Some (Anti-) Self Duality Solutions on Six Manifolds

The bundle of the 2-forms on 6-manifold decomposes into three subbundles such that Λ² (ℝ⁶) = Λ² ₁ ⊕ Λ² ₆ ⊕ Λ² ₈ with dimensions 1, 6 and 8, respectively. The self and anti self duality solutions of the 2-forms, called ϕ-duality are handled and these solutions show that the anti self dual gauge fields live on the subbundles Λ² ₁ and Λ² ₆ while the self ones equations on Λ² ₈. Also the solution on Λ² ₁ presents a flat connection. In addition, the curvatures of the connections on Λ² ₆ and Λ² ₈ have $Tr\left[F^3\right]=0$, and so the topological invariants determined by the Chern classes, i.e. topological charge, consist only on the second Chern class. In the result of this case, the anti self and self ϕ-dual gauge invariant Lagrangians of defined on both subbundles are bounded by the same topological charge. Also, one gives a quantization case to be relating to the instanton number. (paper)