$G_2$-monopoles are special solutions to the Yang--Mills--Higgs equation on (noncompact) $G_2$-manifolds, similar to the 3-dimensional BPS monopoles.

Donaldson and Segal conjectured that these gauge theoretic objects have a close relationship to the geometry of the underlying $G_2$-structure. Intuitively, $G_2$-monopoles with “large mass” are predicted to concentrate around coassociative submanifolds.

In this talk, I will introduce the relevant concepts to this conjecture, in particular, $G_2$-monopoles and coassociative submanifolds, in more detail. Then I will explain what the state-of-art is, with an emphasis on the work I have done (or planning on doing) with my collaborators.

This is a joint project with Gonçalo Oliveira, Daniel Fadel, Saman Habibi Esfahani, and Lorenzo Foscolo.