A note on anti self dual SU(3) gauge theory over a six-dimensional manifold

The bundle of the 2-forms over a 6-dimensional base manifold decomposes to three sub bundles such that with dimensions 1,6 and 8, respectively. A duality notion for the 2-forms called Φ-duality is given by equation η = λ*Φ( η Φ) and an anti self dual SU (3) Yang–Mills theory is studied on the sub bundle Λ62 . The curvature 2-form in such a theory is closed and its components are constants. The integral of the total action is bounded by the second Chern class of the bundle such that ∫M ( F ) ≥ –8π2∫M ch 2⋀Φ. This bound created a stability case for the total action integral. This stability case is satisfied by the coupling constant. Thus the total pseudo energy becomes proportional to that of Yang–Mills action integral, and at the same time to the Φ-topological charge. Also one sees that, when the base manifold is M = S 4⊂ , this stability case serves the quantization condition like in the sense of Dirac. At the origin and infinity of the four dimensional sphere S 4, the connection becomes flat. Key words: Six-dimensional manifold, SU(3) Yang–Mills theory, Φ-duality, topological bound, Φ-topological charge