Flavor symmetries and unitarity bounds in ${\mathcal N}=2$ SCFTs

TL;DR

This work derives stringent unitarity constraints on flavor central charges for four-dimensional ${\mathcal N}=2$ SCFTs with reductive flavor symmetry by exploiting the VOA structure of Schur operators. By analyzing the norms of Higgs-branch singlets built from squared moment maps and their relation to Segal-Sugawara operators, the author obtains a complete matrix of inner products for the flavor-singlet sector and translates positivity into bounds on the Kac–Moody levels and central charges. A key finding is that, unless there are critical flavor factors, at most one sub-critical factor can appear, and if a sub-critical factor saturates the Sugawara bound, the true stress tensor coincides with the total Sugawara stress tensor; in the presence of critical factors, all quadratic Higgs singlets are identified and sub-critical factors are forbidden. These results tightly constrain permissible continuous flavor symmetries in unitary ${\mathcal N}=2$ SCFTs and refine the landscape of allowed central charges and affine levels, with connections to previous bounds and implications for class ${\mathcal S}$ theories and their VOAs.

Abstract

In this letter I analyze the constraints imposed by unitarity on the flavor central charges of four-dimensional ${\mathcal N}=2$ SCFTs with general reductive global symmetry groups. I derive several general and far-reaching consequences of unitarity by computing the norms of flavor singlet Higgs branch operators appearing in the squares of "moment map" operators via the associated vertex operator algebra, and imposing the requirement that they be non-negative.