Continuous spin gauge field in (A)dS space

Totally symmetric continuous spin field propagating in (A)dS is studied. Lagrangian gauge invariant formulation for such field is developed. Lagrangian of continuous spin field is constructed in terms of double traceless tensor fields, while gauge transformations are constructed in terms of traceless gauge transformation parameters. de Donder like gauge condition that leads to simple gauge fixed Lagrangian is found. Gauge-fixed Lagrangian invariant under global BRST transformations is presented. The BRST Lagrangian is used for computation of a partition function. It is demonstrated that the partition function of the continuous spin field is equal to one. Various decoupling limits of the continuous spin field are also studied.


Introduction
Continuous spin field has attracted some interest in recent time. Such field can be considered as a field theoretical realization of continuous spin representation of Poincaré algebra which was studied many years ago in Ref. [1]. For extensive list of references on this theme see Refs. [2,3]. Interesting feature of continuous spin field is that this field is decomposed into infinite chain of coupled scalar, vector, and tensor fields which consists of every field just once. We note then that a similar infinite chain of fields enters higher-spin gauge field theories in AdS space [4]. Note however that fields in Ref. [4] are decoupled as coupling constant tends to zero. Also it turns out that some regimes in string theory are related to continuous spin field [5]. We think that further progress in understanding dynamics of continuous spin field requires, among other things, better understanding of gauge invariant Lagrangian formulation of continuous spin field in (A)dS and flat spaces. This is what we are doing in this paper.
Gauge invariant formulation for bosonic continuous spin field in four-dimensional flat space, R 3,1 , was developed in Ref. [6], while gauge theory of fermionic continuous spin field in R 3,1 was studied in Ref. [7]. So far Lagrangian formulation of continuous spin field propagating in (A)dS space has not been discussed in the literature. Our major aim in this paper is to develop Lagrangian gauge invariant formulation of continuous spin bosonic field in (A)dS d+1 space with arbitrary d ≥ 3. We use our gauge invariant Lagrangian for derivation of gauge-fixed BRST Lagrangian of continuous spin field which is invariant under global BRST and anti-BRST transformations. We use our BRST Lagrangian for computation of a partition function and demonstrate that such partition function is equal to 1. Also we analyse various limits of gauge invariant Lagrangian for continuous spin field in (A)dS space. We demonstrate that such limits lead to appearance of massless, massive and partial-massless fields. By product, considering limit of flat space, we obtain Lagrangian gauge invariant formulation of continuous spin field in flat R d,1 with arbitrary d ≥ 3. We note that, so far, Lagrangian formulation of continuous spin field in flat space R d,1 with arbitrary d ≥ 3 was discussed only in the framework of light-cone gauge approach [2].

Lagrangian and gauge transformations of continuous spin field
We start with a discussion of a field content entering our gauge invariant formulation of continuous spin field. To discuss a continuous spin field propagating in AdS d+1 space, we introduce scalar, vector and tensor fields of the so(d, 1) Lorentz algebra, φ a 1 ...an , n = 0, 1, . . . , ∞ . (2.1) In (2.1), fields with n = 0 and n = 1 are the respective scalar and vector fields of the so(d, 1) algebra, while fields with n ≥ 2 are the totally symmetric tensor fields of the Lorentz so(d, 1) algebra. Fields φ a 1 ...an (2.1) with n ≥ 4 are taken to be double-traceless, φ aabba 5 ...an = 0 , n = 4, 5, . . . , ∞.
Fields in (2.1) subject to constraint (2.2) constitute a field content of our approach.
To streamline our presentation we introduce a set of creation operators α a , υ, and the respective set of annihilation operators,ᾱ a ,ῡ which we will refer to as oscillators. Using the α a , υ, we collect fields (2.1) into a ket-vector |φ defined as In terms of the ket-vector |φ , constraint (2.2) can be represented as (ᾱ 2 ) 2 |φ = 0. Gauge invariant action and Lagrangian of continuous spin field we found can be presented as φ| ≡ (|φ ) † , where e = det e a m , while e a m stands for vielbein in (A)dS space. The notation ✷ (A)dS in (2.5) is used for the D'Alembert operator in (A)dS space. Quantities m 1 , m 2 , and e 1 ,ē 1 appearing in (2.5)-(2.7) are defined by relations where, in (2.8),(2.11), µ 0 , µ 1 stand for dimensionfull constants, while ρ is defined as ρ = − 1 R 2 for AdS space; ρ = 0 for flat space; ρ = 1 R 2 for dS space, (2.12) and R is a radius of (A)dS space. Quantities N υ , αD, α 2 are defined in Appendix. We note the relation e φ|LL|φ = −e L φ|L|φ (up to total derivative). The following remarks are in order. i) Our Lagrangian depends on ρ given in (2.12) and two arbitrary dimensionfull parameters µ 0 , µ 1 .
ii) On space of double-traceless ket-vector |φ , operator E (2.5) can alternatively be represented as 14) iii) Two-derivative contributions to operator E (2.5) coincide with two-derivative contributions to the standard Fronsdal operator that enters Lagrangian of free massless field in (A)dS space. iv) Representation for gauge invariant Lagrangian given in (2.4)-(2.7) is universal and is valid for arbitrary theory of gauge fields propagating in (A)dS space. Various (A)dS field theories are distinguished by operators m 1 , m 2 , e 1 ,ē 1 entering the operator E. Namely, the operators E of massless, massive, conformal, and continuous spin fields propagating in (A)dS space depend on the covariant derivative D a and the oscillators α a ,ᾱ a in the same way as the operator E given in (2.4). This is to say that operators E for massless, massive, conformal, and continuous spin field in (A)dS space are distinguished only by the operators m 1 , m 2 , e 1 , andē 1 . It is finding the operators m 1 , m 2 , e 1 , andē 1 that provides real difficulty. For the reader convenience we note that, for massless fields in (A)dS d+1 , the operators m 1 , m 2 , e 1 , andē 1 take the form Explicit expressions for the operators m 1 , m 2 , e 1 , andē 1 corresponding to the massive and conformal fields in (A)dS can be found in Refs. [11,12].
v) It is the use of operators L,L (2.6),(2.7) that considerably simplifies our Lagrangian. We refer to the operatorL as modified de Donder divergence. Equating e 1 = 0,ē 1 = 0 gives the standard de Donder divergence. For massless continuous spin field in R 3,1 , i.e., the case d = 3, µ 0 = 0, ρ = 0, the operatorL was introduced in Ref. [6]. For d = 3, µ 0 = 0, ρ = 0, our Lagrangian (2.4) coincides with the one in Ref. [6]. Idea to use modified de Donder divergence to simplify Lagrangian of massive field in flat and (A)dS space was first exploited in Refs [10,12]. Alternative representation for Lagrangian of massive field without use of de Donder was first obtained in Ref. [13]. Discussion of the standard de Donder divergence for studying various aspects of higher-spin field theory may be found in Refs. [14]. Gauge symmetries. To discuss gauge symmetries of continuous spin field we introduce the following gauge transformation parameters: To streamline presentation of gauge symmetries we use the α a , υ and collect gauge transformation parameters in ket-vector |ξ defined as Note also that, in terms of the |ξ , algebraic constraints (2.18) take the formᾱ 2 |ξ = 0.

BRST Lagrangian and partition function of continuous spin field
BRST invariant Lagrangian of continuous spin field. In this section we obtain gauge-fixed BRST invariant Lagrangian for continuous spin field. We use then such Lagrangian to compute a partition function of the continuous spin field. As we have already said a general structure of our Lagrangian (2.4) and gauge transformations (2.20) for continuous spin (A)dS field is similar to the one for massive (A)dS field. Derivation of BRST Lagrangian and use of such Lagrangian for a computation of partition function of massive (A)dS field may be found in Ref. [8]. In this section we demonstrate how the method in Ref. [8] can be applied to the case of continuous spin field. Lagrangian of continuous spin field with local BRST symmetries was discussed in Ref. [9].
To built gauge-fixed BRST invariant Lagrangian we introduce Faddeev-Popov fields c a 1 ...an , c a 1 ...an and Nakanishi-Lautrup fields b a 1 ...an , n = 0, 1, . . . , ∞. We use the oscillators to collect all Faddeev-Popov fields into ket-vectors |c , |c , while all Nakanishi-Lautrup fields are collected into ket-vector |b . Using notation |χ for |c , |c , |b , we note that representation of the ket-vectors |c , |c , |b in terms of scalar, vector, and tensor fields of the so(d, 1) algebra takes the form Fields in (3.1) with n = 0, n = 1, and n ≥ 2 are the respective scalar, vector, and traceless totally symmetric tensor fields of the so(d, 1) algebra.
Using ket-vector |φ (2.3) and ket-vectors in (3.1), we note that gauge-fixed Lagrangian L tot in arbitrary α-gauge can be presented as where gauge invariant Lagrangian L is given in (2.4), while the modified de Donder operatorL is defined in (2.6). One can verify that, up to total derivative, gauge-fixed Lagrangian (3.2) is invariant under the following BRST and anti-BRST transformations where gauge transformation operator G is defined in (2.20). It easy to check then that BRST and anti-BRST transformations given in (3.4), (3.5) are off-shell nilpotent: s 2 = 0,s 2 = 0, ss +ss = 0. Lagrangian (3.2) can be cast into the form that is more convenient for practical calculations. This is to say that fixing the α = 1 gauge and integrating out Nakanishi-Lautrup fields, we find that Lagrangian (3.2) leads to the following gauge-fixed Lagrangian Alternatively, Lagrangian (3.6) can be represented in terms of M 1 , M 2 (2.14),(2.15) as where M FP is given in (3.3). Thus we see that it is the use of representation for gauge invariant Lagrangian in (2.5)-(2.7) and the α = 1 gauge that simplify considerably the expression for gaugefixed Lagrangian given in (3.6),(3.7). Gauge-fixed Lagrangian given in (3.6) and (3.7) is also invariant under BRST and anti-BRST transformations given by s|φ = G|c , s|c = 0 , s|c =L|φ ,s|φ = G|c ,s|c = −L|φ ,s|c = 0 , (3.8) whereL and G are given in (2.6) and (2.20). Note however that, in contrast to transformations given in (3.4),(3.5), BRST and anti-BRST transformations given in (3.8) are nilpotent, s 2 = 0, s 2 = 0, ss +ss = 0, only for on-shell Faddeev-Popov fields. Partition function of continuous spin field. In order to compute a partition function of continuous spin field we decompose double-traceless ket-vector |φ (2.3) into two traceless ket-vectors denoted by |φ I , |φ II , Using relations (2.3),(3.9), it easy to understand that a decomposition of the traceless ket-vectors |φ I , |φ II into scalar, vector, and traceless tensor fields of the so(d, 1) algebra can be presented as Plugging |φ (3.9) into gauge-fixed Lagrangian (3.7), we get 14)   From (3.12), (3.18), (3.19), we see that the partition function of continuous spin field is given by where in relation (3.23) the determinant of D'Alembert operator in (A)dS is evaluated on space of traceless rank-n tensor field. Using (3.22), we see that partition function of continuous spin field (3.21) is indeed equal to 1, Z = 1. Note that the partition function of continuous spin field turns out to be equal to 1 without the use of any special regularization procedure required for a computation of partition functions in higher-spin gauge field theory (see, e.g., Ref. [15]). For continuous spin field in (A)dS, we note the same mechanism of cancellation as for higher-spin fields in flat space (see Eq.(2.2) in Ref. [15]). Namely, using (3.18)-(3.20), we check the cancellation of determinant of the physical spin-n field and ghost determinant of spin-(n + 1) field. To this end, we note that partition function Z for Lagrangian L tot (3.12) can alternatively be represented as (3.24) We now use the following relations: where D ⊥ n (M n ) takes the form as in (3.23) with a prescription that the determinant of D'Alembert operator in (A)dS is evaluated on space of traceless and divergence-free rank-n tensor field in (A)dS d+1 . From Z n (3.25), we see the cancellation of determinant of the physical spin-n field and ghost determinant of spin-(n + 1) field in expression for Z in (3.24).
To summarize, in this paper, we developed gauge invariant Lagrangian formulation for continuous spin field in (A)dS and applied our result for a computation of partition function. In this paper, we used metric-like Lagrangian formulation of gauge fields. 5 In the literature, there are many interesting approaches to Lagrangian formulation of gauge fields. We mention frame-like formulation and BRST approach (see, e.g., Refs. [19,20]). It will be interesting to study continuous spin field in the framework of such formulations and establish their connection with a vector-superspace formulation in Refs. [6,21]. We note also that use of extended hamiltonian approach could be helpful for better understanding of physical d.o.f for continuous spin field. Recent discussion of extended hamiltonian approach may be found in Refs. [22,23]. Applications of various methods, which were developed for analysis of interaction vertices of gauge fields in Refs. [24]- [30], to the study of interaction vertices of continuous spin field could of some interest. Light-cone gauge methods in Refs. [31]- [34] can also be helpful for this purpose. Recent developments in light-cone approach to field dynamics may be found in Refs. [35].