Monopoles on the Bryant-Salamon G₂ Manifolds

الأصل · EN

G₂-Monopoles are solutions to gauge theoretical equations on noncompact 7-manifolds of G₂ holonomy. We shall study this equation on the 3 Bryant-Salamon manifolds. We construct examples of G₂-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the S⁴ and CP². These are the first nontrivial examples of G₂-monopoles. Associated with each monopole there is a parameter m ∈ R+, known as the mass of the monopole. We prove that under a symmetry assumption, for each given m ∈ R+ there is a unique monopole with mass m. We also find explicit irreducible G₂-instantons on Λ²-(S⁴) and on Λ²-(CP²). The third Bryant-Salamon G₂-metric lives on the spinor bundle over the 3-sphere. In this case we produce a vanishing theorem for monopoles.