Wall Crossing Structures and Application to SU(3) Seiberg-Witten Integrable system

We apply the wall crossing structure formalism of Kontsevich and Soibelman to Seiberg-Witten integrable systems associated to pure $SU(3)$. This gives an algorithm for computing the Donaldson-Thomas invariants, which correspond to BPS degeneracy of the corresponding BPS states in physics. The main ingredients of this algorithm are the use of split attractor flows and Kontsevich Soibelman wall crossing formulas. Besides the known BPS spectrum in pure $SU(3)$ case, we obtain new family of BPS states with BPS-invariants equal to 2.