Vanishing OPE Coefficients in 4d N=2 SCFTs

TL;DR

This work analyzes how 4d ${\cal N}=2$ SCFTs are constrained by superconformal symmetry, focusing on the Schur sector and its VOA/CFT duals to identify vanishing short multiplets and OPE coefficients in Argyres-Douglas theories. By computing superconformal characters, deriving selection rules, and using Schur/Macdonald indices together with Virasoro and $W$-algebra structures, the authors demonstrate explicit vanishing patterns in $(A_1,A_{2n})$ and related AD theories, and generalize to $(A_{k-1},A_{n-1})$ with coprime $(k,n)$. The results tie the absence of certain multiplets to null states in associated VOAs and to constraints on central charges and flavor central charges, providing precise OPE selection rules and conjectural Macdonald-index formulas. The findings have potential implications for the ${\cal N}=2$ conformal bootstrap, exact OPE data extraction, and deeper connections between 4d SCFT spectra and chiral algebras.

Abstract

We compute the superconformal characters of various short multiplets in 4d N=2 superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the $(A_1, A_{2n})$ Argyres-Douglas (AD) theory. This implies that the operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge $c$ is identical to that of the AD theory. Similarly, by considering the n-th power of the current multiplet, we show that a particular short multiplet and OPE coefficients vanish for a class of AD theories with ADE flavor symmetry. We also consider the generalized AD theory of type $(A_{k-1}, A_{n-1})$ for coprime k, n and compute its Macdonald index using the associated W-algebra under a mild assumption. This allows us to show that a number of short multiplets and OPE coefficients vanish in this theory. We also provide a Mathematica file along with this paper, where we implement the algorithm by Cordova-Dumitrescu-Intriligator to compute the spectrum of 4d N=2 superconformal multiplets as well as their superconformal character.