Ivancevic_Applied-Diff-Geom

722 Applied Differential Geometry: A Modern Introductionaction further in terms of conventional fields, we get [Flume et. al. (1996)]∫A = Tr∫+ Tr{ 1d 4 x2 (φ† D 2 φ) + ¯ψDψ + g o (φ[¯ψ, γ 5 ψ]) + go[φ 2 † , φ] 2}{ 1d 4 x4go2 F µν F µν + θ o32 ˜F}µν F µν . (4.208)The action (4.208) can be recognized as the standard action for a Quark–Gluon–Higgs system in which all the fields are in the adjoint representationand the coupling constants are reduced to g and θ by the supersymmetry.Thus it is not very exotic. Indeed it could be the QCD action except forthe fact that the quarks are in the adjoint and presence of the scalar field.The action (4.208) embodies all the properties of quantum gauge theorythat have surfaced over the past thirty years and could even be used as amodel to teach quantum gauge theory. It might be worthwhile to list theseproperties [Flume et. al. (1996)]:1. It contains a gauge–field coupled to matter2. It is asymptotically free3. It is scale–invariant, but with a scale–anomaly4. It has spontaneous symmetry breaking5. It has central charges (Z and ¯Z)6. It admits both instantons (see [Belavin et. al. (1975)]) and monopolesBecause of the supersymmetry it has some further special properties, whosesignificance will become clear later, namely,7. It generalizes the Montonen–Olive mass formula [Montonen and Olive(1977)] for gauge–fields and monopoles:from(M = |v| N e + 1 )g 2 n mwhere Z = (an e + a d n m ),toM = |Z|,where n e and n m denote the gauge–field and monopole charges respectively,while the so–called prepotential coefficients a and a d will be explained later.8. It is symmetric with respect to a Z 4 symmetry, which is the relic ofthe R−symmetry: (θ α → e iɛ θ α ), which itself survives the axial anomalybreakdown.