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

A bstract We reformulate Kontsevich-Soibelman wall-crossing formulae for 4d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 class $$ \mathcal{S} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> theories and corresponding BPS quivers, including those of wild type, as identities for generating series of symmetric quivers that represent dualities of 3d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.