Abstract
Given an \( \mathcal{N}=2 \) superconformal field theory, we reconsider the Schur index ℐL(q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐL(q) admits an expansion in terms of characters of the chiral algebra \( \mathcal{A} \) introduced by Beem et al., with simple coefficients υL,β(q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υL,β(q) is linearly related to the vacuum expectation values 〈L〉 in U(1)r -invariant vacua of the theory compactified on S1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1)r -invariant vacua, and a Verlindelike algebra associated to \( \mathcal{A} \). Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type (A1, A2), (A1, A4), (A1, A6), (A1, D3) and (A1, D5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.
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Neitzke, A., Yan, F. Line defect Schur indices, Verlinde algebras and U(1)r fixed points. J. High Energ. Phys. 2017, 35 (2017). https://doi.org/10.1007/JHEP11(2017)035
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DOI: https://doi.org/10.1007/JHEP11(2017)035