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

Abstract

Abstract Given an $$ \mathcal{N}=2 $$ 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 $$ \mathcal{A} $$ A introduced by Beem et al., with simple coefficients υ L,β (q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υ L,β (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 Verlindelike algebra associated to $$ \mathcal{A} $$ 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.