Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1137obtained will correspond to those of the dimensionally reduced version ofthe 4D theory up to these terms, i.e., topological numbers.From the gauge fixing conditionG a i = 0, ν = 0, ∂ i A i = 0, −D i ψ i + i (NM − MN) = 0,2the gauge fermion will beΨ = −χ i G i − µν − µν + ρ∂ i A i − λThe anti–fields are then given by[−D i ψ i + i 2 (NM − MN) ].G ∗ i = −χ i , χ ∗ i = −G i , ν ∗ = −µ, ν ∗ = −µ, µ ∗ = −ν, µ = −ν,M ∗ = − i 2 λN, M ∗ = i 2 λN, N ∗ = i 2 λM, N ∗ = − i 2 λM,ρ ∗ = ∂ i A i , A ∗ i = −∂ i ρ + i[λ, ψ i ], b ∗ = c ∗ = ξ ∗ = φ ∗ = ζ ∗ (ζ ∗ ) = 0,λ ∗ = −[−D i ψ i + [b, ξ] + i ]2 (NM − MN) , ψ ∗ i = −D i λ.Therefore we find the quantum action∫ ( )S q = S c + Tr ˜∆Sn d 3 x, where (6.229)Y˜∆S n = −[−D i ψ i + [b, ξ] + i ]2 (NM − MN) η − λ(D i D i φ + iD i {ψ i , c}),+iλ{ψ i , D i c + ψ i } + (φMM − iNN)λ,−χ[i[c, i G i ] + ɛ ijk D j ψ k + D k ξ + [ψ k , ξ] + i ]2 (Nσij T a T a M + Mσ ij T a T a N) ,−µ(iγ µ D µ N + γ µ ψ µ M + icν) + (iγ i D i N + γ µ ψ µ M + icν)µ, (6.230)+2iφµµ − i 2 {χ i, χ i }φ + ρ(∂ i D i c + ∂ i ψ i ) − d i G i − ζν − νζ + e∂ i A i .,In this quantum action, settingM(M) = N(N) = µ(µ) = ν(ν) = 0,we can find that the resulting action coincides with that of Bogomol’nyimonopoles [Birmingham et. al. (1989)].Finally, in order to get the off–shell quantum action, both the auxiliaryfields should be integrated out by the similar technique presented in Abeliancase.