Nonlinear Operator Superalgebras and BFV-BRST Operators for Lagrangian Description of Mixed-symmetry HS Fields in AdS Spaces

0.24 MB·English

by  A. A. Reshetnyak

Nonlinear Operator Superalgebras and
BFV–BRST Operators for Lagrangian
9
0 Description of Mixed-symmetry HS
0
2
Fields in AdS Spaces
n
a
J

9
A.A. Reshetnyak
1

] Institute of Strength Physics and Materials Science
h
t of SB of Russian Academy of Sciences, 634021 Tomsk, Russia
-
p
e
h
[

4 We study the properties of nonlinear superalgebras A and algebras A
b
v
9 arising from a one-to-one correspondence between the sets of relations that
2
extract AdS-group irreducible representations D(E ,s ,s ) in AdS -spaces
3 0 1 2 d
2 and the sets of operators that form A and A , respectively, for fermionic,
b
.
2 s = n + 1, and bosonic, s = n , n ∈ N , i = 1,2, HS fields characterized
1 i i 2 i i i 0
8 by a Young tableaux with two rows. We consider a method of constructing
0
the Verma modules V , V for A, A and establish a possibility of their
: A Ab b
v
Fock-space realizations in terms of formal power series in oscillator operators
i
X
which serve to realize an additive conversion of the above (super)algebra (A)
r
a A , containing a set of 2nd-class constraints, into a converted (super)algebra
b
A =A +A(cid:48) (A =A+A(cid:48)), containing a set of 1st-class constraints only. For
bc b b c
the algebra A , we construct an exact nilpotent BFV–BRST operator Q(cid:48)
bc
having nonvanishing terms of 3rd degree in the powers of ghost coordinates
anduseQ(cid:48) toconstructagauge-invariantLagrangianformulation(LF)forHS
fields with a given mass m (energy E (m)) and generalized spin s=(s ,s ).
0 1 2
LFs with off-shell algebraic constraints are examined as well.
PACS Numbers: 11.10.Ef, 11.10.Kk, 11.15.-q, 03.65.Fd, 04.20.Fy, 03.70.+k
1. INTRODUCTION
The growing interest in field-theoretical models of higher dimensions is due to the
2
problem of a unified description of the known interactions and the variety of elementary
particles, which becomes especially prominent at high energies (partially accessible to the
LHC), thus stimulating the present-day development of a mixed-symmetry higher-spin
(HS) field theory in view of its close relation to superstring theory in constant-curvature
spaces [1,2], which operates with an infinite set of bosonic and fermionic HS fields (that
correspond to arbitrary tensor representations of the Wigner little algebra) subject to a
multi-row Young tableaux (YT) Y(s ,...,s ), k ≥ 1, whose study has been initiated by [3]
1 k
and continued in [4]; for a review on HS field theory, see [5,6]. The theory of free and
interacting mixed-symmetry HS fields has been developed in the framework of various
approaches, which may be classified as the light-cone formalism [7], Vasiliev’s frame-like
formalism [8–10] using the unfolded approach [11], and Fronsdal’s [12], both constrained
[13] and unconstrained [14], metric-like formalism. While the results of constructing a
Lagrangian formulation (LF) for free bosonic mixed-symmetry HS fields in the flat space
are well-known within all of these approaches, see for instance [15–18], the corresponding
results for the AdS -space have been developed in the light-cone, for the AdS -space [19],
d 5
and in the frame-like, for an integer spin [20,21], formulations and remain unknown in a
more involved case of half-integer spins with a YT Y(s ,...,s ), s = n + 1,n ∈ N .
1 k i i 2 i 0
The present article is devoted to solving this problem for free integer and half-integer
HS fields in the AdS -space that are subject to a YT with two rows, in unconstrained and
d
constrained metric-like formulations, on a basis of the (initially elaborated for a Hamilto-
nian quantization of gauge theories, and being universal for all of the above constructions)
BFV–BRSTformalism[22,23]; seethereview[24]aswell. Thebasicideahereconsistsina
solution of a problem inverse to that of the method [22], just as in string field theory [25],
in the sense of constructing a gauge LF with respect to a nilpotent BFV–BRST operator
Q, constructed, in turn, from a system {O } of 1st-class constraints that include a spe-
α
cial nonlinear non-gauge operator symmetry (super)algebra (A )A for (half-)integer HS
c bc
fields {O }: {O } ⊃ {O }. These quantities {O } correspond to the initial AdS -group
I I α I d
irreducible representation (irrep) relations extracting the spin-tensors of a definite mass m
(includingm = 0)andspin(exceptforsuchalgebraicconditionsasthegamma-andtrace-
lessconditionsinthecaseofaconstraineddescription)andrealizedasoperatorconstraints
for a vector of a special Fock space whose coefficients are (spin-)tensors related to the spin
of the basic HS field. As a result, the final action and the sequence of reducible gauge
transformations are reproduced by means of the simplest operations of decomposing the
resulting gauge vectors of the Hilbert space that contain the initial HS (spin-)tensors and
the gauge parameters with respect to the initial oscillator and ghost variables, subject to
the spin and ghost number conditions, and also by means of calculating the corresponding
scalar products, first realized in [26,27]. Due to the required presence of auxiliary (spin-)
tensors with a lesser spin, in order to have a closed LF for the basic (spin-)tensor with a
3
given spin (mentioned in the pioneering works of Fierz–Pauli [28] and Singh–Hagen [29]
as a crucial part in the definition of a correct number of physical degrees of freedom),
there arises a necessity of converting the sub(super)algebra of the total HS symmetry
(super)algebra corresponding to the subset of 2nd-class constraints into that of 1st-class
constraints. This conversion procedure is realized as an additive version of [30,31], by
means of constructing the Verma modules [32] for Lie (super)algebras corresponding to
HS fields in the flat case and for specially deformed nonlinear (super)algebras (A(cid:48))A(cid:48) in
b
the AdS -space for HS fields with Y(s ); see [33,34]. A transition to mixed-symmetry HS
d 1
fieldswithY(s ,...,s ),k ≥ 2meetsasignificantobstacletoanapplicationofaCartan-like
1 k
decomposition for (A(cid:48))A(cid:48), which is one of the goals of the present article.
b
Another aspect concerns the structure of the BFV–BRST operator Q, being more in-
volvedinthecaseofaconvertednonlinear(super)algebraforHSfieldssubjecttoY(s ,s ),
1 2
(A )A ,inviewofthepresenceofnonvanishingtermsof3rdorderinthepowersofghosts,
c bc
because of a nontrivial character of the Jacobi identity for O , in comparison, first, with Q
I
for (half-)integer HS fields in the flat space [16,27], second, with Q for totally-symmetric
(half-)integer HS fields in the AdS -space [33,34], and, third, with Q for special classical
d
quadratic (super)algebras investigated in [36,37], because of a partially nonsupercommut-
ing character of the operators O .
I
The paper is organized as follows. In Section 2, we examine the initial operator
(super)algebra (A)A . In Section 3, we consider Proposition, which determines a way to
b
obtain algebraic relations for the (super)algebras of the parts (A(cid:48))A(cid:48) additional to those
b
foraspeciallymodified(super)algebra(A )A ,andexamineaconstructionofVerma
mod bmod
modules that realize the highest-weight representation of (A(cid:48))A(cid:48) and their realization in
b
an auxiliary Fock space. An exact BFV–BRST operator for a converted (super)algebra
(A )A is obtained in Section 4, on the basis of a solution of the Jacobi identity, due to
c bc
the absence of non-trivial higher-order relations for (A )A . The action and the sequence
c bc
ofreduciblegaugetransformations,mainlyforbosonicHSfieldsofafixedspins = (s ,s ),
1 2
are deduced in Section 5. In the conclusion, we summarize the results of this article and
discuss some open problems.
We mainly use the conventions of Refs. [16,34].
2. NONLINEAR (SUPER)ALGEBRA FOR MIXED-SYMMETRY
HS FIELDS IN ADS SPACE-TIME
A massive spin s = (s ,s ), s = n + 1, n ≥ n , representation of the AdS group in
1 2 i i 2 1 2
an AdS space is realized in a space of mixed-symmetry spin-tensors with a suppressed
d
4
Dirac index, and is characterized by Y(s ,s ),
1 2
µ µ · · · · · · · · · µ
Φ ≡Φ (x)←→ 1 2 n1 , (1)
(µ)n1,(ν)n2 µ1...µn1,ν1...νn2 ν ν · · · · · · · ν
1 2 n2
subject to the following equations (β = (2;3) ⇐⇒ (n > n ;n = n ); r being the inverse
1 2 1 2
(cid:8) (cid:9)
squared AdS radius, and Dirac’s matrices satisfying the relation γ ,γ = 2g (x)):
d µ ν µν
(cid:16) (cid:17)
(cid:2)iγµ∇µ−r21(n1+ d2 −β)−m(cid:3), γµ1, γν1 Φ(µ)n1, (ν)n2 = Φ{(µ)n1,ν1}ν2...νn2 = 0. (2)
For a simultaneous description of all half-integer HS fields, one introduces a Fock space H,
generated by 2 pairs of creation ai(x) and annihilation aj+(x) operators, i,j = 1,2,µ,ν =
µ µ
0,1...,d−1: [ai,aj+] = −g δ , and a set of constraints for an arbitrary string-like vector
µ ν µν ij
|Φ(cid:105) ∈ H,
√
t˜(cid:48)|Φ(cid:105) = (cid:2)−iγ˜µD +γ˜(cid:0)m+ r(g1−β)(cid:1)(cid:3)|Φ(cid:105) = 0, (3)
0 µ 0
(cid:0)ti,t(cid:1)|Φ(cid:105) = (cid:0)γ˜µai,a1+a2µ(cid:1)|Φ(cid:105) = 0, (4)
µ µ
|Φ(cid:105) = (cid:80)∞ (cid:80)n1 Φ (x)a+µ1... a+µn1a+ν1... a+νn2|0(cid:105), (5)
n1=0 n2=0 (µ)n1,(ν)n2 1 1 2 2
givenintermsoftheoperatorD = ∂ −ωab(x)(cid:0)(cid:80) a+a −1γ˜ γ˜ (cid:1),a(+)µ(x) = eµ(x)a(+)a,
µ µ µ i ia ib 8 [a b] a
equivalent to the covariant derivative ∇ in its action in H, with a vielbein eµ, a spin
µ a
connection ωab, and tangent indices a,b = 0,1...d − 1. The scalar fermionic operators
µ
t˜(cid:48),ti are defined with the help of an extended set γ˜µ,γ˜ of Grassmann-odd gamma-matrix-
0
like objects [34], {γ˜µ,γ˜ν} = 2gµν, {γ˜µ,γ˜} = 0, γ˜2 = −1, related to the conventional
gamma-matrices by an odd non-degenerate transformation: γµ = γ˜µγ˜. The validity of
relations (3), (4) is equivalent to a simultaneous fulfilment of Eqs. (2) for all the spin-
tensors Φ .
(µ)n1,(ν)n2
The construction of a Hermitian BFV–BRST charge Q, whose special cohomology
in the zero-ghost-number subspace of a total Hilbert space H = H ⊗ H(cid:48) ⊗ H will
tot gh
coincide with the space of solutions of Eqs. (2), implies constructing a set of 1st-class
quantities O , {O } ⊂ {O }, closed under the operations of a) Hermitian conjugation
I α I
√
with respect to an odd scalar product, (cid:104)Ψ|Φ(cid:105) ≡ (cid:104)Ψ˜|Φ(cid:105) [16], with a measure ddx −detg,
1
and b) supercommutator multiplication [ , }. As a result, the final massive (massless for
m = 0) half-integer HS symmetry superalgebra in a space AdS with Y(s ,s ), A = {o }
d 1 2 I
= {t˜(cid:48),ti,ti+,t,t+,li,li+,l ,l+,gi,˜l(cid:48)}, i ≤ j;i,j = 1,2,
0 ij ij 0 0
(cid:0)ti+;gi;t+;li,l+i;l (cid:1)=(cid:0)γ˜µai+;−ai+aµi+ d;aµ1a2+;−i(aµi,a+µi)D ; 1aµa (cid:1), (6)
0 ij µ µ 2 µ µ 2 i µj
˜l(cid:48) = gµν(D D −Γσ D )−r(cid:16)(cid:88)(gi +ti+ti)+ d(d−5)(cid:17)+(cid:0)m+√r(g1−β)(cid:1)2, (7)
0 ν µ µν σ 0 4 0
i
5
[↓,→} t ti ti+ t t+ l li li+ lij lij+ gi
0 0 0
t −2l 2li 2li+ 0 0 0 Mi −Mi+ 0 0 0
0 0
tk 2lk 4lki Aki −t2δk1 −t1δk2 2Mk 0 −t δik 0 Bk,ij tiδki
0
tk+ 2lk+ Aik 4lki+ t1+δk2 t2+δk1 −2Mk+ t δik 0 −Bk,ij+ 0 −ti+δki
0
t 0 t2δi1 −t1+δi2 0 g1−g2 0 l2δi1 −l1+δi2 Dij −Gij+ Fi
0 0
t+ 0 t1δi2 −t2+δi1 g2−g1 0 0 l1δi2 −l2+δi1 Gij −Dij+ −Fi+
0 0
l 0 −2Mi 2Mi+ 0 0 0 −rKbi+ rKbi 0 0 0
0 1 1
lk −Mk 0 −t δik −l2δk1 −l1δk2 rKbk+ Wki Xki 0 Jk,ij liδik
0 1
lk+ Mk+ t δik 0 l1+δk2 l2+δk1 −rKbk −Xik −Wki+ −Jk,ij+ 0 −li+δik
0 1
lkl 0 0 Bi,kl+ −Dkl −Gkl 0 0 Ji,kl+ 0 Lkl,ij li{kδl}i
lkl+ 0 −Bi,kl 0 Gkl+ Dkl+ 0 −Ji,kl 0 −Lij,kl 0 −l+ δ
i{k l}i
gk 0 −t δik tk+δik −Fk Fk+ 0 −lkδik lk+δik −lk{iδj}k l+ δ 0
0 k k{i j}k
Table 1: The superalgebra of the modified initial operators.
√
willcontainacentralchargem˜ = (m−β r),asubsetof(4+12)differential{l ,l+} ⊂ {o }
i i a
and algebraic {t ,t+,t,t+,l ,l+} ⊂ {o } 2-class constraints, as well as some particle-
i i ij ij a
number operators gi, composing, along with m˜2, an invertible supermatrix (cid:107)[o ,o }(cid:107) =
0 a b
(cid:107)∆ (gi,m˜)(cid:107) + O(o ), and obeys some non-linear algebraic relations w.r.t. [ , }. To
ab 0 a
construct an appropriate LF, it is sufficient to have a simpler (so-called modified) super-
algebra A , obtained from A by a linear nondegenerate transformation of o to another
mod I
√
basis o˜ , o˜ = uJo , γ˜ ∈/ {o˜ }, so that the AdS-mass term m = (m+ r(n + d −β))
I I I J I D 1 2
factors out of t˜(cid:48),˜l(cid:48), which change only to t = −iγ˜µD , l = −t2.
0 0 0 µ 0 0
As a result, the operators o˜ , given by (4), (6), with the central charge m˜, satisfy the
I
relations given by Table 1, where the quantities Aik, Bk,ij, Dij, Fi, Gij, Jk,ij, Lkl,ij are
defined as follows:
Aik = −2(giδik −tδi2δk1−t+δi1δk2), Dij = l{i2δj}1, Gij = l1{iδj}2, (8)
0
Jk,ij = −1l{i+δj}k, Bk,ij = −1t{i+δj}k, Fi = t(δi2−δi1), (9)
2 2
6
Lkl,ij = −Lkl,ij+ = 1(cid:8)δikδlj(cid:2)2gkδkl +gk +gl(cid:3)−δik(cid:2)t(cid:0)δl2(δj1+δk1δkj) (10)
4 0 0 0
+δk2δj1δlk(cid:1)+t+(cid:0)1 ←→ 2(cid:1)(cid:3)−δlj(cid:2)t(cid:0)δk2(δi1+δl1δli)+δl2δi1δkl(cid:1)+t+(cid:0)1 ←→ 2(cid:1)(cid:3)(cid:9),
whereas the nonlinear operators Mi, Wij, Kbi, Xij are given by (εij = −εji, ε12 = 1)
1
(cid:16) (cid:17)
Mi = r 2(cid:80) tk+lki+giti− 1ti−tt1δi2−t+t2δi1 , (11)
k 0 2
Wij = Wij + rt[jti] = 2rεij(cid:2)(g2−g1)l12−tl11+t+l22(cid:3)+ rt[jti], (12)
b 4 0 0 4
(cid:16) (cid:17)
Kbi = 4(cid:80) lik+lk +li+(2gi −1)−2l2+tδi1−2l1+t+δi2 , (13)
1 k 0
(cid:110) (cid:16) (cid:16) (cid:17) (cid:17)(cid:111) (cid:110)
Xij = l +r (cid:80) gk +tk+tk − 5gi +gi2+t+t δij +r 1tj+ti−4(cid:80) ljk+lik
0 k 0 2 0 0 2 k
(cid:111)
−(g1+g2− 3)tδj1δi2−t+(g1+g2− 3)δj2δi1+(g1−g2)δj1δi1 . (14)
0 0 2 0 0 2 0 0
It should be noted that for r = 0 the superalgebras A, A are Lie superalgebras [16],
mod
ALie, ALie , which obey the condition (1.2) mentioned for Lie superalgebras in Ref. [38]
mod
for ALie and do not obey it for ALie.1 In their turn, the original A and modified A
mod b bmod
nonlinearmassive(massless,m = 0)integer-spinalgebrasforbosonicHSfields,i.e.,tensors
in (1), for s = n , contain only the respective bosonic elements o , with o˜ being Lorentz
i i I I
b b
scalar having no γ-matrices in the definition of Dµ as compared to the (cid:0)2[d2]×2[d2](cid:1)-matrix
structure of o , o˜ (with [x] being fractional part of a number x ∈ R) for the superalgebras
I I
A,A ,andobey,inthecaseofo˜ ,thesamealgebraicrelationsasthosegivenbyTable1
mod Ib
without the fermionic operators t ,t ,t+. Only some of the nonlinear relations (12), (14)
0 i i
are changed: first, ti must be removed from (12), second, Eqs. (14), along with the new
definition of l = (cid:0)D2−rd(d−6)(cid:1) ∈ {o˜ },˜l ∈ {o }, acquire the form
ob 4 Ib ob Ib
(cid:110) (cid:16) (cid:17)(cid:111) (cid:110)
Xij = l +r (gi −1−δi1)gi −(1+δi2)g2+t+t δij −r 4(cid:80) ljk+lik
b 0b 0 0 0 k
(cid:111)
+(g1+g2−2)tδj1δi2+t+(g1+g2−2)δj2δi1 , (15)
0 0 0 0
˜l = l +m˜2+r(cid:0)(g1−2β−2)g1−g2(cid:1), m˜2 = m2+rβ(β +1), (16)
ob ob b 0 0 0 b
being a consequence of the AdS-group irrep equations for the tensor Φ , see [19],
(µ)s1, (ν)s2
(cid:2)∇2+r[(s −β −1+d)(s −β)−s −s ]+m2(cid:3)Φ = 0, (17)
1 1 1 2 (µ)s1, (ν)s2
(cid:0) gµ1µ2, gν1ν2, gµ1ν1(cid:1)Φ = Φ = 0, (18)
(µ)s1, (ν)s2 {(µ)s1,ν1}ν2...νs2
realized in H, with a standard scalar product (cid:104) | (cid:105), as constraints: (˜l ,l ,t)|Φ(cid:105) = 0.
ob ij
1Indeed, there are anticommutators, [t˜(cid:48)0,tk} = 2lk −γ˜r21t1δk1, that violate the requirement
Ci =0 if ε(i)+ε(m)+ε(n)(cid:54)=0 for the Grassmann parities ε(i)=ε(χ ) of the quantities χ in
mn i i
Eq. (1.1): [χm,χn}=Cmi nχm in [38], because C[[t˜t(cid:48)01]][tk] =−γ˜r12δk1 for γ˜2 =−1.
7
3. ADDITIVE CONVERSION FOR NONLINEAR
SUPERALGEBRAS AND VERMA MODULE CONSTRUCTION
To convert additively non-linear superalgebras with a subset of 2nd class constraints,
we need the following easily verified
Proposition: If a set of operators {o˜ },{o˜ } : H → H is subject to n-th order polyno-
I I
mial supercommutator relations (with the Grassmann parities ε =ε(o )=0,1)
I I
n m
(cid:88) (cid:89)
[o˜ ,o˜ } = fK1o˜ + fK1···Km o˜ , fK1···Km = −(−1)εIεJfK1···Km, (19)
I J IJ K1 IJ Kl IJ JI
m=2 l=1
then due to the requirement of the composition law for a direct sum,
(cid:92)
o˜ −→ O˜ = o˜ +o˜(cid:48) : {o˜(cid:48)} : H(cid:48) → H(cid:48), [o˜ ,o˜(cid:48) } = 0, H H(cid:48) = ∅,
J J J J I I J
such that the set of enlarged operators {O˜ } must obey involution relations, [O˜ ,O˜ } =
I I J
F˜K(o˜(cid:48),O˜)O˜ , the sets {o˜(cid:48) }, {O˜ } form nonlinear superalgebras A(cid:48), A , given in H(cid:48) and
IJ K J J c
H⊗H(cid:48) with the corresponding explicit multiplication laws
n l
[o(cid:48),o(cid:48) } = fK1o(cid:48) +(cid:88)(−1)l−1+εK(l)fKl···K1 (cid:89)o(cid:48) , (20)
I J IJ K1 IJ Ks
l=2 s=1
n n−1 n
[O˜ ,O˜ }=(cid:16)fK +(cid:88)F(l)K(o(cid:48),O˜)(cid:17)O˜ , ε = (cid:88)ε (cid:16) (cid:88) ε (cid:17), (21)
I J IJ IJ K K(n) Ks Kl
l=2 s=1 l=s+1
l−1 l−1 s l−1
FI(Jl)Kl=fIKJ1···Kl (cid:89) O˜Km +(cid:88)(−1)s+εK(s)fiKj(cid:92)s···K1Ks(cid:92)+1···Kl (cid:89)o(cid:48)Kp (cid:89) O˜Km,
m=1 s=1 p=1 m=s+1
(cid:92) (cid:92)
fKs···K1Ks+1···Kl = fKs···K1Ks+1···Kl +fKs···Ks+1K1Ks+2···Kl(−1)εKs+1εK1 +···+
ij ij ij
fKs+1Ks···K1Ks+2···Kl(−1)εKs+1Psl=1εKl +(cid:16)fKs+1Ks···Ks+2K1Ks+3···Kl(−1)εKs+2εK1 +
ij ij
···+fKs+1Ks+2Ks···K1Ks+3···Kl(−1)εKs+2Psl=1εKl(cid:17)(−1)εKs+1Psl=1εKl +···+
ij
(−1)Plm=s+1εKmPsl=1εKlfiKjs+1···KlKs···K1, (22)
wherethesum(22)contains l! termswithallthepossiblewaysoforderingtheindices
s!(l−s)!
(K ,...,K ) among the indices (K ,...,K ) in fKs···K1Ks+1···Kl.
s+1 l s 1 ij
As a consequence, for n = 2 in Proposition, as well as for the algebraic relations
given by Table 1 for (A)A , we can obtain relations (for the first time deduced in [34]
b
for quadratic superalgebras) for the (super)algebras (A(cid:48))A(cid:48) of the additional o(cid:48) and
b I
for the (super)algebras (A )A of the converted operators O˜ . These relations remain
c bc I
the same for the linear (Lie) part of the superalgebras, with the only respective change
o˜ → (o(cid:48),O˜ ), whereas the quadratic ones (11)–(15) take the form (with a preservation of
I I I
8
Table 1, except for the replacement (Kbi,Kbi+,Mi,Mi+) → −(K(cid:48)bi,K(cid:48)bi+,M(cid:48)i,M(cid:48)i+), for
1 1 1 1
A(cid:48), (rKbi,rKbi+,Mi,Mi+) → (−Vi+,−Vi ,Mˆi ,Mˆi+) for A
1 1 W W W W c
(cid:16) (cid:17)
M(cid:48)i = −r 2(cid:80) t(cid:48)k+l(cid:48)ki+g(cid:48)it(cid:48)i− 1t(cid:48)i−t(cid:48)t(cid:48)1δi2−t(cid:48)+t(cid:48)2δi1 , (23)
k 0 2
W(cid:48)ij = W(cid:48)ij − rt(cid:48)[jt(cid:48)i] = −2rεij(cid:2)(g(cid:48)2−g(cid:48)1)l(cid:48)12−t(cid:48)l(cid:48)11+t(cid:48)+l(cid:48)22(cid:3)− rt(cid:48)[jt(cid:48)i], (24)
b 4 0 0 4
(cid:16) (cid:17)
K(cid:48)bi = 4(cid:80) l(cid:48)ik+l(cid:48)k +l(cid:48)i+(2g(cid:48)i−1)−2l(cid:48)2+t(cid:48)δi1−2l(cid:48)1+t(cid:48)+δi2 , (25)
1 k 0
(cid:110) (cid:16) (cid:17)(cid:111) (cid:110)(cid:104)
X(cid:48)ij = l(cid:48) −r (cid:80) K(cid:48)1k +K(cid:48)0i+ 1K(cid:48)1i+K(cid:48)12 δij +r 4(cid:80) l(cid:48)1k+l(cid:48)k2−1t(cid:48)1+t(cid:48)2
0 k 0 0 2 0 0 k 2
(cid:105) (cid:104) (cid:105)(cid:111)
+((cid:80) g(cid:48)k − 3)t(cid:48)δj1δi2+ 4(cid:80) l(cid:48)k2+l(cid:48)1k − 1t(cid:48)2+t(cid:48)1+t(cid:48)+((cid:80) g(cid:48)k − 3) , (26)
k 0 2 k 2 k 0 2
(cid:110) (cid:16) (cid:17)(cid:111) (cid:110)(cid:104) (cid:105)
X(cid:48)ij = l(cid:48) −r K(cid:48)0i+K(cid:48)12 δij +r 4(cid:80) l1k+lk2+((cid:80) gk −2)t δj1δi2
b 0 0 0 k k 0
(cid:104) (cid:105) (cid:111)
+ 4(cid:80) lk2+l1k +t+((cid:80) gk −2) δj2δi1 . (27)
k k 0
In their turn, the only modified relations, for instance, in the converted algebra A , have
bc
the form (with the choice of Weyl’s ordering of O˜ in the r.h.s. of the commutators),
I
b
which implies, as in [34], an exact expression for the BRST operator,
(cid:16)
Vi+=−r 2(Lii+−2l(cid:48)ii+)Li+2(Li−2l(cid:48)i)Lii++(Li+−2l(cid:48)i+)Gi +(Gi −2g(cid:48)i)Li+
bW 0 0 0
+2(cid:2)(cid:0)(L12+−2l(cid:48)12+)L{1+(L{1−2l(cid:48){1)L12+(cid:1)δ2}i− 1δ1i(cid:0)(L2+−2l(cid:48)2+)T
2
(cid:17)
+(T −2t(cid:48))L2+(cid:1)−δ2i(cid:0)(L1+−2l(cid:48)1+)T++(T+−2t(cid:48)+)L1+(cid:1)(cid:3) , (28)
(cid:110)
Wij =rεij (cid:80) (−1)k(Gk −2g(cid:48)k)L12+(L12−2l(cid:48)12)(cid:80) (−1)kGk −[(T −2t(cid:48))L11
bW k 0 0 k 0
(cid:111)
+(L11−2l(cid:48)11)T]+(T+−2t(cid:48)+)L22+(L22−2l(cid:48)22)T+ , (29)
(cid:110) (cid:111) (cid:110)
Xˆij = L +r(cid:0)(Gi −2g(cid:48)i)Gi + 1{T+,T}−(t(cid:48)+T +t(cid:48)T+)(cid:1) δij −r 2(cid:80) [(Ljk+
bW 0 0 0 0 2 k
−2l(cid:48)jk+)Lik +(Lik −2l(cid:48)ik)Ljk+]+ 1(cid:2)(cid:80) (Gk −2g(cid:48)k)T +(T −2t(cid:48))×
2 k 0 0
(cid:111)
×(cid:80) Gk(cid:3)δj1δi2+ 1(cid:2)(T+−2t(cid:48)+)(cid:80) Gk +(cid:80) (Gk −2g(cid:48)k)T+(cid:3)δj2δi1 . (30)
k 0 2 k 0 k 0 0
In Eqs. (26), (27), we have presented the quantities K0i, K0i = (gi2 −2gi −4l+), being
0 0 0 0 ii
Casimir operators for the bosonic subalgebras so(2,1) generated by l ,l+,gi for each
ii ii 0
i = 1,2. The operators K1i, K1i = (gi +ti+ti) extend K0i up to the Casimir operators
0 0 0 0
Ki, Ki = (K0i +K1i), of the Lie subsuperalgebras in A generated by (ti,ti+,l ,l+,gi)
0 0 0 0 c ii ii 0
for each i = 1,2, and the quantity K12 extend (cid:80) K0i, (cid:80) Ki up to the respective Casimir
0 i 0 i 0
operators K , Kb,
0 0
K = Kb +(cid:88)K1i = (cid:88)(cid:0)K0i+K1i(cid:1)+2K12, K12 = t(cid:48)+t(cid:48)−g(cid:48)2−4l(cid:48)12+l(cid:48)12, (31)
0 0 0 0 0 0 0 0
i i
of the maximal (in A) Lie superalgebra ALie generated by (ti,ti+,l ,l+,gi,t,t+),i,k =
ik ik 0
1,2, and of its so(3,2) subalgebra.
9
These operators appear to be crucial to realize the operators of the (super)algebra
(A(cid:48))A(cid:48) in terms of the creation and annihilation operators of a new Fock space (H(cid:48))H(cid:48),
b b
whose number of pairs is equal to that of the converted 2nd-class constraints o , which
a
allowsonetoobtainthecorrectnumberofphysicaldegreesoffreedomdescribingthebasic
spin-tensor (1) in the final LF, after an application of the BFV–BRST procedure to the
resulting first-class constraints {O˜ } ⊂ {O˜ }.
α I
Among the two variants of an additive conversion for the non-linear superalgebras [35]
of {o } into the 1st-class system {O } [first, for the total set of {o }, resulting in an
I α a
unconstrained LF, second, for the differential and partly algebraic constraints l ,l+,t,t+,
i i
restricting the (super)algebra A to the surface {or} ≡ {o }\{l ,l+,t,t+} at all the stages
a a i i
of the construction, resulting in an LF with off-shell γ-traceless (only for the fermionic
HS field Φ ) and (only) traceless conditions for the fields and gauge parameters],
(µ)n1,(ν)n2
we consider in detail the former case. To find o(cid:48) explicitly, we need, first, to construct an
I
auxiliary representation, known as the Verma module [32], on the basis of a Cartan-like
decomposition, extended from the one for ALie,
A(cid:48) = {{t(cid:48)+,l(cid:48)ij+,t(cid:48)+;l(cid:48)i+}⊕{g(cid:48)i;t(cid:48),l(cid:48)}⊕{t(cid:48),l(cid:48)ij,t(cid:48);l(cid:48)i} ≡ E−⊕H ⊕E+, (32)
i 0 0 0 i
and then to realize the above Verma module as an operator-valued formal power series
√
(cid:80) rn P [(a,a+) ] in a new Fock space H(cid:48) generated by (a,a+) = f ,f+b , b+,b ,
n≥0 n a a i i i i ij
b+,b,b+ (for a constrained LF, {or} ↔ (a,a+)r = {b ,b+,b,b+}).
ij a a i i
A solution of these problems is more involved than the analysis made for a non-linear
A(cid:48), see [34], and for a Lie superalgebra A(cid:48)Lie, see [16], due to a nontrivial entanglement of
the triplet of non-commuting negative root vectors (cid:0)l(cid:48)+t(cid:48)+l(cid:48)+(cid:1) and their ordered products
1 2
(cid:0)(l1(cid:48)+)n1(t(cid:48)+)n(l2(cid:48)+)n2(cid:1), ni,n ∈ N0 (composing the non-commuting part of an arbitrary
vector of the Verma module V ), as follows from Table 1. This task should be effectively
A(cid:48)
solved iteratively, e.g., for an action of the operator t(cid:48) on (l(cid:48)2+)n2,
t(cid:48)(cid:0)l(cid:48)2+(cid:1)n2 → (cid:88) (cid:0)l(cid:48)2+(cid:1)n2−1−2mjl(cid:48)1++... → t(cid:48) (cid:88) (cid:88)(cid:0)l(cid:48)2+(cid:1)n2−2−2m−2m(cid:48) +...,
mj=0 m(cid:48)=0m=0
where the remaining summand does not contain any incorrectly ordered terms, thus, ex-
tendingtheknownresultsofVermamoduleconstruction[33]anditsFockspacerealization
in H(cid:48). First, note that there is no nontrivial entanglement of the above triplet of negative
rootvectors,duetoarestrictionofTable1tothesurfacedeterminedbythenon-converted
second-classconstraintslij,lij+,ti,ti+,and,therefore,thefindingofanoperatorrealization
of the restricted A(cid:48) follows the known way [34]. Second, within our conversion procedure
r
the enlarged central charge M˜ = m˜ +m˜(cid:48) vanishes, whereas explicit expressions for o(cid:48) in
I
terms of (a,a+) and the new constants m ,hi, (l(cid:48),g(cid:48)i) = (m2,hi) +... [they are to be
a 0 0 0 0
determined later from the condition of reproducing the correct form of Eqs. (3)] are found
by partially following [33,34].
10
4. BFV-BRST OPERATOR FOR CONVERTED (SUPER)ALGEBRA
To construct a BRST operator for a non-linear non-gauge (super)algebra (A )A ,
c bc
we shall use an operator version of finding a BRST operator, described in Ref. [23], and
classically in Ref. [24]. Due to the quadratic algebraic relations (28)–(30) and their Her-
mitian conjugates, we must check a nontrivial existence of new structure relations and
new structure functions of 3rd order [24], implied by a resolution of the Jacobi identities
(−1)εIεK[[O˜I,O˜J},O˜K}+cycl.perm.(I,J,K) = 0, for (Ac)Abc, n = 2 in (21), (22),
(cid:16) (cid:17)
(−1)εIεK (cid:0)fM +F(2)M(cid:1)(cid:0)fP +F(2)P(cid:1)+(−1)εPεK[F(2)P,O˜ }
IJ IJ MK MK IJ K
+cycl.perm.(I,J,K)− 1FRS (fP +F(2)P) = FRP O˜ ,
2 IJK RS RS IJK R
F(2)K(o(cid:48),O˜) = −(cid:0)fMK +(−1)εKεMfKM(cid:1)o(cid:48) +fMKO˜ , (33)
IJ IJ IJ M IJ M
with the 3rd order structure functions FRS (o(cid:48),O˜) satisfying the properties of generalized
IJK
antisymmetry with respect to a permutation of any two of the lower indices (I,J,K) and
the upper indices R,S.2 If the 4th-, 5th- and 6th- order structure functions FPRS (o(cid:48),O˜),
IJKL
FPRST (o(cid:48),O˜), FPRSTU (o(cid:48),O˜) are zero, the BRST operator Q(cid:48) has the form of the one
IJKLM IJKLMN
for a formal 2nd-rank “gauge” theory [24], i.e., it has an exact form for the (CP)-ordering
of the ghost coordinates CI, bosonic, q ,q ,q+, and fermionic, η , η+, η , η+, η , η, η+,
0 i i 0 i i ij ij
ηi , and their conjugated momenta operators P : p , p+, p , P , P , P+, P , P+, P, P+,
G I 0 i i 0 i i ij ij
Pi , see [16], with the Grassmann parities opposite to those of O˜ and the values of ghost
G I
number gh(CI) = −gh(P ) = 1,
I
Q(cid:48) = CI(cid:2)O˜I + 12CJ(fJPI +FJ(2I)P)PP(−1)εI+εP + 112CJCKFKRJPIPRPP(−1)εIεK+εJ+εR(cid:3). (34)
Therequirementof(CP)-orderingfortheequationsfollowingfromthenilpotencycondition
for Q(cid:48) in the nth- order in CI, n = 3,4,5,6, leads, for instance, for n = 3 to the necessity
or of the separate fulfillment the relations which do not appear in the classical case [24]:
(−1)εIεK(cid:2)F(2)M,F(2)P(cid:9)+cycl.perm.(I,J,K)− 1(cid:2)FRS ,F(2)P(cid:9) = (cid:2)FRP ,O˜ (cid:9), (35)
IJ MK 2 IJK RS IJK R
or of a corresponding extension of the 3rd structural relations (33) by these terms.
InthecaseofabosonicalgebraA , thereare3typesofnontrivialJacobiidentitiesfor
bc
6 triplets (L ,L ,L ), (L+,L+,L ), (L ,L+,L ), with the existence of 3rd-order structure
1 2 0 1 2 0 i j 0
functions. For instance, one of the solutions for (L ,L+,L ) after a reduction of L+ has
i j 0 11
2Given by Eqs. (33), the resolution of Jacobi identities for a nonlinear superalgebra is more
generalthantheonepresentedinRef.[37]foraclassical(super)algebra,becausewedonotexamine
a more restrictive vanishing of all the coefficients at the 1st, 2nd and 3rd degrees in O˜
I