$\mathcal{N} = 2$ Schur index and line operators

TL;DR

This paper analytically computes the flavored Schur index for various 4d $ =2$ SCFTs, including theories with line operators, revealing connections to chiral algebra characters using new elliptic function integration formulas.

Contribution

Introduces new integration formulas for elliptic functions and Eisenstein series to evaluate Schur indices, including for theories with line operators and closed-form expressions for specific classes.

Findings

Analytic evaluation of Schur index for $A_2$ class-$ ext{S}$ theories.

Closed-form expressions for SU(2) Wilson and 't Hooft line indices.

Demonstrates relation between line operator indices and chiral algebra characters.

Abstract

4d $\mathcal{N} = 2$ SCFTs and their invariants can be often enriched by non-local BPS operators. In this paper we study the flavored Schur index of several types of N = 2 SCFTs with and without line operators, using a series of new integration formula of elliptic functions and Eisenstein series. We demonstrate how to evaluate analytically the Schur index for a series of $A_2$ class-$\mathcal{S}$ theories and the $\mathcal{N} = 4$ SO(7) theory. For all $A_1$ class-$\mathcal{S}$ theories we obtain closed-form expressions for SU(2) Wilson line index, and 't Hooft line index in some simple cases. We also observe the relation between the line operator index with the characters of the associated chiral algebras. Wilson line index for some other low rank gauge theories are also studied.