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

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 $\mathfrak{n} >2$.