Mirror symmetry for circle compactified 4d $\mathcal{N}=2$ SCFTs

TL;DR

This paper proposes a mirror symmetry connecting the Schur sector VOAs of 4d $ abla=2$ SCFTs compactified on a circle with the Coulomb branch geometry of the resulting 3d theories, verified for Argyres-Douglas theories.

Contribution

It introduces a novel mirror symmetry relating VOAs and Coulomb branches for circle-compactified 4d $ abla=2$ SCFTs, with detailed verification for Argyres-Douglas theories.

Findings

VOA properties match geometric features of Coulomb branches

Verification for Argyres-Douglas theories with W-algebras and Hitchin moduli spaces

Connections to 3d symplectic duality

Abstract

We propose a mirror symmetry for 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) compactified on a circle with finite size. The mirror symmetry involves vertex operator algebra (VOA) describing the Schur sector (containing Higgs branch) of 4d theory, and the Coulomb branch of the effective 3d theory. The basic feature of the mirror symmetry is that many representational properties of VOA are matched with geometric properties of the Coulomb branch moduli space. Our proposal is verified for a large class of Argyres-Douglas (AD) theories engineered from M5 branes, whose VOAs are W-algebras, and Coulomb branches are the Hitchin moduli spaces. VOA data such as simple modules, Zhu's algebra, and modular properties are matched with geometric properties like $\mathbb{C}^*$-fixed varieties in Hitchin fibers, cohomologies, and some DAHA representations. We also mention relationships to 3d symplectic duality.