Ivancevic_Applied-Diff-Geom

1126 Applied Differential Geometry: A Modern Introductionreducible theory, we can find zero–eigenvectors Z α a satisfying R i αZ α a = 0.Moreover, when the theory is on–shell reducible, we can find such eigenvectorsby using equations of motion.For the case at hand, under the identificationsθ = Λ, ɛ i = −∂ i Λ, ϕ = −iΛM,and τ = 0, so that (6.200) will be (6.201)δA i = 0, δb = 0, δM = 0, δG i = 0,δν = iΛ(ν − iγ i D i M − γ 0 bM)| on−shell = 0. (6.202)Then for δA i , for example, the R coefficients and the zero–eigenvectors arederived fromδA i = R AiθZθ Λ + Rɛ AijZ ɛjΛ= 0, that isR Aiθ= ∂ i , R Aiɛ j= δ ij , Z θ Λ = 1, Z ɛjΛ = −∂ j.Obviously, similar relations hold for other fields. The reader may think thatthe choice (6.201) is not suitable as a first stage reducible theory, but notethat the zero–eigenvectors appear on every point where the gauge equivalenceand the topological shift happen to coincide. In this three dimensionaltheory, b(A 0 ) is invariant for the usual infinitesimal gauge transformationbecause of its ‘time’ independence, so (6.201) means that the existence ofthe points on spinor space where the topological shift trivializes indicatesthe first stage reducibility.If we carry out BRST quantization via Faddeev–Popov procedure in thissituation, the Faddeev–Popov determinant will have zero modes. Thereforein order to fix the gauge further we need a ghost for ghost. This reflects onthe second generation gauge invariance (6.202) realized on–shell. However,since b is irrelevant to Λ, the ghost for τ will not couple to the secondgeneration ghost. With this in mind, we use Batalin–Vilkovisky algorithmin order to make BRST quantization (for details, see [Ohta (1998)] andreferences therein).Let us assign new ghosts carrying opposite statistics to the local parameters.The assortment is given byθ −→ c, ɛ i −→ ψ i , τ −→ ξ, ϕ −→ N, Λ −→ φ. (6.203)Ghosts in (6.203) are first generations, in particular, c is Faddeev–Popovghost, whereas φ is a second generation ghost. Their Grassmann parity and