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

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