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The quiver approach to the BPS spectrum of a 4d N=2 gauge theory

We present a survey of the computation of the BPS spectrum of a general four-dimensional N=2 supersymmetric gauge theory in terms of the Representation Theory of quivers with superpotential. We focus on SYM with a general gauge group G coupled to standard matter in arbitrary representations of G (consistent with a non--positive beta--function). The situation is particularly tricky and interesting when the matter consists of an odd number of half-hypermultiplets: we describe in detail SU(6) SYM coupled to half a 20, SO(12) SYM coupled to half a 32, and E_7 SYM coupled to half a 56.