Wall-Crossing and Quiver Invariants

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We start with a one-slide review of the Kontsevich-Soibelman
(KS) solution to the wall-crossing problem and then proceed to direct  and comprehensive physics counting of BPS states that eventually  connects to KS. We also asks what input data is needed for either  approaches to produce complete BPS spectra, and this naturally  leads to the BPS quiver representation of BPS states and the new notion of quiver invariants.

We propose a simple geometrical conjecture that can segregate BPS states in Higgs phases of the BPS quiver dynamics to those that experience wall-crossing and those that do not, and give

proofs for all cyclice Abelian quivers. We close with explanation of how physics distinguishes two such classes of BPS states.