Constrained BRST-BFV and BRST-BV Lagrangians for half-integer HS fields on R 1,d−1
CConstrained BRST-BFV and BRST-BVLagrangians for half-integer HS fields on R ,d − ∗ Alexander A. Reshetnyak † Laboratory of Computer-Aided Design of Materials, Institute ofStrength Physics and Materials Science of SB RAS, 634055 Tomsk, Russia
Abstract
Gauge invariant Lagrangian descriptions of irreducible and reducible half-integerhigher-spin mixed-symmetric massless and massive representations of the Poincaregroup with off-shell algebraic constraints are constructed within a metric-like formula-tion in a d -dimensional flat space-time on the basis of a suggested constrained BRSTapproach. A Lorentz-invariant resolution of the BRST complex within the constrainedBRST formulations produces a gauge-invariant Fang-Fronsdal Lagrangian entirely interms of the initial triple gamma-traceless spin-tensor field Ψ ( µ ) n with gamma-tracelessgauge parameter. The triplet and quartet formulations are derived. The minimal(un)constrained BRST–BV actions for above formulations are obtained, from proposedconstrained BRST–BV approach to be by appropriate tools to construct interactingconstrained Lagrangians. Many modern issues of high-energy physics are related to higher-spin (HS) field theory, re-maining by the part of the LHC experiment program. The tensionless limit of superstringtheory [1] with help of BRST operator includes an infinite set of HS fields with integer andhalf-integer generalized spins and incorporates HS field theory into superstring theory andturns it into a method of studying the classical and quantum structure of the latter (forthe present status of HS field theory, see the reviews [2], [3], [4]). Whereas (un)constrainedBRST-BFV (see original papers to quantize constrained dynamical systems [5], [6]) andconstrained BRST-BV approaches to construct respectively gauge-invariant Lagrangian for-mulations (LFs) and BV field-antifield actions for integer HS fields on constant curvaturespace-times with(out) off-shell holonomic constraints are known, see e.g. [7]. [8], [9], [10]and [11], [12], [13] constrained BRST-BFV and BRST-BV methods to be applied to thesame aims for the half-integer HS fields have not been till developed. The paper suggestsconstrained BRST-BFV and BRST-BV approaches for construction of LFs and BRST–BVactions in the minimal sector of the field-antifield formalism for free (ir)reducible Poincaregroup representations with half-integer spins in a flat R ,d − -space-time subject to an arbi-trary Young tableaux (YT) with k rows, Y ( s , s , ..., s k ), with spin s = ( n + , ..., n k + ) ∗ Talk presented at SQS’17, 31 July 05 August, 2017, at JINR, Dubna, Russia † [email protected] a r X i v : . [ h e p - t h ] O c t or n ≥ n ≥ ... ≥ n k [14], in a metric-like formalism (for the study of the LFs in themetric-like and the frame-like formalisms beyond the BRST methods, see e.g. [15] and [16],[17]). The latter constrained BRST–BV action presents a natural ground for the procedureof consistent construction of the interacting LFs for such HS fields.The paper is based on the research [18], [19] and organized as follows. In Section 2, wesuggest the constrained BRST–BFV LFs for half-integer mixed-symmetric (MS) HS fields.In Section 3 a new formalism is applied for the case of totally-symmetric (TS), s = n + ,HS field in various representations. The construction of a minimal field-antifield actions ona base of natural extension of BRST–BFV approach up to a constrained BRST-BV methodfor half-integer HS fields is considered in Section 4.The convention η µν = diag (+ , − , ..., − ) for the metric tensor, with the Lorentz indices µ, ν = 0 , , ..., d −
1, the relations { γ µ , γ ν } = 2 η µν for the Dirac matrices γ µ ,and the notation (cid:15) ( A ), [ gh H , gh L , gh tot ]( A ) for the respective values of Grassmann parity, BFV, gh H , BV, gh L and total, gh tot = gh H + gh L , ghost numbers of a quantity A are used. The supercommutator[ A, B } of quantities A, B with definite values of Grassmann parity is given by [
A , B } = AB − ( − (cid:15) ( A ) (cid:15) ( B ) BA . There exists two equivalent ways of derivation constrained BRST-BFV approach for LFsfor half-integer HS fields [18], first, from unconstrained BRST-BFV method, developed forarbitrary half-integer HS fields in Minkowski space R ,d − [20] (see therein for the referenceson unconstrained BRST-BFV approach for half-integer HS fields), second, in self-consistentway. We consider here in details the second possibility. To do so remind, a massless half-integer spin irreducible representation of the Poincare group in R ,d − is described by arank- (cid:80) ki =1 n i spin-tensor field Ψ ( µ ) n ,..., ( µ k ) nk ≡ Ψ µ ...µ n ,...,µ k ...µ knk A with generalized spin s (forsuppressed Dirac index A ), subject to a YT, Y ( s , ..., s k ) with k rows of length n , ..., n k .The field (being symmetric with respect to permutations of each type of Lorentz indices µ i ) satisfies differential (Dirac) equation (1) and algebraic equations ( γ -traceless and mixed-antisymmetry ones) (2) : ıγ µ ∂ µ Ψ ( µ ) n ,..., ( µ k ) nk = 0 , (1) γ µ ili Ψ ( µ ) n ,..., ( µ k ) nk = 0 , Ψ ( µ ) n ,..., { ( µ i ) ni , ..., µ j ... (cid:124) (cid:123)(cid:122) (cid:125) µ jlj } ...µ jnj ,... ( µ k ) nk = 0 , (2)(for 1 ≤ l i ≤ n i , ≤ i ≤ k , i < j ) where the underlined figure bracket means that theindices inside do not take part symmetrization.Equivalently, the relations for general state (Dirac spinor) | Ψ (cid:105) from Fock space H gener-ated by k pairs of bosonic (symmetric case) oscillators a iµ i ( x ) , a j + ν j ( x ): [ a iµ i , a j + ν j ] = − η µ i ν j δ ij : t | Ψ (cid:105) = t i | Ψ (cid:105) = t rs | Ψ (cid:105) = 0 , g i Ψ (cid:105) = ( n i + d ) | Ψ (cid:105) (3)for (cid:0) t , t i , t rs , g i (cid:1) = (cid:0) − i ˜ γ µ ∂ µ , ˜ γ µ a µi , a + rµ a µs , − { a µi , a + iµ } (cid:1) , r < s. (4) | Ψ (cid:105) = ∞ (cid:88) n =0 n (cid:88) n =0 · · · n k − (cid:88) n k =0 ı (cid:80) i n i n ! × ... × n k ! Ψ ( µ ) n ,..., ( µ k ) nk k (cid:89) i =1 n i (cid:89) l i =1 a + µ ili i | (cid:105) , (5)with the generalized spin constraints imposed on | Ψ (cid:105) in terms of number particle operator g i describe the irreducible massless of spin s = n + Poincare group representation. In (4) we2ave used of d + 1 Grassmann-odd gamma-matrix-like objects { ˜ γ, ˜ γ µ } to be equivalent to thestandard ones: γ µ , whose explicit realization differs in even, d = 2 N , and odd, d = 2 N + 1, N ∈ N dimensions (see for details [18]). :For even-valued dimension we have { ˜ γ µ , ˜ γ ν } = 2 η µν , { ˜ γ µ , ˜ γ } = 0 , ˜ γ = − , so that γ µ = ˜ γ µ ˜ γ, . (6)whereas for odd-valued one, the second and third relations in (6) are changed on [˜ γ µ , ˜ γ ] = 0,˜ γ = 1 with unchanged others. The set of primary constraints { t , t i , t ij , g i } , (cid:15) ( t ) = (cid:15) ( t i ) = 1, (cid:15) ( t ij ) = (cid:15) ( g i ) = 0 will be closed with respect to the [ , } –multiplication if we add to themdivergentless, l i = (1 / t , t i } , traceless, l ij = (1 / t i , t j } , i ≤ j and D’alamber operators, l = − t : ( l , l i , l ij ) = (cid:0) ∂ µ ∂ µ , − ia iµ ∂ µ , a µi a jµ (cid:1) . (7)The reality of the Lagrangian with consistent off-shell holonomic constraints requires aclosedness for subset of differential constraints, o A , with respect to the appropriate hermitianconjugation defined by means of odd scalar product in H : (cid:104) ˜Φ | Ψ (cid:105) = (cid:90) d d x ∞ (cid:88) n =0 n (cid:88) n =0 · · · n k − (cid:88) n k =0 ∞ (cid:88) p =0 p (cid:88) p =0 · · · p l − (cid:88) p l =0 ı (cid:80) i n i ( − ı ) (cid:80) j p j n ! × ... × n k ! p ! × ... × p l ! ×(cid:104) | l (cid:89) j =1 p j (cid:89) m j =1 a ν jmj j Φ +( ν ) p ,..., ( ν l ) pl ˜ γ Ψ ( µ ) n ,..., ( µ k ) nk k (cid:89) i =1 n i (cid:89) l i =1 a + µ ili i | (cid:105) = ∞ (cid:88) n =0 n (cid:88) n =0 · · · n k − (cid:88) n k =0 k (cid:89) i =1 ( − n i n i ! (cid:90) d d x Φ +( µ ) n ,..., ( µ k ) nk ˜ γ Ψ ( µ ) n ,..., ( µ k ) nk , (8)that means the set: o A = { t , l , l i , l + i } (for l + i = − ia + iµ ∂ µ ) composes the first-class con-straints subsystem. The holonomic constraints t i , t rs itself generate the superalgebra oftotal set of constraints { o a } = { t i , t rs , l lm } in H . The algebraically independent subset of { o ¯ a } ⊂ { o a } is given by: { o ¯ a } = { t , t , t , t , . . . , t ( k − k } . (9)As the result the superalgebra A fc ( Y ( k ) , R ,d − ) to be named as constrained half-integer HSsymmetry algebra in Minkowski space with a YT having k rows with off-shell set of algebraicconstraints { t i , t rs } and g i appears by necessary objects to construct constrained LF forHS fields of spin s . The nilpotent constrained BRST operator for the system of { o A } inthe Hilbert space H c , H c = H ⊗ H o A gh , BRST-extended independent algebraic constraints,constrained spin operator determined as Q c ( o A ) = q t + η l + η + i l i + l i + η i + ı (cid:0)(cid:88) l η + l η l − q (cid:1) P , (10) (cid:0) (cid:98) T i , (cid:98) T rs , (cid:98) σ ic ( g ) (cid:1) = (cid:0) t i + C A t BiA P B , t rs + C A t BrsA P B , g i + C A g iBA P B (cid:1) + o ( CP ) , (11)with gh H ( Q c , (cid:98) T i , (cid:98) T rs , (cid:98) σ ic ( g )) = (1 , , , C A ; P B ) = ( q , η , η + i , η i ; p , P , P j , P + j ) with[ q , p ] = { η , P } = ı , { η i , P + j } = δ ij ] should satisfy to the consistency conditions:[ Q c , (cid:98) T i } = 0 , [ Q c , (cid:98) T rs } = 0 , [ Q c , (cid:98) σ ic ( g ) } = 0 , (12)3nown [18] as the generating equations for superalgebra of the constrained BRST, Q c spinoperators (cid:98) σ ic ( g ) and extended off-shell constraints (cid:98) T i , (cid:98) T rs . The exact solutions of (12) forunknown (cid:98) σ ic ( g ), (cid:98) T i , (cid:98) T rs exists in the form: C A (cid:16) t BiA , t
BrsA , g iBA (cid:17) P B = (cid:16) − ıη i p − q P i , − η + r P s − P + r η s , η + i P i − η i P + i (cid:17) . (13)Presenting the general state vector | χ c (cid:105) ∈ H c (for ( gh H , gh L ) | Ψ( a + i ) n b n f ;( n ) fi ( n ) pj (cid:105) = (0 , η i , P i , p , P ) | (cid:105) = 0): | χ c (cid:105) = (cid:88) n q n b η n f (cid:89) i,j ( η + i ) n fi ( P + j ) n pj | Ψ( a + i ) n b n f ;( n ) fi ( n ) pj (cid:105) . (14)and decomposing H c as H c = lim M →∞ ⊕ Ml = − M H lc for gh H ( | χ lc (cid:105) ) = − l , | χ lc (cid:105) ∈ H lc from theBRST condition, Q c | χ c (cid:105) = 0, off-shell constraints, (cid:0) (cid:98) T i , (cid:98) T rs (cid:1) | χ c (cid:105) = 0 and (cid:98) σ ic ( g ) | χ c (cid:105) = R i | χ c (cid:105) for some R i ∈ R . we have the spectral problem analogous to one for unconstrained case [20]: Q c | χ lc (cid:105) = δ | χ l − c (cid:105) , (cid:98) σ ic | χ lc (cid:105) = (cid:0) m i + d − (cid:1) | χ lc (cid:105) , ( (cid:15), gh H ) ( | χ lc (cid:105) ) = ( l + 1 , − l ) , (15) (cid:0) (cid:98) T i , (cid:98) T rs (cid:1) | χ lc (cid:105) = 0 , l = 0 , , ..., s c , (16)with | χ − c (cid:105) ≡ δ | χ s c c (cid:105) = 0 for some s c . Because of the representation (14) the physicalstate | Ψ (cid:105) (5) is contained in | χ c (cid:105) = | χ c (cid:105) = | Ψ (cid:105) + | Ψ Ac (cid:105) for | Ψ Ac (cid:105)| C =0 = 0.The system (15), (16) is compatible, due to the closedness of the superalgebra { Q c , (cid:98) σ ic , (cid:98) T i , (cid:98) T rs } . Therefore, its resolution for the joint set of proper eigen-vectors permits one, first, todetermine from the middle set and the spin and ghost numbers distributions for | χ lc (cid:105) : n i = p i + n fi + n pi , i = 1 , . . . , k, | χ lc (cid:105) ( n ) k : n b + n f + (cid:88) i (cid:0) n fi − n pi (cid:1) = − l. (17)that m i = n i (with n i from s = n + ) and proper eigen-vectors | χ lc (cid:105) ( n ) k . The solution ofthe rest equations is written as the second-order equations of motion and sequence of thereducible gauge transformations (18) with off-shell constraints (19): Q c | χ c (cid:105) ( n ) k = 0 , δ | χ c (cid:105) ( n ) k = Q c | χ c (cid:105) ( n ) k , ... , δ | χ s c − c (cid:105) ( n ) k = Q c | χ s c c (cid:105) ( n ) k , δ | χ s c c (cid:105) ( n ) k = 0 , (18) (cid:16) (cid:98) T i , (cid:98) T rs (cid:17) | χ lc (cid:105) ( n ) k = 0 , l = 0 , , ..., s c , for s c = k. (19)The corresponding BRST-like constrained gauge-invariant action (as for integer HS field) S (2) c | ( n ) k = (cid:90) dη n ) k (cid:104) ˜ χ c | Q c | χ c (cid:105) ( n ) k , (20)contains second order operator l , but less terms in comparison with its unconstrained ana-log [20]. Repeating the procedure of the removing the dependence on l , η , q from theBRST operator Q c (10) and from the whole set of the vectors | χ lc (cid:105) ( n ) k as it was done forunconstrained case by means of partial gauge-fixing we come to the: Statement : The first-order constrained gauge-invariant Lagrangian formulation for half-integer HS field, Ψ ( µ ) n ,..., ( µ k ) nk ( x ) with generalized spin ( s ) k = ( n ) k +( , ..., ), is determinedby the action, S c | ( n ) k = (cid:0) ( n ) k (cid:104) ˜ χ | c | ( n ) k (cid:104) ˜ χ | c | (cid:1) (cid:18) t ∆ Q c ∆ Q c t η + i η i (cid:19) (cid:18) | χ | c (cid:105) ( n ) k | χ | c (cid:105) ( n ) k (cid:19) for ∆ Q c = η + i l i + l + i η i , (21)4nvariant with respect to the sequence of the reducible gauge transformations (for s c − k − δ (cid:32) | χ l (0)0 | c (cid:105) ( n ) k | χ l (1)0 | c (cid:105) ( n ) k (cid:33) = (cid:18) ∆ Q c t η + i η i t ∆ Q c (cid:19) (cid:32) | χ l +1(0)0 | c (cid:105) ( n ) k | χ l +1(1)0 | c (cid:105) ( n ) k (cid:33) , δ (cid:32) | χ k (0)0 | c (cid:105) ( n ) k | χ k (1)0 | c (cid:105) ( n ) k (cid:33) = 0 (22)(for l = − , , ..., k − | χ − m )0 | c (cid:105) = 0, m = 0 ,
1) with off-shell algebraically independentBRST-extended constraints imposed on the whole set of field and gauge parameters: (cid:98) T i (cid:16) | χ l (0) c (cid:105) ( n ) k + q | χ l (1) c (cid:105) ( n ) k (cid:17) = 0 , (cid:98) T rs | χ l ( m ) c (cid:105) ( n ) k = 0 l = 0 , , ..., k ; m = 0 , . (23)The first constraints in q -independent form read t i | χ l (0)0 | c (cid:105) − η i | χ l (1)0 | c (cid:105) = 0 , t i | χ l (1)0 | c (cid:105) − P i | χ l (0)0 | c (cid:105) = 0 , P i | χ l (1)0 | c (cid:105) = 0 . (24)In the massive case for d = 2 N , N ∈ N we should to add k pairs of additional evenoscillators in differential constraints: ( L i , L + i ) = ( l i + mb i , l + i + mb + i ) and in decompositionfor | χ l (1)0 | c (cid:105) (14) in accordance with [18]. For odd dimensions, d = 2 N + 1 the Lagrangianformulation for massive HS fields may be extrapolated from one given in even dimension interms of the ghost-independent or spin-tensor forms, which depends only on the standardGrassmann-even matrices γ µ , and does not depend on ˜ γ µ , ˜ γ , due to the presence of thelatter only as even degrees inside the Lagrangian, and due to the homogeneity of the gaugetransformations w.r.t. ˜ γ (see [18] for details). In [18] it is shown the constrained Lagrangianformulation (21)–(23) is equivalent to unconstrained one for the same HS field and thereforeequivalent to the dynamics to be determined by the initial irreps conditions (1), (2). We demonstrate the above general results on the example of TS ( k = 1) spin-tensor field,Ψ ( µ ) n of spin n + , which is subject to Dirac (1) and γ -traceless equation only from (2) andtherefore is described by 2 Grassman-odd operators t = − ı ˜ γ µ ∂ µ , t = ˜ γ µ a µ ( a µ ≡ a µ ) and g ≡ g = − { a µ , a + µ } acting on the basic vector | Ψ (cid:105) = ∞ (cid:88) n =0 ı n n ! Ψ ( µ ) n a + µ . . . a + µ n | (cid:105) . (25)with respective proper eigen-values (0 , , ( n + d )).The corresponding nilpotent constrained BRST operator for the differential first-class { t , l , l , l +1 } , off-shell independent BRST extended constraint (cid:98) T and constrained spin op-erator (cid:98) σ c ( g ) in H c have the form: Q c = q t + η l + η +1 l + l +1 η + ı (cid:0) η +1 η − q (cid:1) P , (26) (cid:8) (cid:98) T , (cid:98) σ c ( g ) (cid:9) = (cid:8) t − ıη p − q P , g + η +1 P − η P +1 (cid:9) (27)whose algebra satisfy to the relations (12). The first-order constrained irreducible gauge-invariant LF for the field, Ψ ( µ ) n are given by the relations (21), (22) for l = − , (cid:98) T imposed on the fields, | χ m | c (cid:105) n ,5 = 0 , | χ | c (cid:105) n according to (23), (24) for | χ | c (cid:105) n ≡
0. The fieldvectors and gauge parameter being proper for (cid:98) σ c ( g ) have the decomposition in ghosts η +1 , P +1 : | χ | c (cid:105) n = | Ψ (cid:105) n + η +1 P +1 | χ (cid:105) n − = | Ψ (cid:105) n + ı n − ( n − η +1 P +1 χ ( µ ) n − n − (cid:89) k =1 a + µ k | (cid:105) , (28) (cid:0) | χ | c (cid:105) n , | χ | c (cid:105) n (cid:1) = P +1 (cid:0) ˜ γ | χ (cid:105) n − , | ξ (cid:105) n − (cid:1) = ı n − ( n − P +1 (cid:0) ˜ γχ ( µ ) n − , ξ ( µ ) n − (cid:1) n − (cid:89) k =1 a + µ k | (cid:105) , (29)The constraints (24) are resolved as the γ -traceless constraint for the gauge parameter ξ ( µ ) n − and triple γ -traceless one for Ψ ( µ ) n : (cid:0) ( t ) | Ψ (cid:105) n , t | ξ (cid:105) n − (cid:1) = 0 , ˜ γ | χ (cid:105) n − = t | Ψ (cid:105) n , | χ (cid:105) n − = − ( t ) | Ψ (cid:105) n (30)and therefore, (cid:81) i =1 γ µ i Ψ ( µ ) n = 0, γ µ ξ ( µ ) n − = 0.The action (21)) and the gauge transformations in terms of independent field vector | Ψ (cid:105) n in the ghost-free form look as S c | ( n ) = n (cid:104) ˜Ψ | (cid:0) t − ( t +1 ) t t − t +1 t t + l +1 t + t +1 l + t +1 l +1 t + ( t +1 ) l t (cid:1) | Ψ (cid:105) n , (31) δ | Ψ (cid:105) n = l +1 | ξ (cid:105) n − . (32)The gauge invariance for the action S c | ( n ) is easily checked with use of the Noether identity: δ S c | ( n ) ←− δδ | ξ (cid:105) = n (cid:104) ˜Ψ | (cid:0) t − ( t +1 ) t t − t +1 t t + l +1 t + t +1 l + t +1 l +1 t + ( t +1 ) l t (cid:1) l +1 = 0 , (33)modulo the operators L ( t +1 , t , l , l +1 ) t vanishing when acting on the γ -traceless vectors. Thevariational derivative of the functional δ S c | ( n ) = (cid:104) ˜Ψ | L ( t , t , ... ) | ξ (cid:105) + (cid:104) ˜ ξ | L + ( t , t , ... ) | Ψ (cid:105) (withthe kernel L ( t , t , ... ) in (33)) with respect to the vector | ξ (cid:105) was introduced above.In the spin-tensor form the action and the gauge transformations take the familiar form[21] with accuracy up to the common coefficient ( n !) − : S c | ( n ) (Ψ) = ( − n (cid:90) d d x Ψ ( ν ) n (cid:110) − ıγ µ ∂ µ Ψ ( ν ) n + n ( n − η ν n − ν n ( ıγ µ ∂ µ ) η µ n − µ n Ψ ( ν ) n − µ n − µ n (34) − nγ ν n ( ıγ µ ∂ µ ) γ µ n Ψ ( ν ) n − µ n + n ( ı∂ ν n ) γ µ n Ψ ( ν ) n − µ n + n ( ı∂ µ n ) γ ν n Ψ ( ν ) n − µ n − n ( n − (cid:16) γ ν n − ( ı∂ ν n ) η µ n − µ n Ψ ( ν ) n − µ n − µ n + η ν n − ν n γ µ n − ( ı∂ µ n )Ψ ( ν ) n − µ n − µ n (cid:17)(cid:111) , = ( − n (cid:90) d d x (cid:110) Ψ p/ Ψ − n ( n − (cid:48)(cid:48) p/ Ψ (cid:48)(cid:48) + n Ψ (cid:48) p/ Ψ (cid:48) − n Ψ · p Ψ (cid:48) − n Ψ (cid:48) p · Ψ (35)+ n ( n − (cid:16) Ψ (cid:48) · p Ψ (cid:48)(cid:48) + Ψ (cid:48)(cid:48) p · Ψ (cid:48) (cid:17)(cid:111) ,δ Ψ ( µ ) n = − n (cid:88) i =1 ∂ µ i ξ µ ...µ i − µ i +1 ...µ n , (36)where each term in (34) and (35) corresponds to the respective summand in (31), whereas forthe last expression we have used Fang-Fronsdal notations [21] with identifications, p µ = − ı∂ µ , − ıγ µ ∂ µ = p/ and p · Ψ = p µ Ψ ( µ ) n . 6he triplet formulation to describe Lagrangian dynamic of the field Ψ ( µ ) n with help of thetriplet of spin-tensors Ψ ( µ ) n , χ ( µ ) n − , χ ( µ ) n − and gauge parameter ξ ( µ ) n − subject to the off-shell 3 constraints on the field vectors, | Ψ (cid:105) n , | χ (cid:105) n − , | χ (cid:105) n − (30) and γ -traceless constrainton | ξ (cid:105) n − in the ghost-independent form S c | ( n ) (Ψ , χ , χ ) = n (cid:104) ˜Ψ | t | Ψ (cid:105) n − n − (cid:104) ˜ χ | t | χ (cid:105) n − + n − (cid:104) ˜ χ | ˜ γt ˜ γ | χ (cid:105) n − − (cid:0) n − (cid:104) ˜ χ | ˜ γ (cid:8) l | Ψ (cid:105) n − l +1 | χ (cid:105) n (cid:9) + h.c. (cid:1) , (37) δ (cid:0) | Ψ (cid:105) n , | χ (cid:105) n − , | χ (cid:105) n − (cid:1) = (cid:0) l +1 , l , ˜ γt (cid:1) | ξ (cid:105) n − (38)coincides with one suggested in [22]. Without off-shell constraints the triplet formulationdescribes the free propagation of couple of massless particles with respective spins ( n + ),( n − ),..., . It was shown in [23] that this formulation maybe described within unconstrainedquartet formulation with additional, to the triplet, compensator field vector | ς (cid:105) n − , whosegauge transformation is proportional to the constraint on | ξ (cid:105) n − : δ | ς (cid:105) n − = ˜ γt | ξ (cid:105) n − and thewhole off-shell constraints (30) are augmented by the terms proportional to | ς (cid:105) to providetheirs total gauge invariance with respect to (38) and above gauge transformations for | ς (cid:105) : (cid:8) t | Ψ (cid:105) − ˜ γ | χ (cid:105) + l +1 ˜ γ | ς (cid:105) , | χ (cid:105) + t ˜ γ | χ (cid:105) + t ˜ γ | ς (cid:105) , t | χ (cid:105) + l ˜ γ | ς (cid:105) (cid:9) = (cid:8) , , (cid:9) . (39)Introducing the respective Lagrangian multipliers: fermionic n − (cid:104) ˜ λ | , bosonic n − (cid:104) ˜ λ | , fermio-nic n − (cid:104) ˜ λ | with trivial gauge transformations, the equations (39) and theirs hermitian con-jugated may be derived from the action functional S add | ( n ) ( λ ) = n − (cid:104) ˜ λ | (cid:0) t | Ψ (cid:105) n − ˜ γ | χ (cid:105) n − + l +1 ˜ γ | ς (cid:105) n − (cid:1) + n − (cid:104) ˜ λ | (cid:0) | χ (cid:105) n − (40)+ 12 t ˜ γ | χ (cid:105) n − + 12 t ˜ γ | ς (cid:105) n − (cid:1) + n − (cid:104) ˜ λ | (cid:0) t | χ (cid:105) n − + l ˜ γ | ς (cid:105) n − (cid:1) + h.c., so that, the gauge-invariant functional, S ( n ) = S c | ( n ) (Ψ , χ , χ ) + S add | ( n ) ( λ ) (41)determines the unconstrained LF for massless spin-tensor of spin ( n + ) in terms of quartetof spin-tensor fields Ψ ( µ ) n , χ ( µ ) n − , χ ( µ ) n − , ς ( µ ) n − with help of three Lagrangian multipliers λ ( µ ) n − i i , i = 1 , , Here we follow to the research [19], where the constrained BRST-BV approach to formulateminimal BV action for integer and half-integer HS fields on R .d − is suggested and, in part, to[24] for mixed-antisymmetric integer higher-spin fields. First of all, we weaken the vanishingof gh L on the component spin-tensors in the decomposition (14), when considering insteadof field vector | χ c (cid:105) ∈ H c the generalized field-antifield vector | χ g | c (cid:105) ∈ H g | c = H g ⊗ H o A gh with Z -grading for H g | c = lim M →∞ ⊕ Ml = − M H lg | c for gh tot ( | χ lg | c c (cid:105) ) = − l , | χ lg | c (cid:105) ∈ H lg | c . For simplicity,we consider the case of TS HS fields. The total configuration space in the minimal sector7ontains (with off-shell constraints) in addition to the triplet Ψ ( µ ) n , χ | ( µ ) n − , χ ( µ ) n − the ghostspin-tensor field | C c (cid:105) n introduced by the rule: ξ ( µ ) n − ( x ) = C ( µ ) n − ( x ) µ = ⇒ | χ | c (cid:105) n = | C c (cid:105) n µ, (42)with ( (cid:15), gh tot , gh H , gh L ) (cid:8) C ( µ ) n − , | C c (cid:105) n (cid:9) = (cid:8) (0 , , , , (1 , , − , (cid:9) , (43)which due to the vanishing of the total ghost number and Grassmann parity may be combinedwith | χ c (cid:105) n in generalized field vector : | χ | c (cid:105) n = | χ c (cid:105) n + | C c (cid:105) n , (cid:0) (cid:15), gh tot (cid:1) | χ | c (cid:105) = (1 , . (44)with untouched | χ | c (cid:105) n = | χ | c (cid:105) n , (cid:0) (cid:15), gh tot (cid:1) | χ | c (cid:105) = (0 , − ∗ ( µ ) n , χ ∗ ( µ ) n − , χ ∗ ( µ ) n − , C ∗ ( µ ) n − with( (cid:15), gh L )Ψ ∗ = ( (cid:15), gh L ) χ ∗ = ( (cid:15), gh L ) χ ∗ = (0 , −
1) and ( (cid:15), gh L ) C ∗ = (1 , −
2) (45)are combined into generalized antifield vectors as follows: | χ ∗ | c (cid:105) n = | χ ∗ c (cid:105) n + | C ∗ c (cid:105) n = ˜ γ (cid:16) | Ψ ∗ ( a + ) (cid:105) n + P +1 η +1 | χ ∗ ( a + ) (cid:105) n − (cid:17) + ˜ γη +1 | C ∗ (cid:105) n − , (46) | χ ∗ | c (cid:105) n = | χ ∗ c (cid:105) n = η +1 | χ ∗ ( a + ) (cid:105) n − , (cid:0) (cid:15), gh tot (cid:1) | χ ∗ e gen | c (cid:105) = (1 , e − , e = 0 , (cid:0) gh L , gh H (cid:1) | A ∗ (cid:105) = ( − , , for A ∈ { Ψ , χ, χ } , (cid:0) gh L , gh H (cid:1) | C ∗ (cid:105) = ( − ,
0) (48)for the ghost- and ˜ γ - independent antifield vectors | Ψ ∗ ( a + ) (cid:105) n , | χ ∗ ( a + ) (cid:105) n − , | χ ∗ ( a + ) (cid:105) n − , | C ∗ (cid:105) n − having the decompositions in powers of a + µ as for the respective field vectors (25),(28), (29) and for the ghost vector | C ( a + ) (cid:105) n − instead of the gauge parameter. The gen-eralized field and antifield vectors (44), (46), (47) can be uniquely written in terms of thegeneralized field-antifield vector: | χ | c (cid:105) n = (cid:88) e =0 (cid:110) ( q ) e | χ e gen | c (cid:105) n + ( q ) − e | χ ∗ e gen | c (cid:105) n (cid:111) , (cid:0) (cid:15), gh tot (cid:1) | χ | c (cid:105) = (1 , . (49)The minimal BV action for the massless spin-tensor field Ψ ( µ ) n in R ,d − takes the form S c | ( n ) = S c | ( n ) + (cid:8)(cid:0) n (cid:104) ˜ χ ∗ c | (cid:0) η l +1 + η +1 l (cid:1) + n (cid:104) ˜ χ ∗ c | t (cid:1) | C c (cid:105) n + h.c. (cid:9) . (50)The functional S c | ( n ) is invariant with respect to the Lagrangian BRST-transformations δ B S c | ( n ) = 0 for δ B (cid:0) | χ | c (cid:105) n , | χ | c , (cid:105) n , | C c (cid:105) n (cid:1) = .µ (cid:0) η l +1 + η +1 l , t , (cid:1) | C c (cid:105) n , (51)where the field, antifield vectors are subject to the off-shell BRST extended constraintsaccording to (23), (24) (cid:98) T (cid:88) m =0 q m | χ ec (cid:105) n = 0 , (cid:98) T ∗ (cid:16) (cid:88) e =0 q − e | χ ∗ ec (cid:105) n (cid:17) (cid:30) { q P | χ ∗ c (cid:105) n = 0 } = 0 , (52) (cid:98) T | C c (cid:105) n = 0 , (cid:98) T ∗ | C ∗ c (cid:105) n (cid:30) { q P | χ ∗ c (cid:105) n = 0 } = 0 . (53)The resolution of (52), (53) is reduced due to (30) to the form( t ) ( | Ψ (cid:105) , | Ψ ∗ (cid:105) ) = 0 , t ( | Ψ (cid:105) n , | Ψ ∗ (cid:105) n ) = ˜ γ ( | χ (cid:105) n − , −| χ ∗ (cid:105) n − ) , (54) t ( | C c (cid:105) n − , | C ∗ c (cid:105) n − ) = 0 , ( | χ (cid:105) n − , | χ ∗ (cid:105) n − ) = ( t ) ( −| Ψ (cid:105) n , | Ψ ∗ (cid:105) n ) . (55)8n the ghost-independent form the expressions (50), (51) in terms of the triplets of field | Ψ (cid:105) n , | χ (cid:105) n − , | χ (cid:105) n − and antifield | Ψ ∗ (cid:105) n , | χ ∗ (cid:105) n − , | χ ∗ (cid:105) n − vectors and singlets | C c (cid:105) n − , | C ∗ c (cid:105) n − ( BV triplet formulation ) read S c | ( n ) (cid:0) Ψ ( ∗ ) , χ ( ∗ )1 , χ ( ∗ ) (cid:1) = S c | ( n ) (cid:0) Ψ , χ , χ (cid:1) + (cid:110)(cid:16) n (cid:104) ˜Ψ ∗ | ˜ γl +1 + n − (cid:104) ˜ χ ∗ | ˜ γl (56)+ n − (cid:104) ˜ χ ∗ | t (cid:17) | C (cid:105) n − + h.c. (cid:111) ,δ B ( | Ψ (cid:105) n , | χ (cid:105) n − , | χ (cid:105) n − , | C (cid:105) n − ) = µ (cid:0) l +1 , l , ˜ γt , (cid:1) | C (cid:105) n − , (57)and with independent field | Ψ (cid:105) n , ghost | C c (cid:105) n − and antifield | Ψ ∗ (cid:105) n , | C ∗ c (cid:105) n − vectors S c | ( n ) = S c | ( n ) (cid:0) | Ψ (cid:105) (cid:1) + (cid:26) n (cid:104) ˜Ψ ∗ | ˜ γ (cid:18) l +1 + 12 ( t +1 ) l − t +1 t (cid:19) | C (cid:105) n − + h.c. (cid:27) , (58) δ B ( | Ψ (cid:105) n , | C (cid:105) n − ) = µ (cid:0) l +1 , (cid:1) | C (cid:105) n − , (59)with the classical action given by (37) with the constrained (anti)fields (54), (55) and (31), γ -traceless (anti)field | C ( ∗ ) c (cid:105) n − and triple γ -traceless basic (anti)field | Ψ ( ∗ ) (cid:105) n in (50).On the language of the component field Ψ ( ν ) n , ghost C ( ν ) n − spin-tensors and theirsantifields Ψ ∗ ( ν ) n , C ∗ ( ν ) n − the constrained minimal BV action and the Lagrangian BRST-transformations take the form in accordance with (34), (36) : S c | ( n ) (Ψ , C, Ψ ∗ ) = S c | ( n ) (Ψ) + ( − n (cid:90) d d x (cid:104) Ψ ( ν ) n (cid:110) n (cid:16) γ ν n γ µ ∂ µ − ∂ ν n (cid:17) C ( ν ) n − (60)+ n ( n − η ν n − ν n ∂ µ n C ( ν ) n − µ n (cid:111) + h.c. (cid:105) = S c | ( n ) (Ψ) + ( − n (cid:90) d d x (cid:110) n Ψ (cid:48) p/ ( ıC ) − n Ψ · p ( ıC ) + n ( n − (cid:48)(cid:48) p · ( ıC ) + h.c (cid:111) , (61) δ B (cid:0) Ψ ( µ ) n , C ( µ ) n − (cid:1) = µ (cid:0) − (cid:88) ni =1 ∂ µ i C µ ...µ i − µ i +1 ...µ n , (cid:1) . (62)The minimal BV action and Lagrangian BRST-transformations for unconstrained quartetformulations for massless spin-tensor field are easily obtained from the respective BRST-BFVformulations. E.g. we get for the unconstrained minimal BV action S n : S n = S c | ( n ) (cid:0) Ψ ( ∗ ) , χ ( ∗ )1 , χ ( ∗ ) (cid:1) + S add | ( n ) ( λ ) + ( n − (cid:104) ˜ ς ∗ ( a ) | t | C (cid:105) n − + h.c. ) , (63) δ B ( | Ψ (cid:105) n , | χ (cid:105) n − , | χ (cid:105) n − , | ς (cid:105) n − , | C (cid:105) n − ) = µ (cid:0) l +1 , l , ˜ γt , ˜ γt , (cid:1) | C (cid:105) n − , (64)with account of (56), (41) respectively for S c | ( n ) (cid:0) Ψ ( ∗ ) , χ ( ∗ )1 , χ ( ∗ ) (cid:1) , S add | ( n ) ( λ ) and antifieldvector | ς ∗ (cid:105) n − = ˜ γ | ς ∗ ( a + ) (cid:105) n − . The antifield vectors in (63) are considered without off-shellconstraints (54), (55) as well as the antifield spin-tensors λ ∗ ( µ ) n − i i , i = 1 , ,
3, determiningthe total field-antifield space, do not entered in S n .The obtained minimal BV actions for TS massless spin-tensor field Ψ ( µ ) n present thebasic results of the Section.The actions serves to construct quantum actions under an appropriate choice of a gaugecondition (e.g. for the TS field l | Ψ (cid:105) n = 0), as well as to find an interacting theory, includingboth only the TS half-integer HS field Ψ ( µ ) n , with a vertex at least cubic in Ψ ( µ ) n and TSinteger HS field Φ ( µ ) s ., as well as Ψ ( µ ) n interacting with an external electromagnetic fieldand some other HS fields. The consistency of deformation is to be controlled by the master9quation for the deformed action with the interaction terms, thus producing a sequence ofrelations for these terms.Notice, the metric-like LF (21)–(24) may be deformed to describe dynamic of both MSHS field with spin s = n + on the AdS(d) space and, independently, dynamic of MSconformal HS field on R ,d − which, in turn maybe used to study AdS/CFT correspondenceproblem. Acknowledgements
The author is grateful to the organizers of the International Work-shop SQS’17 for their hospitality. I also thank I.L. Buchbinder for the collaboration whensolving the problem and to E.D.Skvortsov, D. Francia, A. Campoleoni for valuable cor-respondence. The work has been done in the framework of the Program of fundamentalresearch of state academies of Sciences for 2013-2020.