The Nilpotency Index for 4d $\mathcal{N}=2$ SCFTs
Anirudh Deb, Carlo Meneghelli, Leonardo Rastelli
TL;DR
This work introduces the nilpotency index ${\mathfrak n}$ of the VOA stress tensor as a refined organizing principle for 4d ${\cal N}=2$ SCFTs, linking the Higgs-branch and Coulomb-branch data through the associated variety. It establishes foundational definitions, links to modular differential equations, and practical computation methods (direct VOA nulls, free-field realizations, and MLDE/Macdonald-index signals). The authors compute or bound ${\mathfrak n}$ across broad families (SQCD, Deligne rank-one, Argyres-Douglas, rank-one ${\cal N}=3/4$, class S) and find patterns such as ${\rm rank} \le {\mathfrak n}-1$ and RG monotonicity ${\mathfrak n}_{\rm IR} \le {\mathfrak n}_{\rm UV}$, supporting a more nuanced classification than rank alone. The results illuminate deep VOA-CB connections, propose a program to classify VOAs by ${\mathfrak n}$, and provide concrete constraints for plausible VOAs of less-understood theories (e.g., certain rank-one and IV$^*$-type theories). Overall, ${\mathfrak n}$ offers a quantitative, RG-sensitive, algebraic lens on the landscape of 4d ${\cal N}=2$ SCFTs with potential for guiding both VOA classifications and holographic/CB analyses.
Abstract
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 $\mathfrak{n} >2$.