Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1131It is now easy to show that the global supersymmetry can be recoveredfrom (6.213). In Witten type theory, Q B can be interpreted as a supersymmetricBRST charge. We define the supersymmetry transformation as[Ohta (1998)]Off–Shell Actionδ S Φ := δ B Φ| c=0 .As was mentioned before, the quantum action of Witten type TQFT can berepresented by BRST commutator with nilpotent BRST charge Q B . However,since our BRST transformation rule is on-shell nilpotent, we shouldintegrate out ν and G i in order to get off–shell BRST transformation andoff–shell quantum action.For this purpose, let us consider the following terms in (6.210),12 (G i − X i ) 2 + 1 2 |ν − A|2 − iµcν + icνµ − ζν − νζ − d i G i , (6.214)whereHere, let us defineX i = ∂ i b − 1 2 ɛ ijkF jk + iMσ i0 M, A = iγ i D i M + γ 0 bM.ν ′ = ν − A, B = −icµ − ζ.ν ′ (ν ′ ) and G i can be integrated out and then (6.214) will be− 1 2 d id i − d i X i − 2|B| 2 + BA + BA.Consequently, we get the off–shell quantum action{S q = Q, ˜Ψ}, where (6.215)˜Ψ ( = −χ i X i + α )2 d i − µ(iγ i D i M + γ 0 bM − βB)−µ(iγ i D i M + γ 0 bM − βB) + ρ∂ i A i − λ[−∂ i ψ i + i ]2 (NM − MN) .α and β are arbitrary gauge fixing paramaters. Convenience choice forthem is α = β = 1. The BRST transformation rule for X i and B fields canbe easily obtained, although we do not write down here.