Quivers and BPS states in 3d and 4d

Authors: Piotr Kucharski, Pietro Longhi, Dmitry Noshchenko, Sunghyuk Park, Piotr Sulkowski

Pre-print: UUITP–22/25

Abstract: We propose a symmetrization relation between BPS quivers encoding 4d =2 theories and symmetric quivers associated to 3d =2 theories. We analyse in detail the symmetrization of BPS quivers for a series of Am 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 Am 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.