Schur Indices, BPS Particles, and Argyres-Douglas Theories

TL;DR

The paper proposes a precise link between the Schur limit of the 4d N=2 superconformal index and the Coulomb-branch BPS spectrum, mediated by the Kontsevich-Soibelman operator. By defining and evaluating the KS trace, the authors conjecture I(q) = (q)_∞^{2r} Tr[O(q)], with refinements for flavor symmetries, and perform extensive tests in free theories, QED, and SU(2) with matter, finding exact agreement. They apply the conjecture to Argyres-Douglas theories, showing that AD Schur indices reproduce vacuum characters of certain non-unitary W_kminimal models or affine algebras, and proposing a general rule for (A_{k-1},A_{N-1}) theories: the Schur index equals the vacuum character of the (k,k+N) W_k minimal model when gcd(k,N)=1. This builds a robust 4d–2d bridge between operator counting, BPS spectra, wall-crossing, and chiral algebras, with broad checks across A, D, E series and consistency with recent independent results.

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.

Figures (12)

• Figure 1: The BPS quiver for the $\mathcal{N}=2$ pure $SU(2)$ gauge theory.
• Figure 2: The BPS quiver for the $\mathcal{N}=2$$SU(2)$ gauge theory with one hypermultiplet in the fundamental representation.
• Figure 3: The BPS quiver for the $\mathcal{N}=2$$SU(2)$ gauge theory with two hypermultiplets in the fundamental representation.
• Figure 4: The BPS quiver for the $\mathcal{N}=2$$SU(2)$ gauge theory with three hypermultiplets in the fundamental representation.
• Figure 5: The BPS quiver for the $A_{2n}$ Argyres-Douglas theory in the sink/source chamber.