Classification of complete N=2 supersymmetric theories in 4 dimensions

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We define the notion of a complete N=2 supersymmetric theory in 4 dimensions as a UV complete theory for which all the BPS central charges can be arbitrarily varied as we vary its Coulomb branch parameters, masses, and coupling constants. We classify all such theories whose BPS spectrum can be obtained via a quiver diagram. This is done using the 4d/2d correspondence and by showing that such complete N=2 theories map to quivers of finite mutation type. The list of such theories is given by the (generalized) Gaiotto theories consisting of two 5-branes wrapping Riemann surfaces with punctures, as well as 11 additional exceptional cases, which we identify.