Monopoles on the Bryant-Salamon $G_2$ Manifolds

Abstract

$G_2$-Monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant-Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the ${\mathbb{S}}^4$ and ${\mathbb{CP}}^2$. These are the first nontrivial examples of $G_2$-monopoles. Associated with each monopole there is a parameter $m\in {\mathbb{R}}^{+}$, known as the mass of the monopole. We prove that under a symmetry assumption, for each given $m\in {\mathbb{R}}^{+}$ there is a unique monopole with mass $m$. We also find explicit irreducible $G_2$-instantons on ${\Lambda }_{-}^2({\mathbb{S}}^4)$ and on ${\Lambda }_{-}^2({\mathbb{CP}}^2)$. The third Bryant-Salamon $G_2$-metric lives on the spinor bundle over the $3$-sphere. In this case we produce a vanishing theorem for monopoles.

Final accepted version to appear on the Journal for Geometry and Physics (2014)

References in corpus (3)

• Gauge Theory in higher dimensions, II
• Instantons on the exceptional holonomy manifolds of Bryant and Salamon
• Deformation theory of G_2 conifolds

Cited by in corpus (15)

• Instantons on the exceptional holonomy manifolds of Bryant and Salamon
• $SU(2{)}^2$-invariant $G_2$-instantons
• Deformations of nearly Kähler instantons
• Calabi-Yau Monopoles for the Stenzel Metric
• Gauge theory on Aloff-Wallach spaces
• Dimension of monopoles on asymptotically conic 3-manifolds
• Deformations of Asymptotically Conical $G_2$-Instantons
• Moduli of $G_2$ structures and the Strominger system in dimension 7
• The limit of large mass monopoles
• Deformation of singular connections I: $G_2-$instantons with point singularities
• Deformation Theory of Asymptotically Conical Spin(7)-Instantons
• $G_2$-monopoles with singularities (examples)
• Gerbes on $G_2$ Manifolds
• Hyperbolic monopoles with continuous symmetries
• New examples of $G_2$-instantons on ${\mathbb{R}}^4\times S^3$