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)=Λ₁²⊕Λ₆²⊕Λ₈² 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 Λ₁² and Λ₆² while the self ones equations on Λ₈². Also the solution on Λ₁² presents a flat connection. In addition, the curvatures of the connections on Λ₆² and Λ₈² have TrF³]=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.