Ivancevic_Applied-Diff-Geom

1130 Applied Differential Geometry: A Modern Introduction˜∆S| c=0 = (−△φ + φMM − iNN)λ −[−∂ i ψ i + i ]2 (NM − MN) η− µ(iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)+ (iγ [ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM)µ + 2iφµµ]− χ i ∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) + ρ∂ i ψ i + e∂ i A i .It is easy to find that (6.211) is consistent with the action found by thedimensional reduction of the 4D topological action [Zhang et. al. (1995)].BRST TransformationThe Batalin–Vilkovisky algorithm also facilitates to construct BRST transformationrule. The BRST transformation rule for a field Φ is defined byδ B Φ = ɛ ∂ ∣rS ′ ∣∣∣Φ∗∂Φ ∗ , (6.212)= ∂r Ψ∂Φwhere ɛ is a constant Grassmann odd parameter. With this definition for(6.210), we getδ B A i = −ɛ(∂ i c + ψ i ), δ B b = −ɛξ, δ B M = −ɛ(icM + N),[]δ B G i = −ɛ ∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) ,δ B ν = −ɛ(icν + iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM − iµφ),δ B c = ɛφ, δ B ψ i = −ɛ∂ i φ, δ B ρ = ɛe, δ B λ = −ɛη,δ B µ = ɛζ, δ B N = −iɛ(φM + cN), δ B χ i = ɛd i ,δ B φ = δ B ξ = δ B d i = δ B e = δ B ζ = δ B η = 0. (6.213)It is clear at this stage that (6.213) has on-shell nilpotency, i.e., the quantumequation of motion for ν must be used in order to have δ 2 B = 0. This is dueto the fact that the gauge algebra has on–shell reducibility. Accordingly, theBatalin–Vilkovisky algorithm gives a BRST invariant action and on–shellnilpotent BRST transformation. Note that the equations∂ i ξ − ɛ ijk ∂ j ψ k + i(Nσ i0 M + Mσ i0 N) = 0,iγ i D i N + γ i ψ i M + γ 0 bN + γ 0 ξM = 0can be recognized as linearizations of the 3D monopole equations and thenumber of linearly independent solutions gives the dimension of the modulispace M.