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

TL;DR

This paper explores the relationship between line defect Schur indices, Verlinde algebras, and $U(1)_r$-invariant vacua in 4D $ ext{N}=2$ superconformal theories, revealing new algebraic structures through explicit computations.

Contribution

It uncovers a novel algebraic relation connecting line defect indices, vacua, and Verlinde-like algebras, with new deformations in theories with flavor symmetries.

Findings

The $q o 1$ limit of defect index coefficients relates to vacuum expectation values.

Identifies a commutative diagram linking defect algebras, vacua, and Verlinde-like algebras.

Discovers new deformations of Verlinde-like algebras in theories with flavor symmetries.

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 $v_{L,\beta}(q)$. We report a puzzling new feature of this expansion: the $q \to 1$ limit of the coefficients $v_{L_,\beta}(q)$ is linearly related to the vacuum expectation values $\langle L \rangle$ 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.