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

Given an N=2 superconformal field theory, we reconsider the Schur index I_L(q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that I_L(q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients v_L,β(q). We report a puzzling new feature of this expansion: the q→1 limit of the coefficients v_L,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 Verlinde-like algebra associated to 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.