Some (Anti-) Self Duality Solutions on Six Manifolds

Abstract

The bundle of the 2-forms on 6-manifold decomposes into three subbundles such that Λ2 (ℝ6) = Λ21 ⊕ Λ26 ⊕ Λ28 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 Λ21 and Λ26 while the self ones equations on Λ28. Also the solution on Λ21 presents a flat connection. In addition, the curvatures of the connections on Λ26 and Λ28 have , 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.