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Higher Spin Symmetries, Star-Product and Relativistic Equations in AdS Space

发布时间:

hep-th/0002xxx FIAN/TD/7–00 February 2000

arXiv:hep-th/0002183v1 22 Feb 2000

HIGHER SPIN SYMMETRIES, STAR-PRODUCT AND RELATIVISTIC EQUATIONS IN ADS SPACE

M.A. VASILIEV I.E.Tamm Department of Theoretical Physics, Lebedev Physical Institute, Leninsky prospect 53, 117924, Moscow, Russia

Abstract We discuss general properties of the theory of higher spin gauge ?elds in AdS4 focusing on the relationship between the star-product origin of the higher spin symmetries, AdS geometry and the concept of space-time locality. A full list of conserved higher spin currents in the ?at space of arbitrary dimension is presented.

Based on the talks given at QFTHEP’99 (Moscow, May 1999), Supersymmetries and Quantum Symmetries (Dubna, July 1999), Bogolyubov Conference Problems of Theoretical and Mathematical Physics (Moscow-Dubna-Kyiv, Sept-Oct. 1999) and Symmetry and Integrability in Mathematical and Theoretical Physics, the Conference in Memory of M.V.Saveliev (Protvino, January 2000) 1

1

Introduction

A theory of fundamental interactions is presently identi?ed with still mysterious M-theory [1] which is supposed to be some relativistic theory having d=11 SUGRA as its low-energy limit. M-theory gives rise to superstring models in d ≤ 10 and provides a geometric explanation of dualities. A particularly interesting version of the M-theory is expected to have anti-de Sitter (AdS) geometry explaining duality between AdS SUGRA and conformal models at the boundary of the AdS space [2]. More recently it has been realized that the star-product (Moyal bracket) plays important role in a certain phase of M-theory with nonvanishing vacuum expectation value of the antisymmetric ?eld Bnm [3, 4]. In the limit α′ → 0, Bnm = const string theory reduces to the noncommutative Yang-Mills theory [5]. The most intriguing question is: “what is M-theory?”. It is instructive to analyze the situation from the perspective of the spectrum of elementary excitations. Superstrings describe massless modes of lower spins s ≤ 2 like graviton (s = 2), gravitino (s = 3/2), vector bosons (s = 1) and matter ?elds with spins 1 and 1/2, as well as certain antisymmetric tensors. On the top of that there is an in?nite tower of massive excitations of all spins. Since the corresponding massive parameter is supposed to be large, massive higher spin (HS) excitations are not directly observed at low energies. They are important however for the consistency of the theory. Assuming that M-theory is some relativistic theory that admits a covariant perturbative interpretation we conclude that it should contain HS modes to describe superstring models as its particular vacua. There are two basic alternatives: (i) m = 0: HS modes in M-theory are massive or (ii) m = 0: HS modes in M-theory are massless while massive HS modes in the superstring models result from compacti?cation of extra dimensions. Each of these alternatives is not straightforward. For the massive option, no consistent superstring theory is known beyond ten dimensions and therefore there is no good guiding principle towards M-theory from that side. For the massless case the situation is a sort of opposite: there is a very good guiding principle but it looks like it might be too strong. Indeed, massless ?elds of high spins are gauge ?elds [6]. Therefore this type of theories should be based on some HS gauge symmetry principle with the symmetry generators corresponding to various representations of the Lorentz group. It is however a hard problem to build a nontrivial theory with HS gauge symmetries. One argument is due to the Coleman-Mandula theorem and its generalizations [7] which claim that symmetries of S-matrix in a non-trivial (i.e., interacting) ?eld theory in a ?at space can only have su?ciently low spins. Direct arguments come [8] from the explicit attempts to construct HS gauge interactions in the physically interesting situations (e.g. when the gravitational interaction is included). However, some positive results [9] were obtained on the existence of consistent interactions of HS gauge ?elds in the ?at space with the matter ?elds and with themselves but not with gravity. Somewhat later it was realized [10] that the situation changes drastically once, instead of the ?at space, the problem is analyzed in the AdS space with nonzero curvature Λ. This generalization led to the solution of the problem of consistent HS-gravitational interactions in the cubic order at the action level [10] and in all orders in interactions at the level of equations of motion [11]. The role of AdS background in HS gauge theories is very important. First it cancels the Coleman-Mandula argument which is hard to implement in the AdS background. 2

From the technical side the cosmological constant allows new types of interactions with higher int derivatives, which have a structure ?Sp,n,m,k ? Λp ? n φ? m φ? k φ , where φ denotes any of the ?elds involved and p can take negative powers to compensate extra dimensions carried by higher derivatives of the ?elds in the interactions (an order of derivatives which appear in the cubic interactions increases linearly with spin [9, 10]). An important general conclusion is that Λ should necessarily be nonzero in the phase with unbroken HS gauge symmetries. In that respect HS gauge theories are analogous to gauged supergravities with charged gravitinos which also require Λ = 0 [12]. HS gauge theories contain in?nite sets of spins 0 ≤ s < ∞. This implies that HS symmetries are in?nite-dimensional. If HS gauge symmetries are spontaneously broken by one or another mechanism, then, starting from the phase with massless HS gauge ?elds, one will end up with a spontaneously broken phase with all ?elds massive except for a subset corresponding to an unbroken subalgebra. (The same time a value of the cosmological constant will be rede?ned because the ?elds acquiring a nonvanishing vacuum expectation value may contribute to the vacuum energy.) A most natural mechanism for spontaneous breakdown of HS gauge symmetries is via dimensional compacti?cation. It is important that in the known d=3 and d=4 examples the maximal ?nite-dimensional subalgebras of the HS superalgebras coincide with the ordinary AdS SUSY superalgebras giving rise to gauged SUGRA models. Provided that the same happens in higher dimensional models, this opens a natural way for obtaining superstring type theories in d ≤ 10 starting from some maximally symmetric HS gauge theory in d ≥ 11.

2

Higher Spin Currents

Usual inner symmetries are related via the Noether theorem to the conserved spin 1 current that can be constructed from di?erent matter ?elds. For example, a current constructed from scalar ?elds in an appropriate representation of the gauge group ? ? J n i j = φi ? n φj ? ? n φi φj is conserved on the solutions of the scalar ?eld equations
n ? ? ?n Ji j = φi(2 + m2 )φj ? (2 + m2 )φi φj .

(1)

(2)

(Underlined indices are used for di?erential forms and vector ?elds in d-dimensional space-time, i.e. n = 0, . . . , d ? 1, while i and j are inner indices.) Translational symmetry is associated with the spin 2 current called stress tensor. For scalar matter it has the form 1 (3) T mn = ? m φ? n φ ? η mn ?r φ? r φ ? m2 φ2 . 2 Supersymmetry is based on the conserved current called supercurrent. It has fermionic statistics and is constructed from bosons and fermions. For massless scalar φ and massless spinor ψν it has the form (4) J n ν = ?m φ(γ m γ n ψ)ν , 3

where γ n ? ν are Dirac matrices in d dimensions. Non-underlined indices m, n, . . . = 0 ÷ d ? 1 are treated as vector indices in the ?ber. (The di?erence between underlined and non-underlined indices is irrelevant in the ?at space). The conserved charges, associated with these conserved currents, correspond, respectively, to generators of inner symmetries T i j , space-time translations P n and supertransformations Qi . The conserved current associated with Lorentz rotations can be constructed from the ν symmetric stress tensor S n ; ml = T nm xl ? T nl xm , T nm = T mn . (5)

These exhaust the standard lower spin conserved currents usually used in the ?eld theory. The list of lower spin currents admits a natural extension to HS currents containing higher derivatives of the physical ?elds. The HS currents associated with the integer spin s J n ;m1 ...mt ,n1 ...ns?1 (6)

are vector ?elds (index n) taking values in all representations of the Lorentz group described by the traceless two-row Young diagrams s-1
q q q q q q

t

(7)

with 0 ≤ t ≤ s ? 1. This means that the currents J n;m1 ...mt ,n1 ...ns?1 are symmetric both in the indices n and m, satisfy the relations (s ? 1)(s ? 2)J n ;m1 ...mt ,r r n3 ...ns?1 = 0 , t(s ? 1)J n ;rm2 ...mt , r n2 ...ns?1 , and obey the antisymmetry property J n ;m2 ...mt {ns ,n1 ...ns?1 }n = 0 , (10) t(t ? 1)J n ;m3 ...mt r r ,n1 ...ns?1 = 0 , (8) (9)

implying that symetrization over any s indices n and/or m gives zero. Let us now explain notation, which simpli?es analysis of complicated tensor structures and is useful in the component analysis. Following [13] we combine the Einstein rule that upper and lower indices denoted by the same letter are to be contracted with the convention that upper (lower) indices denoted by the same letter imply symmetrization which should be carried out prior contractions. With this notation it is enough to put a number of symmetrized indices in brackets writing e.g. Xn(p) instead of Xn1 ...np . Now, the HS currents are J n ;m(t) ,n(s?1) (1 ≤ t ≤ s ? 1) while the conditions (8)-(10) take the form (11) J n ;m(t) ,n(s?2) n = 0 , J n ;m(t?1) n ,n(s?1) = 0 , J n ;m(t) m ,n(s?1) = 0 , 4

and J n ;m(t?1)n ,n(s?1) = 0 . The HS supercurrents associated with half-integer spins J n ;m1 ...mt ,n1 ...ns?3/2 ;ν (13) (12)

are vector ?elds (index n) taking values in all representations of the Lorentz group described by the γ -transversal two-row Young diagrams s-3/2
q q q q q q

t i.e., the irreducibility conditions for the HS supercurrents J n ;m(t) ,n(s?3/2);ν read tJ n ;m(t?1)n,n(s?3/2);ν = 0 and (s ? 3/2) γ n ? ν J n ;m(t),n(s?3/2);ν = 0 . From these conditions it follows that (s ? 3/2)γ m ? ν J n ;m(t),n(s?3/2);ν = 0

(14)

(15) (16)

(17)

and all tracelessness conditions (8) and (9) are satis?ed. To avoid complications resulting from the projection to the space of irreducible (i.e. traceless or γ? transversal) two–row Young diagrams we study the currents J n (ξ) = J n ; m(t) , n(s?1) ξm(t),n(s?1) , J n (ξ) = ξm(t),n(s?3/2); ν J n ; m(t) , n(s?3/2) ;ν , (18)

where ξm(t),n(s?1) and ξm(t),n(s?3/2); ν are some constant parameters which themselves satisfy analogous irreducibility conditions. The conservation law then reads ?n J n (ξ) = 0 . (19)

The currents corresponding to one-row Young diagrams (i.e. those with t = 0) generalize the spin 1 current (1), supercurrent (4) and stress tensor (3). An important fact is that they can be chosen in the form J m ; , n(s?1) ξn(s?1) = T m n(s?1) ξn(s?1) J m ; , n(s?3/2) ;ν ξn(s?3/2); ν = T m n(s?3/2) ;ν ξn(s?3/2); ν , with totally symmetric conserved currents T n(s) or supercurrents T n(s?1/2) ;ν , ξn(s?1) ?n T n(s) = 0 , ξn(s?3/2); ν ?n T n(s?1/2) ;ν = 0 5 (22) (20) (21)

(ξ n n(s?2) = 0, (ξn(s?3/2) γ n )ν = 0). Analogously to the formula (5) for the angular momenta current, the symmetric (super)currents T allow one to construct explicitly x-dependent HS “angular” currents. An observation is that the angular HS (super)currents J n (ξ) = T nn(s?1) xm(t) ξm(t),n(s?1) , J n (ξ) = T n n(s?3/2) ;ν xm(t) ξm(t),n(s?3/2); ν , where we use the shorthand notation xm(s) = xm . . . xm ,
s

(23)

(24)

also conserve as a consequence of (22) because when the derivative in (19) hits a factor of xm , the result vanishes by symmetrization of too many indices in the parameters ξ forming the two-row Young diagrams. Since the parameters ξm(t),n(s?1) and ξm(t),n(s?3/2); ν are traceless and γ?transversal, only the double traceless part of T nn(s?1) T n(2) n(s?2) = 0 , and triple γ?transversal part of T n n(s?3/2) ;ν γ n T n n(s?3/2) = 0 , s ≥ 7/2 . (26) s≥4 (25)

contribute to (23). These are the (super)currents of the formalism of symmetric tensors (tensorspinors) [6, 14]. The currents with integer spins T n(s) were considered in [15, 16] for the particular case of massless matter ?elds. Integer spin currents built from scalars of equal masses (2 + m2 )φi = 0 have the form k T n(2k)ij = (? n(k) φi ? n(k) φj ? η nn ? n(k?1) ?m φi ? n(k?1) ? m φj 2 k 2 nn n(k?1) i n(k?1) j + mη ? φ? φ )+i?j 2 for even spins and T n(2k+1)ij = (? n(k+1) φi? n(k) φj ) ? i ? j for odd spins, where we use notation analogous to (24) ? n(s) = ? n . . . ? n .
s

(27)

(28)

(29)

(30)

HS supercurrents built from scalar and spinor with equal masses (2 + m2 )φ = 0 , 6 (i?n γ n + m)ψν = 0 (31)

k+1 (γ n γ l ψ)ν ? n(k) ?l φ + im(γ n ψ)ν ? n(k) φ . (32) 2 Inserting these expressions into (23) we obtain the set of conserved “angular” HS currents of even, odd and half-integer spins. The usual angular momentum current corresponds to the case s = 2, t = 1. Remarkably, the conserved HS currents listed above are in one–to–one correspondence with the HS gauge ?elds (1-forms) ωn;m(t),n(s?1) and ωn;m(t),n(s?3/2);ν introduced for the boson [17] and fermion [18] cases in arbitrary d. To the best of our knowledge, the fact that any of the HS gauge ?elds has a dual conserved current is new. Of course, such a correspondence is expected because, like the gauge ?elds of the supergravitational multiplets, the HS gauge ?elds should take their values in a (in?nite-dimensional) HS algebra identi?ed with the global symmetry algebra in the corresponding dynamical system (this fact is explicitly demonstrated below for the case of d = 4). The HS currents can then be derived via the Noether theorem from the global HS symmetry and give rise to the conserved charges identi?ed with the Hamiltonian generators of the same symmetries. A few comments are now in order. HS currents contain higher derivatives. Therefore, HS symmetries imply, via the Noether procedure, the appearance of higher derivatives in interactions. The immediate question is whether HS gauge theories are local or not. As we shall see the answer is “yes” at the linearized level and “probably not” at the interaction level. Nontrivial (interacting) theories exhibiting HS symmetries are formulated in AdS background rather than in the ?at space. Therefore an important problem is to generalize the constructed currents to the AdS geometry. This problem was solved recently [19] for the case d=3. Explicit form of the HS algebras is known for d ≤ 4 although a conjecture was made in [18] on the structure of HS symmetries in any d. The knowledge of the structure of the HS currents in arbitrary dimension may be very useful for elucidating a structure of the HS symmetries in any d. T n(k+1) ;ν = ? n(k+1) φψν ?

read

3

D=4 Higher Spin Algebra

The simplest global symmetry algebra of the 4d HS theory can be realized as follows. Consider the associative algebra of functions of the auxiliary Majorana spinor variables Yν (ν = 1 ÷ 4) endowed with the star-product law (f ? g)(Y ) = 1 (2π)2
i

d4 Ud4 V exp(iU? V ? )f (Y + U)g(Y + V ) f (Y + Y 1 )B(Y + Y 2 )|Y 1 =Y 2 =0 . (33)

= e

? ? 1 ?Y? ?Y 2?

Here f (Y ) and g(Y ) are functions (polynomials or formal power series) of commuting variables Y? (spinor indices are raised and lowered by the 4d charge conjugation matrix C?ν : U ? = C ?ν Uν , U? = U ν Cν? ). This formula de?nes the associative algebra with the de?ning relation 7

Y? ? Yν ? Yν ? Y? = 2iC?ν . The star-product de?ned this way describes the product of Weyl ordered (i.e. totally symmetric) polynomials of oscillators in terms of symbols of operators [20]. This Weyl product law (called Moyal bracket [21] for commutators constructed from (33)) is obviously nonlocal. This is the ordinary quantum-mechanical nonlocality. Note that the integral formula (33) is in most cases more convenient for practical computations than the di?erential Moyal formula. The pure Weyl algebra can only be used as the HS algebra in the bosonic case. The algebra shsa(1) [22] that works also in presence of fermions is obtained by adding the “Klein operators” ? k and k having the properties k ?k = 1, k ? yα = ?yα ? k , ? ? k ?k = 1, ? ? k?k = k?k, (34)

? ?˙ k ? yα = ??α ? k . y˙ ? (35) ? (Here Yν = (yα , yα ) with α = 1, 2, α = 1, 2; in other words, moving k or k through Yν is ?˙ ˙ equivalent to multiplication by ±γ5 .) The HS gauge ?elds are k ? yα = yα ? k , ?˙ ?˙ W (y, y; k, k|x) = ? ? 1 ˙ ˙ ? dxn wn A Bα1 ...αn α1 ...αm (x)k A k B yα1 . . . yαn yα1 . . . yαm . ?˙ ?˙ 2im!n! A,B=0,1 n,m=0


? ? k ? yα = yα ? k ,

(36)

? According to [13, 23, 22] the ?elds W (y, y; k, k|x) = W (y, y; ?k, ?k|x) describe the HS ?elds ? ? ? ? ? while the ?elds W (y, y; k, k|x) = ?W (y, y; ?k, ?k|x) are auxiliary, i.e. do not describe non? ? ˙ ˙ trivial degrees of freedom. Therefore we have two sets of HS potentials w A Aα1 ...αn α1 ...αm (x) , 1 A=0 or 1. The subsets associated with spin s are ?xed by the condition [13] s = 1 + 2 (n + m). For a ?xed value of A we therefore expect a set of currents J n ;α1 ...αn ,α1 ...αm (x) . In accordance ˙ ˙ with the results of the section 2, it indeed describes in terms of two-component spinors the set of all two-row Young diagrams (7) both in the integer spin case (n + m is even) and in the 1 half-integer spin case (n + m is odd), with the identi?cation s = 2 (n + m) + 1, t = 1 |n ? m| , 2 where [a] denotes the integer part of a. We see that HS algebras give rise to the sets of gauge ?elds which exactly match the sets of conserved HS currents of the section 2. The HS curvature 2-form is R(Y ; K|x) = dw(Y ; K|x) + (w ∧ ?w)(Y ; K|x) , (37)

? where d = dxn ?n (n, m = 0 ÷ 3 are space-time (base) indices) and K = (k, k) denotes the pair of the Klein operators. Let us stress that star-product acts on the auxiliary coordinates Y while the space-time coordinates xn are commuting. The HS gauge transformations have a form δw(Y ; K|x) = d?(Y ; K|x) + [w, ?]? (Y ; K|x), where [a, b]? = a ? b ? b ? a. A structure of HS algebras h is such that no HS (s > 2) ?eld can remain massless unless it belongs to an in?nite chain of ?elds with in?nitely increasing spins. Indeed, from the de?nition of the star-product it follows that the gauge ?elds having spins s1 and s2 contribute to the spin s1 + s2 ? 2 curvature if at least one of the spins is integer and to s1 + s2 ? 1 curvature if both s1 and s2 are half-integer. Bilinears in the oscillators form a ?nite-dimensional subalgebra sp(4|R) 8

isomorphic to AdS4 algebra o(3, 2). In fact, the model has N = 2 SUSY [22] associated with a ?nite-dimensional subalgebra osp(2, 4). Unbroken HS symmetries require AdS background. One can think however of some spontaneous breakdown of the HS symmetries followed by a ?at contraction via a shift of the vacuum energy in the broken phase. In a physical phase with λ = 0 and m ? mexp for HS ?elds, h should break down to a ?nite-dimensional subalgebra giving rise to usual lower spin gauge ?elds. From this perspective the Coleman-Mandula type theorems can be re-interpreted as statements concerning a possible structure of g rather than the whole HS algebra h which requires AdS geometry. These arguments are based on the d ≤ 4 experience but we expect them to large extend to be true for higher dimensions.

4

AdS Vacuum

A structure of the full nonlinear HS equations of motion is such that any solution w0 of the zero-curvature equation dw = w ? ∧w (38) solves the HS equations. Such a vacuum solution has a pure gauge form w0 (Y ; K|x) = ?g ?1 (Y ; K|x) ? dg(Y ; K|x) (39)

with some invertible element g(Y ; K|x), i.e. g ? g ?1 = g ?1 ? g = I. It breaks the local HS symmetry to its stability subalgebra with the in?nitesimal parameters ?0 (Y ; K|x) satisfying the equation D0 ?0 ≡ d?0 ? w0 ? ?0 + ?0 ? w0 = 0 which solves as ?0 (Y ; K|x) = g ?1 (Y ; K|x) ? ?0 (Y ; K) ? g(Y ; K|x) . (40) ? In the HS theories no further symmetry breaking is induced by the ?eld equations, i.e. ?0 (Y ; k, k) parametrizes the global symmetry of the theory. Therefore, the HS global symmetry algebra identi?es with the Lie superalgebra constructed from the (anti)commutators of the elements of the Weyl algebra and its extension with the Klein operators. Note that the ?elds carrying odd numbers of spinor indices are anticommuting thus inducing a structure of a superalgebra into (38). AdS background plays distinguished role in the HS theories because functions bilinear in Yν form a closed subalgebra with respect to commutators. This allows one to look for a solution of the vacuum equation (38) in the form w0 = 1 ˙ ˙ αβ ω0 (x)yα yβ + ω0˙ β (x)?α yβ + λhαβ (x)yα yβ ?α y ˙ ?˙ ?˙ 0 4i . (41)

Inserting these formulae into (38) one ?nds that the ?elds ω0 , ω0 and h0 identify with the ? Lorentz connection and the frame ?eld of AdS4 , respectively, provided that the 1-form h0 is invertible. The parameter λ = r ?1 is identi?ed with the inverse AdS radius. Thus, the fact that the HS algebras are star-product (oscillator) algebras leads to the AdS geometry as a natural vacuum solution. 9

A particular solution of the vacuum equation (38) corresponding to the stereographic coordinates has a form ˙ ˙ (42) hn αβ = ?z ?1 σn αβ , ωn αβ = ?
˙

λ2 ˙ ˙ σn αβ xβ β + σn β β xα β , ˙ ˙ 2z

ωn α β = ? ? ˙

˙

λ2 ˙ ˙ ˙ ˙ σn αα xα β + σn αβ xα α , 2z

(43)

where σn αβ is the set of 2 × 2 Hermitian matrices and we use notation xαβ = xn σn αβ ,
˙ ˙

1 ˙ x2 = xαβ xαβ , ˙ 2

z = 1 + λ2 x2 .

(44)

Let us note that z → 1 in the ?at limit and z → 0 at the boundary of AdS4 . The form of the gauge function g reproducing these vacuum background ?elds (with all s = 2 ?elds vanishing) turns out to be remarkably simple [24] √ z iλ ˙ √ exp[ √ xαβ yα yβ ] g(y, y|x) = 2 ? ?˙ (45) 1+ z 1+ z with the inverse g ?1 (y, y|x) = g(?y, y|x). In many cases, g plays a role of some kind of evolution ? ? operator (cf eq. (49)). From this perspective λ is analogous to the inverse of the Planck constant, λ ? h?1 . This parallelism indicates that the ?at limit λ → 0 may be essentially ? singular.

5

Free Equations

HS symmetries mix derivatives of physical ?elds of all orders. To have HS symmetries linearly realized, it is useful to introduce in?nite multiplets rich enough to contain dynamical ?elds along with all their higher derivatives. Such multiplets admit a natural realization in terms of the Weyl algebra. Namely, the 0-forms C(Y |x) = C(y, y|x) = ? 1 ˙ ˙ C α1 ...αn , α1 ...αm (x)yα1 . . . yαn yα1 . . . yαm , ?˙ ?˙ n,m=0 m!n!


(46)

taking their values in the Weyl algebra form appropriate HS multiplets for lower spin matter ?elds and Weyl-type HS curvature tensors. The free equations of motion have a form [25] D0 C ≡ dC ? w0 ? C + C ? w0 = 0 , ? (47)

where tilde denotes an involutive automorphism of the HS algebra which changes a sign of the AdS translations ? ? f(y, y) = f (y, ??) . y (48) As a result, the covariant derivative D0 corresponds to some representation of the HS algebra which we call twisted representation. The consistency of the equation (47) is guaranteed by the vacuum equation (38). Since in this “unfolded formulation” dynamical ?eld equations have a 10

form of covariant constancy conditions, one can write down a general solution of the free ?eld equations (47) in the pure gauge form C(Y |x) = g ?1(Y |x) ? C0 (Y ) ? g (Y |x) , ? (49)

where C0 (Y ) is an arbitrary x?independent element of the Weyl algebra. Let us now explain in more detail a physical content of the equations (47). Fixing the AdS4 form (41) for the vacuum background ?eld, (47) reduces to D0 C(y, y|x) ≡ D L C(y, y|x) + λ{hαβ yα yβ , C(y, y|x)}? = 0 , ? ? ?˙ ? where {a, b}? = a ? b + b ? a and i ˙ D L C(y, y|x) = dC(y, y|x) + ([ω αβ yα yβ + ω αβ yα yβ , C(y, y|x)]? ) . ? ? ? ˙ ?˙ ?˙ ? 4 Inserting (46) one arrives at the following in?nite chain of equations D LCα(m), β(n) = iλhγ δ Cα(m)γ, β(n)δ ? inmλhαβ Cα(m?1), β(n?1) , ˙ ˙ ˙ ˙ ˙
˙ ˙ ˙

(50)

(51)

(52)

where D L is the Lorentz-covariant di?erential D L Aαβ = dAαβ + ωα γ ∧ Aγ β + ωβ δ ∧ Aαδ . Here we ?˙ ˙ ˙ ˙ ˙ skip the subscript 0 referring to the vacuum AdS solution and use again the convention with symmetrized indices denoted by the same letter and a number of symmetrized indices indicated in brackets. The system (52) decomposes into a set of independent subsystems with n ? m ?xed. It turns out [25] that the subsystem with |n ? m| = 2s describes a massless ?eld of spin s (note that the ?elds Cα(m), β(n) and Cβ(n), α(m) are complex conjugated). ˙ ˙ For example, the sector of s = 0 is associated with the ?elds Cα(n), β(n) . The equation (52) ˙ 1 n L at n = m = 0 expresses the ?eld Cα,β via the ?rst derivative of C as Cα,β = 2iλ hαβ Dn C , where ˙ ˙ ˙
γ δ hαβ is the inverse frame ?eld (hαβ hγ δ = 2δα δβ with the normalization chosen to be true for ˙ ˙ ˙ n α hαβ = σαβ and hαβ = σn β ). The second equation with n = m = 1 contains more information. ˙ ˙ n L First, one obtains by contracting indices with the frame ?eld that hαβ (Dn C α , β + 8iλC) = 0. ˙ This reduces to the Klein-Gordon equation in AdS4 n ˙ n n ˙ ˙ n n ˙ ˙

2C ? 8λ2 C = 0 .

(53)

The rest part of the equation (52) with n = m = 1 expresses the ?eld Cαα,β β via second ˙˙ n 1 L m L derivatives of C: Cαα ,ββ = (2iλ)2 hαβ Dn hαβ Dm C. All other equations with n = m > 1 either ˙˙ ˙ ˙ reduce to identities by virtue of the spin 0 dynamical equation (53) or express higher components in the chain of ?elds Cα1 ...αn , β1 ...βn via higher derivatives in the space-time coordinates. The ˙ ˙ value of the mass parameter in (53) is such that C describes a massless scalar in AdS4 . For spins s ≥ 1 it is more useful to treat the equations (52) not as fundamental ones but as consequences of the HS equations formulated in terms of gauge ?elds (potentials). To illustrate this point let us consider the example of gravity. As argued in section 3, Lorentz connection 11

1?forms ωαβ , ωαβ and vierbein 1?form hαβ can be identi?ed with the sp(4)?gauge ?elds. The ?˙ ˙ ˙ sp(4)?curvatures read Rα1 α2 = dωα1 α2 + ωα1 γ ∧ ωα2 γ + λ2 hα1 δ ∧ hα2 δ , ˙ ?˙ ˙ Rα1 α2 = d? α1 α2 + ωα1 γ ∧ ωα2 γ + λ2 hγ α1 ∧ hγ α˙2 , ω˙ ˙ ?˙ ˙ ?˙ ˙ ˙ rαβ = dhαβ + ωα γ ∧ hγ β + ωβ δ ∧ hαδ . ?˙ ˙ ˙ ˙ ˙
˙ ˙

(54) (55) (56)

The zero-torsion condition rαβ = 0 expresses the Lorentz connection ω and ω via derivatives of ? ˙ ? h. After that, the λ?independent part of the curvature 2?forms R (54) and R (55) coincides with the Riemann tensor. Einstein equations imply that the Ricci tensor vanishes up to a constant trace part proportional to the cosmological constant. This is equivalent to saying that only those components of the tensors (54) and (55) are allowed to be non-zero which belong to the Weyl tensor. The Weyl tensor is described by the fourth-rank mutually conjugated ?˙ ˙ ˙ ˙ totally symmetric multispinors Cα1 α2 α3 α4 and Cα1 α2 α3 α4 . Therefore, Einstein equations with the cosmological term can be cast into the form rαβ = 0 , ˙ Rα1 α2 = hγ1 δ ∧ hγ2 δ Cα1 α2 γ1 γ2 , ˙
˙ ˙ ˙ ?˙ ˙ ?˙ ˙ ˙ ˙ Rβ1 β2 = hηδ1 ∧ hη δ2 Cβ1 β2 δ1 δ2 .

(57) (58)

?˙ It is useful to treat the 0?forms Cα(4) and Cα(4) as independent ?eld variables which identify with the Weyl tensor by virtue of the equations (58). From (58) it follows that the 0?forms ?˙ Cα(4) and Cα(4) should obey certain di?erential restrictions as a consequence of the Bianchi ? identities for the curvatures R and R. It is not di?cult to make sure that these di?erential restrictions just have the form of the equations (52) with n = 4, m = 0 and n = 0, m = 4 ? ˙ with the ?elds Cα(5),δ and Cγ,β(5) describing unrestricted components of the ?rst derivatives of ˙ the Weyl tensor. The consistency conditions for these relations are expressed by the equations (52) with n = 5, m = 1 and n = 1, m = 5. Continuation of this process leads in the linearized approximation to the in?nite chains of di?erential relations (52) with |n ? m| = 4. All these relations contain no new dynamical information compared to that contained in the original Einstein equations in the form (57), (58), merely expressing the highest 0?forms Cα(n+4),β(n) ˙ ? ?˙ and Cα(n),β(n+4) via derivatives of the lowest 0?forms Cα(4) and Cβ(4) . As a result, the system ˙ of equations obtained in such a way turns out to be equivalent to the Einstein equations with the cosmological term. As shown in [13, 25] this construction extends to all spins s ≥ 1. In terms of the linearized HS curvatures R1 (y, y | x) ≡ dw(y, y | x)?w0 (y, y | x) ? w(y, y | x)+w(y, y | x) ? w0 (y, y | x) ? ? ? ? ? ? the linearized HS equations read
˙ R1 (y, y|x) = hγ β ∧ hγ α ? ˙

(59)

? ? ? ? αγ ˙ ? ∧ hβ γ α β C(y, 0|x) ˙ ˙ C(0, y |x) + h α ˙ β ?y ?y ? ? ?y ?y 12

(60)

together with (50). This statement, which plays a key role from various points of view, we call Central On-Mass-Shell Theorem. Eqs. (60) and (50) contain [13] the usual free HS equations in AdS4 , which follow from the standard actions proposed in [6], and express all auxiliary components via higher derivatives of the dynamical ?elds Cα(n) ,β(m) = ˙ (2iλ) 2
2 h 1˙ D L . . . hαβ 1 ˙ (n+m?2s) αβ n1

1

n

n 1 (n+m?2s)

L Dn 1
2

(n+m?2s)

Cα(2s)

n≥m

(61)

(and complex conjugated) for the 0-forms and by analogous formulae for the gauge 1-forms [13]. A spin s ≥ 1 dynamical massless ?eld is identi?ed with the 1-form (potential) wα(s?1),β(s?1) for ˙ integer s ≥ 1 or wα(s?3/2),β(s?1/2) and wα(s?1/2),β(s?3/2) for half-integer s ≥ 3/2. The matter ˙ ˙ ?elds are described by the 0-forms Cα(0) ,α(0) for s = 0 or Cα(1) ,α(0) and Cα(0) ,α(1) for s = 1/2. ˙ ˙ ˙ The in?nite set of the 0?forms C forms a basis in the space of all on-mass-shell nontrivial combinations of the covariant derivatives of the matter ?elds and (HS) curvatures. Note that the relationships (60) and (50) link derivatives in the space-time coordinates xn with those in the auxiliary spinor variables yα and yα . In accordance with (61), in the sector of ?˙ 0-forms the derivatives in the auxiliary spinor variables can be viewed as a square root of the space-time derivatives, ? ? ˙ ? ? C(y, y|x) ? λhn αβ α β C(y, y|x) . ? n ?x ?y ? y ˙ ? (62)

As a result, the nonlocality of the star-product (33) acting on the auxiliary spinor variables indicates a potential nonlocality in the space-time sense. The HS equations contain starproducts via terms C(Y |x) ? X(Y |x) with some operators X constructed from the gauge and matter ?elds. Once X(Y |x) is at most quadratic in the auxiliary variables Y ν , the resulting expressions are local, containing at most two derivatives in Y ν . This is the case for the AdS background gravitational ?elds and therefore, in agreement with the analysis of this section, the HS dynamics is local at the linearized level. But this may easily be not the case beyond the linearized approximation. Another important consequence of the formula (62) is that it contains explicitly the inverse AdS radius λ and becomes meaningless in the ?at limit λ → 0. This happens because, when resolving these equations for the derivatives in the auxiliary variables ? y and y , the space-time derivatives appear in the combination λ?1 ?xn that leads to the inverse ? powers of λ in front of the terms with higher derivatives in the HS interactions. This is the main reason why HS interactions require the cosmological constant to be nonzero as was ?rst concluded in [10]. To summarize, the following facts are strongly correlated: (i) HS algebras are described by the (Moyal) star-product in the auxiliary spinor space; (ii) relevance of the AdS background; (iii) potential space-time nonlocality of the HS interactions due to the appearance of higher derivatives at the nonlinear level. These properties are in many respects reminiscent of the superstring picture with the parallelism between the cosmological constant and the string tension parameter. The fact that unbroken HS symmetries require AdS geometry may provide an explanation why the symmetric HS phase is not visible in the usual superstring picture with the ?at background space-time. 13

The fact that C(Y |x) describes all derivatives of the physical ?elds compatible with the ?eld equations allows us to solve the dynamical equations in the form (49). The arbitrary parameters C0 (Y ) in (49) describe all higher derivatives of the ?eld C(Y |x0 ) at the point x0 with g(Y |x0 ) = I (x0 = 0 for the gauge function (45)). In other words, (49) describes a covariantized Taylor expansion in some neighborhood of x0 . To illustrate how the formula (49) can be used to produce explicit solutions of the HS equations in AdS4 let us set C0 (Y ) = exp i(y α ηα + y α ηα ) , ? ˙ ?˙ (63)

where ηα is an arbitrary commuting complex spinor and ηα is its complex conjugate. Taking ?˙ ?1 into account that g (Y |x) = g(Y |x) , inserting g(Y |x) into (49) and using the product law ? (33) one performs elementary Gaussian integrations to obtain [24] C(Y |x) = z 2 exp i ?λ(yα yβ + ηα ηβ )xαβ + z(y α ηα + y α ηα ) , ?˙ ?˙ ? ˙ ?˙
1 where z = 1 + λ2 2 xαβ xαβ . From Cα1 ...αn (x) = ˙ the matter ?elds and HS Weyl tensors ˙ ? ?y α1 ˙

(64)

? . . . ?y? n C(y, y|x)|y=?=0 , it follows then for y α
˙

Cα...α2s (x) = z 2(s+1) ηα1 . . . ηα2s exp ikγ β xγ β , ˙

(65)

where kαβ = ?ληα ηβ is a null vector expressed in the standard way in terms of commuting ?˙ ˙ spinors. (Expressions for the conjugated Weyl tensors carrying dotted indices are analogous). Since z → 1 in the ?at limit, the obtained solution describes plane waves in the ?at limit λ → 0 ?˙ provided that the parameters ηα and ηα are rescaled according to ηα → λ?1/2 ηα , ηα → λ?1/2 η α . ?˙ ? ?˙ ? On the other hand, z → 0 at the boundary of AdS4 and therefore the constructed AdS plane waves tend to zero at the boundary. Let us note that although the equation (60) does not have a form of a zero-curvature equation, it also can be solved explicitly [24] using a more sophisticated technics inspired by the analysis of the nonlinear HS dynamics.

6

Nonlinear Higher Spin Equations

Now we discuss the full nonlinear system of 4d HS equations following [11, 26, 27]. The resulting formulation, amounts to certain non-commutative Yang-Mills ?elds and is interesting on its own right. The key element of the construction consists of the doubling of auxiliary Majorana spinor variables Yν in the HS 1-forms w(Y ; K|x) ?→ W (Z; Y ; K|x) and 0-forms C(Y ; K|x) ?→ B(Z; Y ; K|x). The dependence on the additional variables Zν is determined in terms of “initial data” w(Y ; K|x) = W (0; Y ; K|x) , C(Y ; K|x) = B(0; Y ; K|x). (66) by appropriate equations and e?ectively describes all nonlinear corrections to the ?eld equations. To this end we introduce a compensator-type spinor ?eld Sν (Z; Y ; K|x) which does not carry its own degrees of freedom and plays a role of a covariant di?erential along the additional 14

Zν directions. It is convenient to introduce anticommuting Z?di?erentials dZ ν dZ ? = ?dZ ? dZ ν to interpret Sν (Z; Y ; K|x) as a Z 1-form S = dZ ν Sν . The nonlinear HS dynamics is formulated in terms of the star-product (f ? g)(Z; Y ) = 1 (2π)4 d4 U d4 V exp [iU ? V ν C?ν ] f (Z + U; Y + U)g(Z ? V ; Y + V ) , (67)

where U ? and V ? are real integration variables. It is a simple exercise with Gaussian integrals to see that this star-product is associative f ? (g ? h) = (f ? g) ? h and is normalized such that 1 is a unit element of the star-product algebra, i.e. f ? 1 = 1 ? f = f . The star-product (67) again yields a particular realization of the Weyl algebra1 . The following simple formulae are true ?f ?f [Y? , f ]? = 2i ? , [Z? , f ]? = ?2i ? , (68) ?Y ?Z for any f (Z, Y ). From (67) it follows that functions f (Y ) independent of Z form a proper subalgebra with the Weyl star-product (33). An important property of the star-product (67) is that it admits the inner Klein operators υ = exp izα y α and υ = exp i?α y α, having the properties ? z ˙ ?˙ υ ? υ = υ ? υ = 1 and ? ? υ ? f (z, z ; y, y) = f (?z, z ; ?y, y ) ? υ , ? ? ? ? υ ? f (z, z ; y, y) = f (z, ??; y, ??) ? υ . ? ? ? z y ? (69)

The star-product (67) is regular: given two polynomials f and g, f ? g is also some polynomial. The special property of the star-product (67) is that it is de?ned for the class of nonpolynomial functions [28, 29] which appear in the process of solution of the nonlinear HS equations and contains the Klein operators υ and υ . ? The full system of 4d equations has the form dW = W ? W , dB = W ? B ? B ? W , dS = W ? S ? S ? W , (70) (71)

S?B =B?S,

S ? S = dZ ν dZ ? (?iCν? + 4Rν? (B)) ,

where Cν? is the charge conjugation matrix and Rν? (B) is a certain star-product function of the ?eld B and some central elements of the algebra. The function Rν? (B) that encodes all information about the HS dynamics has the form dZ ν dZ ? Rν? (B) = 1 ? ? dzα dz α (ν + ηF (B)) ? k ? υ + d?α d?α (? + η F (B)) ? k ? υ . z˙ z˙ ν ? ? 4i (72)

Here ν, ν , η and η are arbitrary parameters. The parameters η and η play a role of the coupling ? ? ? constants while the auxiliary parameters ν and ν are introduced for the future convenience and ? can be set equal to zero at least for the most symmetric vacuum solution. The function F describes the ambiguity in the HS interactions. The simplest choice F (B) = B leads to the nontrivial (nonlinear) dynamics. The case with ν = ν = F = 0 leads to the free ?eld equations. ?
The star-product (67) corresponds to the normal ordering of the Weyl algebra with respect to the creation 1 and annihilation operators a+ = 2 (Y? ? Z? ) and a? = 1 (Y? + Z? ) satisfying the commutation relations ? 2 + + + [a? , aν ]? = [a? , aν ]? = 0, [a? , aν ]? = iC?ν .
1

15

? Note that the exterior Klein operator k (k) anticommutes with all left (right) spinors including α α ˙ the di?erentials dz (d? ). z The equations (70) and (71) are invariant under the gauge transformations δW = dε + [ε, W ]? , δS = [ε, S]? , δB = [ε, B]? . (73)

The space-time di?erential d only emerges in the equations (70) which have a form of zerocurvature and covariant constancy conditions and therefore admit explicit solution in the pure gauge form analogous to (39) and (40) W = ?g ?1 (Z; Y ; K|x) ? dg(Z; Y ; K|x) , B(Z; Y ; K|x) = g ?1 (Z; Y ; K|x) ? b(Z; Y ; K) ? g(Z; Y ; K|x) , S(Z; Y ; K|x) = g ?1 (Z; Y ; K|x) ? s(Z; Y ; K) ? g(Z; Y ; K|x) (74) (75) (76)

with some invertible g(Z; Y ; K|x) and arbitrary x?independent functions b(Z; Y ; K) and s(Z; Y ; K). Due to the gauge invariance of the whole system one is left only with the equations (71) for b(Z; Y ; Q) and s(Z; Y ; Q). These encode in a coordinate independent way all information about the dynamics of massless ?elds of all spins. In fact, the “constraints” (71) just impose appropriate restrictions on the quantities b and s to guarantee that the original space-time equations of motion are satis?ed. In the analysis of the HS dynamics, a typical vacuum solution for the ?eld S is S0 = dZ ν Zν . From (68) it follows that [S0 , f ]? = ?2i?f , ? = dZ ν ? . ?Z ν (77)

Interpreting the deviation of the full ?eld S from the vacuum value S0 as a Z?component of the gauge ?eld W , S = S0 + 2idZ ν Wν , (78) one rewrites the equations (70), (71) as R = dZ ν dZ ? Rν? (B) , DB = 0 , (79)

where the generalized curvatures and covariant derivative are de?ned by the relations R = (d + ?)(dxn Wn + dZ ν Wν ) ? (dxn Wn + dZ ν Wν ) ∧ (dxn Wn + dZ ν Wν ) , D(A) = (d + ?)A ? (dxn Wn + dZ ν Wν ) ? A + A ? (dxn Wn + dZ ν Wν ) . (80) (81)

(dxn dZ ν = ?dZ ν dxn .) We see that the function Rν? (B) in (71) identi?es with the ZZ components of the generalized curvatures, while xx and xZ components of the curvature vanish. The equation DB = 0 means that the curvature Rν? (B) is covariantly constant. In fact, it is the compatibility condition for the equations (80) and (72). The consistency of the system of equations (70), (71) guarantees that it admits a perturbative solution as a system of di?erential equations with respect to Zν . A natural vacuum solution 16

is W0 (Z; Y ; K|x) = w0 (Y |x), B0 = 0 and S0ν = Zν (ν = ν = 0) with the ?eld w0 (41) describing ? the AdS4 vacuum. All ?uctuations of the ?elds can be expressed modulo gauge transformations in terms of the “initial data” (66) identi?ed with the physical HS ?elds. Inserting thus obtained expressions into (70) one reconstructs all nonlinear corrections to the free ?eld equations. In our approach, non-commutative gauge ?elds appear in the auxiliary spinor space associated with the coordinates Z ν . The dynamics of the HS gauge ?elds is formulated entirely in terms of the corresponding non-commutative gauge curvatures. For the ?rst sight, it is very di?erent from the non-commutative Yang-Mills model considered recently in [5] in the context of the new phase of string theory, in which star-product is de?ned directly in terms of the original space-time coordinates xn . However, the di?erence may be not that signi?cant taking into account the relationships like (62) between space-time and spinor derivatives, which are themselves consequences of the equations (70). From this perspective, the situation with the HS equations is reminiscent of the approach developed in [30] to solve the problem of quantization of symplectic structures in which the complicated problem of quantization of some (base) manifold (coordinates xn ) is reduced to a simpler problem of quantization in the ?bre endowed with the Weyl star-product structure (analog of coordinates Z n ). The di?erence between the Fedosov’s approach and the structures underlying the HS equations is that the former is based on the vector ?ber coordinates Z n , while the HS dynamics chooses spinor coordinates Z ν . The same di?erence is obvious in the context of a possible relationship of the HS theories with M theory. However, such a relationship should be reconsidered in presence of non-zero vacuum antisymmetric tensor ?elds. Antisymetric tensor ?elds are indeed present in the HS theories in the sector of auxiliary ?elds B containing even combinations of the Klein operators. Most likely, the corresponding vacuum solution of the HS equations will be non-polynomial in the spinor variables Yν in the sector of HS gauge 1-forms. If so, the resulting HS equations will become space-time non-local even at the free ?eld level. An interesting problem for the future is therefore to investigate the explicit character of this non-locality in presence of the non-zero antisymmetric tensor ?elds to check whether or not it develops the non-commutative structure in the space-time sense. Let us note that a relationship between non-commutative Yang-Mills theory and Fedosov approach has been discussed in the recent paper [31]. An important property of the d4 equations is the existence of the ?ows with respect of the coupling constants η and η that commute to the whole system (70), (71) and to each other ? [26, 27], ?X ?X ?X ?X ? = ? F (B) , = ? F (B) (82) ?η ?ν ?η ? ?ν ? for X = W , S or B (other forms of the ?ows suggested in [27] are equivalent to (82) modulo gauge transformations). The integrating ?ows (82) manifest the simple fact that B behaves like a constant in the system (70)-(71): it commutes to Sν and satis?es the covariant constancy condition. Indeed, the meaning of (82) is that the derivative with respect to ηF (B) is the same as that with respect to ν. The parameter η can be identi?ed with the coupling constant. The ?ows (82) therefore describe the evolution with respect to the coupling constant. The ?ows (82) reduce the problem of solving the nonlinear HS equations to the ordinary di?erential equations with respect to η and η provided that the “initial data” problem with ? B = 0 is solved for arbitrary constants ν and ν . Remarkably, the latter problem admits explicit ? 17

solution [29]. We believe that this indicates some sort of integrability of the full nonlinear system of 4d HS equations. Such an approach is very e?cient at least perturbatively allowing one to solve equations by iterating the ?ows (82). In particular this technics was used in [24] to reconstruct the form of plane wave AdS4 HS potentials in terms of the ?eld strengths. Although the mapping induced by the ?ow (82) does not manifestly contain space-time derivatives, it contains them implicitly via highest components Cν(n) of the generating function which are identi?ed with the highest derivatives of the dynamical ?elds according to (61). For example, the equation (82) in the zero order in η reads in the sector of B ? ?C(Z, Y ) B1 (Z, Y ) = ? C(Y ) . ?η ?ν (83)

Because of the nonlocality of the star-product, for each ?xed rank multispinorial component of the left hand side of this formula there appears, in general, an in?nite series involving bilinear combinations of the components Cν(n) with all n on the right hand side of (83). Therefore, the right hand side of (83) e?ectively involves space-time derivatives of all orders, i.e. may describe some nonlocal transformation. Therefore, the system (70), (71) cannot be treated as locally equivalent to the free system (η = η = 0). Instead we can only claim that there exists ? a nonlocal mapping between the free and nonlinear system. For the ?rst sight the existence of the 4d ?ows is paradoxical because it establishes a connection between the full nonlinear problem and the free system with vacuum ?elds in the sector of the gauge ?elds W . For example, in the gravity sector, the appropriate version of the Einstein equations has the form (57), (58) with the Weyl tensor components Cα(4) and Cα(4) replaced by ηCα(4) and ηCα(4) , respectively. In the limit η = 0, Einstein equations ? ˙ ˙ therefore reduce to the vacuum equations of the AdS space. The dynamical equations of the massless spin 2 ?eld reappear as equations on the Weyl tensor contained in (50). What happens is that the integrating ?ow generates a sort of a normal coordinate expansion of the form W = W0 + ηα1 (x)C + η 2 α2 (x)C 2 + . . . , providing a systematic way for the derivation of the coe?cients of the expansions in powers of (HS) Weyl tensors C. The procedure is purely algebraic at any given order in η (equivalently C). In particular, such an expansion reconstructs the metric tensor in terms of the curvature tensor.

7

Conclusions

HS gauge theories are based on the in?nite-dimensional HS symmetries realized as the algebras of oscillators carrying spinorial representations of the space-time symmetries [33]. These starproduct algebras exhibit the usual quantum-mechanical nonlocality in the auxiliary spinor spaces. An important point is that the dynamical HS ?eld equations transform this nonlocality into space-time nonlocality, i.e. the quantum mechanical nonlocality of the HS algebras may imply some space-time nonlocality of the HS gauge theories at the interaction level. The same time the HS gauge theories remain local at the linearized level. The relevant geometric setting is provided by the Weyl bundles with space-time base manifold and Weyl algebras with spinor generating elements as the ?bre. The star-product acts in the ?ber rather than directly in the space-time. The noncommutative Yang-Mills theory structure also appears in the ?ber sector. 18

An important implication of the star-product origin of the HS algebras is that the spacetime symmetries are simple and therefore correspond to AdS geometry rather than to the ?at one. The space-time symmetries are realized in terms of bilinears in spinor oscillators according to the isomorphism o(3, 2) ? sp(4; R). This phenomenon has two consequences. On the one hand, it explains why the theory is local at the linearized level. The reason is that bilinears in the non-commuting auxiliary coordinates can lead to at most two derivatives in the star-products. On the other hand, the fact that HS models require AdS geometry is closely related to their potential nonlocality at the interaction level because it allows expansions with arbitrary high space-time derivatives, in which the coe?cients carry appropriate (positive or negative) powers of the cosmological constant ?xed by counting of dimensions. As a result, HS symmetries link together such seemingly distinct concepts as AdS geometry, space-time nonlocality of interactions and quantum mechanical nonlocality of the star-products in auxiliary spinor spaces. Another consequence of the star-product origin of the HS symmetries is that HS theories are based on the associative structure rather than on the Lie-algebraic one. As a result, the construction can be extended [25] to the case with inner symmetries by endowing all ?elds with the matrix indices. HS gauge theories with non-Abelian symmetries classify [32] in a way analogous to the Chan-Paton symmetries in oriented and non-oriented strings. Some of them exhibit N-extended space-time supersymmetries [33, 34]. Acknowledgments. This research was supported in part by INTAS, Grant No.96-0308 and by the RFBR Grant No.99-02-16207.

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