a s?1,b1...b t with t≥2,one has to express the extra?elds in terms of the dynamical HS?elds because they contribute beyond the linearized approximation.The ?eldsωa1...a s?1,b1...b t with t>0express via up to order t derivatives of the dynamical?eld by virtue of certain constraints[21,22]analogous to the zero torsion condition in gravity.As a result,the condition that the free action is independent of the extra?elds is equivalent to the condition that it does not contain higher derivatives.

3(A)dS d bosonic higher spin algebra

From the analysis of section2it is clear that,to reproduce the correct set of HS gauge?elds,one has to?nd such an algebra g which contains h=o(d?1,2) or h=o(d,1)as a subalgebra and decomposes under the adjoint action of h in g into a sum of irreducible?nite-dimensional modules over h described by various two-row rectangular traceless Young tableaux.Such an algebra was described recently by Eastwood in[25]as the algebra of conformal HS symmetries of the free massless Klein-Gordon equation in d?1dimensions. Here we give a slightly di?erent de?nition of this algebra which is more suitable for the analysis of the HS interactions.

Consider oscillators Y A i with i=1,2satisfying the commutation relations [Y A i,Y B j]?=εijηAB,εij=?εji,ε12=1,(3.1) whereηAB is the invariant symmetric form of o(n,m).For example,one can interpret these oscillators as conjugated coordinates and momenta Y A1=P A, Y B2=Y B.ηAB andεij will be used to raise and lower indices in the usual manner A A=ηAB A B,a i=εij a j,a i=a jεji.

We use the Weyl(Moyal)star product

(f?g)(Y)=

Y iA Y B i,t ij=t ji=Y A i Y jA.(3.4)

Consider the subalgebra S spanned by the sp(2)singlets f(Y)

[t ij,f(Y)]?=0.(3.5)

Eq.(3.5)is equivalent to Y Ai ?Y A

i f (Y )=0.For the expansion (3.3)this condition implies that the coe?cients f A 1...A m ,B 1...B n are nonzero only if

n =m and that symmetrization over any m +1indices of f A 1...A m ,B 1...B m gives

zero,i.e.they have the symmetry properties of a two-row rectangular Young tableau.As a result,the gauge ?elds of S are

ω(Y |x )=∞ l =0ωA 1...A l ,B 1...B l (x )Y A 11...Y A l 1Y B 12...Y B l 2(3.6)

with the component gauge ?elds ωA 1...A l ,B 1...B l (x )taking values in all two-row rectangular Young tableaux of o (n +m ).The algebra S is not simple.It contains the two-sided ideal I spanned by the elements of the form g =t ij ?g ij ,(3.7)

where g ij transforms as a symmetric tensor with respect to sp (2),i.e.,[t ij ,g kl ]?=δk i g j l +δk j g i l +δl i g j k +δl j g i k .(Note that t ij ?g ij =g ij ?t ij .)Actually,from (3.5)it follows that f ?g,g ?f ∈I ?f ∈S ,g ∈I .Due to the de?nition (3.4)of t ij ,the ideal I contains all traces of the two-row Young tableaux.As a result,the algebra S/I has only traceless two-row tableaux in the expansion (3.6).(Let us note that this factorization is not optional because some of the traces of two-row rectangular tableaux are not themselves two-row rectangular tableaux and may not admit a straightforward interpretation in terms of HS ?elds.)The algebra S/I was identi?ed by Eastwood in [25]as conformal HS algebra in d ?1dimensions.

For the complex algebra S/I we will use notation hgl (1/sp (2)[n +m ]|C ).Its real form corresponding to a unitary HS theory in the AdS case of n =2will be called hu (1/sp (2)[n,m ]).The meaning of this notation is as follows.According to [26],hgl (p,q |2r )is the superalgebra of (p +q )×(p +q )matrices whose elements are arbitrary even (odd)power polynomials of 2r pairs of oscillators in the diagonal (o?-diagonal)blocks.Because of the sp (2)invariance condition (3.5),in our case only even functions of oscillators appear.So we discard the label q in the notation hgl (p,q |2r ).The label sp (2)[n,m ]means that the appropriate quotient of the centralizer in hgl (1|2(n +m ))with respect to the sp (2)subalgebra,which commutes with the o (n,m )spanned by bilinears of oscillators,is taken.o (n,m )is the subalgebra of hu (1/sp (2)[n,m ]).

Note that the described construction of the HS algebra is analogous to that of the AdS 7HS algebra given by Sezgin and Sundell in [27]in terms of spinor oscillators with the symmetric 7d charge conjugation matrix in place of the metric tensor in (3.1).Also let us note that the key role of the algebra

sp(2)in the analysis of HS dynamics explained below is reminiscent of the role of sp(2)in the two-time approach developed by Bars[28].In[29],the sp(2) invariant technics was applied to the description of interacting massless?elds. The important di?erence is that in our case the sp(2)invariance condition acts in the?ber space described by polynomials of the auxiliary variables Y A i, reducing the set of?elds appropriately,while the sp(2)algebra in the models of[28,29]acts on the base.

4Twisted adjoint representation and Central On-Mass-Shell theorem

Now we are in a position to de?ne the twisted adjoint representation which describes the HS Weyl0-forms.Let a HS algebra admit such an involutive automorphismτ(i.e.,τ(f?g)=τ(f)?τ(g),τ2=1)that its action on the elements of the(A)dS d subalgebra is

τ(P a)=?P a,τ(L ab)=L ab.(4.1) Once the Lorentz algebra is singled out by the compensator,the automorphism τdescribes the re?ection with respect to the compensator vector.In particular, for the HS algebra under investigation we setτ(Y A i)=?Y A i,where

?A A=A A?2

V A V B A B i,⊥A A i=A A i?1

Central On-Mass-Shell theorem formulated in[5]in terms of Lorentz com-ponents of C(Y|x)states that the equations for totally symmetric free massless ?elds in(A)dS d can be formulated in the form

R1( Y,⊥Y)=1

?Y A i?Y B j

εij C(0,⊥Y),(4.6)

(C)=0,(4.7)

where R1(Y)=dω(Y)+ω0?∧ω+ω?∧ω0,?D0(C)=dC+ω0?C?C??ω0 andω0=ωAB0(x)T AB whereωAB0(x)satis?es(2.9)to describe the(A)dS d background.

The components of the expansion of the0-forms C(0,⊥Y)on the r.h.s. of(4.6)in powers of Y A i are V transversal.These are the HS Weyl0-forms

c a1...a s,b1...b s describe

d by th

e traceless two-row rectangular Lorentz Young tableaux o

f length s.They parametrize those components of the HS?eld strengths that remain nonvanishin

g when the?eld equations and constraints on extra?elds are satis?ed.For example,the Weyl tensor in gravity(s=2)parametrizes the components of the Riemann tensor allowed to be nonvanishing when the zero-torsion constraint and Einstein equations(requiring the Ricci tensor to vanish)are imposed.The equation(4.7)describes the consistency conditions for the HS equations and also dynamical equations for spins0and1.(Dy-namics of a massless scalar was described this way in[30].)In addition they express an in?nite set of auxiliary?elds contained in C in terms of derivatives

of the dynamical HS?elds.

The key fact is that the equations(4.6)and(4.7)are consistent,i.e.,the application of the covariant derivative to the l.h.s.of(4.6)and(4.7)does not lead to new conditions.The only nontrivial point is to check that

εij D0 E A0∧E B0?2

? Y B i

? Y Bi?

4εji

where the Lorentz covariant derivative is D L 0=d +ωL AB 0⊥Y Ai ?

?Y A i ?Y B j εij

.As a result,it remains

to check the frame dependent terms.These all vanish either because the

r.h.s.of (4.6)is Y A i –independent or because of the identity E A 0∧E B 0∧E C 0?3?Y A i ,N tw =⊥Y A i ?

? Y A i .(4.11)

This means that the free ?eld equations (4.6)and (4.7)decompose into inde-pendent subsystems for the sets of ?elds satisfying

N ad ω=2(s ?1)ω,N tw C =2sC ,s ≥0.(4.12)

(Note that the operator N tw does not have negative eigenvalues when acting on tensors with the symmetry property of a two-row rectangular Young tableau because having more than a half of vector indices aligned along V A would imply symmetrization over more than a half of indices,thus giving zero.)

The set of ?elds singled out by (4.12)describes a massless spin s .As expected,the massless scalar ?eld is described only in terms of the 0-form C (Y |x ),having no associated gauge ?eld.In terms of Lorentz irreducible components,the spin s gauge connections take values in the representations (2.5)with various 0≤t ≤s ?1while the spin s Weyl tensors take values in the Lorentz representations p s with various p ≥s .(Note that the missed cells compared to the rectangular diagram of the length of the upper row cor-respond to the Lorentz invariant direction along V A .)We see that the twisted adjoint action of the (A )dS d algebra on the HS algebra decomposes into an in?nite set of in?nite-dimensional submodules associated with di?erent spins,while its adjoint action decomposes into an in?nite set of ?nite-dimensional submodules.This ?ts the fact (see e.g.[31]),that the space of physical states is described by the 0-form sector which therefore has to form an in?nite module over a space-time symmetry for any dynamical system with in?nite number of degrees of freedom.

5Nonlinear equations

Following to the standard approach in HS theory[32]we will look for a non-linear deformation of the free?eld equations(4.6)and(4.7)in the form of a free di?erential algebra dWα=Fα(W),satisfying the consistency condition FβδFα

π2(d+1) dSdT f(Z+S,Y+S)g(Z?T,Y+T)exp?2S A i T i A,(5.1) which is associative,normalized so that1?f=f?1=f and gives rise to the commutation relations

[Y A i,Y B j]?=εijηAB,[Z A i,Z B j]?=?εijηAB,[Y A i,Z B j]?=0.(5.2)

The star product(5.1)describes a normal-ordered basis in A2(n+m)with respect to creation and annihilation operators Z?Y and Y+Z,respectively.For Z independent elements(5.1)coincides with(3.2).The following useful formulae are true

Y A i?=Y A i+1?Y i

??2 ??Z i A ,(5.3)

?Y A i=Y A i?1?Y i

A +

2 ??Z i A

.(5.4)

Important property of the star product(5.1)is that it admits inner Klein operators.Indeed,it is elementary to see with the aid of(5.1)that the element

K=exp?2z i y i,(5.5)

where

y i=1

V B Y B i,z i=

V B Z B i(5.6)

has the properties

K?f=?f?K,K?K=1,(5.7) where?f(Z,Y)=f(?Z,?Y).This follows from the following formulae

K?f=exp?2z i y i f(Z A i?1V2V A V B(Z B i?Y B i)),

(5.8)

f?K=exp?2z i y i f(Z A i?1V2V A V B(Z B i+Y B i)).

(5.9)

W(Z,Y|x)=dx n(Z,Y|x),S(Z,Y|x)=dZ A i S i A(Z,Y|x).(5.10) The?eldsωand C are identi?ed with the“initial data”for the evolution in Z variables

ω(Y|x)=W(0,Y|x),C(Y|x)=B(0,Y|x).(5.11) The Z-connection S will be determined modulo gauge ambiguity in terms of B.The di?erentials satisfy the standard anticommutation relations dx n=?dx m,dZ A i dZ B j=?dZ B j dZ A i,dx n and commute to all other variables(from now on we discard the wedge symbol).

The full nonlinear system of HS equations is

dW+W?W=0,(5.12)

dS+W?S+S?W=0,(5.13)

dB+W?B?B??W=0,(5.14)

S?B=B??S,(5.15)

S?S=?

√

(dZ A i dZ i A+4Λ?1dz i dz i B?K),W?B=B??W.(5.17)

We see that dz i dz i B?K is the only nonzero component of the curvature.Note that B has dimension cm?2to match the Central On-Mas-Shell theorem(4.6) upon identi?cation of B with C in the lowest order.So,Λ?1B is dimensionless. Since the B dependent part of the equation(5.16)is responsible for interac-tions,this indicates that taking the?at limit may be di?cult in the interacting theory if there is no other contributions to the cosmological constant due to some condensates which break the HS gauge symmetries.

The system is formally consistent in the sense that the associativity re-lations W?(W?W)=(W?W)?W and(W?W)?B=B?(W?W), equivalent to Bianchi identities,are respected by the equations(5.12)-(5.16). The only nontrivial part of this property might be that for the relationship (S?S)?S=S?(S?S)in the sector of(dz i)3due to the second term on the r.h.s.of(5.16)since B?K commutes with everything except dz i to which it anticommutes.However,this does not break the consistency of the system because(dz i)3≡0.As a result,the equations(5.12)-(5.16)are consistent as “di?erential”equations with respect to x and Z variables.A related statement is that the equations(5.12)-(5.16)are invariant under the gauge transforma-tions

δW=[ε,W]?,δB=ε?B?B??ε(5.18) with an arbitrary gauge parameterε(Z,Y|x).

Let us show that the condition(3.5)admits a proper deformation to the full nonlinear theory,i.e.that there is a proper deformation t int ij of t ij that allows us to impose the conditions

D(t int ij)=0,[S,t int ij]?=0,B??t int ij?t int ij?B=0,(5.19)

which amount to the original conditions[t ij,W]?=[t ij,B]?=0in the free?eld limit.Indeed,let us introduce the generators of the diagonal sp tot(2)algebra

t tot ij=Y A i Y Aj?Z A i Z jA.(5.20) In any sp(2)covariant gauge(in which S i A is expressed in terms of B with no external sp(2)noninvariant parameters-see below)sp tot(2)acts on S i A as [t tot ij,S A n]?=εin S A j+εjn S A i provided that B is a sp(2)singlet.

Setting S=dZ A i S i A,the equation(5.16)gets the form[S i A,S j B]?=?εij(ηAB?4V A V B B?K).From(5.15)and(5.7)it follows that

S i A?B?K=B?K?(S i A?2

√

2{S A i,S Aj}?form sp(2)and[T ij,S A n]?=εin S A j+εjn S A i.

As a result,the operators

t int ij=t tot ij?T ij(5.23) form sp(2)and commute with S A i.The conditions(5.19)are equivalent to the usual sp(2)invariance conditions for the?elds W and B and are identically satis?ed on S A i,which essentially means that nonlinear corrections due to the evolution along Z-directions in the noncommutative space do not a?ect sp(2). In the free?eld limit with S A i=Z A i,t int ij coincides with(3.4).One uses t int ij in the nonlinear model the same way as t ij(3.4)in the free one to impose(5.19) and to factor out the respective ideal I int.This guarantees that the nonlinear ?eld equations described by Eqs.(5.12)-(5.16)and(5.19)make sense for the HS?elds associated with hu(1/sp(2)[n,m]).

6Perturbative analysis

The perturbative analysis of the equations(5.12)-(5.16)is analogous to that carried out in spinor notations in[3]for the4d case.Let us set

W=W0+W1,S=S0+S1,B=B0+B1(6.1)

with the vacuum solution

B0=0,S0=dZ A i Z i A,W0=

The generic solution of this equation is

?z i

s j1=?jε1+2Λ?1z j 10dt tC(?t Z,⊥Y)exp?2tz i y i.(6.5) The ambiguity in the functionε1=ε1(Z,Y|x)manifests invariance under the gauge transformations(5.18).It is convenient to?x a gauge by requiring ?iε1=0in(6.5).(This gauge is covariant because it involves no external parameters carrying nontrivial representations of sp(2).)This gauge?xing is not complete as it does not?x the gauge transformations with Z independent parameters

ε1(Z,Y|x)=ε1(Y|x).(6.6) As a result,the?eld S is expressed in terms of B.It is not surprising of course that the noncommutative gauge connection S is reconstructed in terms of the noncommutative curvature B modulo gauge transformations.The leftover gauge transformations with the parameter(6.6)identify with the HS gauge transformations acting on the physical HS?elds.

Now,let us analyze the equation(5.13).In the?rst order,one gets

?i W1=ds i1+W0?s i1?s i1?W0.(6.7) Using that generic solution of the equation?

χi(z)≡0and i=1,2,one?nds

?z i

W1(Z,Y)=ω(Y)?Z j A V A 10dt(1?t)e?2tz i y i E B?

(note that the terms with z i dS1i vanish because z i z i≡0).Since,perturba-tively,the system as a whole is a consistent system of di?erential equations

with respect to?

?x it is enough to analyze the equations(5.12)and

(5.14)at Z=0.Thus,to derive dynamical HS equations,it remains to insert

(6.8)into(5.12)and(6.3)into(5.14),interpretingω(Y|x)and C(Y|x)as gen-erating functions for the HS?elds.The elementary analysis of(5.12)at Z=0 with the help of(5.3)and(5.4)gives(4.6).For B=C,the equation(5.14) amounts to(4.7)in the lowest order.Thus it is shown that the linearized part of the HS equations(5.12)-(5.16)reproduces the Central On-Mass-Shell theorem for symmetric massless?elds.The system(5.12)-(5.16)allows one to derive systematically all higher-order corrections to the free equations.

7Discussion

The system of gauge invariant nonlinear dynamical equations for totally sym-metric massless?elds of all spins in AdS d presented in this paper can be generalized to a class of models with the Yang-Mills groups U(p),USp(p)or

O(p).This results from the observation that,analogously to the case of d=4 [32,26],the system(5.12)-(5.16)remains consistent for matrix valued?elds

W→Wαβ,S→Sαβand B→Bαβ,α,β=1...p.Upon imposing the reality conditions

W?(Z,Y|x)=?W(?iZ,iY|x),S?(Z,Y|x)=?S(?iZ,iY|x),(7.1)

B?(Z,Y|x)=??B(?iZ,iY|x)(7.2) this gives rise to a system with the global HS symmetry algebra hu(p/sp(2)[n,m]). Here all?elds,including the spin1?elds which correspond to the Z,Y-independent part of Wαβ(Z,Y|x),take values in u(p)which is the Yang-Mills algebra of the theory.

Combining the antiautomorphism of the star product algebraρ(f(Z,Y))=

f(?iZ,iY)with some antiautomorphism of the matrix algebra generated by

a nondegenerate formραβone can impose the conditions

Wαβ(Z,Y|x)=?ρβγρδαWγδ(?iZ,iY|x),Sαβ(Z,Y|x)=?ρβγρδαSγδ(?iZ,iY|x),

(7.3)

Bαβ(Z,Y|x)=?ρβγρδα?Bγδ(?iZ,iY|x),(7.4) which truncate the original system to the one with the Yang-Mills gauge group USp(p)or O(p)depending on whether the formραβis antisymmetric or sym-metric,respectively.The corresponding global HS symmetry algebras are

called husp(p/sp(2)[n,m])and ho(p/sp(2)[n,m]),respectively.In this case all ?elds of odd spins take values in the adjoint representation of the Yang-Mills group while?elds of even spins take values in the opposite symmetry second rank representation(i.e.,symmetric for O(p)and antisymmetric for USp(p)) which contains singlet.The graviton is always the singlet spin2particle in the theory.Color spin2particles are also included for general p however.1 The minimal HS theory is based on the algebra ho(1/sp(2)[n,m]).It describes even spin particles,each in one copy.(Odd spins do not appear because the adjoint representation of o(1)is trivial.)

All HS models have essentially one dimensionless coupling constant g2= |Λ|d?2

1Let us note that this does not contradict to the no-go results of[36,37]because the theory under consideration does not allow a?at limit with unbroken HS symmetries and color spin2symmetries.

Acknowledgments

The author is grateful to Lars Brink for hospitality at the Chalmers University of Technology where a part of this work was done.This research was supported in part by INTAS,Grant No.00-01-254,and the RFBR,Grant No.02-02-17067. References

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Nonlinear Equations for Symmetric Massless Higher Spin Fields in $(A)dS_d$

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