  Inspired by Cachazo, Katz and Vafa (``Geometric transitions and $\mathcal
{N}=1$ quiver theories'' (hep-th/0108120)), we examine representations of
``${N}=1$ quivers'' arising from string theory. We derive some mathematical
consequences of the physics, and show that these results are a natural
extension of Gabriel's ADE theorem. Extending the usual ADE case that relates
quiver representations to curves on surfaces, we relate these new quiver
representations to curves on threefolds.
