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Phases of N=1 Supersymmetric Gauge Theories and Matrices

N=1 supersymmetric U(N) gauge theory with adjoint matter $Φ$ and a polynomial superpotential $\Tr W(Φ)$ has been much studied recently. The classical theory has several vacua labeled by integers $(N_1,N_2,...,N_k)$, with the classical unbroken gauge group $\prod_i U(N_i)$. Quantum mechanically, each classical vacuum leads to $\prod_i N_i$ different vacua. As the parameters of $W(Φ)$ are varied, these vacua change in a continuous (and holomorphic) fashion. We find that vacua associated with $(N_1,N_2,...,N_k)$ can be continuously transformed to vacua with $(\tilde N_1,\tilde N_2,...,\tilde N_k)$, thus leading to a new kind of duality. Traditional order parameters, like the Wilson loop and 't Hooft loop, sometimes distinguish different phases. We also find phases that are not distinguished by conventional order parameters. The whole picture of the phase diagram is reminiscent of the phase diagram of $M$-theory.