Fermionic continuous spin gauge field in (A)dS space

Fermionic continuous spin field propagating in (A)dS space-time is studied. Gauge invariant Lagrangian formulation for such fermionic field is developed. Lagrangian of the fermionic continuous spin field is constructed in terms of triple gamma-traceless tensor-spinor Dirac fields, while gauge symmetries are realized by using gamma-traceless gauge transformation parameters. It is demonstrated that partition function of fermionic continuous spin field is equal to one. Modified de Donder gauge condition that considerably simplifies analysis of equations of motion is found. Decoupling limits leading to arbitrary spin massless, partial-massless, and massive fermionic fields are studied.


Introduction
In view of many interesting features of continuous spin gauge field theory this topic has attracted some interest in recent time [1]- [3]. For a list of references devoted to various aspects of continuous spin field, see Refs. [2,4,5]. It seems likely that some regime of the string theory can be related to continuous spin field theory (see, e.g., Refs. [6]). Other interesting feature of continuous spin field theory, which triggered our interest in this topic, is that the bosonic continuous spin field can be decomposed into an infinite chain of coupled scalar, vector, and totally symmetric tensor fields which consists of every field just once. We recall then that a similar infinite chain of scalar, vector and totally symmetric fields enters the theory of bosonic higher-spin gauge field in AdS space [7].
Supersymmetry plays important role in string theory and higher-spin gauge field theory. We expect that supersymmetry will also play important role in theory of continuous spin field. For a discussion of supersymmetry, we need to study bosonic and fermionic continuous spin fields. Lagrangian formulation of bosonic continuous spin field in flat space R 3,1 was studied in Ref. [2], while Lagrangian formulation of bosonic continuous spin field in flat space R d,1 and (A)dS d+1 space with arbitrary d was discussed in Ref. [3]. Lagrangian formulation of fermionic continuous spin field in flat space R 3,1 was obtained in Ref. [1]. So far Lagrangian formulation of fermionic continuous spin field in (A)dS d+1 and R d,1 with arbitrary d has not been discussed in the literature. Our major aim in this paper is to develop Lagrangian formulation of continuous spin fermionic field in flat space R d,1 and (A)dS d+1 space with arbitrary d ≥ 3. 1 As by product, using our Lagrangian formulation, we compute partition functions of fermionic continuous spin fields in (A)dS and flat spaces and show that such partition functions are equal to 1. Considering various decoupling limits, we demonstrate how massless, partial-massless, and massive fermionic fields appear in the framework of Lagrangian formulation of fermionic continuous spin (A)dS field.
Dirac fields (2.1) subject to constraints (2.2) constitute a field content in our approach. In order to obtain a gauge invariant description of the continuous spin fermionic field in an easy-to-use form, we introduce bosonic creation operators α a , υ. Using the α a , υ, we collect all fields appearing in (2.1) into a ket-vector |ψ , We note that triple gamma-tracelessness constraint (2.2) can be represented asᾱ 2 γᾱ|ψ = 0. Lagrangian. Gauge invariant action and Lagrangian of the fermionic continuous spin field we found take the form ψ| ≡ (|ψ ) † γ 0 , where e = det e a m and e a m stands for vielbein in (A)dS space. In (2.6) and below, the notation D / is used for the Dirac operator in (A)dS space. A definition of quantities like αD, γα, α 2 may be found in Appendix. Quantities e Γ 1 , e 1 , andē 1 entering E (0) (2.7) are defined as where the R stands for a radius of (A)dS space. The following remarks are in order. i) Our Lagrangian depends on two arbitrary dimensionfull parameters κ 0 , µ 0 and on ρ (2.12).
ii) The one-derivative operator E (1) (2.6) coincides with the one-derivative contribution to the Fang-Fronsdal operator that enters Lagrangian of fermionic massless field in (A)dS space. Gauge symmetries. In order to describe gauge transformation of continuous spin field we use the following gauge transformation parameters: where α stands for spinor index which will be implicit in what follows. In (2.13), gauge transformation parameters with n = 0 and n = 1 are the respective spin-1 2 and spin-3 2 Dirac fields of the Lorentz so(d, 1) algebra, while gauge transformation parameters with n ≥ 2 are totally symmetric γ-traceless spin-(n + 1 2 ) Dirac fields of the Lorentz so(d, 1) algebra, γ a ξ aa 2 ...an = 0 , n = 1, 2, . . . , ∞ . (2.14) In order to simplify the presentation of gauge transformation we use the oscillators α a , υ and collect all gauge transformation parameters appearing in (2.13) into ket-vector |ξ defined by the relation |ξ = ∞ n=0 υ n+1 n! (n + 1)! α a 1 . . . α an ξ a 1 ...an |0 .
We now find that our Lagrangian (2.4)-(2.7) is invariant under a gauge transformation given by where the operators e Γ 1 , e 1 ,ē 1 entering derivative independent part of gauge transformation (2.16) are defined in (2.8)- (2.12).
Representation for gauge invariant Lagrangian (2.4)-(2.7) and the corresponding gauge transformation (2.16) is universal and valid for arbitrary theory of fermionic totally symmetric gauge (A)dS fields. Various theories of fermionic totally symmetric gauge (A)dS fields are distinguished only by explicit form of the operators e Γ 1 , e 1 ,ē 1 entering E (0) (2.7) and gauge transformation (2.16). This is to say that, operators E and G for fermionic totally symmetric massless, massive, conformal, and continuous spin fields in (A)dS depend on the oscillators α a ,ᾱ a , the Dirac γ-matrices, and on the derivative D a in the same way as operators E (2.5)-(2.7) and G (2.16). Namely, operators E and G for fermionic totally symmetric massless, massive, conformal, and continuous spin fields in (A)dS are distinguished only by the explicit form of the operators e Γ 1 , e 1 , andē 1 . For the reader convenience we note that, for massless fields in (A)dS d+1 , the e Γ 1 , e 1 , andē 1 take the form For spin-(s + 1 2 ) and mass-m massive (A)dS field, the operators e Γ 1 , e 1 , andē 1 can be read from expressions (2.18)-(2.21) in Ref. [8]. 2 For fermionic conformal fields in flat space, the operators e Γ 1 , e 1 , andē 1 can be read from expressions (3.31)-(3.34) in Ref. [13]. For fermionic conformal fields in (A)dS space, explicit expressions for the operators e Γ 1 , e 1 , andē 1 are still to be worked out.

(Ir)reducible classically unitary fermionic continuous spin field
Our Lagrangian (2.4) depends on the two parameters κ 0 and µ 0 . In this Section, we find restrictions imposed on the κ 0 and µ 0 for irreducible and reducible classically unitary dynamical systems. Let us start with our definition of classically unitary reducible and irreducible systems. i) Lagrangian (2.4) is constructed out of complex-valued Dirac fields. In order for the action be hermitian the κ 0 (2.8) should be real-valued, while the quantity F υ (2.10) should be positive for all eigenvalues N υ = 0, 1, . . . , ∞. Introducing the notation we note then that, depending on behaviour of the F υ (n), we use the following terminology is positive for all n, then we will refer to fields (2.1) as classically unitary system. From (3.3), we learn that, if F υ (n) (3.1) has no roots, then we will refer to fields (2.1) as irreducible system. For this case, Lagrangian (2.4) describes infinite chain of coupling fields (2.1). From (3.4), we learn that, if F υ (n) (3.1) has roots, then we will refer to fields (2.1) as reducible system. For the reducible system, Lagrangian (2.4) and gauge transformation (2.16) are factorized and describe finite and infinite decoupled chains of fields. From now on we assume that κ 0 is real-valued. Using definitions above-given in (3.2)-(3.4), we define (ir)reducible classically unitary systems as follows F υ (n) > 0 for all n = 0, 1, . . . , ∞, irreducible classically unitary system; (3.5) F υ (n r ) = 0 for some n r ∈ 0, 1, . . . , ∞ , F υ (n) > 0 for all n = 0, 1, . . . , ∞ and n = n r reducible classically unitary system.

Partition function and modified de Donder gauge condition
In this section, we are going to demonstrate that a partition function of the fermionic continuous spin field is equal to one. Namely, using gauge invariant Lagrangian (2.4), we find the following expression for the partition function Z of the fermionic continuous spin field where, in (4.2), a determinant is computed on a space of the Lorentz so(d, 1) algebra spin-(n + 1 2 ) Dirac field subject to γ-tracelessness constraint. Note also that in (4.1) we assume the convention D −2 (M 2 ) = 1, D −1 (M 2 ) = 1. From (4.1), we see immediately that the partition function of the fermionic continuous spin field is equal to one, Z = 1.
Alternatively, Z (4.1) can be cast into more convenient form by using general formula for the D n (M 2 ) with arbitrary M 2 (4.2), where D ⊥ n (M 2 ) appearing in (4.4) takes the same form as in (4.2), with assumption that the determinant is computed on space of the gamma-traceless and divergence-free spin-(n + 1 2 ) Dirac field of the Lorentz so(d, 1) algebra. For M 2 n as in (4.3), relation (4.4) takes the form Using (4.5), we see that Z −1 n (4.1) can be represented as . (4.6) Using (4.6), we find then, for fermionic continuous spin field in (A)dS and flat spaces, the same cancellation mechanism as for higher-spin fields in flat space [17]. Namely, from Z (4.1) and Z n (4.6), we see the cancellation of determinant of the physical spin-(n + 1 2 ) field and ghost determinant of spin-(n+ 3 2 ) field. Note that Z is equal to 1 without the use of special regularization procedure required for a computation of partition functions of higher-spin (A)dS gauge fields (see, e.g., Ref. [17]).
Representation for the partition function given in (4.1) can be obtained from Lagrangian (2.4) by using the well-known technique discussed in the earlier literature (see, e.g., Refs. [18]- [20]). Here, instead of a repetition of the well-known technicalities, we prefer to demonstrate how the expression for partition function of continuous spin fermionic field (4.1) leads to partition functions of massless, partial-massless and massive fermionic fields. Also we present our modified de Donder gauge for fermionic fields which allows us to obtain in a straightforward way the mass terms M 2 n (4.3) entering the determinants in (4.1),(4.2). Partial-massless and massless fields. As we said above, if κ 0 and µ 0 take values given in (3.40), (3.41), then field |ψ s+1,S appearing in (3.43) describes spin-(S + 1 2 ), depth-(S − s − 1) partialmassless field. Partition function for such field is obtained from (4.1) by considering contribution of Z n with n = s + 1, s + 2, . . . , S. Doing so, we get where Z n entering (4.7) takes the same form as in (4.1), while M 2 n is obtained by inserting µ 0 (3.41) into (4.3).