N=1supersymmetric product group theories in the Coulomb phase

We study the low-energy behavior of $N=1\mathrm{}$ supersymmetric gauge theories with product gauge groups $\mathrm{SU}{\mathrm{}\left(N\right)\mathrm{}}^M$ and $M$ chiral superfields transforming in the fundamental representation of two of the $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ factors. These theories are in the Coulomb phase with an unbroken $\mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}^{N-1}$ gauge group. For $N>~3,$ $M>~3$ the theories are chiral. The low-energy gauge-kinetic functions can be obtained from hyperelliptic curves which we derive by considering various limits of the theories. We present several consistency checks of the curves including confinement through the addition of mass perturbations and other limits.