Light-cone continuous-spin field in AdS space

We develop further the general light-cone gauge approach in AdS space and apply it for studying continuous-spin field. For such field, we find light-cone gauge Lagrangian and realization of relativistic symmetries. We find a simple realization of spin operators entering our approach. Generalization of our results to the gauge invariant Lagrangian description is also described. We conjecture that, in the framework of AdS/CFT, the continuous-spin AdS field is dual to light-ray conformal operator. For some particular cases, our continuous-spin field leads to reducible models. We note two reducible models. The first model consists of massive scalar, massless vector, and partial continuous-spin field involving fields of all spins greater than one, while the second model consists of massive vector, massless spin-2 field, and partial continuous-spin field involving all fields of spins greater than two.


Introduction
In view of the aesthetic features, continuous-spin field has attracted some interest in recent time. For review, see Refs. [1,2,3]. Extensive list of references on earlier studies of this theme may be found in Refs. [4,5]. Alternative points of view on the role of continuous-pin field in string theory are presented in Refs. [6,7]. Interrelation of continuous-spin field and massive higher-spin field is discussed in Ref. [8]. Interacting continuous-spin fields are considered in Refs. [10]- [12], while various BRST Lagrangian formulations are studied in Refs. [13,14,15]. Continuous-spin field in AdS space was investigated in Refs. [16]- [19]. Other various important aspects of continuous-spin field were discussed in Refs. [21]- [28].
Continuous-spin field is decomposed into infinite chain of scalar, vector, and tensor fields which consists of every field just once. A similar infinite chain of fields appears in higher-spin gauge field theories in AdS space [29]. Other example of dynamical system involving infinite number of fields is a string theory. Light-cone gauge formulation simplifies considerable superstring action in AdS space [30,31,32]. We think that light-cone gauge formulation will simplify study of continuous-spin field and therefore will be useful for better understanding of various aspects of continuous-spin field.
In this paper, we develop further our light-cone gauge formulation of AdS fields in Refs. [33,34]. Namely, we obtain representation of the 4th-order Casimir operator of the so(d, 2) algebra in terms of spin operators entering light-cone gauge Lagrangian. This allows us to express two constant parameters entering light-cone gauge Lagrangian of continuous-spin field entirely in terms of the eigenvalues of the 2nd-and 4th-order Casimir operator of the so(d, 2) algebra. Such representation for the Lagrangian and a suitable parametrization of eigenvalues of the Casimir operators make the whole study more transparent and straightforward and considerably simplify analysis of classical unitarity and irreducibility of continuous-spin field. We obtain simple representation for spin operators entering our light-cone gauge approach. Also we make conjecture about duality between continuous-spin field and light-ray conformal operator. Interrelations of light-cone gauge and gauge invariant approaches allow us to extend all our light-cone gauge results to the gauge invariant Lagrangian of continuous-spin field in a rather straightforward way. In due course, we discuss two models of continuous-spin fields which, besides partial continuous-spin fields, involve interesting spectrum of low-spin fields.
2 General light-cone gauge approach in AdS d+1 space General light-cone gauge approach in AdS space was developed in Refs. [33,34]. In this section, first, we review the formulation obtained in Ref. [34] and, second, we present our new result regarding the light-cone gauge representation for 4th-order Casimir operator of the so(d, 2) algebra.
Let φ(x) be arbitrary spin and type of symmetry bosonic fields. Collecting the fields into a ket-vector |φ , we present a light-cone gauge action in the following form [33] where x + is considered as a light-cone time. Our conventions for the coordinates and derivatives are as follows: where we use the indices i, j = 1, . . . , d−2; I, J = 1, . . . , d−2, d. Vectors of so(d−1) algebra are decomposed as X I = (X i , X z ), φ| = (|φ ) † , where operator A being independent of space-time coordinates and their derivatives is acting only on spin indices of |φ . In general, fields entering the |φ are complex-valued.
The choice of the light-cone gauge spoils the relativistic so(d, 2) symmetries of fields in AdS d+1 . Therefore in order to demonstrate that so(d, 2) symmetries are still present we find the Noether charges which generate them. For free fields, Noether charges (or generators) have the following representation in terms of the |φ : where G diff stands for differential operators acting on |φ . These operators are given by 3) where C 2 stands for an eigenvalue of the 2nd-order Casimir operator of the so(d, 2) algebra. The operators B I , M IJ satisfy the commutators where, in (2.13),(2.14) and below, we use the notation  Note also that we use the following hermicity conjugation rules for the spin operators: From (2.13), we see that the M IJ are spin operators of the so(d −1) algebra, while the operator B I transforms as vector operator under transformations of the so(d − 1) algebra. It is the commutators (2.14) that are basic equations of light-cone gauge formulation of relativistic dynamics in AdS d+1 . We now ready to formulate our new result in this Section. We find that the 4th-order Casimir operator of the so(d, 2) algebra is expressed in terms of the spin operators B I , M IJ as follows where we use notation given in (2.15),(2.16). Our conventions for the Casimir operators are given in Appendix. Action (2.1) is invariant under the transformations δ|φ = G diff |φ .
The following remarks are in order. a) For solution (3.20), there are no restrictions on ℑp, ℑq (3.20). We note that the same happens for light-ray conformal operator. Namely, such operator is realized as unitary representation of the so(d, 2) algebra and labelled by conformal dimension ∆ = E 0 and continuous-spin s given in (3.14) with ℜp = 0, ℜq = 0 and no restrictions on ℑp, ℑq. We recall that our solution (3.20) describes irreducible classically unitary continuous-spin field. We then conjecture that, in the framework of AdS/CFT correspondence, our continuous-spin AdS field with E 0 and s in (3.14) and solution (3.20) is dual to light-ray conformal operator having ∆ = E 0 and s as in (3.14) with ℜp = 0, ℜq = 0. Also we expect that continuous-spin AdS fields associated with the solutions (3.21)- (3.24) are also dual to the respective conformal operators having conformal dimension ∆ = E 0 and s as in (3.14). b) Using decomposition (3.43) for n = 0, n = 1, we note two models of reducible classically unitary continuous-spin field with interesting spectrum for low-spin finite-component fields 2), while f υ ,f υ (3.24) involve a square root of degree-4 polynomials in the N υ . Therefore it seems preferable to study the continuous-spin field by using f υ ,f υ given in (3.20)-(3.23). It is the use of complex-valued fields in (3.1) that allows us to introduce simple representations for solutions in (3.20)-(3.23). Flat space. We use the chance to discuss new representation for operators f υ ,f υ entering continuous-spin fields in flat space. Light-cone gauge formulation of massless and massive continuousspin fields in flat space R d−1,1 , d-arbitrary, was obtained in Refs. [9,12] . In the flat space R d−1,1 , equation for f υ ,f υ (3.10) takes the form where the mass parameter m and the continuous-spin parameter κ are related to 2nd-and 4th-order Casimir operators of the Poincaré algebra (for details see Ref. [12]). Restrictions on the m and κ (3.54) are obtained by requiring the classical unitarity and irreducibility. In Ref. [12], we discussed solution given by Such solution is realized on space of real-valued continuous-spin field. We now note a new solution given by The new solution (3.55) is realized on space of complex-valued continuous-spin field. Our new solution enters the spin operator of massive continuous-spin field as follows. In expression for g andḡ in (2.21) in Ref. [12], we make the respective substitutions √ F υ → f υ and √ F υ →f υ .

Gauge invariant action for continuous-spin AdS field
Gauge invariant action for continuous-spin field was obtained in Ref. [16]. The gauge invariant action depends on two parameters. In this section, our aim is twofold. First, motivated by our result for light-cone gauge continuous-spin field we are going to express the two parameters in terms of the Casimir operators. Second, we discuss extension of our new representation for solutions given in (3.20)-(3.23) to the gauge invariant Lagrangian formulation. Gauge invariant action is formulated in terms of ket-vector given by We note that, in (4.1), fields with n = 0 and n = 1 are the respective scalar and vector fields of the Lorentz algebra so(d, 1), while fields with n ≥ 2 are the totally symmetric tensor fields of the Lorentz algebra so(d, 1). Also, we note that, in (4.1), fields with n ≥ 4 are considered to be double-traceless, φ aabba 5 ...an = 0. All fields in (4.1) are taken to be complex-valued. Gauge invariant action and Lagrangian of continuous-spin field we found can be presented as 3 where φ| ≡ (|φ ) † , while quantities m 1 , m 2 ,e 1 , andē 1 are defined by the relations where µ 0 , µ 1 stand for constant parameters. The quantity F υ appearing in (4.10) is the same as the one appearing in light-cone gauge approach in (3.11). Therefore comparing (4.10) and (3.11),(3.12), we can entirely express the parameters µ 0 and µ 1 in terms of the eigenvalues of the 2nd and 4th-order Casimir operators Using (3.15), (3.16), and (4.11), we can represent F υ (4.10) as in (3.17). This implies that our whole analysis we carried out for classically (ir)reducible light-cone gauge continuous-spin field in Section 3 is automatically extended to the gauge invariant formulation in this Section. In (4.12), gauge parameters with n = 0 and n = 1 are the respective scalar and vector fields of the Lorentz algebra so(d, 1), while the gauge parameters with n ≥ 2 are totally symmetric traceless tensor fields of the Lorentz so(d, 1) algebra. Gauge transformation parameters with n ≥ 2 are taken to be traceless, ξ aaa 3 ...an = 0. Use of ket-vectors |φ and |ξ and operator e 1 ,ē 1 (4.8) allows us to write gauge transformations in the following form: To summarize, in this paper, we developed further the general light-cone formulation in Ref. [33,34] and applied it for studying free continuous-spin field in AdS space. Extension of our approach to interacting continuous-spin AdS field could be of great interest. In this respect we note that, using methods in Ref. [36], we studied interacting vertices of light-cone gauge continuous-spin field in flat space in Refs. [9,12], while, in Ref. [37], we developed method for studying finitecomponent light-cone gauge AdS fields. We believe therefore that results in this paper and the ones in Refs. [9,12,37] will be helpful for studying interacting continuous-spin AdS fields. For reader convenience we note that various BRST methods for studying interacting finite-component fields may be found in Refs. [38,39]. Other interesting methods for investigation of interacting finite-component fields were developed in Refs. [40,41]. Study of continuous-spin field along the line of group-theoretical methods in Refs. [42] could also be of some interest.
Appendix A Casimir operators of the so(d, 2) algebra In this Appendix, we explain our conventions for Casimir operators of the so(d, 2) algebra. To this end we start with a manifestly (d + 2)-dimensional covariant approach. Generators of the so(d, 2) algebra denoted by J AB satisfy the commutators where vector indices of the so(d, 2) algebra take values A, B, C, E = 0 ′ , 0, 1, 2, . . . , d. In terms of the J AB , the 2nd-and 4th-order Casimir operators of the so(d, 2) algebra are defined to be In Ref. [33], we shown that the first relation in (A.2) allows us to find the relation for the operator A in (2.11). We now note that by using the light-cone gauge generators of the so(d, 2) algebra given in relations (2.3)-(2.12), we verified that the the 4th-order Casimir operator C 4 (A.2) takes the form given in (2.18). For doing so, we should relate generators in (A.1) with the lightcone generators (2.3)-(2.10). To this end we decompose (d + 2) coordinates x A as where η ⊕⊖ = 1, η ⊖⊕ = 1. Now, using notation of the conformal algebra considered in the base of the algebra so(d − 1, 1) spanned by J ab , we identify generators (A.4) as: Generators (A.5) and J ab satisfy the commutators given in (A6),(A7) in Ref. [35], while relation (A8) in Ref. [35] provides the light-cone gauge decomposition of the generators (A.5). Now using (A.2) and relations (2.3)-(2.10), we verified that C 4 in (A.2) amounts to C 4 in (2.18).
To parametrize eigenvalues of the Casimir operators for totally symmetric representations of the so(d, 2) algebra we use labels E 0 , s. In conformal algebra notation, E 0 ≡ ∆, where ∆ is conformal dimension of a conformal operator, while s is associated with spin. In general, the E 0 and s are complex-valued. Eigenvalues of the Casimir operators (A.2) are given by