Schur sector of Argyres-Douglas theory and $W$-algebra

Abstract

We study the Schur index, the Zhu’s C_2C2 algebra, and the Macdonald index of a four dimensional \mathcal{N}=2𝒩=2 Argyres-Douglas (AD) theories from the structure of the associated two dimensional WW-algebra. The Schur index is derived from the vacuum character of the corresponding WW-algebra and can be rewritten in a very simple form, which can be easily used to verify properties like level-rank dualities, collapsing levels, and S-duality conjectures. The Zhu’s C_2C2 algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu’s C_2C2 algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the WW-algebra.