Coulomb branch of $N=2\mathrm{}$ supersymmetric product group theories from branes

We determine the low-energy description of $N=2\mathrm{}$ supersymmetric ${\Pi }_i$ $\mathrm{SU}{(k}_i)$ gauge theories with bifundamental and fundamental matter based on $M$-theory five-brane configurations. The dependence on moduli and scales of the coefficients in the nonhyperelliptic Seiberg-Witten curves for these theories are determined by considering various field theory and brane limits. A peculiarity in the interpretation of these curves for the vanishing β-function case is noted.