Minimally Unbalanced Quivers

We develop a classification of \emph{minimally unbalanced} $3d~\mathcal{N}=4$ quiver gauge theories. These gauge theories are important because the isometry group $G$ of their Coulomb branch contains a single factor, which is either a classical or an exceptional Lie group. Concurrently, this provides a classification of hyperk\"ahler cones with isometry group $G$ which are obtainable by Coulomb branch constructions. HyperK\"ahler cones such as Coulomb branches of $3d~\mathcal{N}=4$ quivers are indispensable tools for describing Higgs branches of different theories in various dimensions. In particular, they are used to describe Higgs branches of $5d~\mathcal{N}=1$ SQCD with gauge group $SU(N_c)$ and $6d~\mathcal N = (1,0)$ SQCD with gauge group $Sp(N_c)$ at the respective UV fixed points.