New symplectic singularities from $SU(2)$ gauge theories

Abstract

Families of $\mathrm{Sp}(1)\simeq \mathrm{SU}(2)$ gauge theories with eight supercharges are found to have a Higgs branch which is an isolated symplectic singularity. These are, in some sense, the most ``minimal'' gauge theories as they only Higgs to a trivial theory. The matter content is $N$ fundamental half-hypermultiplets and one half-hypermultiplet in the ${\mathrm{Sym}}^k$ representation where $k=1,3,5,7$. The cases of $k=1,3$ reproduce the known Kraft--Procesi construction for minimal nilpotent orbit closures of $\mathrm{SO}(N+1)$ and the $g\mathrm{SO}(N)$ singularities of \cite{Bourget:2025wsp}, respectively. The cases $k=5,7$ are new isolated symplectic singularities which are termed $h\mathrm{SO}(N)$ and $i\mathrm{SO}(N)$, respectively. The classification of these isolated symplectic singularities is argued for through the Higgs mechanism, with Hilbert series and highest weight generating (HWG) functions computed for some cases. Each of the $g\mathrm{SO}(N)$, $h\mathrm{SO}(N)$, and $i\mathrm{SO}(N)$ families has a (quaternionic) one-dimensional member; these are the Klein $A_3$, $E_6$, and $E_8$ singularities, respectively. Our construction hence provides realisations of these Klein singularities as Higgs branches (hyper-Kähler quotients) of $\mathrm{Sp}(1)$ gauge theories, complementary to Kronheimer's construction \cite{Kronheimer:1989zs} using the ${\hat{A}}_3$, ${\hat{E}}_6$, and ${\hat{E}}_8$ affine quivers. The Klein $E_7$ singularity is also realised as a Higgs branch (hyper-Kähler quotient) of an $\mathrm{Sp}(1)\times \mathrm{O}(1)$ gauge theory.