The asymptotic geometry of G2-monopoles
Abstract: G2-monopoles are special solutions to the
Yang-Mills-Higgs equation on G2-manifolds, similar to
3-dimensional BPS monopoles.
Donaldson and Segal conjectured that these gauge theoretic objects
have a close relationship to the geometry of the underlying
G2-structure. Intuitively, G2-monopoles with
"large mass" are predicted to detect coassociative submanifolds.
One of the first steps in proving this claim is understanding the
analytic behavior of G2-monopoles. In this talk, I will first
introduce the proper analytic setup for the problem. Then I present
results about the asymptotic form of G2-monopoles with structure
group SU(2) on Asymptotically Conical manifolds. These are
joint results with Gonçalo Oliveira and Daniel Fadel. Finally, I will
also talk about further plans in this project, in particular:
(1) Generalizations of these results to manifolds with fibered end and higher rank gauge groups.
(2) Glue-in construction of monopoles.
(3)
A
concentration result for G2-monopoles with large mass.
These are ongoing research directions with Gonçalo Oliveira, Daniel
Fadel, and Saman Habibi Esfahani.