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arXiv · 1904.09094

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

Dan Xie·Wenbin Yan
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Abstract

We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of a four dimensional $\mathcal{N}=2$ Argyres-Douglas (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 level-rank dualities, collapsing levels, and S-duality 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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Dan Xie, Wenbin Yan. 2021-03-08. Schur sector of Argyres-Douglas theory and $W$-algebra. https://doi.org/10.21468/scipostphys.10.3.080

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