G2-instantons on the ALC members of the \(\mathbb {B}_7\) family

Using co-homogeneity one symmetries, we construct a two-parameter family of non-abelian \(G_2\)-instantons on every member of the asymptotically locally conical \(\mathbb {B}_7\)-family of \(G_2\)-metrics on \(S^3 \times \mathbb {R}^4 \), and classify the resulting solutions. These solutions can be described as perturbations of a one-parameter family of abelian instantons, arising from the Killing vector-field generating the asymptotic circle fibre. Generically, these perturbations decay exponentially to the model, but we find a one-parameter family of instantons with polynomial decay. Moreover, we relate the two-parameter family to a lift of an explicit two-parameter family of anti-self-dual instantons on Taub-NUT \(\mathbb {R}^4\), fibred over \(S^3\) in an adiabatic limit.