III. SLAVNOV-TAYLOR IDENTITIES FOR THE GENERATING FUNCTIONALS: FOR- MAL ASPECTS Following the standard procedure adopted in commutative gauge theories, we start by consider- ing the generating functional for the Green functions of the basic fields and their BRST variations, where S was given in the previous section and The invariance of the functional integral (13) under the field-coordinate transformation (5) and the nilpotency of that variations imply the ST identity where W = − i ln Z is the generating functional for the connected Green functions. Furthermore, by subjecting the functional integral to an arbitrary variable change δ ¯ C , we may derive that As usual, the generating functional Γ of proper (one-particle-irreducible) vertex functions is obtained by a Legendre transformation where we have introduced the classical fields From these definitions, it follows that In terms of Γ the identities (15) and (16) become Z [ J, η, ¯ η, ζ, ¯ ζ ; K, v, ω, ¯ ω ] = ∫ DA$_{µ}$DψD ¯ ψDCD ¯ Ce i ( S + S$_{source}$ $^{)}$, (13) S$_{source}$ = ∫ d $^{4}$x [ J$_{µ}$ ⋆ A µ + ¯ η ⋆ ψ + ¯ ψ ⋆ η + ¯ ζ ⋆ C + ¯ C ⋆ ζ + K$_{µ}$ ⋆ 1 e ( ∂ $^{µ}$C − ie [ A $^{µ}$, C ]$_{⋆}$) + iv ⋆ C ⋆ C + i ¯ ω ⋆ C ⋆ ψ + i ¯ ψ ⋆ C ⋆ ω ] . (14) ∫ d $^{4}$x ( J$_{µ}$ ⋆ δW δK$_{µ}$ − ¯ ζ ⋆ δW δv − 1 αe ∂$_{µ}$ δW δJ$_{µ}$ ⋆ ζ − ¯ η ⋆ δW δ ¯ ω + δW δω ⋆ η ) = 0 , (15) ζ = e∂$_{µ}$ δW δK$_{µ}$ . (16) W [ J, η, ¯ η, ζ, ¯ ζ ; K, v, ω, ¯ ω ] = Γ[ A$_{cl}$, ψ$_{cl}$, ¯ ψ$_{cl}$, C$_{cl}$, ¯ C$_{cl}$ ; K, v, ω, ¯ ω ] + ∫ d $^{4}$x ( J$_{µ}$ ⋆ A µ cl + ¯ η ⋆ ψ$_{cl}$ + ¯ ψ$_{cl}$ ⋆ η + ¯ ζ ⋆ C$_{cl}$ + ¯ C$_{cl}$ ⋆ ζ ) , (17) A µ cl = δW δJ$_{µ}$ , ψ$_{cl}$ = δW δ ¯ η , ¯ ψ$_{cl}$ = − δW δη , C$_{cl}$ = δW δ ¯ ζ , ¯ C$_{cl}$ = − δW δζ . (18) δ Γ δA$_{cl µ}$ = − J $^{µ}$, δ Γ δψ$_{cl}$ = ¯ η, δ Γ δ ¯ ψ$_{cl}$ = − η, δ Γ δC$_{cl}$ = ¯ ζ, δ Γ δ ¯ C$_{cl}$ = − ζ. (19) ∫ d $^{4}$x ( δ Γ δA µ cl ⋆ δ Γ δK$_{µ}$ + δ Γ δC$_{cl}$ ⋆ δ Γ δv − 1 αe ( ∂$_{µ}$A µ $_{cl}$) ⋆ δ Γ δ ¯ C$_{cl}$ + δ Γ δψ$_{cl}$ ⋆ δ Γ δ ¯ ω + δ Γ δω ⋆ δ Γ δ ¯ ψ$_{cl}$ ) = 0 (20) 5