AI 中文总结
该研究构造了SU(2)规范理论的新孤立辛奇点，将Klein A3、E6、E8、E7奇点实现为对应规范理论的希格斯分支，补充了Kronheimer的仿射箭图构造。
AI 中文摘要
具有八个超荷的$\boldsymbol{\rm Sp}(1)\backsimeq\boldsymbol{\rm SU}(2)$规范理论族，其希格斯分支被发现是一个孤立辛奇点。从某种意义上说，这些是最“极小”的规范理论，因为它们仅希格斯化为平凡理论。其物质内容为$N$个基本半超多重态，以及一个处于$\boldsymbol{\rm Sym}^k$表示中的半超多重态，其中$k=1,3,5,7$。$k=1,3$的情况分别重现了已知的$\boldsymbol{\rm Kraft--Procesi}$构造，该构造对应$\boldsymbol{\rm SO}(N+1)$的极小幂零轨道闭包，以及文献\boldsymbol{\rm [Bourget:2025wsp]}中的$\boldsymbol{gSO}(N)$奇点。$k=5,7$的情况是新的孤立辛奇点，分别被命名为$\boldsymbol{hSO}(N)$和$\boldsymbol{iSO}(N)$。通过希格斯机制对这些孤立辛奇点进行了分类，并计算了部分情况的希尔级数和最高权生成（HWG）函数。$\boldsymbol{gSO}(N)$、$\boldsymbol{hSO}(N)$和$\boldsymbol{iSO}(N)$族各有一个（四元数）一维成员；它们分别是克莱因$\boldsymbol{A_3}$、$\boldsymbol{E_6}$和$\boldsymbol{E_8}$奇点。因此，我们的构造将这些克莱因奇点实现为$\boldsymbol{\rm Sp}(1)$规范理论的希格斯分支（超凯勒商），这与$\boldsymbol{\rm Kronheimer}$利用$\boldsymbol{\rm \textit{A}_3}$、$\boldsymbol{\rm \textit{E}_6}$和$\boldsymbol{\rm \textit{E}_8}$仿射箭图的构造（文献\boldsymbol{\rm [Kronheimer:1989zs]}）互为补充。克莱因$\boldsymbol{E_7}$奇点也被实现为$\boldsymbol{\rm Sp}(1)\times\boldsymbol{\rm O}(1)$规范理论的希格斯分支（超凯勒商）。
英文摘要
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 $\widehat A_3$, $\widehat E_6$, and $\widehat E_8$ 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.
