Finite representations of a quiver arising from string theory

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.