Phases of N=1 supersymmetric gauge theories -: art. no. 042

N=1 supersymmetric U(N) gauge theory with adjoint matter Phi and a polynomial superpotential TrW(Phi) 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 Pi(i) U(N-i). Quantum mechanically, each classical vacuum leads to Pi(i) N-i different vacua. As the parameters of W(Phi) 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 tranformed to vacua with ((N) over tilde (1),(N) over tilde (2),...,(N) over tilde (k)), thus leading to anew 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.