Mirror symmetry for circle compactified 4d $A_1$ class-$S$ theories

TL;DR

This paper proposes a new 4d mirror symmetry for class-S theories that links Higgs branch representation theory with Coulomb branch geometry, using VOA correspondence and Hitchin moduli spaces.

Contribution

It introduces a novel mirror symmetry connecting Higgs and Coulomb branches of class-S theories via representation theory and Hitchin systems, extending previous Argyres-Douglas results.

Findings

Matching data with Hitchin moduli space fixed points

Using VOA and modular differential equations for representation analysis

Establishing a systematic link between Higgs and Coulomb branches

Abstract

In this letter, we propose a 4d mirror symmetry for the class-$\mathcal{S}$ theories which relates the representation theory of the chiral quantization of the Higgs branch and the geometry of the Coulomb branch. We study the representation theory by using the 4d/VOA correspondence, (defect) Schur indices and (flavor) modular differential equations, and match the data with the fixed manifolds of the Hitchin moduli spaces. This correspondence extends the connection between Higgs and Coulomb branch of Argyres-Douglas theories, and can provide systematic guidance for the study of the representation theory of vertex operator algebras by exploiting results from Hitchin systems.