) Factorization of Seiberg-witten Curves and N = 1 Vacua

N = 2 gauge theories broken down to N = 1 by a tree level superpotential are necessarily at the points in the moduli space where the Seiberg-Witten curve factorizes. We find exact solution to the factorization problem of Seiberg-Witten curves associated with the breaking of the U(Nc) gauge group down to two factors U(N1) × U(N2). The result is a function of three discrete parameters and two continuous ones. We find discrete identifications between various sets of parameters and comment on their relation to the global structure of N = 1 vacua and their various possible dual descriptions. In an appendix we show directly that integrality of periods leads to factorization.