Schur Indices, BPS Particles, and Argyres-Douglas Theories

TL;DR

This paper proposes a conjecture linking the superconformal index of 4D $ =2$ theories to BPS spectra, verifies it in specific cases, and uses it to compute indices for Argyres-Douglas theories, revealing connections to 2D chiral algebras.

Contribution

It introduces a conjecture relating the Schur index to BPS spectra, verifies it for certain theories, and computes indices for Argyres-Douglas theories, connecting them to 2D chiral algebras.

Findings

Verification of the conjecture for free theories and specific gauge theories.

Computation of Schur indices for Argyres-Douglas theories.

Identification of the chiral algebra as the vacuum sector of a W-minimal model.

Abstract

We conjecture a precise relationship between the Schur limit of the superconformal index of four-dimensional $\mathcal{N}=2$ field theories, which counts local operators, and the spectrum of BPS particles on the Coulomb branch. We verify this conjecture for the special case of free field theories, $\mathcal{N}=2$ QED, and $SU(2)$ gauge theory coupled to fundamental matter. Assuming the validity of our proposal, we compute the Schur index of all Argyres-Douglas theories. Our answers match expectations from the connection of Schur operators with two-dimensional chiral algebras. Based on our results we propose that the chiral algebra of the generalized Argyres-Douglas theory $(A_{k-1},A_{N-1})$ with $k$ and $N$ coprime, is the vacuum sector of the $(k,k+N)$ $W_{k}$ minimal model, and that the Schur index is the associated vacuum character.