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Quivers and BPS states in 3d and 4d

Piotr Kucharski, Pietro Longhi, Dmitry Noshchenko, Sunghyuk Park, Piotr Sułkowski

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.

Quivers and BPS states in 3d and 4d

Abstract

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

Paper Structure

This paper contains 10 sections, 10 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Physics and topology of 3d-4d systems that lead to symmetrization of quivers.
  3. Two constructions of the symmetrization map.
  4. Quivers and DT invariants.
  5. Isomorphism between wall-crossing and unlinking.
  6. Path polytopes.
  7. Properties of the symmetrization map.
  8. Schur index and symmetric quivers.
  9. Topology of 2d and 3d manifolds and associated quivers
  10. Sequence of signed flips

Figures (2)

  • Figure 1: The path polytopes corresponding to $A_2$, $A_3$ and $A_4$ quivers with linear orientation. On one hand, they can be viewed as a set of unlinking operators (where every directed edge is some unlinking $U(ij)$ and consecutive arrows define composition), and define the symmetrization maps for the respective 4d BPS quivers. On the other hand, they are in a way dual to associahedra $K_4$, $K_5$ and $K_6$, respectively.
  • Figure 2: