Gauge theory on Cohomogeneity One Spin(7)-Manifolds from Aloff--Wallach Spaces

We begin a systematic study of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-instantons on complete non-compact cohomogeneity one $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-manifolds with Aloff–Wallach spaces $N_{k,l}=\mathrm{S}\mathrm{U}(3)\mathrm{/}\mathrm{U}(1{)}_{k,l}$ as principal orbits. We provide a complete classification of admissible principal $\mathrm{U}(1)$- and $\mathrm{S}\mathrm{O}(3)$-bundles and smooth bounded solutions to the $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-instanton equation on them. We study bubbling phenomena for the constructed $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-instantons and discuss the compactification of the moduli space of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$-instantons in both the ALC and AC settings.