Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1127ghost number (U number) are given byc ψ i ξ N φ1 − 1 − 1 − 1 − 2 + ,(6.204)where the superscript of ghost number denotes the Grassmann parity. Notethat the ghost number counts the degree of differential form on the modulispace M of the solution to the 3D monopole equations. The minimal setΦ min of fields consists ofA i b M G i ν0 + 0 + 0 + 0 + 0 + ,and (6.204).On the other hand, the set of anti–fields Φ ∗ min carrying opposite statisticsto Φ min is given byΦ ∗ minA ∗ i b ∗ M ∗ G ∗ i ν ∗ c ∗ ψ ∗ i N ∗ φ ∗−1 − −1 − −1 − −1 − −1 − −2 + −2 + −2 + −3 − .Next step is to find a solution to the master equation with Φ min and, given by∂r S ∂ l S∂Φ A ∂Φ ∗ − ∂ rS ∂ l SA∂Φ ∗ = 0, (6.205)A∂ΦA where r(l) denotes right (left) derivative.The general solution for the first stage reducible theory at hand can beexpressed byS = S c +Φ ∗ i RαC i 1 α +C1α(Z ∗ β α C β 2 +T βγC α γ 1 Cβ 1 )+C∗ 2γA γ βα Cα 1 C β 2 +Φ∗ i Φ ∗ j Bα ji C2 α +· · · ,(6.206)where C1 α (C2 α ) denotes generally the first (second) generation ghost andonly relevant terms in our case are shown. We often use Φ A min =(φ i , C1 α , C β 2 ), where φi denote generally the fields. In this expression, theindices should be interpreted as the label of fields. Do not confuse withspace–time indices. The coefficients Zβ α, T βγ α , etc can be directly determinedfrom the master equation. In fact, it is known that these coefficientssatisfy the following relationsR i αZ α β C β 2 − 2∂ rS c∂φ j Bji α C α 2 (−1) |i| = 0,∂ r R i αC α 1∂φ j R j β Cβ 1 + Ri αT α βγC γ 1 Cβ 1 = 0,∂ r Z α β Cβ 2∂φ j R j γC γ 1 + 2T α βγC γ 1 Zβ δ Cδ 2 + Z α β A β δγ Cγ 1 Cδ 2 = 0, (6.207)