Wild wall-crossing and symmetric quivers in 4d and 3d $\mathcal{N}=2$ field theories

TL;DR

This paper reformulates wall-crossing formulas for 4d $ abla=2$ theories using symmetric quivers, revealing dualities in 3d boundary theories and computing Donaldson-Thomas invariants for wild type quivers.

Contribution

It introduces a new symmetric quiver framework for wall-crossing in 4d $ abla=2$ theories, connecting 3d boundary dualities and calculating invariants for wild quivers.

Findings

Identified symmetric quivers for both sides of wall-crossing.

Derived a formula relating 4d and 3d Donaldson-Thomas invariants.

Computed invariants for wild type quivers.

Abstract

We reformulate Kontsevich-Soibelman wall-crossing formulae for 4d $\mathcal{N}=2$ class $\mathcal{S}$ theories and corresponding BPS quivers, including those of wild type, as identities for generating series of symmetric quivers that represent dualities of 3d $\mathcal{N}=2$ boundary theories. We identify such symmetric quivers for both sides of the wall-crossing formulae. In the finite chamber such a quiver is captured by the symmetrized BPS quiver, whereas on the other side of the wall we find an infinite quiver with an intricate pattern of arrows and loops. Invoking diagonalization, for $m$-Kronecker quivers we find a wall-crossing type formula involving trees of unlinkings that expresses closed Donaldson-Thomas invariants of the corresponding 4d theories in terms of open Donaldson-Thomas invariants of the 3d theories and invariants of $m$-loop quivers. Using this formula, we determine a number of closed Donaldson-Thomas invariants of wild type.