Higher rank DT/PT wall-crossing in Bridgeland stability

We prove that the Gieseker moduli space of stable sheaves on a smooth projective threefold $X$ of Picard rank 1 is separated from the moduli space of PT stable objects by a single wall in the space of Bridgeland stability conditions on $X$, thus realizing the higher rank DT/PT correspondence as a wall-crossing phenomenon in the space of Bridgeland stability conditions. In addition, we also show that only finitely many walls pass through the upper $(\beta ,\alpha )$-plane parametrizing geometric Bridgeland stability conditions on $X$ which destabilize Gieseker stable sheaves, PT stable objects or their duals when $\alpha >{\alpha }_0$.