BRST structure of non-linear superalgebras
aa r X i v : . [ h e p - t h ] S e p BRST structure of non-linear superalgebras
M. Asorey a ∗ , P.M. Lavrov a,b † , O.V. Radchenko b ‡ and A. Sugamoto c § a Departamento de F´ısica Te´orica,Facultad de Ciencias Universidad de Zaragoza,50009 Zaragoza, Spain b Department of Mathematical Analysis,Tomsk State Pedagogical University,634041 Tomsk, Russia c Department of Physics, Ochanomizu University,Otsuka, Bunkyo-ku,Tokyo 112-8610, Japan
In this paper we analyse the structure of the BRST charge of nonlinear superalgebras.We consider quadratic non-linear superalgebras where a commutator (in terms of (super)Poisson brackets) of the generators is a quadratic polynomial of the generators. We findthe explicit form of the BRST charge up to cubic order in Faddeev-Popov ghost fieldsfor arbitrary quadratic nonlinear superalgebras. We point out the existence of constraintson structure constants of the superalgebra when the nilpotent BRST charge is quadraticin Faddeev-Popov ghost fields. The general results are illustrated by simple examples ofsuperalgebras.
The nilpotent BRST charge as the Noether charge of the global Becchi-Rouet-Stora-Tyutin super-symmetry [1, 2] is a crucial element in both Lagrangian [3] and Hamiltonian [4] quantization methodsof gauge theories (see also the reviews [5]). For general gauge theories, the existence theorem for thenilpotent BRST charge has been proven [5]. It proceeds by the construction of the BRST charge byan infinite, in general, series expansion in the Faddeev-Popov ghost fields. Sometimes these series aretruncated and reduce to finite polynomials. The most remarkable examples are given by the Yang-Mills theories when the nilpotent BRST charge is a quadratic function of Faddeev-Popov ghost fields.Another interesting examples are given by some quadratic nonlinear Lie algebras [6, 7]. The intereston nonlinear algebras was initiated by discovery of conformal field theories [8] which led to a newclass of gauge theories with the nonlinear gauge algebras, the so-called W N algebras [9]. The BRSTconstruction for such algebras was discussed in [10, 6]. This is closely related to the problem of theBRST construction for quantum groups with quadratic nonlinear algebras [11]. Recently, it was shownthat a special class of nonlinear gauge algebras arises in the Lagrangian BRST approach to higherspin theories on anti de Sitter (AdS) space [12]. Note that non-linear algebras of supersymmetry arisefor some quantum mechanical systems with periodic finite-gap potentials [13]. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] Q A , P A ) , ǫ ( Q A ) = ǫ ( P A ) = ǫ A ( ǫ ( X ) denotes the Grassmann parity of a quantity X ) forany two functions F, G { F, G } = ∂F∂Q A ∂G∂P A − ∂G∂Q A ∂F∂P A ( − ǫ ( F ) ǫ ( G ) (1)where the derivatives with respect to momenta P A stand for left derivatives, and those with respectto corresponding coordinates Q A stand for right derivatives. The Poisson superbracket (1) obeys thefollowing properties:(1) Generalized antisymmetry { F, G } = − ( − ǫ ( F ) ǫ ( G ) { G, F } , (2)(2) Generalized Jacobi identity { F, { G, H }} ( − ǫ ( F ) ǫ ( H ) + cyclic perms. ( F, G, H ) ≡ , (3)(3) Grassmann parity ǫ ( { F, G } ) = ǫ ( F ) + ǫ ( G ) , (4)(4) By-linearity { F + H, G } = { F, G } + { H, G } , ( ǫ ( F ) = ǫ ( H )) , (5)(5) Leibniz rule { F H, G } = F { H, G } + { F, G } H ( − ǫ ( H ) ǫ ( G ) , (6) { F, GH } = { F, G } H + G { F, H } H ( − ǫ ( F ) ǫ ( G ) . In the present paper, we study the nilpotent BRST charge for quadratic nonlinear superalgebrasand find some special restrictions on structure constants when the nilpotent BRST charge is given inthe simplest form.The paper is organized as follows. In Section 2 the Jacobi identities for quadratic nonlinearsuperalgebras are derived and some simple examples of such a kind of superalgebras are constructed. InSection 3 the classical nilpotent BRST charge for quadratic nonlinear superalgebras with some specialrestrictions on structure constants is constructed. In Section 4 we consider some simple examples ofsuperalgebras for which general approach can be applied. In Section 5 we present some concludingremarks.
Let us consider a phase space M with local coordinates { ( q i , p i ) , i = 1 , , .., n ; ( ǫ ( q i ) = ǫ ( p i ) = ǫ i ) } andlet { T α = T α ( q, p ) , ǫ ( T α ) = ǫ α } be a set of independent functions on M . We suppose that T α satisfythe involution relations in terms of the Poisson superbracket { T α , T β } = T γ F γαβ + T δ T γ V γδαβ , (7)where the Grassmann parities ǫ ( F γαβ ) = ǫ α + ǫ β + ǫ γ , ǫ ( V γδαβ ) = ǫ α + ǫ β + ǫ γ + ǫ δ and structure constants F γαβ and V γδαβ possess the symmetry properties F γαβ = − ( − ǫ α ǫ β F γβα , V γδαβ = − ( − ǫ α ǫ β V γδβα = ( − ǫ δ ǫ γ V δγαβ . (8)2he Jacobi identities for (7) read F µασ F σβγ ( − ǫ α ǫ γ + cyclic perms. ( α, β, γ ) = 0 , (9) (cid:16) V µνασ F σβγ + F µασ V σνβγ ( − ǫ α ǫ ν + F νασ V σµβγ ( − ǫ µ ( ǫ α + ǫ ν ) (cid:17) ( − ǫ α ǫ γ + cyclic perms. ( α, β, γ ) = 0 , (10) (cid:16) V µνασ V σλβγ ( − ǫ λ ( ǫ α + ǫ µ ) + cyclic perms. ( µ, ν, λ ) (cid:17) ( − ǫ α ǫ γ + cyclic perms. ( α, β, γ ) = 0 . (11)The simplest case of superalgebras really involving fermionic functions is a superalgebra with threegenerators T, G , G where T is bosonic ( ǫ ( T ) = 0) and G , G are fermionic ( ǫ ( G ) = ǫ ( G ) = 1). Inparticular, we have that G = G = 0. The most general relations for the Poisson superbrackets ofgenerators preserving the Grassmann parities have the form { T, G } = a ( T ) G + a ( T ) G , { T, G } = b ( T ) G + b ( T ) G , { G , G } = α ( T ) + α ( T ) G G , { G , G } = β ( T ) + β ( T ) G G , { G , G } = γ ( T ) + γ ( T ) G G . Here a i , b i , α i , β i , γ i , i = 1 , T . Jacobi identities for this algebra requirethe fulfilment of equations α ′ a + α γ = 0 , β ′ b + β β = 0 , α ′ a − α α = 0 , β ′ b + β γ = 0 ,a γ + a β + b α + b γ = 0 , b ′ a − a ′ + a ′ b + a b ′ − b α + a β = 0 , γ ′ a + α ′ b + 2 γ γ + α β = 0 , γ ′ a + α ′ b − γ α − α γ = 0 ,β ′ a + 2 γ ′ b + β γ + 2 γ β = 0 , β ′ a − γ ′ b − α β − γ γ = 0 , where f ′ denotes the derivative of f = f ( T ) with respect to T . We have nine first order differentialequations and one algebraic nonlinear equation with ten unknowns a i , b i , α i , β i , γ i , i = 1 ,
2. We willnot study the general solution to this system and will just list below some special cases. We have thefollowing examples:1 . { T, G } = 0 , { T, G } = 0 , { G , G } = α ( T ) , (12) { G , G } = β ( T ) , { G , G } = γ ( T ) . . { T, G } = a ( T ) G , { T, G } = a ( T ) G , { G , G } = 0 , (13) { G , G } = β ( T ) G G , { G , G } = γ ( T ) G G . . { T, G } = a ( T ) G , { T, G } = b ( T ) G , { G , G } = α ( T ) G G , (14) { G , G } = 0 , { G , G } = γ ( T ) G G . . { T, G } = a ( T ) G , { T, G } = 0 , { G , G } = α ( T ) G G , (15) { G , G } = β ( T ) G G , { G , G } = γ ( T ) G G . . { T, G } = 0 , { T, G } = b ( T ) G + b ( T ) G , { G , G } = 0 , (16) { G , G } = β ( T ) G G , { G , G } = γ ( T ) G G . If we restrict ourselves to the case of quadratic nonlinear superalgebras the examples (12)-(16) ofsuperalgebras (7) reduce to1 . α ( T ) = A T + A T , β ( T ) = B T + B T , γ ( T ) = D T + D T , (17)3 = A , F = B , F = D , V = A , V = B , V = 12 D . . a ( T ) = A + A T, β ( T ) = B , γ ( T ) = D , (18) F = A , F = A , V = 12 A , V = 12 A , V = 12 B , V = 12 D . . a ( T ) = A + A T, b ( T ) = B + B T, α ( T ) = C , γ ( T ) = D , (19) F = A , F = B , V = 12 A , V = 12 B , V = 12 C , V = 12 D . . a ( T ) = A + A T, α ( T ) = C , β ( T ) = B , γ ( T ) = D , (20) F = A , V = 12 A , V = 12 C , V = 12 B , V = 12 D . . b ( T ) = B + B T, b ( T ) = B + B T, β ( T ) = B , γ ( T ) = D , (21) F = B , F = B , V = 12 B , V = 12 B , V = 12 B , V = 12 D . where we introduce the notation T = T , G = T , G = T , ǫ = 0 , ǫ = ǫ = 1. Note that inthe example (12) there are superalgebras which appear in quantum systems with periodic finite-gap potentials [13] if we identify the Hamiltonian with H = T , the two supersymmetry generators Q = G , Q = G and γ ( T ) = 0 , α ( T ) = β ( T ) = P n +1 ( H ). The example (12), (17) contains thesuperalgebra for dynamical systems with Hamiltonian H = T which is invariant under BRST Q = G and anti-BRST ¯ Q = G symmetry (the canonical quantization method based on this supersymmetrywas proposed in [15]) if we identify A = 1 , A = B = B = D = G = 0 : { Q, Q } = 0 , { ¯ Q, ¯ Q } =0 , { H, Q } = 0 , { H, ¯ Q } = 0 , { Q, ¯ Q } = H . In the example (13), (18) there exists the so-called self-reproducing superalgebras (for self-reproducing algebras within BRST formalism see [7]). Indeed, inthe example (13) it is enough to choose a ( T ) = A T, β ( T ) = 0 , γ ( T ) = D to get the self-reproducingsuperalgebra. The main quantity in the generalized canonical formalism [3, 4] for dynamical systems with the firstclass constraints T α = T α ( q, p ) , ǫ ( T α ) = ǫ α fulfilling the property { T α , T β } ≈
0, where ≈ denotesequality on the surface T α ( q, p ) = 0, is the BRST charge Q . Nonlinear superalgebras (7) belong tothis class. The BRST charge require to introduce for each constraint T α an anti-commuting ghost c α and an anticommuting momenta P α having the following Grassmann parities ǫ ( c α ) = ǫ ( P α ) = ǫ α + 1and ghost numbers gh ( c α ) = − gh ( P α ) = 1. They have to obey the relations { c α , P β } = δ αβ , { c α , c β } = 0 , {P α , P β } = 0 , { c α , T β } = 0 , {P α , T β } = 0 . (22)The BRST charge Q is defined as a solution to the equation {Q , Q} = 0 (23)which is an odd function of the variables ( p, q, c, P ), has ghost number gh ( Q ) = 1 and satisfies theboundary condition ∂ Q ∂c α (cid:12)(cid:12)(cid:12) c =0 = T α . (24)A solution to the problem can be obtained in terms of power-series expansions in the ghost variables Q = T α c α + X k ≥ P β k · · · P β P β U ( k ) β β ..β k α α ..α k +1 c α k +1 · · · c α c α = Q + X k ≥ Q k +1 , (25)4here the symmetry properties of U ( k ) in lower indices coincide with the symmetries of monomials c α k +1 c α k · · · c α while in upper indices they are defined by the symmetries of P β k P β k − · · · P β . Inparticular U ( k ) β β ..β k α α ..α k +1 = ( − ( ǫ α +1)( ǫ α +1) U ( k ) β β ..β k α α ..α k +1 = ( − ( ǫ β +1)( ǫ β +1) U ( k ) β β ..β k α α ..α k +1 . Let us now apply the BRST construction to nonlinear superalgebras (7). In lower order, thenilpotency of Q implies that P β (cid:16) ( − ǫ α [ F β α α + T β V β β α α ] − U (1) β α α (cid:17) c α c α = 0 . Thus, the structure function has to be of the form U (1) U (1) γαβ = 12 (cid:16) F γαβ + T δ V δγαβ (cid:17) ( − ǫ α , U (1) γαβ = U (1) γβα ( − ( ǫ α +1)( ǫ β +1) (26)and the contribution Q of second order in ghosts c α to Q is Q = 12 P γ (cid:16) F γαβ + T δ V δγαβ (cid:17) c β c α ( − ǫ α . (27)Using Jacobi identities (9), (10), (11), the condition of nilpotency for Q in the third order can berewritten as( − ǫ β ǫ β P β T β (cid:16) T β V β β α σ V σβ α α ( − ǫ α + ǫ α ǫ β + 4 U (2) β β α α α ( − ǫ β (cid:17) c α c α c α = 0 . (28)Let us introduce the following quantities X β β β α α α = V β β α σ V σβ α α ( − ǫ α + ǫ β ǫ α , (29) X β β β α α α = X β β β α α α ( − ǫ β ǫ β = X β β β α α α ( − ( ǫ α +1)( ǫ α +1) which define the nilpotency equation in the third order (28). Symmetrization of this quantity withrespect to lower indices can be done using the rule obtained in Appendix A (see (A.3)) X β β β [ α α α ] = X β β β α α α + X β β β α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) + X β β β α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) . (30)Then the nilpotency condition (28) can be written in the form( − ǫ β ǫ β P β T β (cid:16) T β X β β β [ α α α ] + 12 U (2) β β α α α ( − ǫ β (cid:17) c α c α c α = 0 . (31)From the Jacobi identities (11) it follows that X β β β [ α α α ] + X β β β [ α α α ] ( − ǫ β ( ǫ β + ǫ β ) + X β β β [ α α α ] ( − ǫ β ( ǫ β + ǫ β ) = 0 . (32)Consider now the quantities N αα α α N αα α α = T β T β X β αβ [ α α α ] ( − ǫ α ǫ β . (33)Due to (32) they satisfy the relations T α N αα α α = 0 . (34)Therefore, N αα α α can be rewritten in the form N αα α α = T β N { αβ } α α α , N { αβ } α α α = − N { βα } α α α ( − ǫ α ǫ β . (35)5aking into account (33), (35) we can show that N { αβ } α α α has a linear dependence on T α N { αβ } α α α = T σ N { αβ } σα α α . (36)In terms of these quantities the structure functions U (2) are given by U (2) β β α α α = − T σ N { β β } σα α α ( − ǫ β + ǫ β ǫ β , U (2) β β α α α = U (2) β β α α α ( − ( ǫ β +1)( ǫ β +1) . (37)Using (33), (35) and the Jacobi identities (32) we obtain the following equations N { β β } β α α α + N { β β } β α α α ( − ε β ε β = X β β β [ α α α ] + X β β β [ α α α ] ( − ǫ β ǫ β (38)which define an explicit form of N { αβ } σα α α . In particular the structure of (38) allows us to suggest theform of N { β β } β α α α N { β β } β α α α = C { β β } β µ ( µ µ ) X µ µ µ [ α α α ] (39)where C { β β } β µ ( µ µ ) is a matrix constructed from the unit matrices δ αµ obeying the following symmetryproperties C { β β } β µ ( µ µ ) = − C { β β } β µ ( µ µ ) ( − β β = C { β β } β µ ( µ µ ) ( − ǫ µ ǫ µ . (40)It is not difficult to find a general structure of C { β β } β µ ( µ µ ) with the required symmetry properties C [ β β ] β µ ( µ µ ) = C (cid:16) δ β µ δ β µ δ β µ − δ β µ δ β µ δ β µ ( − ǫ β ǫ β + (41)+ δ β µ δ β µ δ β µ ( − ǫ µ ǫ µ − δ β µ δ β µ δ β µ ( − ǫ β ǫ β + ǫ µ ǫ µ (cid:17) , where C is a constant. From (39) and (41) it follows that N { β β } β α α α = 2 C (cid:16) X β β β [ α α α ] − X β β β [ α α α ] ( − ǫ β ǫ β (cid:17) . (42)Inserting this result into (38) one obtains4 CX β β β [ α α α ] = (2 C + 1) X β β β [ α α α ] ( − ǫ β ǫ β + (2 C + 1) X β β β [ α α α ] ( − ǫ β ǫ β + ǫ β ǫ β . (43)Taking into account the relations (32) and the symmetry of X β β β [ α α α ] we have X β β β [ α α α ] ( − ǫ β ǫ β + ǫ β ǫ β = − X β β β [ α α α ] − X β β β [ α α α ] ( − ǫ β ǫ β and therefore one can rewrite (43) in the form (cid:16) C + 1 (cid:17) X β β β [ α α α ] = 0 (44)in full agreement with bosonic case [14]. Then we have two solutions to the nilpotency equation atcubic order in ghost variables c α . In the first case there is no restriction on the structure constants V αβγδ of a quadratic nonlinear algebra C = − / . (45)6t leads to N { β β } β α α α = − (cid:16) X β β β [ α α α ] − X β β β [ α α α ] ( − ǫ β ǫ β (cid:17) . (46)Therefore U (2) β β α α α = − T β (cid:16) X β β β [ α α α ] − X β β β [ α α α ] ( − ǫ β ǫ β (cid:17) ( − ǫ β (47)and Q = − P P T β V β β α σ V σβ α α ( − ǫ α + ǫ β + ǫ α ǫ β c α c α c α . (48)The second possibility corresponds to restriction on structure constants of nonlinear superalgebras X β β β [ α α α ] = 0 (49)or V β β α σ V σβ α α ( − ǫ α ( ǫ α + ǫ β ) + cyclic perms. ( α , α , α ) = 0 . (50)It means that N { β β } β α α α = 0 and U (2) β β α α α = 0 , Q = 0 . (51)In what follows we restrict ourselves to the case of superalgebras where the restrictions (50) are fulfilled.In that case the Jacobi identities (11) are satisfied.Now let us analyse the constraints generated for the condition of nilpotency at forth order of ghostfields c α . It has the form( − ǫ β P β P β T β h(cid:16) F β γσ + T β V β β γσ (cid:17) V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) +24 U (3) β β β α α α α i c α c α c α c α = 0 . (52)Let us introduce the following quantities Y β β β α α α α = F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) , (53) X β β β β α α α α = V β β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) (54)which have the symmetry properties Y β β β α α α α = Y β β β α α α α ( − ( ǫ α +1)( ǫ α +1) = Y β β β α α α α ( − ( ǫ α +1)( ǫ α +1) (55) X β β β β α α α α = X β β β β α α α α ( − ( ǫ α +1)( ǫ α +1) = X β β β β α α α α ( − ( ǫ α +1)( ǫ α +1) . (56)In order to define the structure function U (3) correctly we have to symmetrize quantities which appearin (53) and (54) in the indices α , α , α , α . Using the symmetrization (A.4), (A.6) (A.7) and ofsymmetry properties (55) and (56) we get( − ǫ β P β P β T β (cid:16) Y β β β [ α α α α ] + T β X β β β β [ α α α α ] + 144 U (3) β β β α α α α (cid:17) c α c α c α c α = 0 . (57)From (53) and (54) and symmetry properties of structure constants F αβγ , V αβγδ it follows Y β β β [ α α α α ] = Y β β β [ α α α α ] ( − ( ǫ β +1)( ǫ β +1) , X β β β β [ α α α α ] = X β β β β [ α α α α ] ( − ( ǫ β +1)( ǫ β +1) (58)7t is possible to show that X β β β β [ α α α α ] = 0 (59)as consequence of restrictions (49) or (50). Indeed, using definitions (54) and restrictions (50) weobtain the equations X β β β β α α α α − V β β α σ V σβ α γ V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ α ( ǫ α + ǫ β )+ ǫ β ( ǫ α + ǫ α ) ++ V β β α σ V σβ α γ V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ α ǫ β + ǫ β ( ǫ α + ǫ α ) = 0 . (60)Symmetrization of (60) in indices α , α , α , α gives rise to X β β β β [ α α α α ] − V β β α σ X σβ β [ α α α ] ( − ( ǫ α +1)( ǫ β +1)+ ǫ α ǫ β −− V β β α σ V σβ α γ V γβ α α ( − ǫ α + ǫ α + ǫ β ǫ α ( ǫ α + ǫ β )+ ǫ β ( ǫ α + ǫ α ) − (61) − V β β α σ V σβ α γ V γβ α α ( − ǫ α + ǫ α + ǫ β ǫ α ( ǫ α + ǫ β )+ ǫ β ( ǫ α + ǫ α )+( ǫ α +1)( ǫ α + ǫ α ) −− V β β α σ V σβ α γ V γβ α α ( − ǫ α + ǫ α + ǫ β ǫ α ( ǫ α + ǫ β )+ ǫ β ( ǫ α + ǫ α )+( ǫ α +1)( ǫ α + ǫ α ) = 0 . Multiplying these equations by c α c α c α c α we have (cid:16) X β β β β [ α α α α ] − V β β α σ X σβ β [ α α α ] ( − ( ǫ α +1)( ǫ β +1)+ ǫ α ǫ β (cid:17) c α c α c α c α = 0 (62)or due to (49) X β β β β [ α α α α ] c α c α c α c α = 0 (63)which proves (59).Solutions of the nilpotency equation (57) are given by the quantities Y β β β [ α α α α ] with symmetryproperties (58). We shall prove that these quantities satisfy the following symmetries Y β β β [ α α α α ] = Y β β β [ α α α α ] ( − ( ǫ β +1)( ǫ β +1) . (64)In that case we will have for U (3) U (3) β β β α α α α = − Y β β β [ α α α α ] (65)and for contribution to BRST charge in the forth order Q = − P β P β P β F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) c α c α c α c α . (66)This result can be considered as a supersymmetric generalization of BRST charge in the forth orderfor quadratic nonlinear Lie algebras [6].To prove (64) we start with Jacobi identities (10) (cid:16) V β β γσ F σα α + F β γσ V σβ α α ( − ǫ γ ǫ β + F β γσ V σβ α α ( − ǫ β ( ǫ γ + ǫ β ) (cid:17) ( − ǫ γ ǫ α + cyclic perms. ( γ, α , α ) = 0 . (67)Multiplying these equations from right by V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ β ( ǫ α ǫ α )+ ǫ α ǫ γ , Y β β β α α α α − Y β β β α α α α ( − ( ǫ β +1)( ǫ β +1) −− F β α σ X σβ β α α α ( − ǫ α + ǫ β + ǫ α ( ǫ β + ǫ β )+ ǫ α ǫ α −− F β α σ X σβ β α α α ( − ǫ α + ǫ β + ǫ α ( ǫ β + ǫ β )+ ǫ α ǫ α + ǫ β ǫ β ++ F β α σ X σβ β α α α ( − ǫ α + ǫ β + ǫ α ( ǫ β + ǫ β ) + (68)+ F β α σ X σβ β α α α ( − ǫ α + ǫ β + ǫ α ( ǫ β + ǫ β )+ ǫ β ǫ β ++ V β β γσ F σα α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ β + ǫ γ )( ǫ α + ǫ α ) ++ V β β α σ F σγα V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ β ( ǫ α + ǫ α )+ ǫ α ( ǫ α + ǫ γ ) ++ V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ β ( ǫ α + ǫ α ) = 0 . Multiplying these equation form the right by c α c α c α c α and taking into account the symmetrizationin indices α α α α , we get h (cid:16) Y β β β [ α α α α ] − Y β β β [ α α α α ] ( − ( ǫ β +1)( ǫ β +1) (cid:17) + (69)+ 23 (cid:16) F β α σ X σβ β [ α α α ] ( − ǫ β + ǫ α ǫ β + F β α σ X σβ β [ α α α ] ( − ǫ β + ǫ α ǫ β + ǫ β ǫ β (cid:17) ( − ǫ α (1+ ǫ β ) ++ V β β γσ F σα α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α )( ǫ β + ǫ γ ) ++2 V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β i c α c α c α c α = 0 . or due to (49) h (cid:16) Y β β β [ α α α α ] − Y β β β [ α α α α ] ( − ( ǫ β +1)( ǫ β +1) (cid:17) + (70)+ V β β γσ F σα α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α )( ǫ β + ǫ γ ) ++2 V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β i c α c α c α c α = 0 . Consider now the Jacobi identities (10) (cid:16) V σβ α γ F γα α + F σα γ V γβ α α ( − ǫ α ǫ β + F β α γ V γσα α ( − ǫ σ ( ǫ α + ǫ β ) (cid:17) ( − ǫ α ǫ α + cyclic perms. ( α , α , α ) = 0 . Multiplying these equation form the right by c α c α c α c α and from left by V β β α σ ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α + ǫ α ǫ α , we obtain (cid:16) V β β α σ V σβ α γ F γα α ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α ++ V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ β ( ǫ α + ǫ α ) + (71)+ V β β α σ F β α γ V γσα α ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α + ǫ σ ( ǫ α + ǫ β ) (cid:17) c α c α c α c α = 0 . Notice that V β β α σ F β α γ V γσα α ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α + ǫ σ ( ǫ α + ǫ β ) c α c α c α c α = (72)= − X β β γ [ α α α ] F β γα ( − ( ǫ γ +1)( ǫ α + ǫ α + ǫ α )+ ǫ β + ǫ α ǫ β c α c α c α c α = 09ue to (49). Then from (71) it follows (cid:16) V β β α σ V σβ α γ F γα α ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α ++ V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ β ( ǫ α + ǫ α ) (cid:17) c α c α c α c α = 0 . (73)Let us now consider some additional relations which can be derived from (50) V β β α σ V σβ α γ ( − ǫ α ( ǫ γ + ǫ β ) + V β β γσ V σβ α α ( − ǫ γ ( ǫ α + ǫ β ) + V β β α σ V σβ γα ( − ǫ α ( ǫ α + ǫ β ) = 0Multiplying these equation form the right by F γα α ( − ǫ α + ǫ α + ǫ β + ǫ γ ǫ α c α c α c α c α we obtain (cid:16) V β β α σ V σβ α γ F γα α ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α + (74)+ V β β γσ V σβ α α F γα α ( − ǫ α + ǫ α + ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) (cid:17) c α c α c α c α = 0 . Now, we can take into account the following relations V β β γσ V σβ α α F γα α ( − ǫ α + ǫ α + ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) c α c α c α c α = (75)= − V β β σγ F γα α V σβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ σ + ǫ β )( ǫ α + ǫ α ) c α c α c α c α and V β β α σ V σβ α γ F γα α ( − ǫ α + ǫ α + ǫ β + ǫ β ǫ α c α c α c α c α = (76)= − V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β + ǫ β ( ǫ α + ǫ α ) c α c α c α c α . which can be derived with the help of (73). From (74), (75) and (76) it follows that (cid:16) V β β γσ F σα α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α )( ǫ β + ǫ γ ) + (77)+2 V β β α σ F σα γ V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β (cid:17) c α c α c α c α = 0 . Therefore we derive from (70) (cid:16) Y β β β [ α α α α ] − Y β β β [ α α α α ] ( − ( ǫ β +1)( ǫ β +1) (cid:17) c α c α c α c α = 0 , (78)that proves the symmetry properties of Y β β β [ α α α α ] (64).If we additionally require the fulfilment of restrictions on structure constants of superalgebra (7) Y β β β [ α α α α ] = 0 (79)or F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β + ǫ γ ( ǫ α + ǫ α + ǫ β ) ++ F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β +( ǫ α +1)( ǫ α + ǫ α )+ ǫ γ ( ǫ α + ǫ α + ǫ β ) ++ F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β +( ǫ α +1)( ǫ α + ǫ α )+ ǫ γ ( ǫ α + ǫ α + ǫ β ) + (80)+ F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β +( ǫ α +1)( ǫ α + ǫ α )( ǫ α +1)( ǫ α + ǫ α )+ ǫ γ ( ǫ α + ǫ α + ǫ β ) ++ F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β +( ǫ α + ǫ α )( ǫ α + ǫ α )+ ǫ γ ( ǫ α + ǫ α + ǫ β ) ++ F β γσ V σβ α α V γβ α α ( − ǫ α + ǫ α + ǫ β +( ǫ α + ǫ α ) ǫ β + ǫ α +1)( ǫ α + ǫ α )+ ǫ γ ( ǫ α + ǫ α + ǫ β ) = 0 , Q = T α c α + 12 P γ (cid:16) F γαβ + T δ V δγαβ (cid:17) c β c α ( − ǫ α . (81)for any superalgebras (7) satisfying the additional restrictions (50), (80) on its structure constants F αβγ and V αβγδ . In this Section we approach the construction of the nilpotent BRST charge of the form (81) for thesimple examples listed in Section 2. In what follows we will use the following notation for the ghostvariables ( c , c , c ) = ( c, η , η ) , ( P , P , P ) = ( P , P , P ).1. The explicit form of structure constants (17) implies that the indices β , β , β , σ of the non-trivial relations in restrictions (50) must have the following values β = β = β = σ = 1. Thus, theonly relation that has to be verified is the vanishing of V α V α α ( − ǫ α ǫ α + cyclic perms. ( α , α , α ) = 0 (82)and this relation is satisfied because V α = 0. In order to verify (80) non-trivial relations must besatisfied when γ = σ = 1 and all terms in these relations contain a factor F β = 0 which has to vanish.Therefore the nilpotent BRST charge for this example has to be of the form Q = T c + G η + G η + 12 A P η + 12 B P η + D P η η + (83)+ 12 A P T η + 12 B P T η + 12 D P T η η . In this example there are no restrictions on the parameters ( A , B , D , A , B , D ) which ultimatelydefine superalgebras (17).2. The analysis of the relations (50) require the following restrictions on structure constants ofsuperalgebras (18) V = V = V = 0 , ( A = D = 0) . (84)Due to the vanishing of V αβγ = 0 and F αβ = 0 for all values of α, β, γ , the relations (80) are satisfied.The nilpotent BRST charge can in this case be written as Q = T c + G η + G η + A ( P η + P η ) c + 12 B ( P G − P G ) η . (85)3. Analyzing the relations (50) we obtain the following restrictions for the superalgebra (19) V = V = V = V = 0 , ( A = B = C = D = 0) , (86)which reduces to a linear superalgebra with the usual nilpotent BRST charge for linear superalgebras Q = T c + G η + G η + A P η c + B P η c. (87)11. As in the previous case, the analysis of the relations (50) imposes severe restrictions on thesuperalgebra (20) V = V = V = V = 0 , ( A = B = C = D = 0) (88)which also reduces to a linear superalgebra. The nilpotent BRST charge has the form Q = T c + G η + G η + A P η c. (89)5. The analysis of the relations (50) imposes the following restrictions on structure constants ofthe superalgebras (21) V = V = V = 0 , ( B = D = B = 0) . (90)Imposing the vanishing of V αβγ = 0 for all values of α, β, γ and F = 0 , F = 0, the relations (80) aresatisfied. The nilpotent BRST charge can then be written in the form Q = T c + G η + G η + (cid:16) B P + B P + 12 B ( P G + P T ) (cid:17) η c. (91) In this paper we have investigated the BRST structure of quadratic nonlinear superalgebras of form (7)which are characterized by the structure constant F αβγ and V αβγδ . The explicit form of the BRST chargeboth in the second and third orders was found without any additional restrictions on the structureconstants. In the case when the structure constants verify the constraints (50), the construction of theBRST charge can be achieved up to the fourth order in the ghost fields c α . We have found additionalrestrictions (see (80)) on structure constants of any non-linear quadratic superalgebras when nilpotentBRST charge can be written in a canonical form (81) which is quadratic in ghost fields c α . We haveconstructed simple quadratic nonlinear superalgebras with one bosonic and two fermionic generatorsand have verified all the constraints of the structure constants in order to explicitly construct theBRST charge in the canonical form. Acknowledgements
The work of M.A. is partially supported by CICYT (grant FPA2006-2315) and DGIID-DGA (grant2007-E24/2). P.M.L. acknowledges the MEC for the grant (SAB2006-0153). O.V.R. thanks OchanomizuUniversity for the financial support where the part of this work was done. The work of P.M.L. andO.V.R. was supported by the grant for LRSS, project No. 2553.2008.2. The work of P.M.L. was alsosupported by the RFBR-Ukraine grant No. 08-02-90490.
AppendixA Symmetrization
Let us now consider the procedure of symmetrization used for the correct definition of structurefunctions U ( k ) , k = 2 ,
3. Let X α α α be some quantities appearing in expression X = X α α α c α c α c α .12ue to known symmetry properties of monomials c α c α c α , X can be expressed in terms of X [ α α α ] having required symmetry. We have X [ α α α ] = ∂ X∂c α ∂c α ∂c α == X α α α + X α α α ( − ( ǫ α + ǫ α )( ǫ α +1) + X α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) ++ X α α α ( − ( ǫ α +1)( ǫ α +1) + X α α α ( − ( ǫ α +1)( ǫ α +1) ++ X α α α ( − ( ǫ α + ǫ α )( ǫ α +1)+( ǫ α +1)( ǫ α +1) == (cid:16) X α α α ( − ǫ α + ǫ α ǫ α + cyclic perms. ( α , α , α ) (cid:17) ( − ǫ α + ǫ α ǫ α −− (cid:16) X α α α ( − ǫ α + ǫ α ǫ α + cyclic perms. ( α , α , α ) (cid:17) ( − ǫ α + ǫ α ( ǫ α + ǫ α ) and X = 13! X [ α α α ] c α c α c α . (A.1)If X α α α has additional symmetry properties X α α α = X α α α ( − ( ǫ α +1)( ǫ α +1) (A.2)then X [ α α α ] = 2 (cid:16) X α α α + X α α α ( − ( ǫ α + ǫ α )( ǫ α +1) + X α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) (cid:17) = 2 (cid:16) X α α α ( − ǫ α + ǫ α ǫ α + cyclic perms. ( α , α , α ) (cid:17) ( − ǫ α + ǫ α ǫ α . (A.3)Let us now consider quartic quantities in the ghost fields, Y = Y α α α α c α c α c α c α . One canintroduce the symmetric structure Y [ α α α α ] Y [ α α α α ] = ∂ Y∂c α ∂c α ∂c α ∂c α which can be expressed in terms of three indices symmetric quantities Y [ α α α α ] = Y α [ α α α ] + Y α [ α α α ] ( − ( ǫ α +1)( ǫ α + ǫ α + ǫ α +1) ++ Y α [ α α α ] ( − ( ǫ α + ǫ α )( ǫ α + ǫ α ) + Y α [ α α α ] ( − ( ǫ α +1)( ǫ α + ǫ α + ǫ α +1) . (A.4)Then we have Y = 14! Y [ α α α α ] c α c α c α c α , (A.5)and if Y α α α α has additional symmetry properties Y α α α α = Y α α α α ( − ( ǫ α +1)( ǫ α +1) = Y α α α α ( − ( ǫ α +1)( ǫ α +1) , (A.6)one can finally show that Y [ α α α α ] = 4 (cid:16) Y α α α α + Y α α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) + (A.7)+ Y α α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) + Y α α α α ( − ( ǫ α +1)( ǫ α + ǫ α )+( ǫ α +1)( ǫ α + ǫ α ) ++ Y α α α α ( − ( ǫ α + ǫ α )( ǫ α + ǫ α ) + Y α α α α ( − ( ǫ α +1)( ǫ α + ǫ α ) (cid:17) . eferences [1] C. Becchi, A. Rouet and R. Stora, Renormalization of the Abelian Higgs-Kibble model, Commun.Math. Phys. (1975) 127.[2] I.V. Tyutin, Gauge invariance in field theory and statistical physics in operatorial formulation, Prenrint of Lebedev Physics Institute , No. (1975).[3] I.A. Batalin and G.A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. (1981)27; I.A. Batalin and G.A. Vilkovisky, Quantization of gauge theories with linearly dependentgenerators,
Phys. Rev.
D28 (1983) 2567.[4] E.S. Fradkin and G.A. Vilkovisky, Quantization of relativistic systems with constraints,
Phys.Lett.
B55 (1975) 224; I.A. Batalin and G.A. Vilkovisky, Relativistic S -matrix of dynamical sys-tems with boson and fermion constraints, Phys. Lett. (1977) 309; I.A. Batalin and E.S.Fradkin, A generalized canonical formalism and quantization of reducible gauge theories,
Phys.Lett. (1983) 157.[5] M. Henneaux, Hamiltonian form of the path integral for the theories with gauge degrees offreedom,
Phys. Repts (1985) 1; I.A. Batalin and E.S. Fradkin, Operator quantization methodand abelization of dynamical systems subject to first class constraints,
Riv. Nuovo Cimento No 10 (1986) 1; I.A. Batalin and E.S. Fradkin, Operatorial quantization of dynamical systemssubject to constraints. A further study of the construction,
Ann. Inst. H. Poincare , A49 (1988)145.[6] K. Schoutens, A. Servin and P. van Nieuwenhuizen, Quantum BRST charge for quadratic non-linear Lie algebras,
Commun. Math. Phys. (1989) 87.[7] A. Dresse and M. Henneaux, BRST structure of polynomial Poisson algebras,
J. Math. Phys. (1994) 1334.[8] A.B. Zamolodchikov, Infinite additional symmetries in two dimensional conformal quantum fieldtheory, Theor. Math. Phys. (1986) 1205.[9] V.G. Knizhnik, Superconformal algebras in two dimensions, Theor. Math. Phys. (1986) 68;M. Bershadsky, Superconformal algebras in two dimensions with arbitrary N , Phys. Lett.
B174 so ( N )-extended superconformaloperator algebras, Nucl. Phys.
B314 (1989) 519; C.M. Hull, Higher-spin extended conformalalgebras and W -algebras, Nucl. Phys.
B353 (1991) 707; C. Hull, Classical and quantum W -gravity, Proceedings of the Seminar ”Strings and Gravity” , Stony Brook, 1991, World Scientific,1992, p. 495; K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, properties of covariant W gravity, Int. J. Mod. Phys. A6 (1991) 2891; K. Schoutens, A. Sevrin and P. van Nieuwenhuizen,Nonlinear Yang-Mills theories, Phys. Lett.
B255 (1991) 549.[10] J. Thierry-Meg, BRS-analysis of Zamolodchikov’s spin 2 and 3 current algebra,
Phys. Lett.
B197 (1986) 368; H. Lu, C.N. Pope and X.J. Wang, On higher-spin generalization of string theory,
Int.J. Mod. Phys. A9 (1994) 1527.[11] A.P. Isaev and O.V. Ogievetsky, BRST operator for quantum Lie algebras and differential calculuson quantum groups, Teor. Mat. Phys.
No. 2 (2001) 289; V.G. Gorbounov, A.P. Isaev and O.V.Ogievetsky, BRST Operator for Quantum Lie Algebras: Relation to Bar Complex,
Theor. Math.Phys.
No. 1 (2004) 473; A.P. Isaev, S.O. Krivonos and O.V. Ogievetsky, BRST operators forW algebras,
ArXiv:0802.3781 [math-ph] .[12] I.L. Buchbinder, A. Pashnev and M Tsulaia, Lagrangian formulation of the massless higher integerspin fields in the AdS background,
Phys. Lett.
B523 (2001) 338; I.L. Buchbinder, A. Pashnev and14. Tsulaia, Massless Higher Spin Fields in the AdS Background and BRST Constructions forNonlinear Algebras,
ArXiv: hep-th/0206026 ; I.L. Buchbinder, V.A. Krykhtin and P.M. Lavrov,Gauge invariant Lagrangian formulation of higher massive bosonic field theory in AdS space,
Nucl. Phys.
B762 (2007) 334; I.L. Buchbinder, V.A. Krykhtin and A.A. Reshetnyak, BRSTapproach to Lagrangian construction for fermionic higher spin fields in AdS space,
Nucl. Phys.
B787 (2007) 211.[13] A.A. Andrianov, M.V. Ioffe and V.P. Spiridonov, Higher derivative supersymmetry and the Wit-ten index,
Phys. Lett.
A174 (1993) 273; M. Plyushchay, Supersymmetries in pure parabosonicsystems,
Int. J. Mod. Phys.
A15 (2000) 3679; J. Fernandez, J. Negro and L.M. Nieto, Second-order supersymmeric periodic potentials,
Phys. Lett.
A275 (2000) 338; F. Correa, L.M. Nietoand M. S. Plyushchay, Hidden nonlinear su(2—2) superunitary symmetry of N=2 superextended1D Dirac delta potential problem.
Phys. Lett.
B659 (2008) 746; F. Correa, V. Jakubsky, L.M.Nieto and M. S. Plyushchay, Self-isospectrality, special supersymmetry, and their effect on theband structure,
Phys. Rev. Lett. (2008) 030403; F. Correa, V. Jakubsky and M.S. Plyushcay,Finite-gap systems, tri-supersymmetry and self-isospectrality,
ArXiv:0806.1614 [hep-th] .[14] I. L. Buchbinder and P. M. Lavrov, Classical BRST charge for nonlinear algebras,
J. Math. Phys. No. 8 (2007) 082306-1-15.[15] I. A. Batalin, P. M. Lavrov and I. V. Tyutin, Extended BRST quantization of gauge theories inthe generalized canonical formalism,
J. Math. Phys.31