Monopoles on the Bryant–SalamonG2-manifolds

Abstract

a b s t r a c t 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 S 4 and CP 2 . These are the first nontrivial examples of G 2 -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 G 2 -instantons on Λ 2 − (S 4 ) and on Λ 2 − (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.

Gauge theory on Aloff–Wallach spaces

“…A G 2 -instanton is a solution to a gauge theoretical equation that can be written in an oriented 7-dimensional manifold X 7 equipped with a G 2 -structure ϕ. Even though G 2 -instantons have entered the mathematical literature for over 30 years now [13], it was only in the past few years that the first nontrivial examples appeared, namely in [27], [25], [28], [12], [24] and [22]. This recent interest in G 2 -instantons is mostly due to the suggestion by Donaldson-Thomas [15] and Donaldson-Segal [16] that it may be possible to use G 2 -instantons to construct an enumerative invariant of G 2 -manifolds.…”

Gerbes on G2 manifolds

“…In [Don11] Donaldson and Segal suggested it may be easier, to define an invariant from solutions of a gauge theoretical equation, the G 2 -monopole equation. The authors have further suggested that such monopoles may be somehow related to coassociative submanifolds, and this have been further investigated in [Oli14]. This short note contains a detour, where we consider gauge theoretical objects of another nature, which show to also be related to coassociative submanifolds.…”

The limit of large mass monopoles

“…for some function a : R + → R. Similarly, seeing a Higgs field Φ(r) as a function in the total space with values in su(2) one can show, see the Appendix in [16], that any spherically symmetric Higgs field must be of the form Φ = φ(r) S 1 , with φ : R + → R some function. A computation yields that…”

“…, Some particular solutions are given by the flat connection (a, φ) = (±1, 0), and the Dirac monopole (a, φ) = (0, m − 1/2r), where m ∈ R. However, the regularity conditions so that the configuration (A, Φ) smoothly extends over the origin yield that φ(0) = 0 and a(0) = 1. One can then show, see the Appendix in [16], that any such solution is given by…”