Quivers and BPS states in 3d and 4d

TL;DR

This paper introduces a symmetrization relation linking 4d $ abla=2$ BPS quivers with symmetric quivers for 3d $ abla=2$ theories, analyzing geometric backgrounds and wall-crossing structures in Argyres-Douglas theories.

Contribution

It establishes a novel symmetrization map connecting 4d and 3d BPS quivers, with detailed geometric and algebraic analysis of Argyres-Douglas theories.

Findings

Derived quiver partition functions from skein modules.

Proved wall-crossing structures are isomorphic to unlinking of symmetric quivers.

Showed Schur indices are captured by symmetrized quivers.

Abstract

We propose a symmetrization relation between BPS quivers encoding 4d $\mathcal{N}=2$ theories and symmetric quivers associated to 3d $\mathcal{N}=2$ theories. We analyse in detail the symmetrization of BPS quivers for a series of $A_m$ Argyres-Douglas theories by engineering 3d-4d systems in geometric backgrounds involving appropriate 3-manifolds and Riemann surfaces. We discuss properties of these geometric backgrounds and derive the corresponding quiver partition functions from the perspective of skein modules, which forms the foundation of the symmetrization map for the minimal chamber. We also prove that the structure of wall-crossing in 4d $A_m$ Argyres-Douglas theories is isomorphic to the structure of unlinking of symmetric quivers encoding their partner 3d theories, which allows for a proper definition of the symmetrization map outside the minimal chamber. Finally, we show that the Schur indices of 4d theories are captured by symmetric quivers that include symmetrization of 4d BPS quivers.