Line defect Schur indices, Verlinde algebras and $U(1{)}_r$ fixed points

Abstract

Given an $\mathcal{N}=2$ superconformal field theory, we reconsider the Schur index ${\mathcal{I}}_L(q)$ in the presence of a half line defect $L$. Recently Cordova-Gaiotto-Shao found that ${\mathcal{I}}_L(q)$ admits an expansion in terms of characters of the chiral algebra $\mathcal{A}$ introduced by Beem et al., with simple coefficients $v_{L,{\hat{I}}^2}(q)$. We report a puzzling new feature of this expansion: the $q\to 1$ limit of the coefficients $v_{L_{,}{\hat{I}}^2}(q)$ is linearly related to the vacuum expectation values $⟨L⟩$ in $U(1{)}_r$-invariant vacua of the theory compactified on $S^1$. 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 Verlinde-like algebra associated to $\mathcal{A}$. Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type $(A_1,A_2)$, $(A_1,A_4)$, $(A_1,A_6)$, $(A_1,D_3)$ and $(A_1,D_5)$. In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

64 pages, 21 figures. v2 published version, references updated