Attractor flow versus Hesse flow in wall-crossing structures

We recast the physics discussions in the paper of Van den Bleeken (J High Energy Phys 2012(2):67, 2012) within the context of wall-crossing structure à la Kontsevich and Soibelman (Homol Mirror Symmetry Trop Geom, 2014. https://doi.org/10.1007/978-3-319-06514-4_6). In particular, we compare the Hesse flow given in Van den Bleeken (J High Energy Phys 2012(2):67, 2012) and the attractor flow on the base of the complex integrable system, and show that both can be used in the formalism of wall-crossing structure. We also propose the notions of dual Hesse flow and dual attractor flow, and show that under the rotation of the \(\mathbb {Z}\)-affine structure, the Hesse flow can be transformed into the dual attractor flow, while the attractor flow into the dual Hesse flow. This suggests its possible use in Mirror Symmetry.