BRST approach to Lagrangian construction for bosonic continuous spin field
aa r X i v : . [ h e p - t h ] J un BRST approach to Lagrangian constructionfor bosonic continuous spin field
I.L. Buchbinder a,b,c ∗ , V.A. Krykhtin a † , H. Takata a ‡ a Department of Theoretical Physics, Tomsk State Pedagogical University,Kievskaya St. 60, 634061, Tomsk, Russia b National Research Tomsk State University,Lenin Av. 36, 634050 Tomsk, Russia c Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora,Campus Universit´ario-Juiz de Fora, 36036-900, MG, Brazil
Abstract
We formulate the conditions defining the irreducible continuous spin representa-tion of the four-dimensional Poincar´e group based on spin-tensor fields with dottedand undotted indices. Such a formulation simplifies analysis of the Bargmann-Wignerequations and reduces the number of equations from four to three. Using this for-mulation we develop the BRST approach and derive the covariant Lagrangian for thecontinuous spin fields.
Description of the irreducible representations of the Poincar´e and AdS groups play importantrole in the formulation of the higher spin field models (see e.g. the reviews [1], [2]). One ofsuch representations is representation with continuous spin.Continuous (or infinite) spin representation of the Poincar´e group [3–5] being masslesshas unusual properties such as an infinite number of degrees of freedom and appearance ofthe dimensional parameter µ in the conditions defining the irreducible representation (seefor the review [6]). Lagrangian for the bosonic field in d = 4 was first proposed in ref. [7]and its structure was analyzed in ref. [8]. Later Lagrangian for bosonic continuous fieldwas generalized for d > ∗ [email protected] † [email protected] ‡ [email protected]
1n the works [9] and in terms of two towers of traceless fields in the works [10] (see alsothe approaches to Lagrangian constriction in the works [11–16] for bosonic and fermionicfields and relation to the string theory in [17, 18]). Interaction of continuous spin fields withfinite spin massive fields was considered in the papers [19, 20]. Model of relativistic particlecorresponding to continuous spin field has been constructed in the recent work [21].In the present paper we develop the BRST approach to derive the Lagrangian for thecontinuous spin fields in four-dimensional Minkowski space. This approach is a direct gen-eralization of the general BRST construction which was used in our works for deriving theLagrangians for the free fields of different types in flat and AdS spaces (see e.g. [22–26] andthe references therein, see also the review [27]) . The crucial element of our approach is theimplementation of the two-component spinor description for the continuous spin fields. Wesuppose that the BRST construction in terms of spin-tensor fields, considered in the givenpaper, essentially simplifies an clarifies the derivation of the Lagrangian and its analysis.The paper is organized as follows. In the next section we consider a new, in comparisonwith ref. [3], possibility of realization for the spin momentum operators in terms of two-component spinors and obtain new equations for the field realizing the continuous spinirreducible representation of the Poincar´e group. After this we solve one of these equationsand in section 3 we construct Lagrangian on the base of the BRST method. Then we showthat after removing one auxiliary field and rescaling the other fields and the gauge parameterthe Lagrangian obtained exactly coincides with Lagrangian derived by Metsaev [10]. According to Bargmann and Wigner [3], the continuous spin field are characterized by thefollowing eigenvalues of the Casimir operators P Ψ = 0 W Ψ = µ Ψ . (1)In order to obtain these equations in explicit form we need the explicit expressions for theoperators entering into the Casimir operators. Such expressions for the operators depend onhow Ψ transforms under action of the Poincar´e group.In the works on continuous spin field it usually was done in the following way. Let usintroduce an auxiliary 4-dimensional vector w µ and define ϕ s ( x, w ) = ϕ µ ...µ s ( x ) w µ . . . w µ s , (2)where ϕ µ ...µ s ( x ) is a totally symmetric tensor field. Then one realizes the spin momentumoperator for ϕ s ( x, w ) as follows M µν = w µ i ∂∂w ν − w ν i ∂∂w µ . (3) The various applications of the BRST-BFV construction in the continuous spin field theory have beenstudied in refs. [9, 10, 14, 15, 19, 28, 29]. p Ψ( p, w ) = 0 , ( p ν w ν + µ )Ψ( p, w ) = 0 ,p ν ∂∂w ν Ψ( p, w ) = 0 , (cid:18) ∂∂w ν ∂∂w ν + 1 (cid:19) Ψ( p, w ) = 0 . (4)If the four equations (4) are satisfied, then the field Ψ will satisfy (1) and thus will describethe irreducible representation of the Poincare group with continuous spin.We want to turn attention that there is another possibility to realize the spin momentumoperator and corresponding representation. Instead of usual tensor fields (2) we consider thespin-tensor field ϕ a ...a n ˙ b ... ˙ b k ( x ) with n undotted and k dotted indices and define ϕ n,k ( x, ξ, ¯ ξ ) = ϕ a ...a n ˙ b ... ˙ b k ( x ) ξ a . . . ξ a n ¯ ξ ˙ b . . . ¯ ξ ˙ b k , (5)where we have introduced two auxiliary bosonic 2-dimensional spinors ξ a and ξ ˙ b of theLorentz group . In this representation the spin momentum operator looks like this M µν = σ abµν M ab − ¯ σ ˙ a ˙ bµν M ˙ a ˙ b , (6)where M ab = 12 ξ c ε ca ∂∂ξ b + 12 ξ c ε cb ∂∂ξ a = − i (cid:16) ξ a π b + ξ b π a (cid:17) , (7)¯ M ˙ a ˙ b = 12 ¯ ξ ˙ c ε ˙ c ˙ a ∂∂ ¯ ξ ˙ b + 12 ¯ ξ ˙ c ε ˙ c ˙ b ∂∂ ¯ ξ ˙ a = − i (cid:16) ¯ ξ ˙ a ¯ π ˙ b + ¯ ξ ˙ b ¯ π ˙ a (cid:17) (8) π a = − i ∂∂ξ a ¯ π ˙ a = − i ∂∂ ¯ ξ ˙ a . (9)One can prove that the operator (6) satisfies the commutation relation for the Lorentz groupgenerators.To construct the second Casimir operator for the spin-tensor representation one uses theexplicit expression for the spin operator (6) acting on the spin-tensor fields. In this case thesecond Casimir operator becomes W = − M ab ¯ M ˙ a ˙ b P a ˙ a P b ˙ b + 12 (cid:16) M ab M ab + ¯ M ˙ a ˙ b ¯ M ˙ a ˙ b (cid:17) P (10)After some transformations the second Casimir operator takes the form W = 12 ( ξσ µ ¯ ξ )(¯ π ¯ σ ν π ) P µ P ν + 12 ( ¯ ξ ¯ σ µ π )(¯ π ¯ σ ν ξ ) P µ P ν + 12 (cid:16) M ab M ab + ¯ M ˙ a ˙ b ¯ M ˙ a ˙ b + i ¯ ξ ˙ a ¯ π ˙ a (cid:17) P (11) The two-component spinors have been used for description of the representations with continuous spinin ref. [4]. However they were applied for the other aims. We use the notation as in book [30] ξ ¯ σ µ π )(¯ π ¯ σ ν ξ ) P µ P ν = ( ξσ µ ¯ ξ )(¯ π ¯ σ ν π ) P µ P ν + ¯ ξ ˙ a ¯ π ˙ a ( i + π a ξ a ) P (12)we can write the operator W in two equivalent forms W = ( ξσ µ ¯ ξ )(¯ π ¯ σ ν π ) P µ P ν + 12 (cid:16) M ab M ab + ¯ M ˙ a ˙ b ¯ M ˙ a ˙ b + ¯ ξ ˙ a ¯ π ˙ a ξ a π a (cid:17) P (13)or W = ( ¯ ξ ¯ σ µ π )(¯ π ¯ σ ν ξ ) P µ P ν + 12 (cid:16) M ab M ab + ¯ M ˙ a ˙ b ¯ M ˙ a ˙ b − ¯ ξ ˙ a ¯ π ˙ a π a ξ a (cid:17) P . (14)We will consider the irreducible representation with continuous spin on the fields Ψ( p, ξ, ¯ ξ ),depending on the momentum p µ and spin-tensor variables ξ a and ξ ˙ a . Let the field Ψ( p, ξ, ¯ ξ )satisfies the constraints p Ψ( p, ξ, ¯ ξ ) = 0 , (15)((¯ π ¯ σ ν π ) p ν + iµ )Ψ( p, ξ, ¯ ξ ) = 0 , (16)(( ξσ µ ¯ ξ ) p µ − iµ )Ψ( p, ξ, ¯ ξ ) = 0 . (17)Then on can show that conditions (1) are satisfied and hence the field Ψ( p, ξ, ¯ ξ ) describesthe irreducible representation with continuous spin. Thus we have obtained the equationson the field Ψ in the spin-tensor representation. Equations (15)–(17) are similar to the“modified Wigner’s equations” in the paper [16]. In the case under consideration analogof the fourth equation in (4) are resolved automatically due to the properties of the twocomponent spinors.We will construct the Lagrangian for the continuous spin field. To do that one shouldsomehow decompose Ψ( p, ξ, ¯ ξ ) in a series of ξ and ¯ ξ and get the spin-tensor fields. How-ever, one can see that, because of equation (17), Ψ( p, ξ, ¯ ξ ) such a direct decomposition isimpossible. Therefore we first solve (17) in the formΨ( p, ξ, ¯ ξ ) = δ (( ξσ µ ¯ ξ ) p µ − iµ ) ϕ ( p, ξ, ¯ ξ ) . (18)One can prove that if the field ϕ ( p, ξ, ¯ ξ ) obeys equations ∂ ϕ ( x, ξ, ¯ ξ ) = 0 , (19) (cid:18) ¯ σ µ ˙ aa ∂∂ξ a ∂∂ ¯ ξ ˙ a ∂∂x µ + µ (cid:19) ϕ ( x, ξ, ¯ ξ ) = 0 , (20)then the field Ψ( p, ξ, ¯ ξ ) will satisfy the rest equations (15) and (16). Here we have madeFourier transform from momentum p µ representation into the coordinates x µ representation.Equations (19) and (20) have a solution in the form of an expansion in ξ and ¯ ξϕ ( x, ξ, ¯ ξ ) = ∞ X n,k =0 √ n ! k ! ϕ a ...a n ˙ b ... ˙ b k ( x ) ξ a . . . ξ a n ¯ ξ ˙ b . . . ¯ ξ ˙ b k . (21)Since we are going to construct Lagrangian for real bosonic fields we will consider n = k case in (21). 4 Lagrangian construction
Following the general BRST approach in higher spin field theory we begin with realizationof the equations (19) and (20) in auxiliary Fock space.Let us introduce creation and annihilation operators h | ¯ c ˙ b = h | c a = 0 , ¯ a ˙ b | i = a a | i = 0 , h | i = 1with the following nonzero commutation relations[¯ a ˙ α , ¯ c ˙ β ] = δ ˙ α ˙ β , [ a α , c β ] = δ βα . The states in the auxiliary Fock space are defined as follows | ϕ i = ∞ X k,l =0 | ϕ kl i | ϕ kl i = 1 √ k ! l ! ϕ a ( k ) ˙ b ( l ) ( x ) c a ( k ) ¯ c ˙ b ( l ) | i . (22)We determine the Hermitian conjugation in the Fock space by the rule( a a ) + = ¯ c ˙ a (¯ c ˙ a ) + = a a (¯ a ˙ a ) + = c a ( c a ) + = ¯ a ˙ a Then the state which is Hermitin conjugate to state (22) is written as follows h ¯ ϕ | = ∞ X k,l =0 √ k ! l ! h | ¯ a ˙ a ( k ) a b ( l ) ¯ ϕ b ( l ) ˙ a ( k ) . (23)Let us introduce the following operators l = ∂ l = a a ∂ a ˙ b ¯ a ˙ b l +1 = − c b ∂ b ˙ a ¯ c ˙ a . (24)Here ∂ a ˙ b = σ µa ˙ b ∂ µ . One can show that these operators satisfy the commutation relation[ l +1 , l ] = ( N + ¯ N + 2) l , (25)where N = c a a a ¯ N = ¯ c ˙ a ¯ a ˙ a (26)All other commutators among these operators (24) vanish.One can show that in order for a state | ϕ i describe the continuous spin representation itis necessary that the following constraints on the vector | ϕ i will satisfied l | ϕ i = 0 ( l − µ ) | ϕ i = 0 (27)where k = l in (22) is assumed.Now we turn to construction of the BRST charge and the Lagrangian. Taking intoaccount that the Lagrangian is real, we should get the Hermitian BRST charge. However, the5ystem of constraints (27) is not invariant under the Hermitian conjugation. This situationis similar with BRST Lagrangian construction for free higher spin fields. Construction ofthe BRST charge for such a case was studied in works [22], [24] and we will follow theseworks. First of all we introduce the operator l +1 − µ and then add it to the set of constraints(27). Thus set of operators l , l − µ , l +1 − µ will be invariant under Hermitian conjugation.Moreover this set of operators will form an algebra with the only nonzero commutator (25)[ l +1 − µ, l − µ ] = ( N + ¯ N + 2) l . (28)Now we can apply the procedure described in the works [22], [24] and construct HermitianBRST charge on the base of operators l , l − µ , l +1 − µ . As a result we arrive at the HermitianBRST charge in the form Q = η l + η +1 ( l − µ ) + η ( l +1 − µ ) + η +1 η ( N + ¯ N + 2) P . (29)Here we have extended the Fock space by introducing η , η , η +1 which are the fermionicghost “coordinates” and P , P +1 , P which are their canonically conjugated ghost “momenta”respectively. These operators obey the anticommutation relations { η , P +1 } = {P , η +1 } = { η , P } = 1 (30)and act on the vacuum state as follows η | i = P | i = P | i = 0 . (31)They possess the standard ghost numbers, gh ( η i ) = − gh ( P i ) = 1, providing the property gh ( ˜ Q ) = 1.The operator (29) acts in the extended Fock space of the vectors | Φ i = | ϕ i + η P +1 | ϕ i + η +1 P +1 | ϕ i (32)and realizes the gauge transformations | Φ ′ i = | Φ i + Q | Λ i , (33)for the equation of motion Q | Φ i = 0 , (34)where | Λ i is the gauge parameter | Λ i = P +1 | λ i . (35)The fields | ϕ i , | ϕ i and and the gauge parameter | λ i in relations (32), (35) have similardecomposition like | ϕ i (22) with k = l . In case of µ = 0 the BRST charge (29) becomesBRST charge for the massless higher spin fields [31].6he equations of motion Q | Φ i = 0 and gauge transformations δ | Φ i = Q | Λ i in terms ofstates | ϕ i i and gauge parameter | λ i look like l | ϕ i − l +1 | ϕ i + µ | ϕ i = 0 (36) l | ϕ i − l +1 | ϕ i + ( N + ¯ N + 2) | ϕ i − µ | ϕ i + µ | ϕ i = 0 (37) l | ϕ i − l | ϕ i + µ | ϕ i = 0 (38) δ | ϕ i = l +1 | λ i − µ | λ i δ | ϕ i = l | λ i δ | ϕ i = l | λ i − µ | λ i (39)The Lagrangian for the continuous spin field is constructed in the framework of the BRSTapproach as follows (see e.g. [24]) L = Z dη h Φ | Q | Φ i == h ¯ ϕ | n l | ϕ i − l +1 | ϕ i o − h ¯ ϕ | n l | ϕ i − l +1 | ϕ i + ( N + ¯ N + 2) | ϕ i o − h ¯ ϕ | n l | ϕ i − l | ϕ i o + µ n h ¯ ϕ | ϕ i + h ¯ ϕ | ϕ i − h ¯ ϕ | ϕ i − h ¯ ϕ | ϕ i o (40)Lagrangian (40) consists of the sum of Lagrangians for massless bosonic fields plus µ -dependent terms responsible for the continuous spin.Now we rewrite the Lagrangian (40) in terms of the component fields. Using the equationof motion (37) we remove the field | ϕ i from the Lagrangian (40). Then calculating the “av-erage values” over Fock space vectors, converting the spin-tensor fields into traceless tensorfields and converting the spin-tensor gauge parameters into traceless tensor field parameterswe arrives at the Lagrangian L = ∞ X s =0 s ϕ µ ( s ) h ∂ ϕ µ ( s ) − s ∂ µ ∂ ν ϕ νµ ( s − − s − ∂ µ ∂ µ ϕ µ ( s − + µ s + 1) ( ϕ µ ( s ) − ϕ µ ( s ) ) + µ∂ ν ϕ νµ ( s ) − µ ∂ µ ϕ µ ( s − + µ s + 1 s + 1 ∂ µ ϕ µ ( s − i − ∞ X s =0 s ϕ µ ( s )2 h s + 3 s + 2 ∂ ϕ µ ( s ) + s s + 2 ∂ µ ∂ ν ϕ νµ ( s − + 2( s + 1) ∂ ν ∂ τ ϕ ντµ ( s ) + µ s + 1) ( ϕ µ ( s ) − ϕ µ ( s ) ) + µ s + 3 s + 2 ∂ ν ϕ νµ ( s ) − µ s + 1 s + 2 ∂ ν ϕ νµ ( s ) + µ ss + 1 ∂ µ ϕ µ ( s − i (41)and gauge transformations δϕ µ ( s ) = s ∂ µ s λ µ ( s − − s − η µ (2) ∂ ν λ νµ ( s − − µ λ µ ( s ) , (42) δϕ µ ( s ) = − s + 1) ∂ ν λ νµ ( s ) − µ λ µ ( s ) . (43)7e note that the set of the fields in Lagrangian (41) is the same like in the Metsaev’sLagrangian (2.14) in [10] in the case d = 4, m = 0. These Lagrangians will be exactly thesame if we make the redefinition of the fields ϕ µ ( n ) → A n φ a ( n ) I , ϕ µ ( n )2 → − A n φ a ( n ) II and gaugeparameters λ µ ( n ) → A n +1 ξ a ( n ) where A n = (2 n +1 n !) − / and also µ → κ . As a result weconclude that the BRST Lagrangian construction works perfectly for the continuous spinfields as well as for all other higher spin fields. We have developed the Lagrangian BRST construction for the continuous spin field theory. • We have reformulated the Wigner equations, defining the irreducible representationwith continuous spin, in terms of two auxiliary bosonic 2-component spinor variables.The spin operator (6) for such an representation has been introduced. The represen-tation is realized on fields satisfying the constraints (15), (16), (17). The number ofthe constraints turns out to be less than in case of the usually used representation interms of vectorial auxiliary variable. • The constraints are reformulated in terms of operators acting on the vectors of the aux-iliary Fock space. Extra operator has been introduced to provide the real Lagrangian.The algebra of all operastors has been calculated. • Hermitian BRST charge is constructed (29) for the continuous spin field theory, theLagrangian and gauge transformations in terms of the Fock space vectors (40), andin terms of traceless tensor fields (41), (42), (43) have been derived. The Lagrangiancoincides with Metsaev Lagrangian [10] after some redefinitions of the fields and gaugeparameters.Thus, the BRST Lagrangian construction, developed in refs. [22–26] is generalized for thecontinuous spin field.The results of the paper can be applied for deriving the Lagrangians for fermionic con-tinuous spin field and for supersymmetric continuous spin field theory. It would be alsointeresting to generalize these results for the continuous spin fields in the AdS space.
Acknowledgments
The authors are thankful to R.R. Metsaev for useful comments. I.L.B is appreciative toA.P. Isaev and S. Fedoruk for discussing the various aspects of the continuous spin fields.The research was supported in parts by Russian Ministry of Education and Science, projectNo. 3.1386.2017. The authors are also grateful to RFBR grant, project No. 18-02-00153 forpartial support. 8 eferences [1] X. Bekaert, S. Cnockaert, C. Iazeolla, M. Vasiliev, “Nonlinear higher spin theories onvarious dimensions”, arXiv:hep-th/0503128.[2] V. E. Didenko, E. D. Skvortsov, “Elements of Vasiliev theory”, arXiv:1401.2975 [hep-th].[3] V. Bargmann and E. P. Wigner, “Group Theoretical Discussion of Relativistic WaveEquations,” Proc. Nat. Acad. Sci. (1948) 211.[4] G. J. Iverson and G. Mack, “Quantum Fields and Interactions of Massless Particles:Spin Case ”, Ann. Phys. (1971) 211.[5] L. Brink, A. M. Khan, P. Ramond and X. Xiong, “Continuous spin representations of thePoincare and superPoincare groups,” J. Math. Phys. (2002) 6279, [hep-th/0205145].[6] X. Bekaert and E. D. Skvortsov, “Elementary particles with continuous spin,” Int. J.Mod. Phys. A (2017) no.23n24, 1730019, [arXiv:1708.01030 [hep-th]].[7] P. Schuster and N. Toro, “Continuous-spin particle field theory with helicity correspon-dence,” Phys. Rev. D (2015) 025023, [arXiv:1404.0675 [hep-th]].[8] V. O. Rivelles, “Gauge Theory Formulations for Continuous and Higher Spin Fields,”Phys. Rev. D (2015) no.12, 125035 [arXiv:1408.3576 [hep-th]].[9] R. R. Metsaev, “Continuous spin gauge field in (A)dS space,” Phys. Lett. B (2017)458, [arXiv:1610.00657 [hep-th]].[10] R. R. Metsaev, “BRST-BV approach to continuous-spin field,” Phys. Lett. B (2018)568, [arXiv:1803.08421 [hep-th]].[11] Y. M. Zinoviev, “Infinite spin fields in d = 3 and beyond,” Universe (2017) no.3, 63,[arXiv:1707.08832 [hep-th]].[12] M. V. Khabarov and Y. M. Zinoviev, “Infinite (continuous) spin fields in the frame-likeformalism,” Nucl. Phys. B (2018) 182, [arXiv:1711.08223 [hep-th]].[13] X. Bekaert, M. Najafizadeh and M. R. Setare, “A gauge field theory of fermionicContinuous-Spin Particles, Phys. Lett. B (2016) 320, [arXiv:1506.00973 [hep-th]].[14] R. R. Metsaev, “Fermionic continuous spin gauge field in (A)dS space,” Phys. Lett. B (2017) 135, [arXiv:1703.05780 [hep-th]].[15] R. R. Metsaev,, “Continuous-spin-mixed-symmetry fields in AdS(5),” J. Phys. A (2018) no.21, 2015401, [arXiv:1711.11007 [hep-th]].[16] M. Najafizadeh, “Modified Wigner equations and continuous spin gauge field,” Phys.Rev. D (2018) no.6, 065009, [arXiv:1708.00827 [hep-th]].917] G. K. Savvidy, “Tensionless strings: Physical Fock space and higher spin fields,” Int. J.Mod. Phys. A (2004) 3171 [hep-th/0310085].[18] J. Mourad, “Continuous spin particles from a string theory,” hep-th/0504118.[19] R. R. Metsaev, “Cubic interaction vertices for continuous-spin fields and arbitrary spinmassive fields,” JHEP (2017) 197, [arXiv:1709.08596 [hep-th]].[20] X. Bekaert, J. Mourad and M. Najafizadeh, “Continuous-spin field propagator andinteraction with matter,” JHEP (2017) 113, [arXiv:1710.05788 [hep-th]].[21] I. L. Buchbinder, S. Fedoruk, A. P. Isaev, A. Rusnak, “Model of massless relativisticparticle with continuous spin and its twistorial description”, [arXiv:1805.09706 [hep-th]].[22] I. L. Buchbinder, A. Pashnev, M. Tsulaia, “Lagrangian formulation of the masslesshigher integer spin fields in the AdS background”, Phys, Lett. B (2001) 338,[arXiv:hep-th/0109067].[23] I. L. Buchbinder, V. A. Krykhtin and A. Pashnev, “BRST approach to Lagrangianconstruction for fermionic massless higher spin fields,” Nucl. Phys. B (2005) 367,[arXiv:hep-th/0410215].[24] I. L. Buchbinder and V. A. Krykhtin, “Gauge invariant Lagrangian construction formassive bosonic higher spin fields in D dimensions,” Nucl. Phys. B (2005) 537,[hep-th/0505092].[25] I. L. Buchbinder, V. A. Krykhtin, L. L. Ryskina and H. Takata, “Gauge invariantLagrangian construction for massive higher spin fermionic fields,” Phys. Lett. B (2006) 386, [hep-th/0603212].[26] I. L. Buchbinder, V. A. Krykhtin and P. M. Lavrov, “Gauge invariant Lagrangianformulation of higher spin massive bosonic field theory in AdS space,” Nucl. Phys. B (2007) 344, [hep-th/0608005].[27] A. Fotopoulos, M. Tsulaia, “Gauge Invariant Lagrangians for Free and InteractingHigher Spin Fields. A Review of the BRST formulation”, Int.J.Mod.Phys. A (2009)1 [arXiv:0805.1346 [hep-th]].[28] A. K. H. Bengtsson, “BRST Theory for Continuous Spin,” JHEP (2013) 108,[arXiv:1303.3799 [hep-th]].[29] K. B. Alkalaev and M. A. Grigoriev, “Continuous spin fields of mixed-symmetry type,”JHEP (2018) 030, [arXiv:1712.02317 [hep-th]].[30] I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of Supersymmetry and Super-gravity, Or a Walk Through Superspace , IOP, Bristol, 1995 (Revised Edition 1998).[31] I. L. Buchbinder and K. Koutrolikos, JHEP1512