In this talk we will discuss $G_2$-instantons and their relation to the heterotic $G_2$-system on 7-dimensional 2-step nilpotent Lie groups. We will first consider left-invariant coclosed $G_2$-structures and present necessary and sufficient conditions for their characteristic connection to be a $G_2$-instanton, expressed in terms of the torsion of the $G_2$-structure, the torsion of the connection, and the Lie group structure. We will then explain how these conditions imply that the corresponding left-invariant Riemannian metrics define naturally reductive structures.

As a continuation, we will consider solutions to the heterotic $G_2$-system on 2-step nilmanifolds. We will describe constructions of solutions with non-abelian gauge groups and discuss how the geometry of the underlying $G_2$-structure and the choice of gauge connection enter the equations. In particular, we will present families of solutions arising from the $G_2$-instantons described in the first part.

This is joint work with Andrew Clarke, Andrés J. Moreno, and Udhav Fowdar.