SciPost Phys. 10, 080 (2021) ·
published 30 March 2021

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We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of a four dimensional $\mathcal{N}=2$ ArgyresDouglas (AD) theories from the structure of the associated two dimensional $W$algebra. The Schur index is derived from the vacuum character of the corresponding $W$algebra and can be rewritten in a very simple form, which can be easily used to verify properties like levelrank dualities, collapsing levels, and Sduality conjectures. The Zhu's $C_2$ algebra can be regarded as a ring associated with the Schur sector, and a surprising connection between certain Zhu's $C_2$ algebra and the Jacobi algebra of a hypersurface singularity is discovered. Finally, the Macdonald index is computed from the Kazhdan filtration of the $W$algebra.
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in Submissions  report on Schur sector of ArgyresDouglas theory and $W$algebra