Wang, Qiang2019-10-252019-10-252019-12-01http://hdl.handle.net/2097/40198Wall-crossing structure (WCS) is a formalism proposed and studied by M.Kontsevich and Y.Soibelman that enables us to encode the Donaldson-Thomas (DT) invariants (BPS degeneracies in physics) and to control their “jumps” when certain walls (walls of marginal stability in physics) on the moduli space are being crossed. The celebrated Kontsevich-Soibelman wall-crossing formulas (KSWCF) are the essential ingredients of WCS. WCS formalism is well adapted to the data coming from the complex integrable system. The famous Seiberg-Witten (SW) integrable system is an example. By considering certain gradient flows on the base of the integrable system called the split attractor flows, WCS can produce an algorithm for computing the DT-invariants inductively. This dissertation is about applying the WCS to the SW integrable systems associated to the pure SU(2) and SU(3) supersymmetric gauge theories. We will see that the results via the WCS formalism match perfectly well with those obtained via physics approaches. The main ingredients of this algorithm are the use of split attractor flows and KSWCF. Besides the known BPS spectrum in pure SU(3) case, we obtain new family of BPS states with BPS-invariants equal to 2.en-USWall-crossing structureWall-crossing formulasDonaldson-Thomas invariantBPS stateSeiberg-Witten integrable systemSplit attractor flowDissertation