From Coxeter Higher-Spin Theories to Strings and Tensor Models
aa r X i v : . [ h e p - t h ] A ug FIAN/TD/02-18
From Coxeter Higher-Spin Theories to Strings and Tensor Models
M.A. Vasiliev
I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,Leninsky prospect 53, 119991, Moscow, Russia To my father
Abstract
A new class of higher-spin gauge theories associated with various Coxeter groups isproposed. The emphasize is on the B p –models. The cases of B and its infinitegraded-symmetric product sym ( × B ) ∞ correspond to the usual higher-spin theoryand its multi-particle extension, respectively. The multi-particle B –higher-spin theoryis conjectured to be associated with String Theory. B p –higher-spin models with p > p boundary tensor sigma-models. B p higher-spin models with p ≥ AdS background and stringy/tensor effects, respectively. The brane-like idempotent extension of the Coxeter higher-spin theory is proposed allowing tounify in the same model the fields supported by space-times of different dimensions.Consistency of the holographic interpretation of the boundary matrix-like model inthe B -higher-spin model is shown to demand N ≥ N = 4 SYM upon spontaneous breaking of higher-spin symmetries. The proposedmodels are shown to admit unitary truncations. ontents A N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 General Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 d Coxeter higher-spin models . . . . . . . . . . . . . . . . . . . . . . 186.2.2 4 d Coxeter higher-spin models . . . . . . . . . . . . . . . . . . . . . . 196.3 Vector Coxeter higher-spin models in any dimension . . . . . . . . . . . . . . 21 N = 4 SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10 Unitarity 3011 Field frames of multi-particle theories 32
12 Interpretation 38
13 Conclusion 43 Introduction
Higher-spin (HS) gauge theories, that describe interactions of massless fields of all spins,provide an interesting arena for testing principles of fundamental physics and, in particular,the
AdS/CF T correspondence conjecture [1, 2, 3]. First example of fully nonlinear HStheory was given for the 4 d case in [4], while its modern formulation was worked out in [5](see [6] for a review). A specific property of HS gauge theories is that consistent interactionsof propagating massless fields demand a curved background providing a length scale in HSinteractions that contain higher derivatives. ( A ) dS is the most symmetric curved backgroundcompatible with HS interactions [7]. The AdS HS model is the simplest nontrivial one inthe sense that d = 4 is the lowest dimension where HS massless fields propagate.There were several long-standing problems in HS theories. One was the issue of localityraised originally in [8, 9] where it was shown that current interactions in HS theories canbe removed by a seemingly local field redefinition containing infinite power series in higherderivatives with the coefficients containing inverse powers of the cosmological constant. Thisissue has been further analyzed in a number of papers [10]-[16]. The conclusion of [14,15] was that unfolded HS equations lead to correct local interactions in the lowest order.These results are now extended to higher orders due to the development of the appropriatehomotopy technics in [17, 18] where it is explained in particular how to identify the minimallynonlocal frame of [15] and to decrease the level of nonlocality in HS equations of [5] in higherorders.Once the problem of locality in HS gauge theory has been sorted out, the fundamentalremaining question is how to construct more general HS models that could be related toString Theory [19]. This is the problem addressed in this paper where we propose a newclass of HS-like gauge theories that contain richer spectra of fields than the standard modelsof [5, 8]. The proposed models are argued to be rich enough to lead to String Theory-likemodels with massive HS fields via spontaneous breakdown of HS symmetries. A concreteproposal for the realization of this idea briefly discussed in the end of this paper can shedlight on the origin of the most symmetric phase of String Theory discussed long ago in[20, 21]. (For related discussion see also [22]-[25].) Although this conjecture was supportedby the analysis of high-energy limit of string amplitudes [21] and passed some nontrivialtests [26]-[29], no satisfactory understanding of this relation beyond the free field sectorof the tensionless limit of String Theory [30, 31, 32] was available. Note however that aninteresting idea of the singleton string whose spectrum is represented by multiple tensorproducts of singletons was put forward in [33, 34].The idea of our construction came from the recent development of the AdS/CF T holog-raphy [1, 2, 3]. Indeed, the fact that HS theories are most naturally formulated in the
AdS background was conjectured to play a role in the context of
AdS/CF T correspondence [35]-[39]. This expectation conforms to the fundamental result of Flato and Fronsdal [40] on therelation between tensor products of 3 d conformal fields (singletons) and infinite towers of 4 d massless fields that appear in HS theories.In the important work of Klebanov and Polyakov [41] it was argued that the HS gaugetheory of [5] should be dual to the 3 d O ( N ) sigma-model in the N → ∞ limit. The3lebanov-Polyakov conjecture was checked by Giombi and Yin in [42, 43] where it wasshown in particular how the bulk computation in HS gauge theory reproduces at least someof conformal correlators in the free 3 d theory. (See also important papers [44, 45].)Original Klebanov-Polyakov conjecture [41] related large- N limit of the 3 d boundarysigma-model with the action S = 12 Z d x (cid:0) ∂ ν ϕ i ∂ ν ϕ i + λN ( ϕ i ϕ i ) (cid:1) (1.1)for scalar fields ϕ i in the vector representation of O ( N ) ( i = 1 , , . . . , N ) with the partic-ular HS gauge theories of [5], so-called A and B -models, in the bulk. In [46] Klebanov-Polyakov conjecture was extended to the boundary vector models with Chern-Simons inter-action where the Chern-Simons level was related to the free phase parameter in the modelsof [5]. Fermionic analogues of these conjectures were put forward in [47, 48, 49] and werepartially checked in [50] (for more recent results see [51, 52]), suggesting a nontrivial dualitybetween bosonic and fermionic boundary models, so-called bosonization, [53, 54, 55].Some difficulties of the original analysis of HS holography originated from non-localityof the naive perturbative approach in HS theory leading to divergencies in the holographictests [42]. In [51, 52, 56] it has been checked that unfolded HS equations in the local frameof [14, 57, 15] properly reproduce the anticipated holographic results including the generalcase of Chern-Simons boundary theory, thus opening a way towards further analysis of HSholography in the framework of the nonlinear HS system of [5]. AdS /CF T HS duality was extended to
AdS /CF T holography by Gaberdiel andGopakumar [58] (see [59] for a review) that has been extensively studied in the literature(see e.g. [60]-[69]). Elaborating on AdS /CF T duality, Gaberdiel and Gopakumar arrivedat the interesting conjecture on the relation between certain compactifications of String The-ory and 3 d HS theories [70] as well as on the structure of stringy HS symmetry [71], furtherelaborated in [72]-[75] (see also [76]). Remarkably, symmetries of the string-like HS theo-ries proposed in this paper exhibit interesting similarities with the Gaberdiel-Gopakumarconstruction.The vector boundary models were recently extended to so-called tensor models [77] withthe scalar and spinor fields ϕ i i ...i p carrying several color indices (symmetrized and/or trace-less or not). The Lagrangian for the so-called O ( N ) boundary model is S = 12 Z d d x (cid:0) ∂ n ϕ ijk ∂ n ϕ ijk + gϕ ijk ϕ inm ϕ jnl ϕ kml (cid:1) . (1.2)For O ( N ) p tensor models with any p > p + 1. One canalso consider bosonic and fermionic tensor models of U ( N ) p and U SP (2 N ) p types. Recently,these models have been extensively studied in the literature [78]-[81] in particular due totheir relation [82] to the SYK model [83, 84], providing a new class of models possessingan interesting large N regime [85]-[89]. This suggests in turn that they should admit someholographically dual description. The case of p = 2 corresponds to models with matrix-valued fields like in N = 4 SYM theory. In this case, the holographically dual theory isString Theory. 4o the best of our knowledge no models holographically dual to the tensor sigma-modelswith p > O ( N ) model were expressed e.g. in [89, 90]. It is immediately clear that theHS model of [5] as well as its generalizations to three [8] and any [91] dimensions are notappropriate duals since the spectrum of O ( N ) singlet operators of rank- p tensor modelsincreases with p due to the presence of new multi-particle states [78, 81]. The same time,the sigma-model form of the boundary theory suggests that a bulk dual model has to be ofa HS type.An obvious class of conformal boundary models holographically dual to tensorial HStheories in the bulk first discussed by Becaria and Tseytlin in [78] is provided by the freeboundary conformal fields ϕ i ,i ,...i n with the tensorial currents of the type of the vertexin (1.2), i.e., J = ϕ ijk ϕ inm ϕ jnl ϕ kml , or similar. Analogously to the Klebanov-Polyakovconjecture [41] for vector models, we conjecture that the proposed Coxeter HS theories areholographically dual to such higher currents in the free boundary models. Moreover, the AdS Coxeter HS models admit an extension to complex coupling constants that should beholographically dual to the 3 d boundary conformal models with Chern-Simons interactionsanalogous to the models considered in [46] for the vectorial HS holography. A subtlety ofthe holographic interpretation of the boundary tensor sigma-models is that their conformalstructure analogous to that of critical sigma-model in the Klebanov-Polyakov conjecture [41]is not yet obvious. Note however that, as argued in [92], it is not a priory guaranteed thatthe boundary duals of the class of HS theories proposed in this paper are represented bylocal boundary sigma-models free from additional interactions via boundary HS conformalfields which may effectively induce nonlocal interactions on the boundary (see also [93]).An important problem in the context of HS duality consists of the construction of so-called multi-particle HS theories containing multi-particle composite operators of the originalHS theory as fundamental operators. In [94] a multi-particle extension of the usual HSsymmetry was proposed, conjectured to underly an infinite extension of the conventionalHS theory. The aim of this paper is to propose a class of generalized nonlinear HS systemsto be associated with the tensor boundary sigma-models and their multi-particle extensionsincluding the vector and tensor ones.The proposed models are characterized by an integer p analogous to the rank of the tensor ϕ i i ...i p and have a much richer spectrum than usual HS theory of [5, 8, 91]. They are basedon the deformed oscillator algebras found in [95, 96, 97] in the context of p -particle Calogeromodels, known as Cherednik algebras [98], which in turn are associated with various Coxetergroups. HS-like models of this class could be formulated long ago and potential relevance ofthe Cherednik algebra to HS theory was mentioned already in [97]. However, naive extensionof this class was not formulated explicitly so far because the resulting spectra of states leftno room for a massless spin-two state, i.e., graviton, not allowing the resulting models tofit into the standard paradigm of HS gravity. The same time, the naive extension to HStheories of this kind exhibits interesting features indicating their potential relevance as bulkduals of the tensor boundary sigma-models. 5n this paper we extend the construction of Cherednik algebras to framed Cherednikalgebras in such a way that the related HS theories turn out to be free of the aforementioneddifficulties, containing massless HS fields in the spectrum. This generalization was inspiredby the construction of multi-particle algebras of [94] and, other way around, allows us toformulate nonlinear equations for the multi-particle HS theory. It is based on the q th directproduct degree of a chosen rank- p Coxeter group C , thus containing two positive integers asfree parameters: p and q are associated, respectively, with the tensor degree of the boundarymodel and the degree of multi-particle states. For instance C = A ∼ B and q = 1 or q = ∞ correspond to the conventional HS theory or its multi-particle extension, respectively. Onthe other hand, higher ranks p of the Coxeter group B p are conjectured to be associatedwith the rank- p tensor boundary models. Generalization to other Coxeter groups and rootsystems (for more detail on Coxeter groups see for instance [99]) is also possible and worth tobe investigated. The case of p = 2 , q = ∞ is particularly appealing in connection with the HSinterpretation of String Theory. The B -model possesses two coupling constants responsiblefor HS interactions in AdS background and stringy effects, respectively. It is tempting tospeculate that this model can provide a realization of the most symmetric phase of StringTheory. A possible HS symmetry breaking mechanism is also discussed in the concludingsection of this paper.The main idea of this work was to look for HS bulk models providing generalizations of theusual HS theory to models with richer spectrum that would qualitatively match the patternof operators of the tensor boundary models and their multi-particle extensions. It shouldbe stressed that the formal consistency and gauge symmetries along with simple physicalconditions like, e.g. , Lorentz covariance severely restrict possible higher-rank extensions ofHS theories. The HS-like models based on framed Cherednik algebras provide a distinguishedand, may be, exhaustive possibility for the construction of models fulfilling these properties.The paper is organized as follows. We start with recalling the form of standard nonlinearHS equations in Section 2. In Section 3 we recall the construction of deformed oscillatorsthat underlies the standard formulation of HS equations as well as its Cherednik extensionassociated with an arbitrary Coxeter root system. In Section 4 we explain the difficulty ofthe naive extension of HS algebras via enlargement of the number of species of oscillators andintroduce the notion of the framed oscillator algebra free of this difficulty. Then in Section5 we apply this idea to the construction of framed Cherednik algebras underlying the properextension of HS equations to any Coxeter system presented in Section 6. Further extensionsof the proposed systems to higher differential forms generating invariant functionals, colorsystems with Chan-Paton factors, and idempotents associated with brane-like dynamicalsystems in different dimensions are presented in Sections 7, 8 and 9, respectively. Unitarityof the proposed models is discussed in Section 10. Graded extension of the constructionof multi-particle algebras of [94] along with the discussion of its different frames and newidempotent construction are presented in Section 11. General properties of the proposedsystems, their interpretation in the context of string-like and tensor-like HS systems and apossible HS symmetry breaking mechanism are discussed in Section 12 where it is also arguedthat the proposed scheme allows one to interpret composite fields in the original space-time6s elementary fields in higher dimensions. Section 13 contains brief conclusions.
Nonlinear HS equations of [5, 8, 91] are formulated in terms of three types of fields W ( Z ; Y ; k | x ) = dx n W n ( Z ; Y ; k | x ) , B ( Z ; Y ; k | x ) , S = dZ A S A ( Z ; Y ; k | x ) . (2.1)Here x n are space-time coordinates while the variables Z A and Y A ( A, B, . . . = 1 , . . . M ) areauxiliary coordinates acquiring different interpretation in the spinorial HS models of [5, 8]and vectorial HS model of [91]. dZ A are anticommuting Z − differentials dZ A dZ B = − dZ B dZ A , dZ A dx n = − dx n dZ A , dx m dx n = − dx n dx m . (2.2)HS equations of [5, 8, 91] have the formd W + W ∗ W = 0 , (2.3)d B + W ∗ B − B ∗ W = 0 , (2.4)d S + W ∗ S + S ∗ W = 0 , (2.5) S ∗ B = B ∗ S , (2.6) S ∗ S = − i ( dZ A dZ A + ( dz α dz α F ∗ ( B ) ∗ κ + c.c. )) . (2.7)Here d = dx n ∂∂dx n is the space-time de Rham differential and ∗ denotes a model-dependentstar product. F ∗ ( B ) is some star-product function of the field B . Index α takes two valuesenumerating basis elements of some subspace of the space labelled by A . The complexconjugated term c.c. does not contribute in the real models of [8, 91] but has to be addedin the 4 d model of [5]. κ is an element of the algebra defined to commute with everythingexcept for dz α with which it anticommutes κ ∗ dz α = − dz α ∗ κ . (2.8)The associative star product ∗ acts on functions of two spinor variables( f ∗ g )( Z ; Y ) = 1(2 π ) Z d U d V exp [ iU A V B C AB ] f ( Z + U ; Y + U ) g ( Z − V ; Y + V ) , (2.9)where C AB is a symplectic form and U A , V B are real integration variables. It is normalizedin such a way that 1 is a unit element of the star-product algebra, i.e., f ∗ ∗ f = f . Star product (2.9) provides a particular realization of the Weyl algebra[ Y A , Y B ] ∗ = − [ Z A , Z B ] ∗ = 2 iC AB , [ Y A , Z B ] ∗ = 0 (2.10)7[ a, b ] ∗ = a ∗ b − b ∗ a ) resulting from the normal ordering with respect to the elements b A = 12 i ( Y A − Z A ) , a A = 12 ( Y A + Z A ) , (2.11)which satisfy [ a A , a B ] ∗ = [ b A , b B ] ∗ = 0 , [ a A , b B ] ∗ = C AB . (2.12)The choice of the star-product realization (2.9) of the Weyl algebra is significant from manyperspectives and, in particular, for the analysis of locality in [17, 18]. An important propertyof the star product (2.9) is that it admits the inner Klein operator κ = exp iZ α Y α , (2.13)which behaves as ( − N , where N is the oscillator number operator, i.e., κ ∗ κ = 1 , (2.14) κ ∗ f ( Z ; Y ) = f ( ˜ Z ; ˜ Y ) ∗ κ , (2.15)where ˜ Z A = ( − Z α , Z a ) for any Z A = ( Z α , Z a ) with Z a being the rest components of Z A .The Klein operator k = κ κ , (2.16)on which all fields (2.1) depend, anticommutes with Y α , Z α and dZ α . Hence, κ in Eq. (2.7)is represented as κ = k κ . (2.17)(Note that the fields of the 4 d model of [5] also depend on the conjugated Klein operator ¯ k .) The nontrivial part of the HS equations is represented by Eqs. (2.6), (2.7). Since B is covari-antly constant by (2.4), and commutes with S by (2.6) and κ since it is a dz -independentzero-form, it behaves as a central element. Replacing F ∗ ( B ) by a central element ν anddiscarding for a moment the complex conjugated term we observe that the nontrivial partof the HS equations has the form S ∗ S = − i ( dZ A dZ A + ν ∗ dz α dz α ∗ κ ) , α = 1 , . (3.1)This equation has many important interpretations for particular realizations of S , ∗ and κ (see e.g. [6, 10]). The most important property in the context of this paper is that the r.h.s. of (3.1) commutes with S despite of (2.8) since any trilinear combination of dz α iszero because α = 1 , dz α anticommute. It is this property that guarantees formalconsistency of HS equations (2.3)-(2.7). 8q. (3.1) induces commutation relations of the deformed oscillator algebra. Indeed, let S = dz α q α + . . . , where ellipses denotes terms proportional to components of dZ A differentfrom dz α , and ˜ ν = νk , (3.2)where kdz α = − dz α k , kq α = − q α k , k = 1 . (3.3)So defined k and ˜ ν still commute with S while, setting κ := κk , (3.4)(3.1) implies deformed oscillator commutation relations in the form of [100][ q α , q β ] = − iǫ αβ (1 + ˜ ν κ ) , κ q α = − q α κ , κ = 1 , (3.5)where ˜ ν is central. The enveloping algebra Aq (˜ ν ) of these relations, which is the algebra ofpolynomials f ( q, κ ) with ˜ ν being a parameter, has the fundamental property [100] that t αβ := i { q α , q β } (3.6)generate sp (2) [ t αβ , t γδ ] = ǫ βγ t αδ + ǫ βδ t αγ + ǫ αγ t βδ + ǫ αδ t βγ (3.7)rotating q α as a sp (2) vector [ t αβ , q γ ] = ǫ βγ q α + ǫ αγ q β . (3.8)By these properties, the deformed oscillators (3.5) provide a representation-free realizationof the Wigner oscillator [101] considered by many authors (see, e.g. , [102, 103, 104]).That the deformed oscillator algebra possesses the sp (2) automorphism algebra impliesthat nonlinear HS equations (2.3)-(2.7) are also sp (2) covariant. In 4 d HS theory of [5] and3 d HS theory of [8] the respective sp (2 | C ) and sp (2 | R ) imply usual Lorentz covariance [6].In vectorial models in any dimension of [91], sp (2 | R ) is a Howe dual algebra allowing totruncate away extra states from the spectrum. The deformed oscillator of Section (3.1) can be interpreted as describing the relative motionin the two-body Calogero model. Extension to the N -body Calogero model proposed in[95, 96, 97] consist of N copies of oscillators q α i with α = 1 , i = 1 , . . . N and the generators K ij of the symmetric group S N that obey K ij q α i = q αj K ij , (3.9) K ij = K ji , K ij K ij = I , K ij K jl = K il K ij = K jl K il (3.10)9no summation over repeated indices), and K ij commutes with q α l with i = l , j = l .Commutation relations of the oscillators q α l have the form [95, 96, 97][ q α n , q β m ] = − iǫ αβ (cid:0) δ nm (cid:16) I + ν N X l =1 K ln (cid:17) − νK nm (cid:1) . (3.11)Remarkably, this deformation still respects Jacobi identities since antisymmetrization overany three two-component indices gives zero as is most obvious from the Coxeter realizationpresented in Section 3.3.From (3.11) it follows that the center of mass coordinates Q α := N − N X n =1 q α n (3.12)have undeformed commutation relations with themselves and relative coordinates [ Q α , Q β ] = − iǫ αβ , [ Q α , q β n − q β m ] = 0 . (3.13)The fundamental property of relations (3.11) is that the operators t αβ := i N X n =1 { q α n , q β n } (3.14)obey the sp (2) commutation relations (3.7), properly rotating all indices α by virtue of[ t αβ , q γ n ] = ǫ βγ q α n + ǫ αγ q β n . (3.15)Clearly, t αβ can be represented as a sum of sp (2) generators acting separately on the centerof mass and relative coordinates t αβ = t cmαβ + t relαβ , t cmαβ := i { Q α , Q β } . (3.16)At N = 2, there is only one relative coordinate q α = 1 √ q α − q β ) . (3.17)With K = κ (3.11) amounts to (3.5) which therefore describes the relative coordinatesector of the two-body Calogero model. A rank- p Coxeter group C is generated by reflections with respect to a system of root vectors { v a } in a p -dimensional Euclidean vector space V with the scalar product ( x, y ) ∈ R , x, y ∈ V v a actson x ∈ V as follows R v a x i = x i − v ia ( v a , x )( v a , v a ) , R v a = I (3.18)(no summation over a ). p is the rank of the Coxeter root system { v a } . Note that R v a changesa sign of v a R v a v a = − v a . (3.19)Let q nα ( α = 1 , n = 1; . . . , p ) obey Heisenberg commutation relations[ q nα , q mβ ] = − iǫ αβ δ nm , q nα = δ nm q m α . (3.20)Cherednik deformation of the semidirect product of the Heisenberg algebra with the groupalgebra of C is [ q nα , q mβ ] = − iǫ αβ δ nm + X v ∈R ν ( v ) v n v m ( v, v ) K v ! , (3.21) K v q nα = R vnm q mα K v , (3.22)where K v generate the Coxeter group, and the coupling constants ν ( v ) are invariants of C being constant on the conjugacy classes of root vectors under the action of C . These relationsgeneralize (3.11) from A p = S p +1 to any Coxeter group. By definition, the operators K v obeydefining relation of the Coxeter group C . In particular, each K v is involutive K v K v = I (3.23)and since the reflections generated by v and − v coincide, the respective reflection generatorsare also identified K − v = K v . (3.24)It is not difficult to check that the double commutator of q nα in (3.21) respects the Jacobiidentities for any (not necessarily C -invariant) ν ( v ). Indeed, the non-zero part of the triplecommutator of q nα , q mβ , q kγ resulting from (3.21) is proportional v n v m v k and hence contains thetotal antisymmetrization over three two-component indices α, β, γ giving zero. Covarianceof relations (3.21) under the action of the Coxeter group itself demands ν ( v ) be C -invariant.In the sector of relative coordinates, the commutation relations (3.11) is a particular caseof (3.21) for the A p root system v nm = e n − e m , (3.25)where e n form an orthonormal frame in R p +1 . V is the p -dimensional subspace of relativecoordinates in R p +1 spanned by v nm .Another important case of the Coxeter root system is B p with the roots R = {± e n ≤ n ≤ p } , R = {± e n ± e m ≤ n < m ≤ p } . (3.26)11n addition to permutations, B p contains reflections of any basis axis in V = R p generatedby v n ± = ± e n . In this case R and R form two conjugacy classes of B p . Let K n , K nm and K + nm be associated with e n , e n − e m and e n + e m , respectively.The defining relations of B p are represented by relations (3.10) for the symmetric groupgenerators K nm along with K n K nm = K nm K m , K n K n = I , K + nm = K n K nm K n (3.27)(generators K n , K nm and K + nm with pairwise different indices commute).The Coxeter group underlying 3 d HS theory of [8] is A ∼ B . The case of B is also ofgreat importance since it is argued in Section 12 to be related to the string-like models.For any Coxeter root system the generators t αβ := i p X n =1 { q nα , q nβ } (3.28)obey the sp (2) commutation relations (3.7), properly rotating all indices α by virtue of[ t αβ , q nγ ] = ǫ βγ q nα + ǫ αγ q nβ (3.29)as is easy to see using (3.22) and (3.19). This fact is of fundamental importance for HStheories. A natural way to allow a richer spectrum in the model including, in particular, mixed-symmetry massless and massive states (the latter are known to be present in String Theory[19]) known to allow consistent interactions at least in the lowest orders [105]-[111], is to letmore species of oscillators in the model: y α → y nα . However, the construction of oscillatoralgebras with multiple species of oscillators as well as their Cherednik deformation recollectedin Section 3.3 cannot be directly applied to the construction of physically acceptable HStheories for the following reason.Consider a HS algebra called for brevity hs with the single set of oscillators ( i.e., p = 1).It is well known [23, 112] that the Fock hs -module F describes free boundary conformalfields [113, 114], i.e., Dirac singletons [115, 116, 40, 117, 118, ? ]. Their conformal dimensionis identified with the weight h of the vacuum state in F with respect to the dilatationoperator D represented by a bilinear of oscillators in hs D | i = h | i . (4.1) More precisely the Coxeter root system A ∼ C while the root systems B p and C p differ by thenormalization of root vectors [99] having the same Coxeter group. AdS unitary frame D is replaced by the energy operator E . For details on theoscillator realization of semisimple Lie algebras and their modules see, e.g. , [120].) Corre-spondingly, lowest weight representations of the naively extended algebras hs p built from p copies of oscillators will have multiple weights in the respective Fock vacuum F p , h p = ph , (4.2)hence carrying higher lowest weights.For p = 1, elements of the tensor product F ⊗ F are interpreted either as conservedcurrents on the boundary or as massless fields in the bulk as was originally shown by Flatoand Fronsdal [40]. However for p > F p ⊗ F p turnout to be too high to describe both conserved boundary currents and bulk massless fields.In particular, the p > i.e., graviton.That the respective theory cannot contain gravity makes it impossible the construction ofconsistent HS models via allowing more oscillators, i.e., more generating elements of thestar-product algebra. (Note that this argument does not apply to 3 d HS theories since inthis case the graviton is topological [121, 122], carrying no local degrees of freedom, and,hence, having no nontrivial associated module of the space-time symmetry group. As aresult, the construction of 3 d HS theories turns out to be far less restrictive than in d > i.e., U ( hs ). Fromthis definition it is clear that HS algebra-modules form modules over the multi-particle HSalgebra as well. As observed in [10], an important difference between the construction ofmulti-particle algebra and oscillator algebras with many oscillators is due to appearance ofindependent unit elements associated with each copy of oscillators.Namely, in the usual oscillator algebras the fundamental commutation relations are[ q nα , q mβ ] = 2 iδ nm ǫ αβ I , (4.3)where I is the unit element of the algebra. In the multi-particle HS algebra these relationsare replaced by [ q nα , q mβ ] = 2 iδ nm ǫ αβ I n , (4.4)where “units” I n assigned to the respective species of the oscillators obey I n I n = I n , I n I m = I m I n . (4.5)Such algebras will be called framed oscillator algebras . Note that the framed oscillatoralgebras are nothing else as multiple tensor products of the original associative oscillatoralgebra with no color indices n and m , i.e., of a single pair of oscillators while usual oscillatoralgebras (4.3) with the single unit element are their quotients.The same time multi-particle HS algebra is unital being endowed with the unit element I unrelated to I n . Roughly speaking, I is associated with the physical vacuum with no excitedstates while I n is associated with the minimal energy state in the sector of n -th particles.13n important feature of the multi-particle algebras is that their elements are (graded)symmetrizedover elementary species. In that case, the only single-particle element built from I n is e = p X i =1 I i . (4.6)As such e becomes algebraically independent of individual I n with the consequence that itseigenvalues may be different from multiples of those for I n which may have no sense at all.Technically, in the framed oscillator algebras, the problem with vacuum eigenvalues isavoided as follows. The “units” I n form a set of idempotents allowing to consider Fock-typemodules F ip generated from vacua | i i obeying I j | i i = δ ji | i i , a j − | i i = 0 , (4.7) a j + | i i = 0 i = j , (4.8)where a i ± are the creation and annihilation operators built from q iα . Clearly, the modules ofthe generators t αβ formed by F ip are equivalent to the p = 1 Fock module F . Note that forusual oscillator algebras this construction is applicable to the oscillators with indices α, β taking any even number of values 2 M in which case t αβ (3.28) generate sp (2 M ).Since higher-rank HS theories are anticipated to be associated with the Cherednik-likedeformation of the oscillator algebras, to let them contain gravity it is necessary to find aframed extension of the Cherednik algebra having a room for units I n . A N To illustrate the idea let us first consider the A p − system of Section 3.2. In addition to q αn and K nm with n, m = 1 , . . . p , we introduce elements I n that obey I n I m = I m I n , I n I n = I n , (5.1) I n q αn = q αn I n = q αn , I n q αm = q αm I n . (5.2)In presence of idempotents I n , the proper modification of the deformed oscillator relationsconsistent with (5.2) is[ q α n , q β m ] = − iǫ αβ (cid:0) δ nm (cid:16) I n + ν p X l =1 ˆ K ln (cid:17) − ν ˆ K nm (cid:1) , (5.3)where ˆ K nm = I n I m K nm . (5.4)14ote that operators ˆ K nm obey all relations of the symmetric group S p except for the secondrelation in (3.10) replaced by ˆ K nm ˆ K nm = I n I m . (5.5)ˆ K nm are demanded to obey the analogs of (3.9)ˆ K ij q α i = q αj ˆ K ij (5.6)and to commute with all I l I l ˆ K nm = ˆ K nm I l ∀ l, n, m . (5.7)(Note that by (5.1) this is consistent with the naive expectation that K nm I m = I n K nm .)Also, the following relations held true I n ˆ K nm = I m ˆ K nm = ˆ K nm . (5.8)It should be stressed that the unhatted generators of the symmetric group K nm do notappear in the construction of the framed Cherednik algebra. Relation (5.5) is not of thebraid group since I n I m is not invertible. Hence, ˆ K nm do not generate a group algebra.This modification guarantees consistency of defining relations (5.3). Namely, (5.7) impliesconsistency of (5.3) and (5.2) while the modification (5.5) of the symmetric group relationsplays no role in the Jacobi identity check[ q nα , [ q mβ , q kγ ]] + cycle = 0 . (5.9) Extension of commutation relations (3.21) to framed Cherednik algebra is[ q nα , q mβ ] = − iǫ αβ δ nm I n + X v ∈R ν ( v ) v n v m ( v, v ) ˆ K v ! , (5.10)where I n are idempotents analogous to those of Section 4 andˆ K v := K v Y I i ( v ) . . . I i k ( v ) , (5.11)where the labels i ( v ) , . . . , i k ( v ) enumerate those I n that carry labels affected by the re-flection R v . For instance, in the case of B p , ˆ K v ij = K v ij I i I j for the root v ij generating thepermutation of e i , and e j and ˆ K v i = K v i I i for v i representing the reflection of e i of B p .It is important that framed Cherednik algebra still possesses inner sp (2) automorphismsgenerated by t αβ := i p X n =1 { q nα , q nβ } I n (5.12)obeying (3.29). Evidently, [ t αβ , I n ] = 0 . (5.13)Note that usual Cherednik algebra results from the framed one by quotioning out theideal identifying all I n with the unit element of the algebra.15 Coxeter higher-spin equations
The idea is to let all x -dependent fields W , S and B depend on p sets of variables enumeratedby the label n = 1 , . . . p , that include Y nA , Z nA ( A = 1 , . . . M ), idempotents I n , anticommutingdifferentials dZ An and Klein-like operators ˆ K v associated with all root vectors of a chosenCoxeter group C (at the convention ˆ K − v = ˆ K v ). As usual in HS theory, the fields W , S and B are allowed to be valued in any associative algebra A (see also Section 8). To makecontact with the tensorial boundary theory, it is useful to set A = ( M at N ) p with elementsrepresented by tensorial matrices a u ...u p v ...v p , u i , v i = 1 . . . N . Here p is the tensor degree ofthe boundary model while N is the number of values of color indices with respect to whichthe N → ∞ limit has to be taken on the boundary. In this limit the leading contributionshould be represented by the colorless HS theory in the bulk.The field equations associated with Cherednik algebra (3.21) are formulated in terms ofthe star product analogous to (2.9)( f ∗ g )( Z ; Y ; I ) = 1(2 π ) pM Z d pM S d pM T exp [ iS An T Bm δ nm C AB ] f ( Z i + I i S i ; Y i + I i S i ; I ) g ( Z − T ; Y + T ; I )(6.1)with central elements I n obeying relations I n ∗ Y nA = Y nA ∗ I n = Y nA , I n ∗ Z nA = Z nA ∗ I n = Z nA , I n ∗ I n = I n . (6.2)With this definition star product (6.1) implies[ Y nA , Y mB ] ∗ = − [ Z nA , Z mB ] ∗ = 2 iC AB δ nm I n , [ Y nA , Z mB ] ∗ = 0 . (6.3)Analogously to Eq. (2.13) this star product admits inner Klein operators κ v associated withthe root vectors v κ v := exp i v n v m Z αn Y αm ( v, v ) (6.4)and any subset of indices α among A such that det | C αβ | 6 = 0. It is straightforward to seethat the Klein operators κ v generate the star-product realization of the Coxeter group via κ v ∗ q nα = R vnm q mα ∗ κ v , q nα = Y nα , Z nα . (6.5)Nonlinear equations for the generalized HS theory associated with the Coxeter group C are d W + W ∗ W = 0 , (6.6)d B + W ∗ B − B ∗ W = 0 , (6.7)d S + W ∗ S + S ∗ W = 0 , (6.8) S ∗ B = B ∗ S , (6.9)16 ∗ S = − i dZ An dZ An + X i X v ∈R i F i ∗ ( B ) dZ αn v n dZ α m v m ( v, v ) ∗ ˆ κ v ! , (6.10)where ˆ κ v act trivially on ( i.e., commute with) all elements except for dZ αn on which theyact in the standard fashion ˆ κ v ∗ dZ nα = R vnm dZ mα ∗ ˆ κ v . (6.11) F i ∗ ( B ) is any star-product function of the zero-form B on the conjugacy classes R i of C . Forinstance in the important case of B p equation (6.10) reads as S ∗ S = − i dZ An dZ An + X v ∈R F ∗ ( B ) dZ αn v n dZ α m v m ( v, v ) ∗ ˆ κ v + X v ∈R F ∗ ( B ) dZ αn v n dZ α m v m ( v, v ) ∗ ˆ κ v ! (6.12)with arbitrary F ∗ ( B ) and F ∗ ( B ). In particular, one can set F ∗ ( B ) = 0 keeping F ∗ ( B ) = ηB . In the case of B this gives usual HS equations. F ∗ ( B ) can be nonzero for the B p –HSmodels at p ≥
2. As discussed in more detail in Section 12, the roles of the coupling constantscontained in F ∗ ( B ) and F ∗ ( B ) are different: the former are responsible for the HS featuresof the model in AdS while the latter for the stringy and, more generally, tensorial ones.Perturbative analysis is performed around the vacuum solution B = 0 , S = dZ An Z An , W = W ( Y | x ) , (6.13)where W ( Y | x ) is some solution to (6.6) that usually is taken to describe AdS space-time.(In the 3 d models of [8] the solution with B = const is also important.)The realization of ˆ κ v can be different in different models and usually needs introductionof the outer Klein operators ˆ k v that, by definition, obey (3.22) with q = Y αn , Z αn and dZ αn ,so that ˆ κ v = κ v ˆ k v only affects the differentials dZ αn .Equations (6.6)-(6.10) are formally consistent since relations (3.21) respect the Jacobiidentities which in terms of the field equations are fulfilled due to the property that the r.h.s. of (6.10) is central. That HS-type equations can be consistently formulated based onCoxeter root systems was clear long ago. (This was mentioned already in [97].) However,such equations were never discussed in the literature so far since their interpretation in thestandard HS paradigm was not easy as discussed in Section 4 unless the construction isextended to the framed Cherednik algebras proposed in this paper.Multi-particle extensions of the Coxeter HS systems are associated with the semi-simpleCoxeter groups. The simplest option is associated with the Coxeter root system B N p thatconsists of the product of N B p systems B N p := B p × B p × . . . | {z } N . (6.14)The limit N → ∞ along with the graded symmetrization of the product factors expressingthe spin-statistics relation (for more detail see Section 11) is in many respects most natural.17ne can consider reductions of the Coxeter HS theories by restricting all fields to invari-ants of some group S of automorphisms of C . A natural choice is some subgroup S ⊂ S p × S N with S p ⊂ B p and S N exchanging the product factors in (6.14). The latter reduction is pos-sible in the case with all functions F i ( B ) associated with different product factors in C equalto each other. Restriction to invariants of S may be important for taking the N → ∞ limit.The choice of S containing the full symmetric group S N = ∞ is distinguished from variousperspectives. If S = S ∞ , the resulting algebra is the (graded symmetric) multi-particlealgebra M ( h ( C )) of [94] of the HS algebra h ( C ) associated with the C -framed Cherednikalgebra. As explained in [94], M ( h ( C )) is isomorphic to the universal enveloping algebraof h ( C ) (see also Section 11). This implies in particular that M ( h ( C )) is a Hopf algebra.Analogously, if S = S ∞ × G with G being some subgroup of the simple Coxeter group C , theresulting multi-particle algebra is M ( h G ( C )) where h G ( C ) is the subgroup of invariants of G in h ( C ). Clearly, M ( h G ( C )) is also a Hopf algebra.As discussed in Section 12, the particularly important case of B ∞ multi-particle HS modelassociated with the sym ( B × B × . . . ) system, where the graded symmetrization is withrespect to all elementary B factors, is anticipated to represent a stringy HS model.Now we are in a position to specify different types of the HS models resulting from theproposed construction. Spinor HS models were formulated in three [8] and four [5] dimensions. Naively, thesemodels may look being too far from Superstring living in 10 dimensions [19]. This is indeedtrue for the original 3 d and 4 d HS models but may not be true for their multi-particleextensions. The point is that, as emphasized in [123, 124], the multi-particle states of alower-dimensional model can be identified with elementary states in an appropriate higher-dimensional theory. Such interpretation turns out to be most natural within the matrix-space approach to massless field theories elaborated in [125, 114, 126, 127] (see also [92]and recent reviews [128, 129]). Moreover, the maximal space-time where such models admitinterpretation in terms of local fields is ten-dimensional [125, 126, 127] which is just thesuperstring space-time dimension. (It would be interesting to see whether this phenomenonis related to the twistor-like transform in ten dimensions introduced by Witten in [130].)From this perspective the original 3 d and 4 d theories can be interpreted as certain branes inthe ten-dimensional theory [126] with the 3 d HS model serving as an elementary brick fromwhich the others are composed (see also Section 12.2). d Coxeter higher-spin models
The 3 d Coxeter extension of the models of [8] is straightforward. It still needs introductionof two Clifford elements ψ , as in [8] to induce the doubling of the spectrum of fields as isstandard in 3 d gravity [121, 122] and HS theory [131]. The indices A and α coincide, i.e., A takes just two values representing a 3 d spinor index. sp (2) (5.12) represents the local Lorentz18ymmetry. For definiteness we consider the case of the Coxeter group B p which seems to bemost appropriate for HS applications. Extension to other Coxeter groups is straightforward.The multi-particle extension of the 3 d Coxeter HS theories results from endowing allvariables with the additional flavour index a = 1 , . . . , N I n , z α n , y α n , dz α n , ψ , , ˆ κ n , ˆ κ nm −→ I an , z aα n , y aα n , dz aα n , ψ a , , ˆ κ an , ˆ κ anm . (6.15)One can either consider different functions F a , ∗ ( B ) or to keep them all equal ( i.e., a -independent) to allow the S N reduction. In the latter case HS equations (6.12) read as S ∗ S = − i N X a =1 p X n =1 (cid:16) dz aαn dz αna + F ∗ ( B ) ∗ dz αan dz aα n ∗ ˆ κ n (6.16)+2 F ∗ ( B ) ∗ p X m =1 (cid:16) dz αan ( dz aα n − dz aα m ) ∗ ˆ κ anm + dz αan ( dz aα n + dz aα m )ˆ κ an ∗ ˆ κ anm ∗ ˆ κ an (cid:17)(cid:17) , where κ n and κ nm are the generators of the Coxeter group B p , that act only on dz aα n . Theresulting model contains both bosons and fermions for any N including the N = 1 B p –HStheory. The master fields are demanded to be graded symmetric under the exchange of oddcombinations of spinors z aα n , y aα n at different a (see also Section 11).Further reductions can be performed with S = S p and/or S = B p . Since fermions arerepresented by odd functions of y the latter choice leads to a bosonic reduced model.In the graded symmetrically reduced model usual HS gauge fields are represented bythe z -independent one-form part P p, N i,a ω ( y ai , ψ a i , ˆ k ai | x ) ∗ I ai of W with y aαi anticommuting atdifferent a . In particular, AdS background fields belong to this field (for more detail see [8].)Massless matter fields are represented by the z -independent part P i,a C ( y ai , ψ a i , ˆ k ai | x ) ∗ ψ a i of the zero-form B . Indeed, due to the presence of idempotents I ai (recall that ψ a i = ψ a i ∗ I ai )these obey the same unfolded field equations as the massless fields of [8]. In the absence of I ai the additional terms would contribute changing the pattern of the equations in agreementwith the discussion of Section 4. In the framed system such terms are uplifted to theequations on the two-particle fields containing pairs of units I a i I a i .As usual in HS theory [5, 6], one can consider models with fields valued in matrix algebras i.e., carrying Chan-Paton-like matrix indices. Upon imposing appropriate reality conditionsthe respective HS models possess U ( n ) gauge symmetry in the spin-one sector. Followingthe construction of [139, 8] (see also [6]) the resulting models can be truncated to those with O ( n ) and Sp (2 m ) spin-one gauge groups. (See also Section 8.) d Coxeter higher-spin models
In the Minkowski signature 4 d Coxeter HS model, A = α, ˙ α and one introduces two mutuallycommuting conjugated algebras generated by the left and right elements y nα , z nα , ˆ k v and ¯ y n ˙ α ,¯ z n ˙ α , ˆ¯ k ¯ v , respectively, on which all fields W , S and B depend. In the 4 d B p –HS system,19quation (6.12) is S ∗ S = − i (cid:16) dz An dz An + X v ∈R F ∗ ( B ) dz αn v n dz α m v m ( v, v ) ∗ ˆ κ v + X v ∈R F ∗ ( B ) dz αn v n dz α m v m ( v, v ) ∗ ˆ κ v + X ¯ v ∈R ¯ F ∗ ( B ) d ¯ z ˙ αn ¯ v n d ¯ z ˙ α m ¯ v m (¯ v, ¯ v ) ∗ ˆ¯ κ ¯ v + X ¯ v ∈R ¯ F ∗ ( B ) d ¯ z ˙ αn ¯ v n d ¯ z ˙ α m ¯ v m (¯ v, ¯ v ) ∗ ˆ¯ κ ¯ v (cid:17) . (6.17)In fact, there are several options for the higher-rank extension of the 4 d HS theory. Thesimplest ( B p × B p ) ′ -option is with identified idempotents I n = ¯ I n associated with the left andright oscillators y nα , z nα and ¯ y n ˙ α , ¯ z n ˙ α . We will call this HS model class I . Recall that by (5.7)both I n and ¯ I n commute with the Klein operators ˆ k and¯ˆ k . Hence I n − ¯ I n generates an ideal I of the B p × B p system with independent I n and ¯ I n . The ( B p × B p ) ′ system is ( B p × B p ) / I .In this model the lowest states are associated with 4 d massless fields represented by functionsof a single copy of oscillators y nα , z nα , ¯ y n ˙ α , ¯ z n ˙ α and I n . For instance, in the graded symmetrizedcase the massless fields are described by the Z -independent parts ω and C of W and B ,respectively ω = p, N X i,a ω ( y ai , ˆ k ai ; ¯ y ai , ˆ¯ k ai | x ) ∗ I ai , ω ( y ai , ˆ k ai ; ¯ y ai , ˆ¯ k ai | x ) = ω ( y ai , − ˆ k ai ; ¯ y ai , − ˆ¯ k ai | x ) , (6.18) C = p, N X i,a C ( y ai , ˆ k ai ; ¯ y ai , ˆ¯ k ai | x ) ∗ I ai , C ( y ai , ˆ k ai ; ¯ y ai , ˆ¯ k ai | x ) = − C ( y ai , − ˆ k ai ; ¯ y ai , − ˆ¯ k ai | x ) . (6.19)The 4 d model with different I an = ¯ I an which we call class II is a particular real form of thecomplexified 3 d ( B p ) system which construction is fully analogous to the sl ( C ) realizationof the 4 d Lorentz algebra. In fact this is a simplest manifestation of the fact discussed inthe beginning of Section 6.2 that tensoring of the lower-dimensional models can give higher-dimensional ones. Though this phenomenon is less straightforward for the higher tensorproducts since higher-dimensional Lorentz algebras are not real forms of the direct sums of anumber of sl ( C ) the results of [125]-[129] suggest that a proper interpretation can be avail-able in a certain limit in which the contribution of (delocalized) branes associated with thelower rank (dimensional) models can be discarded. In this model, the lower-rank states startwith ω and C that depend either only on y an , ˆ k an or only on ¯ y an , ˆ¯ k an . These states are analogousto the 3 d massless states to be identified with the boundary states in the AdS /CF T holog-raphy once the background ( i.e., vacuum) fields are chosen appropriately. This opens aninteresting possibility for describing both bulk and boundary degrees of freedom in the samemodel. A natural framework for the realization of this option suggested by the approach of[92] is described in Section 9.Finally, there is an interesting further extension of the 4 d HS system to the B p ( C ) -model,referred to as class III which is a specific real form of the complexified B p d system wherethe ˆ K i ¯ j reflection that exchanges the spinor variables of opposite chiralities is antilinear tobe compatible with the fact that the dotted and undotted spinors exchanged by ˆ K i ¯ j are20omplex conjugated. Note that this leads to a parametric freedom in the model via thephase parameter ϕ in the complex structure relationsˆ K i ¯ j = exp iϕ ˆ¯ K ¯ ij . (6.20)For the 4 d models the natural choice of the symmetric reduction is with S = S p beingthe diagonal subgroup of S p × ¯ S p in the left and right sector, allowing to keep the frame-likefield h α ˙ α p X n =1 y nα ¯ y n ˙ α (6.21) S -invariant, as well as its further extension to S = S p × S N in the multi-particle case. Vector HS model in d dimensions of [91] is formulated in terms of the oscillators Z An , Y An with the double index A = ν, α where ν = 0 , . . . d is the vector index of the d -dimensional AdS algebra o ( d − ,
2) and α = 1 , sp (2) with the symplectic form ǫ αβ so that C AB = η νµ ǫ αβ , A = ν, α , B = µ, β , (6.22)where η νµ is the ( x -independent) o ( d − , ν is decomposedinto ν = ( ν k , ν ⊥ ) where ν ⊥ = 0 , . . . , d − o ( d − ,
1) vector while ν k , taking one value, labels the Lorentz-invariant component. Theindex α of the nontrivial part of Eq. (6.10) is identified with ν = ( ν k , α ) ( i.e., with the d +1thcomponent of the o ( d − ,
2) vector index ν in A = ν, α ) and still takes two values.The original HS system of [91] is formulated in terms of the star product( f ∗ g )( Z, Y ) = 1 π d +1) Z dSdT e − S Aα T αA f ( Z + S, Y + S ) g ( Z − T, Y + T ) , (6.23)which gives rise to the commutation relations[ Z να , Z µβ ] ∗ = − ǫ αβ η νµ , [ Y να , Y µβ ] ∗ = ǫ αβ η νµ , [ Y να , Z µβ ] ∗ = 0 . The analogue of Eq. (2.7) has slightly different normalization S ∗ S = −
12 ( dZ αA dZ Aα + 4 dZ ν k α dZ αν k B ∗ κ ) . (6.24) κ is the product of the inner Klein operator κ and the outer Klein operator k that anticom-mutes with all ν k components but commutes with the ν ⊥ Lorentz components.The nontrivially deformed part of (6.24) is of the B type. To extend the sp (2) algebraof automorphisms of the B Cherednik algebra to the full sp (2) rotations of ( ν, α ) with any21 , for ν = d the action of sp (2) is generated by the sp (2) of undeformed oscillators. The full sp (2) generators are t αβ := − N X n =1 { S nνα , S nµβ η νµ } ∗ . (6.25) t totαβ is the sp (2) generator acting on all indices α, β including those of differentials dZ να .In addition, in the model of [91] it is demanded that the generator t intαβ := t totαβ − t αβ (6.26)obeys relations d t intαβ − [ t intαβ , W ] ∗ = 0 , [ t intαβ , S ] ∗ = 0 , [ t intαβ , B ] ∗ = 0 (6.27)(that restricts the spectrum of dynamical fields to two-row rectangular Young diagrams of o ( d − , t intαβ ∗ f αβ or f αβ ∗ t intαβ have to be factored out, that restricts the spectrum of dynamical fields to tracelesstwo-row rectangular Young diagrams of o ( d − ,
2) appropriate for the description of genuinemassless HS fields in the unfolded formalism [132, 133].The Coxeter extension needs several species of oscillators. Here however is a subtletythat, as in the simplest B case of [91], the Lorentz components ν ⊥ of the AdS d vectorsshould be treated differently from the d + 1th components ν k . The latter are supposed to beaffected by the action of the Coxeter reflections while the former are not to preserve manifestLorentz covariance. (Note that the latter condition is needed in the Lorentz covariant setupbut may be relaxed in a more general situation.) Naively, this suggests that, to use theframed Coxeter algebra construction, one has to introduce different idempotent elements forthe d + 1th components of oscillators and the Lorentz ones. This is however not necessarysince Coxeter HS equations involve hatted Klein operators that obey (5.7) allowing to use o ( d − , Y ναn , Y µβm ] = δ nm ǫ αβ η νµ I n . (6.28)In terms of these oscillators, the o ( d − ,
2) algebra is realized by the generators T νµ = − ǫ αβ N X n =1 Y νn α Y µn β . (6.29)The sp (2) generators are t αβ := − η νµ N X n =1 { S νnα , S µnβ } ∗ . (6.30)The generators t int and t tot are defined as in the original model using (6.26).Analogously to the original HS model, one imposes the sp (2) invariance condition followedby the factorization condition t intαβ ∗ f αβ ∼ f αβ ∗ t intαβ ∼
0. An important remaining question is22hether this is sufficient to remove all unnecessary states from the general Coxeter HS modelmaking it unitary. The natural choice of the symmetric reduction is with the symmetricgroup S p that acts on all indices n of Y An , Z An and dZ An irrespectively of the value of A and/or S N in the multi-particle extension.Finally let us note that, in agreement with the analysis of [134], it is natural to expectthe existence of another HS model in d dimensions which is holographically dual to freeconformal fermions on the boundary, as well as the further supersymmetric extension of thebosonic and fermionic models. Though the explicit form of the nonlinear field equationswas not presented so far, we anticipate that it can be elaborated along the lines of [91].(Recently, some low-order results in the construction of the fermionic model were presentedin [135].) Such models will also generate a class of Coxeter HS theories in d dimensions. HS theories admit an important extension to higher differential forms along the lines of [136].To this end, it is convenient to unify the dZ – and dx –one-forms into the master field W = d + dx n W n ( Z ; Y ; ˆ K | x ) + dZ A S A ( Z ; Y ; ˆ K | x ) , d = dx n ∂∂x n , (7.1)extending it further to all higher differential forms of odd total degrees with respect to both dZ and dx , W = X p =1 , ,... W p . (7.2)Analogously, the zero-form master field B is extended to the field B containing differentialforms of all total even degrees B = X p =0 , ,... B p , B = B . (7.3)In these terms the appropriately extended system (6.6)-(6.10) takes the form
W ∗ W = − i (cid:16) dZ An dZ An + F ∗ ( B , γ i ) (cid:17) , (7.4)[ W , B ] ∗ = 0 , (7.5)where γ i = X v ∈R i dZ αn v n dZ α m v m ( v, v ) ∗ ˆ κ v . (7.6)It is important that elements γ i (7.6) are central, commuting with any element of the algebra.Together with (7.5) this guarantees formal consistency of the system in the sense that (7.4)is compatible with the associativity of the star product[ W ∗ W , W ] ∗ = 0 , (7.7)23nd the consequence of (7.5) [ W ∗ W , B ] ∗ = 0 . (7.8)In the sector of one-forms W = W, S and zero-forms B = B this system with F ∗ ( B , γ i )linear in γ i reproduces (6.6)-(6.10). Higher degrees of two-forms γ i contribute to equationsfor the higher forms and are important for the construction of invariants as discussed below.The construction of equations (7.4), (7.5) applies both to irreducible Coxeter systemsand to the reducible ones, containing their direct products. In the latter case the label i of γ i encodes both the conjugacy classes of the irreducible Coxeter systems and differentfactors in the direct product. This corresponds to the extension i → i, a . To impose oneor another symmetric reduction, F ∗ ( B , γ i ) should be invariant under the chosen symmetrygroup S . The appearance of higher differential forms in the model is important both for itsstring-like interpretation since string theory is known to contain higher degree differentialforms and for the construction of invariants along the lines of [136].Invariant functionals are associated with various x –space forms L that are closed as aconsequence of field equations (7.4), (7.5). As shown in [136], these can be associated withvarious dZ -independent central elements of the star-product algebra. In the case of usual HStheories considered in [136] the only relevant central element was unity of the star-productalgebra. In the Coxeter HS theories of this paper the dZ -independent center of the algebrais generated by the elements I an . In the simplest additive case equation (7.4) is modified to W ∗ W = − i (cid:0) dZ An dZ An + F ∗ ( B , γ i ) (cid:1) + L (7.9)where L := X n ,...,n k ,a ,...,a k I a n ∗ . . . ∗ I a k n k L n ,...,n k a ,...,a k ( x ) (cid:17) (7.10)obeys d L n ,...,n k a ,...,a k ( x ) = 0 . (7.11)The density forms L n ,...,n k a ,...,a k ( x ) are symmetric under the exchange of pairs of indices n l , a l L ...n n ,...,n m ,......,a n ,...,a m ,... ( x ) = L ...n m ,...,n n ,......,a m ,...,a n ,... ( x ) (7.12)and contain differential forms of any positive even degree L = L + L + L + . . . (7.13)generating invariant functionals as integrals over space-time cycles of appropriate dimension S = Z Σ L . (7.14)Let us stress that (7.11) is a consequence of (7.9), i.e., apart from imposing field equationson dynamical fields, Eq. (7.9) expresses L via dynamical fields themselves in such a way that L turns out to be closed. 24ystem (7.9), (7.11) and (7.5) is invariant under the following gauge transformations withthree types of gauge parameters ε , ξ and χ [136]: δ W = [ W , ε ] ∗ + ξ N ∂ F ∗ ( B , γ ) ∂ B N + χ i ∂ L ∂ L i , (7.15) δ B = {W , ξ } ∗ + [ B , ε ] ∗ , (7.16) δ L i = d χ i , (7.17)where N is the multiindex running over all components of B , the gauge parameters ε and ξ are differential forms of even and odd degrees, respectively, being otherwise arbitraryfunctions of x and the generating elements of the star-product algebra, while χ i only dependon x and dx . By (7.17), the functional S (7.14) is gauge invariant.Generalization of this construction to the systems with non-additive contribution of theLagrangian forms L is also possible and can be constructed along the lines of [136]. As usual in HS theory [5, 6], one can let all fields be valued in a matrix algebra, i.e., carryChan-Paton-like [137, 138] matrix indices. Upon imposing appropriate reality conditions therespective HS models possess U ( n ) gauge symmetry in the spin-one sector. Following theconstruction of [139, 8] (see also [6]) the resulting models can be truncated to those with O ( n ) and Sp (2 m ) spin-one gauge groups. (For instance O (1)-model is the so-called minimalHS model that only contains even spins.)The local symmetry algebras of different types of 4 d HS systems of [5] are hu ( n, m | ho ( n, m |
8) and husp ( n, m | Analogously, local symmetry algebras of different types of 3 d HS systems are hu αβ ( n, m | ho αβ ( n, m |
4) and husp αβ ( n, m |
4) ( α, β = 0 , α + β = 0) [8].In the both cases the meaning of these notations is that the master fields are valued inappropriate ( n + m ) × ( n + m ) matrices in which bosons belong to the diagonal n × n and m × m blocks while fermions are in the two off-diagonal n × m blocks. The respective HSmodels possess usual supersymmetry at n = m .For multi-particle HS theories one can use algebras M ( h... ( n, m | l )). Also one can addmatrix indices to M ( h... ( n, m | l )) in the end which option seems to be less interesting,however.One proceeds analogously in the rank- p Coxeter HS models. The respective algebraswill be called C h ··· ( n, m | l ) with C denoting a Coxeter group in question. An interestingoption available in this case is to let the fields W and B be valued in the matrix algebra( × M at N + M ) p , e.g., C u ...u p v ...v p ( Y n ), u i , v i = 1 . . . N + M with the idea to assign any pair ofmatrix indices u n , v n to the variable Y n . From the AdS/CF T perspective, indices u i with These should not be confused with the 4 d global symmetry algebras hu ( n, m | ho ( n, m |
4) and husp ( n, m |
4) of [139]. The difference expresses the doubling of spinor variables Y → Y, Z in the non-linealsystem compared to the Z -independent dynamical fields like ω ( Y ; K | x ). = 1 , . . . N and i = N + 1 , . . . N + M can be associated with the boundary scalar and spinorfields, respectively. More generally, one can consider multi-indexed fields with indices u i and v i associated with rank– p tensors obeying various symmetry and/or tracelessness conditions.Still, the respective C h ··· ( n, m | l ) Coxeter HS theories make sense though the relation of n and m with N and M is more complicated.Usually, indices of this type are not considered in the bulk HS models. The rationale be-hind this is that the singlet contributions due to the field C u ...u p v ...v p ( Y n ) ∼ δ u v . . . δ u p v p C ( Y n )decouple from the traceful components, being in a certain sense dominating. This happensbecause invariants of the color symmetry cannot source components transforming nontriv-ially under the color group. Careful analysis of this phenomenon may be important forunderstanding specificities of the ( O ( N )) p -models associated with boundary fields beingsymmetric and/or traceless with respect to color indices (as well as their unitary and sym-plectic analogues) in which case the respective factors δ u v . . . δ u p v p have to be appropriatelysymmetrized and/or projected to the traceless components within the sets of indices u . . . u p and v . . . v p . In this section we propose a further extension of the construction of HS equations allowingto unify brane-like systems of different space-time dimensions in the same HS model. Thisconstruction is associated with idempotents (projectors) of the star-product algebra under-lying the system in question. Usually, these are Fock idempotents. The idea is illustratedby the fact that the Fock module of the 4 d HS algebra describes 3 d conformal fields [113]associated with the boundary fields in the AdS /CF T correspondence [92]. Let us stressthat in presence of unbroken HS symmetries, the lower-dimensional branes of this type turnout to be delocalized, i.e., reduction of space-time dimension is via a projective identificationof astigmatic type not allowing to observe certain directions. However, if the HS symmetriesare broken (e.g. by boundary conditions), the HS branes can become space-time localized. The general construction is as follows. Let A be some associative algebra with the product ∗ and π i ∈ A be a set of idempotents π i ∗ π i = π i . (9.1)Let A ij ⊂ A be subspaces of A spanned by the elements of the form a ij ∈ A ij : a ij = π i ∗ a ∗ π j , a ∈ A . (9.2)The composition law is of the matrix-like form which, in turn, is a particular case of thisconstruction ( a ∗ b ) ij = X k a ik ∗ b kj . (9.3)26ote that in this construction the idempotents π i need not be orthogonal. The only impor-tant condition is (9.1).The resulting idempotent-extended algebra A { π } is associative and unital with the unitelement Id ij = δ ji π i . (9.4)If the algebra A was unital it can itself be realized as a particular case of the above con-struction with π = Id .A particular case of this scheme with two idempotents was used in [140] for the formula-tion of a d = 2 HS theory. Here we propose its slightly different application directly to thenonlinear equations of Sections 6, 7. In HS theory, A is the algebra of functions of dx, dZ, Z, Y, κ, x with the star product (6.23)underlying the construction of Section 6. Idempotents π i can be identified with the Z -independent Fock projectors of the star-product algebra. We conjecture that exact holo-graphic correspondence can take place for the systems which correspond to consistent non-linear equations containing both bulk and boundary fields via the idempotent constructionof this section. For instance, in the 4 d HS theory π i = 4 I i exp y iα ¯ y αi (9.5)are such idempotents, i.e., π i ∗ π i = π i (9.6)(note that here dotted and undotted indices are on equal footing). π i can indeed be inter-preted as Fock idempotents since( y iα − i ¯ y iα ) ∗ π i = 0 , π i ∗ ( y iα + i ¯ y iα ) = 0 . (9.7)The idempotent-extended system has the same form as the systems of Sections 6, 7 withthe replacement of the original algebra A by A { π } . The unit element of the star-productalgebra is considered as one of the idempotents, say, π . Note that with this convention π isnot orthogonal to π i . Luckily, as emphasized in the previous section, they do not need to beorthogonal for the applicability of the general scheme. The important consistency conditionis that the full set of idempotents has to to be invariant under the Coxeter group in thesystem and the mutual products π i ∗ π j as well as the elements (9.2) should be well-defined.In the original HS system, the vacuum solution (6.13) contains S linear in Z thatsolves (6.10) because Z A obey Heisenberg commutation relations (2.10). In the idempotent-extended version of the Coxeter HS equations, the respective vacuum solution is S = X i π i ∗ dZ An Z An ∗ π i , (9.8)27here the condition that all π i are Z -independent and hence ∗ -commute with Z An impliesthat so defined S solves (7.9) with the unit element (9.4).If HS fields carry matrix indices, the star-product idempotents π stari can be accompaniedby those in color indices π colori π i = π stari π colori . (9.9)The simplest idempotent π colori is given by the diagonal matrix with a single unit elementon the diagonal, like δ u δ v . In that case the left and right modules in this sector can beassociated with the space of rows and columns with respect to the matrix algebra. Thisleads to the vector-like fields in A i and A i .The physical interpretation of the proposed construction is that the fields valued in newidempotent sectors of A { π } live in lower space-time dimensions compared to those valuedin A . This is so since the HS modules supported by A ij -modules are smaller ( i.e., havesmaller Gelfand-Kirillov dimension) than those of the twisted adjoint A -module supportingdynamical fields in the original system. For instance, the 4 d twisted adjoint module describes4 d massless fields of all spins while the A i -module (9.2), (9.5) describes 3 d conformal fields( i.e., d singletons) [113]. This suggests that the idempotent extensions of the Coxeter HSsystems are appropriate to describe lower-dimensional objects analogous to branes in StringTheory.From the AdS/CF T perspective this construction makes it possible to describe in thesame model both fundamental boundary fields as valued in the A i and A i sectors andcomposite current-like operators as fundamental fields in the twisted adjoint A -module. Letus stress again that the simplest extension of the idempotent construction to the matrixsector gives vector-like boundary fields.The condition that the set of idempotents { π i } is invariant under the action of the Coxetergroup C restricts significantly the brane (in particular, boundary) sectors. Since the Coxetergroup action exchanges different arguments Y and Z the condition is that all idempotents π stari that belong to the same orbit of C should be multiplied by the same color idempotent, i.e., π colori = π colori ( O C ( π stari )), where π colori ( O C ( π stari )) depends on the C− orbits of π stari -idempotents. For instance, if there is a C -invariant π stari -idempotent, it can be multiplied byany π colori . On the other hand, if there are many different idempotents related by the actionof C , they should all be multiplied by the same color idempotent.The latter situation is important in the context of usual HS holography. In this caseidempotents π stari associated with the boundary tensor fields have to be of the form π stari = Π i π color , (9.10)where Π i is the Fock projector in the sector of Y i oscillators and π color has to be the samefor any π stari . The most natural option is to choose π color as the rank-one idempotent δ u δ v . . . δ u p δ v p . (9.11)Here all indices u . . . u p and v . . . v p have to take the same number of values N . Corre-spondingly, the boundary fields are scalars or spinors of the type ϕ u ...u p ( x ) with all indicestaking N values. 28 .2.2 N = 4 SUSY
The original and most important example of holographic correspondence is the duality ofSuperstring Theory with the most supersymmetric N = 4 SYM theory [1]. Here we brieflydiscuss how this correspondence can be interpreted from the perspective of the proposalof this paper suggesting that string-like HS theories are associated with the Coxeter group B (see also Section 12). Remarkably, the case of N = 4 supersymmetry turns out to bedistinguished from this perspective as well.Our consideration is based on the unfolded formulation of the linearized N = 4 SYMtheory proposed in [114, 141] where it was shown that 4 d conformal massless fields can bedescribed by the fields valued in certain Fock modules. Let the Fock vacuum π be definedby the relations a α ∗ π = 0 , ¯ b ˙ β ∗ π = 0 , φ i ∗ π = 0 , (9.12) π ∗ ¯ a ˙ α = 0 , π ∗ b α = 0 , π ∗ ¯ φ i = 0 (9.13)with respect to the oscillators obeying the following non-zero relations[ a α , b β ] ∗ = δ βα , [¯ a ˙ γ , ¯ b ˙ β ] ∗ = δ ˙ β ˙ γ , { φ i , ¯ φ j } ∗ = δ ij , (9.14) i, j = 1 , . . . N . Bilinears of these oscillators form su (2 , N ) which therefore acts on the Fockmodule where 4 d massless fields are valued. In fact, in [141] it was shown that, to have aroom for HS potentials for massless fields, one should introduce four Fock modules generatedfrom the Fock vacua π p ¯ p ( p, ¯ p = 0 ,
1) obeying relations analogous to (9.12) with exchangedroles of the creation and annihilation operators a α , b β and ¯ a ˙ α , ¯ b ˙ β . For instance, b α ∗ π = 0 , ¯ a ˙ β ∗ π = 0 , ¯ φ i ∗ π = 0 , (9.15) π ∗ ¯ b ˙ α = 0 , π ∗ a α = 0 , π ∗ φ i = 0 . (9.16)The Clifford oscillators can be interpreted as generating the color matrix algebra M at N where all fields are valued. Let us now explain how this construction appears in the frame-work of the B -HS model and why the case of N = 4 is distinguished.The B -HS theory contains a pair of oscillators y iα , ¯ y i ˙ α . This allows us to build oscillatorsanalogous to (9.7) a α = y α + iy α , b α = 14 i ( y α − iy α ) , ¯ a ˙ α = ¯ y α − i ¯ y α , ¯ b ˙ α = 14 i (¯ y α + i ¯ y α ) (9.17)defining vacuum π (9.12). More precisely, since the commutation relations (6.3) contain I n , to obey (9.12), (9.13) and (9.1) the idempotent π should contain the factor of I ∗ I .The fields valued in the left and right modules generated from this vacuum describe 4 d massless conformal fields. To describe supermultiplets the vacuum π has to be combinedwith the Clifford vacuum also obeying (9.12), (9.13) [114].Although, naively, this construction relates the B –HS theory to the massless conformalsupermultiplets at any N , this is not quite the case. The problem is that the idempotent29 defined this way is not invariant under the B reflection exchanging Y A with Y A orchanging a sign of one of them. Indeed, such reflections map idempotent π to the oppositeidempotent π . In fact, the condition that the set of idempotents has to be invariant underthe action of the Coxeter group B demands all π p ¯ p be present in the model.The problem however is that opposite idempotents may have ill-defined mutual starproduct so that elements π ∗ a ∗ π (9.2) are ill defined. Namely, for purely bosonicopposite idempotents of this type, i.e., at N = 0, π ∗ π is divergent. On the otherhand, in the purely Clifford case the product of opposite Fock idempotents is zero. In thesupersymmetric case the bosonic and fermionic contributions to the vanishing determinantsin the respective Gaussian integrals resulting from the star product of the idempotents haveopposite signs. The full compensation of the bosonic divergency occurs just at N = 4 whenthe numbers of bosonic and fermionic oscillators are equal. (Note that for a similar reason,at N = 4 the conformal SUSY is psu (2 ,
2; 4).)Thus the consistent rank-two conformal boundary system in the B –HS theory musthave N ≥
4. The modules generated from π p ¯ p describe 4 d conformal massless fields (su-permultiplets) of all spins [114, 141]. It is natural to conjecture that all of them except forthose in the lowest-spin N = 4 SYM supermultiplet will become massive upon spontaneousbreakdown of HS symmetries in the multi-particle B –HS theory. Note that simultaneouslythe non-zero string tension should result from the spontaneous breaking of HS symmetriesAt the condition that the resulting theory is free from massless fields of spins s ≥
2, theonly remaining option is the N = 4 SYM multiplet. This is the case of N = 4 SYM which isthe only N = 4 massless conformal system with spins s ≤
1. Thus, a spontaneously broken B -HS theory with non-zero string tension has a chance to be dual to N = 4 SYM.To have a nontrivial gauge group for the N = 4 SYM boundary theory of the spon-taneously broken HS theory one has to introduce color indices additional to the Cliffordalgebra. This can be achieved either directly by letting all fields be valued in the matrix al-gebra (and its further orthogonal/symplectic reductions) or via the construction used for thevector models introducing the fields C u u v v ( Y ) and using the additional color idempotent π col = δ u u δ v v δ u v . (9.18)
10 Unitarity
One of the advantages of the unfolded formulation of dynamical equations is that it makessymmetries manifest operating directly with modules of the symmetry h underlying themodel in question. In particular, linearized equations on the zero-forms in the system havea form of covariant constancy conditions D C I ( x ) = 0 (10.1)with C I ( x ) valued in some h -module V , and D being a flat covariant derivative of h .Though V is not unitary in the Lorentz-covariant frame, the condition that the systemadmits consistent quantization compatible with unitarity demands V be complex equivalent30o some unitary h -module U [114]. This makes it straightforward to analyze the pattern ofone or another unfolded system in terms of space-time symmetry algebra modules includingthe issue of unitarity.Naively, the proposed Coxeter HS systems contain non-unitary sectors associated withthe tensor products of the unitary HS modules with the so-called topological modules of theoriginal HS theory, that describe non-unitary finite-dimensional modules of the AdS algebra.For instance, in the 4 d B -HS theory, consider a rank-two field C ( Y , k ; Y , k ) := C , ( Y , Y ) k proportional to the Klein operator k of the second type and independentof k . The covariant derivative with respect to AdS background connection contains com-mutator with the frame field h ( Y ) with respect to the first argument and anticommutatorwith respect to the second. This is manifested in the form of covariant derivative D ( C , ( Y , Y )) = D L C , ( Y , Y ) − h α ˙ β y α ∂∂ ¯ y ˙ β + ∂∂y α ¯ y β − iy α ¯ y β + i ∂ ∂y α ∂ ¯ y ˙ β ! C , ( Y , Y )(10.2)with the Lorentz covariant derivative D L A = d − X i =1 , ω αβ y iα ∂∂y βi + ¯ ω ˙ α ˙ β ¯ y i ˙ α ∂∂ ¯ y ˙ βi ! . (10.3)The first term in (10.2) acts on homogeneous polynomials of Y of any definite degree whilethe second mixes polynomials of different degrees of Y . This structure is tantamount tothe fact that C , ( Y , Y ) is valued in the tensor product of the adjoint module of the HSalgebra with respect to Y and twisted adjoint module with respect to Y . The twistedadjoint module and its tensor products correspond to unitary particle-like states [142] andtheir multi-particle tensor products, respectively.Having a form of the infinite sum of finite-dimensional modules, the zero-form fieldsvalued in the adjoint module or even containing it as a factor as in (10.2) cannot correspondto a unitary particle-like representation. The only exception is when the field C , ( Y , Y ) isindependent of Y . Then it contains the factor of I and, in accordance with the discussionof Section 6, corresponds to unitary massless states of the usual HS theory in the sector ofvariables Y .Naively, non-singlet states in the adjoint module factors break down unitarity of thesystem. However, these potentially non-unitary states can be consistently truncated away.Indeed, consider for instance product of the factors C , ( Y , Y ) k . There are two options.Either the arguments Y , k and Y , k are permuted, in which case one ends up with theunitary rank-two field, or they are not permuted. In the latter case, the product will stillcontain the singlet component I in the adjoint factor describing nonlinear corrections tousual massless HS equations, i.e., non-singlet elements of the adjoint factor are never gener-ated. Note that, to account properly the contribution of the Klein operators, their explicitappearance in Eq. (7.4) via the factors of γ i should be taken into account.In the original HS theory topological fields were interpreted as moduli (coupling con-stants) rather than propagating degrees of freedom. Similarly, in the Coxeter HS theories,31he fields associated with non-unitary sectors resulting from the tensor products of unitaryand topological modules should be interpreted as non-propagating by imposing appropri-ate boundary conditions in the topological sector somewhat analogously to the analysis ofconformal gravity in [143]. On the other hand assigning non-zero VEVs to the fields in po-tentially non-unitary sectors may significantly affect the interpretation of the bulk HS theorysimilarly to the 3 d HS model of [8] in which the value of the singlet topological field is inter-preted as the mass parameter in the theory. Analogous suggestion for the mass-generatingspontaneous breaking mechanism in multi-particle HS theories is discussed in Section 12.The question whether the non-unitary fluctuations can be consistently truncated away inpresence of non-zero VEVs of topological fields remains to be investigated.A related comment is that the issue of whether a HS model is unitary or not may dependon the choice of reality conditions and vacuum solution. For instance, in the sector of zero-forms C of the 4 d HS theory, condition (6.19) just implies that, from the perspective of the3 d model, the field C is valued in the tensor product of (complexified) 3 d particle modulewith the topological one. Nonetheless, the resulting module of the 4 d HS model turns out tobe unitary due to imposing appropriate reality conditions compatible with the 4 d vacuumsolution.To summarize, to make the proposed models unitary, potentially dangerous fields con-taining products with non-singlet factors of the adjoint HS module have to be truncatedaway. Remarkably, this is possible in all orders so that the reduced model turns out to beunitary. This factorization is somewhat analogous to that of non-singlet color sectors in theHS theory with colorful fields.Let us stress that the HS models dual to the free tensor boundary theories with O ( N p )broken to O ( N ) at the level of observables have to be unitary [78]. This condition has littleto do with potential non-unitarity of interacting tensor models with ill-defined potentials.On the other hand, it would be interesting to see whether the non-unitary boundary tensormodels can be dual to B p -HS theories with non-unitary bulk fields switched on.
11 Field frames of multi-particle theories
In this section we recall the construction of [94] for the multi-particle algebra extending itto the graded-symmetric case and analysis of idempotents.
Let A be an associative algebra with the product law ∗ and basis elements t i obeying t I ∗ t J = f KIJ t K , f KIJ f NKL = f NIK f KJL . (11.1)In the Coxeter HS case, A is (matrix-valued extension of) the algebra of functions of I i , Z Ai , Y Ai , K v , dZ Ai as well as of π i for the idempotent extension of Section 9 (space-timedifferential dx n is not included). 32s a linear space, multi-particle algebra M ( A ) is the direct sum of all graded-symmetrictensor degrees of A M ( A ) = ∞ X n =0 ⊕ Sym A ⊗ . . . ⊗ A | {z } n , (11.2)where the Z -grading is identified with the power in Z Ai and Y Ai , f ( − Z Ai , − Y Ai ) = ( − π f f ( Z Ai , Y Ai ) , (11.3) i.e., odd polynomials in Z Ai and Y Ai are antisymmetrized under the symbol Sym in (11.2).Let us stress that this does not contradict to the fact that Z Ai and Y Ai obey star-productcommutation relations in A . So defined graded symmetrization accounting the total numberof spinorial indices carried by Z Ai and Y Ai expresses the Pauli principle. (Note that in thevectorial model of [91] considered in Section 6.3 only integer-spin states contribute.)Let α I be odd (even) elements of the Grassmann algebra associated with the odd (even)elements t I , respectively, α I · α J = ( − π I π J α J · α I . (11.4)The product · denotes usual (juxtaposition) product in the Grassmann algebra of α intro-duced to distinguish between the graded symmetric tensor product · and the commutativeproduct in the definition of the star-product algebra underlying A of M ( A ). The latter isthe algebra of functions F · ( α ) with the product law F · ( α ) ◦ G · ( α ) = F · ( α ) · exp · (cid:16) ←− ∂∂α I f NIJ α N −→ ∂∂α J (cid:17) · G · ( α ) . (11.5)An elementary computation shows that associativity of A implies associativity of the product ◦ of M ( A ). (For the even case see [94].)Algebra M ( A ) is unital, with the unit element Id identified with F ( α ) = 1. The Z –grading of M ( A ) is induced by that of AF (( − π ( α ) α ) = ( − π ( F ) F ( α ) . (11.6)The graded commutativity (11.4) implies the graded symmetrization in (11.2).The power-series expansion of the functions F · ( α ) F · ( α ) = ∞ X n =0 F I ...I n · α I · . . . · α I n (11.7)will be assumed to have anticommuting coefficients F I ...I n F I ...I n · G J ...J m = ( − ( P nk =1 π Ik )( P ml =1 π Jl ) G J ...J m · F I ...I n . (11.8)The same time it is convenient to demand them to commute with α I F I ...I n · α J = α J · F I ...I n . (11.9)33ince components of F · ( α ) are identified with the physical fields in the multi-particle HStheory, convention (11.8) expresses the Pauli principle.Since A ⊂ M ( A ) is represented by linear functions of α I , the ∗ product acts on linearfunctions of α I according to α J ∗ α K = f MJK α M . (11.10)Note that A ◦ A does not belong to A . M ( A ) is isomorphic to the universal enveloping algebra U ( l ( A )), M ( A ) ∼ U ( l ( A )) , (11.11)where l ( A ) is the Lie superalgebra associated with A ,[ t I , t J ] ∗ = g KIJ t K , g KIJ = f KIJ − ( − π I π J f KJI . (11.12)This is because M ( A ) is isomorphic to U ( l ( A )) as a linear space and α I ◦ α J − ( − π I π J α J ◦ α I = g KIJ α K . (11.13)Concise form of the product law (11.5) is specific for the case of Lie superalgebras l = l ( A )associated with associative algebras A . The following useful property of M ( A ) is a simple consequence of Eq. (11.5) ∀ f, g ∈ A : exp · f ( α ) ◦ exp · g ( α ) = exp · ( f • g )( α ) , (11.14)where f • g := f + g + f ∗ g = ( f + e ∗ ) ∗ ( g + e ∗ ) − e ∗ ∈ A (11.15)and e ∗ is the unit element of A if the latter is unital (recall that f, g ∈ A implies that f ( α )and g ( α ) are linear in α ). Associativity of ∗ implies associativity of • even if A is not unital.Let G · ν = exp · ( ν ) ∈ M ( A ) , ν = ν I · α I , (11.16)where ν I are free Grassmann-odd (even) parameters for odd (even) α I , which however com-mute with α i ν I · ν J = ( − π I π j ν J · ν I , ν I · α J = α J · ν I . (11.17)Eq.(11.14) gives G · ν ◦ G · µ = G · ν • µ . (11.18)This formula is convenient for practical computations via differentiation over ν I with G · ν serving as the generating function for elements of M ( A ).34s any universal enveloping algebra M ( A ) is double filtered. Indeed, let V n be the linearspace of order- n polynomials of α I . From Eqs. (11.5), (11.13) it follows that F n ◦ F m ∈ V n + m , F n ◦ F m − ( − π ( F n ) π ( F m ) F m ◦ F n ∈ V n + m − , F m ∈ V m , F n ∈ V n . (11.19)A linear map of M ( A ) to itself can be represented by κ ( α, a )[ F · ] = κ ( α, −→ ∂∂α ) F · ( α ) , (11.20)where derivatives −→ ∂∂α act on F · ( α ). This formula can be interpreted as representing the actionof the normal-ordered oscillator algebra with the generating elements α I and ¯ α J acting on theFock module spanned by F ( α ) | i with ¯ α I | i = 0. To respect the double filtration propertymapping order- n polynomials to order- n polynomials, κ ( α, ¯ α ) should be filtered obeying κ ( α, ¯ α ) = ∞ X m,n =0 κ I ...I m J ...J n α I · . . . · α I m · ¯ α J · . . . · ¯ α J n , κ I ...I m J ...J n = 0 at m > n . (11.21)In these terms, the unit map is Id = 1 . Consider maps of the form U ( f ) ≡ κ ( α, ¯ α | f ) = φ exp · ( α I · f I · ( ¯ α )) (11.22)with some α -independent coefficients f I · ( ¯ α ) and constant φ . The map U ( f · ) is filtered if f I · ( ¯ α ) is at least linear in ¯ α , i.e., f I · (0) = 0 . (11.23)Interpreting ¯ α as parameters, we can identify any f ( α ) := P I f I α I with f ( t ) = P I f I t I ∈ A . For U ( f ) (11.22) acting on G · ν (11.16) we obtain U ( f )( G · ν ) = exp · ( ˜ f I ( ν ) α I ) , ˜ f I ( ν ) = ν I + f I ( ν ) . (11.24)Hence, Eq. (11.14) gives U ( f )( G · ν ) ◦ U ( g )( G · µ ) = exp · (( ˜ f ( ν ) • ˜ g ( µ ))( α )) , (11.25)where ˜ f ( ν ) and ˜ g ( µ ) are interpreted as elements of A , i.e., ˜ f ( ν ) = ˜ f I ( ν ) α I . Consider map (11.22) with f I ( ¯ α ) α I = f ( a n ) , a n := α I ∗ . . . ∗ α I n · ¯ α I · . . . · ¯ α I n , a = a = α I · ¯ α I , a = e ∗ , (11.26)where f ( a n ) is a linear function of a n ( n ≥ a n ∈ A .A particularly important class of maps (11.22) is represented by U u of the form U u ( a ) =: exp[ u ( a ) − a ] : , (11.27)35here normal ordering is with respect to α I and ¯ α I and for a ∈ Au ( a ) = ( u a + u e ∗ ) ∗ ( u a + u e ∗ ) − ∗ , ( e ∗ + βa ) − ∗ := ∞ X n =0 ( − β ) n a n (11.28)with u ij ∈ R . Composition of such maps gives a map of the same class U u U v = U uv , (11.29)where ( uv ) ij = u ik v kj is the matrix product. The maps U u with det | u | 6 = 0 are invertibleand form Mobius group.From Eq. (11.24) it follows that U u ( G · ν ) = G · u ( ν ) . (11.30)To find the composition law of M ( A ) in the frame T uI ...I n associated with G · u ( ν ) via T uI ...I n = ∂ n ∂ν I . . . ∂ν I n G · u ( ν ) (cid:12)(cid:12)(cid:12) ν =0 (11.31)we compute G · ν ⋄ G · µ := U − u ( G · u ( ν ) ◦ G · u ( µ ) ) . (11.32)Eq. (11.15) gives G · ν ⋄ G · µ = G · u − ( u ( ν ) • u ( µ )) . (11.33)General map (11.27) is not filtered, not respecting condition (11.23). The subgroup P offiltered maps (11.27) is represented by the lower triangular matrices u b,β ( f ) = bf ∗ ( e ∗ + βf ) − ∗ (11.34)with the composition law b , = b b , β , = β + β b . (11.35)Using for these transformations notation U b,β instead of U u we observe that the unitelement is Id = U , (11.36)and U − b,β = U b − , − βb − . (11.37)The map R = U − , (11.38)is involutive R = Id (11.39)describing the principal antiautomorphism of M ( A ) [94].36or the maps (11.34), the composition law (11.33) takes the form [94] G · ν ⋄ G · µ = G · σ b,β ( ν,µ ) , (11.40)where σ b,β ( ν, µ ) = − β − ( e ∗ − ( e ∗ + βµ ) ∗ ( e ∗ − β ( b + β ) ν ∗ µ ) − ∗ ( e ∗ + βν )) . (11.41)Three most important cases include σ , ( ν, µ ) = ν + µ + ν ∗ µ = ν • µ , (11.42) σ − , ( ν, µ ) = ν + µ + µ ∗ ν = µ • ν , (11.43) σ , − ( ν, µ ) = 2( e ∗ − (2 e ∗ − µ ) ∗ (4 e ∗ + ν ∗ µ ) − ∗ ∗ (2 e ∗ − ν )) . (11.44)Here σ , ( ν, µ ) corresponds to the identity map reproducing the basis in M ( A ) resultingjust from the symmetrized tensor product of the framed oscillator algebras. σ − , ( ν, µ )corresponds to the basis resulting from the action of principal antiautomorphism R . Thecase of σ , − ( ν, µ ) is most interesting, reproducing the current operator algebra of [144].Practically, the difference between the frames (11.42) and (11.44) is as follows. Theproduct law associated with (11.42) is the original product law (11.4), from which it iseasy to see that the product ◦ of two polynomials of α of degrees n and m gives degree- k polynomials with max ( n, m ) ≤ k ≤ n + m . This implies that, with the product law (11.4),the higher-rank fields (say, rank-two) will not contribute to the equations for the lower-rankones (say, rank-one). This property is inconsistent both with structure of the boundaryoperator algebra analyzed in [144] and with the idea that VEVs of the higher-rank fields canmodify field equation for the lower-rank ones to yield a Higgs phenomenon. The productlaw associated with (11.43) has analogous properties.However, in [94] it was shown that the multi-particle algebra in the frame (11.44) properlyreproduces the boundary operator algebra inducing such a product law that the product oftwo polynomials of α of degrees n and m gives degree- k polynomials with | m − n | ≤ k ≤ n + m which is consistent both with the structure of the boundary operator algebra of [144] andwith the idea of higgsing by virtue of VEVs of higher-rank fields. Hence, we anticipate thatframe (11.44) is most appropriate for the analysis of the multi-particle Coxeter HS theories.Let us stress that since the multi-particle algebra is infinite dimensional it is not a priori guaranteed that the multi-particle HS theories associated with different frames described inthis section are physically equivalent.In application to nonlinear equations of the multi-particle HS theory, system (7.4), (7.5)preserves its form with the ∗ replaced by ⋄W ⋄ W = − i (cid:16) dZ An dZ An + F ⋄ ( B , γ i ) (cid:17) , [ W , B ] ⋄ = 0 , (11.45)where ⋄ is built via (11.44) from the star product of the Coxeter HS theory underlying itsmulti-particle extension. 37 Multi-particle algebra has an important property that every idempotent π ∈ A induces anidempotent Π ∈ M ( A ). This immediately follows from formulae (11.14), (11.15) withΠ := exp · − π , (11.46) i.e., Π ◦ Π = Π if π ∗ π = π . In particular, application of this construction to π = e givesthe unity idempotent of M ( A ) Π e := exp · − e . (11.47)Note that being built from the central element e , Π e is central in M ( A ).Unity idempotent Π e has an interesting property that it is ◦ -orthogonal to any a ∈ A Π e ◦ a = a ◦ Π e = 0 , a ∈ A , (11.48)which is obvious from (11.5) and (11.47). Analogously one can check that( a · Π e ) ◦ b = ( a ∗ b ) · Π e , ∀ a ∈ A (11.49)and a · Π e ◦ b · Π e = a ∗ b · Π e , ∀ a, b ∈ A . (11.50)This relation provides a homomorphism of A to M ( A ). It should be stressed however thatthis map is not polynomial. (No homomorphism of A to M ( A ) realized as polynomialfunctions of A exists.)Relation (11.50) has the consequence thatΠ e := e · Π e (11.51)is also an idempotent Π e ◦ Π e = Π e . (11.52)From (11.49) it follows Π e ◦ a = a · Π e , ∀ a ∈ A . (11.53)The consequence of this construction is that if A had a Lie (super)algebra l associatedwith some t i ∈ A , then M ( A ) admits a symmetry algebra l ⊕ l generated by t i and t i · Π e with t i generating the diagonal subalgebra l ⊂ l ⊕ l . This fact may have important implicationsfor the space-time interpretation of the string-like HS theories discussed in the next section.
12 Interpretation
Coxeter HS equations and their multi-particle extensions have a number of features indicatingtheir relation to string-like models and their further tensor-like extensions anticipated to beholographic duals of the boundary tensor sigma-models considered in [77, 78].38irst of all, the spectra of fields described by the rank- p Coxeter HS models with p > C ( Y nα ; k v ) depend on p copies of the oscillators Y nα as well as onthe Klein operators k v associated with all roots of the underlying Coxeter system (moduloidentification k − v = k v ). This enlargement of the spectrum is in qualitative agreement withthe observation of [78, 81] that the spectrum of the boundary operators in tensor boundarymodels is far richer than in the vector sigma-model. Most importantly, however, the Kleinoperators generating Coxeter reflections effectively permute the arguments of the elementarymaster fields like C ( Y , . . . Y p ; k | x ).In the absence of the Klein operators, products of the fields would correspond to the tensorproduct of p copies of star-product algebras valued in M at N + M . However, in the process ofsolving Coxeter HS equations, relations of the type (3.22) will permute the variables Y n and Y m with different n and m , not affecting the matrix indices a n , b n and a m , b m . For instance,in the case of p = 2, the star product of two master fields C ( Y , Y | x ) k gives( C ( Y , Y | x ) k ) ∗ ( C ( Y , Y | x ) k ) = C ( Y , Y | x ) ∗ C ( Y , Y | x ) . (12.1)As a result, nonlinear corrections to the p = 2 system will contain products of elementarystrings of master fields with repeatedly permuted arguments Y and Y C nstring := C ( Y , Y | x ) ∗ C ( Y , Y | x ) ∗ C ( Y , Y | x ) . . . | {z } n . (12.2)Such strings are analogous to the product of elementary matrix factors and can be identifiedwith the letters of an infinite Alphabet with n = 0 , , . . . ∞ . These are analogues of thesingle-trace operators in the ordinary AdS/CF T dictionary. General product of operatorsdecomposes into products of elementary letters analogous to the multi-trace operators. Forinstance, operator (12.1) as well as C ( Y , Y | x ) are single-trace while C ( Y , Y | x ) ∗ C ( Y , Y | x )is double-trace. Thus the spectrum of operators of the p = 2 HS model is analogous to thatof String Theory with the infinite set of Regge trajectories. More precisely, to interpretthese nonlinear combinations of operators as corresponding to elementary states of somestring-like model one has to consider a multi-particle HS model associated with the infiniteset of elementary B systems sym ( B × B × . . . ) , (12.3)where the graded symmetrization is with respect to all elementary B factors in the sensethat all respective master fields C ( Y , Y , k v ; Y , Y , k v ; Y , Y , k v ; . . . | x ) are demanded tobe graded symmetric under the exchange of the variables Y a , Y a , k av associated with differentfactors of B ( i.e., index a ).The p = 2 Coxeter HS theory has deep parallels with the analysis of stringy HS models byGaberdiel and Gopakumar [70]-[74]. In particular, the master fields of this theory dependon the two sets of oscillators Y A , which is close to saying that the stringy HS theory isbased on two different HS symmetry algebras being one of the conclusions of Gaberdiel andGopakumar. It should be stressed that these algebras do not commute with each other if the39tringy coupling constant identified with the vacuum expectation value of F ∗ ( B ) in (6.16) isnon-zero inducing nontrivial Cherednik-like deformation (5.10) of the oscillator commutationrelations. (Note that for conformal models considered by Gaberdiel and Gopakumar the VEVof F ∗ ( B ) must be non-zero.)Usual HS theory and its multi-particle extension result from the analogous constructionapplied to B . Rank- p tensor HS theories and their multi-particle extensions result analo-gously from B p with p > p > p = 2 string-like models which is in agreement with [78, 81].To make the holographic correspondence more explicit it is necessary to compare thespectrum of singlet boundary operators with that of the singlet sector of the tensor bulkmodel. As explained in Section 6, due to using framed algebras, these spectra do match atleast in the massless sector containing usual massless states dual to the boundary currents inthe holographic interpretation. The pattern of the full spectrum of the higher-rank CoxeterHS theories remains to be elaborated.Let us stress that the Coxeter HS theories proposed in this paper are formulated in theanti-de Sitter space of appropriate dimension. In particular, this is true for the stringy B -HS models. Hence the construction of this paper is different from that of genuine StringTheory formulated in the flat rather than, say, AdS space. In fact, the reason why itis difficult to formulate String Theory in AdS d is analogous to that discussed in Section 4for HS theory: a naive attempt to deform the string spectrum to AdS would immediatelylead to infinite vacuum energy since in this case all string modes have to contribute to themomentum generators to ensure that their commutator is proportional to the Lorentz ones[ P n , P m ] ∼ − Λ M nm (12.4)that act on all modes. (Recall that the usual string theory momentum operator is builtfrom the zero modes [19] which is not possible in AdS .) Hence, the extension to the framedoscillator algebra can also be crucial to reach a string-like theory in
AdS .An important related feature of the B p –HS models with p ≥
2, is that they have twoindependent coupling constants instead of one in the usual B -HS theories. These are thecoefficients η , in the linearized parts of the functions F , ∗ ( B ) in (6.12). The function F ∗ is analogous to that of the rank-one ( i.e., B ) HS theory. The respective term is importantfor the proper interpretation of the equations in AdS leading to the so-called central on-shell theorem of [142] representing the unfolded equations for massless fields in AdS . Thefunction F ∗ first appears in the rank-two stringy model and, containing the Klein operatorsthat permute different species of Y -variables, is responsible for the appearance of single-trace-like strings of operators. The presence of two different coupling constants is anticipated tohave important implications for establishing relation with usual string theory in flat space.One option is that to reach the latter theory one has to take the limit with η /η → ∞ whichin turn may select String Theory in critical dimension.It should be stressed that it is a distinguishing property of the B p ∼ C p Coxeter groupthat the related Cherednik system has two types of coupling constants responsible for HS40nd stringy effects. It is not clear, in particular, whether the A p and D p systems admit ameaningful HS interpretation since, due to the absence of usual Klein operators reflectingsigns of different species of oscillators Y iA , the free field equations unlikely have a room formassless fields.The full-fledged string theory is conjectured to be related to a rank-two multi-particle B Coxeter HS model. As explained in Section 11 following [94], the form of the multi-particlealgebra significantly depends on the chosen basis. It remains to be analysed to which extentthe formulations in different frames are equivalent to each other. More precisely, differentframes of the Coxeter HS models of finite rank are equivalent. However, for the multi-particle extensions the respective frame changes are in the infinite-dimensional space anddifferent frames may not be equivalent. The details of description of multi-particle algebrasin different frames are presented in Section 11 where the construction of [94] is extended tothe graded-symmetric case expressing the Pauli principle.This question may be related to the fundamental issue of the breaking of HS symmetriesin the Coxeter HS theory. Indeed, it is plausible to expect [145, 146] that spontaneousbreaking of HS symmetries resulting in the appearance of massive HS fields is only possiblein string-like models with the infinite number of Regge trajectories. As conjectured in thispaper the appropriate Coxeter HS theory is the B multi-particle theory. The goal is tobreak down HS symmetries to usual space-time symmetry of AdS or Minkowski type. Thesimplest way to do so is to let a rank-two zero-form topological field B ( Y ; Y ) acquire anon-zero VEV B = Y iA · Y Aj ( γσ ij + ρδ ij ) , (12.5)where the Pauli matrix σ ij and δ ij are the two symmetric matrices invariant under theexchange 1 ↔ i, j . (Note that this proposal is somewhat reminiscent of the Girardello-Porrati-Zaffaroni mechanism [147].) Such B preserves the AdS symmetry but breaks downthe HS one. Note that so defined B is nonzero because elementary oscillators Y iA areanticommuting with respect to the product · (11.4). It remains to be seen how this VEVwould affect (deform) the structure of the AdS modules (field equations) of the originallymassless fields. Spontaneous symmetry breaking would correspond to the mixture betweenthe originally massless rank-one particle module and the rank-two current module. In theinfinite-dimensional case of the multi-particle algebra the mechanism of such mixing maydepend on the choice of the frame in the multi-particle algebra. As argued in Section 11.2, itis natural to anticipate that the proper frame is defined by (11.44), being associated with theboundary current algebra of [144]. Detailed analysis of this issue is one of the most urgentproblems on the agenda.
The formulation of HS equations in the unfolded form expressing the space-time exteriorderivative d via the values of other fields like in equations (2.3)-(2.5) on the fields W , B and S subjected to the constraints (2.6) and (2.7) allows us to unify in the same framework thesystems that live in space-times of different dimensions [124, 92, 128]. This is achieved by41etting the de Rham derivative d be defined in the infinite-dimensional space-time. Then thephysical space-times like Minkowski or AdS appear when the background (vacuum) connec-tion W ( x ) of the respective symmetry group G is nontrivial with non-degenerate frame-likecomponents along x n k associated with the translation (or transvection) generators. In thatcase the coordinates x n k are observable while the rest ones x n ⊥ are not: in the absence ofcomponents of forms W ( x ) along x n ⊥ the respective unfolded equations treated perturba-tively would imply that, locally, all other differential form fields in the system are either x n ⊥ -independent zero-forms or pure gauge ( i.e., exact) p > x n k representing one or another G –invariant space described by W ( x n k ).In [124] it was observed that the higher-rank fields in lower dimension can be interpretedas elementary fields in higher dimension. In the framework of Coxeter HS theory this phe-nomenon acquires a direct realization as we explain now. Consider a spinorial Coxeter HStheory. In this case, the rank-one sector consists of the fields depending on a single spinorvariable Y A with A = 1 , . . . M ( M = 2 in the 3 d model and M = 4 in the 4 d model).Bilinears of Y A form generators of sp ( M ) with respect to the star product. This is the AdS algebra sp (4) in the 4 d model and a sp (2) half of the AdS algebra in the 3 d model(to be doubled via introducing the Clifford element ψ [8]). The diagonal embedding of thestar-product generators of sp diag ( M ) into a rank- p system is t AB = p X i =1 { Y Ai , Y Bi } ∗ . (12.6)Vacuum connection W ( x ) that describes usual space-time geometry like AdS or AdS is aflat sp ( M ) connection ( sp (2) ⊕ sp (2) in the 3 d case).Let Ω = A, i . Operators T ΩΛ = { Y Ω , Y Λ } ∗ ∗ I ∗ . . . ∗ I p (12.7)are generators of sp ( pM ). The diagonal embedding sp diag ( M ) into sp ( pM ) is realized by thegenerators t ′ AB = t AB ∗ I ∗ . . . ∗ I p . (12.8)The same time the sp ( M ) generated by t AB (12.6) acts diagonally on the full framed Chered-nik algebra. Evidently, ( t AB − t ′ AB ) ∗ T ΩΛ = T ΩΛ ∗ ( t AB − t ′ AB ) = 0 . (12.9)Let τ Ω ′ Λ ′ form a basis of sp ( pM ) /sp diag ( M ). Due to (12.9), it is possible to identify thespace-time components of the flat connections of t AB and t ′ AB demanding them to have non-zero components along coordinates x n k of the space-time M originally associated with the sp ( M ) symmetry. Now it is possible to choose a flat connection W ′ ( X ) on some sp ′ ( pM )-invariant space-time M ′ with local coordinates X in such a way that the pushforward of42 ′ ( X ) to M gives W ( x ). This makes it possible to treat the rank- p fields in M as elemen-tary fields in M ′ .One can proceed analogously with the additional species of oscillators in the multi-particlealgebra construction. From the analysis of Section 11.3 it follows that the multi-particle B -HS model has sp (8) ⊕ sp (8) as a finite-dimensional symmetry. The natural homogeneousspace that admits this symmetry is the group manifold Sp (8) which is 36-dimensional. More-over, the conformal-like symmetry of Sp (8) is Sp (16) [114, 148, 149].This has an interesting consequence that the usual space-time interpretation of the Cox-eter HS theories and their multi-particle extensions is likely to have maximal Minkowskidimension ten. This follows from the analysis of the sp (2 M ) invariant theories in [126, 127]where it was shown how usual Minkowski space-time dimensions emerges from the sp (2 M )-invariant equations with the conclusion that the maximal Minkowski space known to emergefrom the sp (16)-invariant equations is ten dimensional. Though it is tempting to conjec-ture that this is how the Superstring dimension ten emerges from the Coxeter HS theory,details of this phenomenon need further investigation. In particular, it would be interestingto see whether this phenomenon is related to the twistor-like transform in ten dimensionsintroduced by Witten in [130].Note that the mechanism explained in this section is based on the extension of the space-time symmetries due to appearance of the additional species of oscillators in the higher-rankCoxeter HS theories and/or multi-particle extension of HS theories. As such it is not quitethe same as the holographic correspondence within the idempotent construction of Section 9where the symmetry algebra g remains the same but the g -module pattern changes dependingon the idempotent sector in question.
13 Conclusion
We propose a class of Coxeter HS models conjectured to underly a symmetric phase ofString Theory (Coxeter group B ) and its further tensor-like extensions (Coxeter group B p ).These models contain two coupling constants one of which is responsible for stringy effects(absent in the conventional HS theory) and exhibit a number of interesting parallelisms withString Theory. In particular, consistency of the holographic interpretation of the boundarymatrix-like model demands the latter to have N = 4 SUSY.The main idea of our construction was to find a formally consistent extension of the knownHS gauge systems [5, 8, 91] possessing a richer spectrum and having a room for masslessfields of all spins including spin-two gravitational field. The former goal was reached viaextension of the Coxeter groups Z or Z × Z generated by the Klein elements of the modelsof [5, 8, 91] to any Coxeter group including the most important case of B p . To let masslessfields be present in the model this construction was further extended to the framed algebrascontaining additional idempotent elements.Though the main emphasize in this paper was on the spinorial HS models somewhatanalogous to Green-Schwarz superstring, bosonic Coxeter HS theories of vectorial type con-sidered in Section 6.3 and their fermionic counterparts to be elaborated are also of interest as43nalogues of the bosonic and fermionic strings. (For more detail see [134].) We believe thatformal consistency in presence massless fields in the spectrum is so restrictive that it hardlyleaves a room for consistent HS models beyond the list presented in this paper, supplementedby the Coxeter extensions of to be constructed fermionic generalizations of the model of [91]in any dimension. In particular it determines the structure of extended HS symmetries aswell as the field pattern.Many of the important aspects of the proposed models, such as detailed analysis of fieldspectra, spontaneous breaking of HS symmetries, holographic interpretation, proper space-time interpretation, the choice of appropriate frame in the multi-particle HS theories andothers were only briefly sketched in this paper, demanding a more detailed study delegatedto the future as well as some other issues including, for instance, the analysis of localityalong the lines of [17, 18].An interesting open question is to give an interpretation to the Coxeter HS theoriesbased on the Coxeter groups different from B p which were of most interest in this paper. Inparticular, it would be interesting to study more carefully the case of Dihedral group I ( n )which is the symmetry group of the n -gone on the plane. In this relation it should be notedthat the case of B is special due to the isomorphism B ∼ I (4). An interesting feature ofthe B ∼ I (4) HS theory is that, as shown recently in [150, 151], for certain linear relationsbetween the coupling constants ν and ν the respective Cherednik algebra acquires ideals.As a result, on this locus of the plane of coupling constants some states in the system shoulddecouple. It would be interesting to investigate this phenomenon in detail, especially in thecontext of unitarity. Acknowledgments
I am grateful to Nima Arkani-Hamed, Matthias Gaberdiel, Olga Gelfond, Igor Klebanov,Arkady Tseytlin, Herman Verlinde and especially Semyon Konstein for useful discussions.This research was supported by the Russian Science Foundation Grant No 18-12-00507.
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