Free Field Realizations from the Higgs Branch

TL;DR

This work develops economical free field realizations for the VOAs associated to a broad class of 4d N=2 SCFTs, tying the VOA generators directly to the Higgs branch EFT data. By employing a small set of chiral bosons (and symplectic bosons or C2-cofinite blocks when needed), the authors realize the simple quotients of the relevant VOAs and illuminate how Higgs branch geometry governs the VOA structure, including an intrinsic R-filtration. The DC exceptional series, N=4 SU(2) SYM, and A1,D odd Argyres-Douglas theories are treated in detail, with explicit constructions of generators, stress tensors, and null relations that match known 4d data and DS reduction behavior. The approach offers both a geometric viewpoint on VOA associated varieties and a potential path toward classifying 4d N=2 theories via their 2d counterparts, highlighting the deep interplay between Higgs branch physics and vertex operator algebras.

Abstract

We present free field realizations for the associated vertex operator algebras of a number of four-dimensional $\mathcal{N}=2$ superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the complex dimensionality of the Higgs branch of the superconformal field theory. In the case of theories whose Higgs branches support additional degrees of freedom (free vector multiplets or decoupled interacting SCFTs), the corresponding "free field realizations" include additional ingredients: symplectic fermions in the case of vector multiplets and a $C_2$ co-finite VOA in the case of a residual interacting SCFT. The resulting picture is that the associated VOA can be constructed from the Higgs branch effective theory via free field realization. Our constructions also provide a natural realization of the $R$-filtration of the associated VOA.