标题： 4d $\mathcal{N}=2$ SCFTs 的幂零指数 显示英文标题

标题： The Nilpotency Index for 4d $\mathcal{N}=2$ SCFTs

摘要： 一个针对4d$\mathcal{N}=2$超共形场论（SCFTs）的完善分类程序利用了真空的Coulomb分支上的Seiberg-Witten几何；理论按其Coulomb分支的复数维数$\mathfrak{rank}$递增排列。 另一种组织方案关注相关的顶点算子代数（VOA），这与Higgs分支更密切相关。 从VOA的角度来看，一种自然的理论排列方式是根据其“幂零指数”，即最小整数$\mathfrak{n}$，使得$T^\mathfrak{n} = 0$在$C_2$代数中成立，其中$T$是VOA的应力张量。 根据希格斯分支重构猜想，对于任何4d ${\cal N}=2$ SCFT，都有 $\mathfrak{n} < \infty$。 从几个例子外推，我们猜想 $\mathfrak{n}$是一个RG单调函数， $\mathfrak{n}_{\rm IR} \leq \mathfrak{n}_{\rm UV}$。 更重要的是，我们在所有情况下都发现 $\mathfrak{rank} \leq \mathfrak{n}-1$。 通过 $\mathfrak{n}$对理论进行排序似乎比通过 $\mathfrak{rank}$进行排序更加精细。 例如，在$\mathfrak{rank}=1$理论列表中，Kodaira SCFT 和$SU(2)$ ${\cal N}=4$ SYM 有$\mathfrak{n} =2$，而其他所有理论都有$\mathfrak{n} >2$。 显示英文摘要

摘要： A well-developed classification program for 4d $\mathcal{N}=2$ super conformal field theories (SCFTs) leverages Seiberg-Witten geometry on the Coulomb branch of vacua; theories are arranged by increasing $\mathfrak{rank}$, the complex dimension of their Coulomb branch. An alternative organizational scheme focusses on the associated vertex operator algebra (VOA), which is more closely related to the Higgs branch. From the VOA perspective, a natural way to arrange theories is by their ``index of nilpotency'', the smallest integer $\mathfrak{n}$ such that $T^\mathfrak{n} = 0$ in the $C_2$ algebra, where $T$ is the VOA stress tensor. It follows from the Higgs branch reconstruction conjecture that $\mathfrak{n} < \infty$ for any 4d ${\cal N}=2$ SCFT. Extrapolating from several examples, we conjecture that $\mathfrak{n}$ is an RG monotone, $\mathfrak{n}_{\rm IR} \leq \mathfrak{n}_{\rm UV}$. What's more, we find in all cases that $\mathfrak{rank} \leq \mathfrak{n}-1$. Theory ordering by $\mathfrak{n}$ appears thus more refined than ordering by $\mathfrak{rank}$. For example, in the list of $\mathfrak{rank}=1$ theories, the Kodaira SCFTs and $SU(2)$ ${\cal N}=4$ SYM have $\mathfrak{n} =2$, while all others have $\mathfrak{n} >2$.