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Large $N_c$ from Seiberg-Witten Curve and Localization

When N = 2 gauge theories are compactified on $S^4$, the large $N_c$ limit then selects a unique vacuum of the theory determined by saddle-point equations, which remains determined even in the flat-theory limit. We show that exactly the same equations can be reproduced purely from Seiberg-Witten theory, describing a vacuum where magnetically charged particles become massless, and corresponding to a specific degenerating limit of the Seiberg-Witten spectral curve where $2N_c-2$ branch points join pairwise giving $a_{Dn}=0$, $n=1,...,N_c-1$. We consider the specific case of N = 2 $SU(N_c)$ SQCD coupled with $2N_f$ massive fundamental flavors. We show that the theory exhibits a quantum phase transition where the critical point describes a particular Argyres-Douglas point of the Riemann surface.