D2-brane probes of non-toric cDV threefolds via monopole superpotentials

We develop a framework to construct worldvolume gauge theories on D2-branes probing compound Du Val (cDV) Calabi-Yau threefold singularities. By viewing these singularities as ADE surface fibrations over the complex $w$-plane, we encode their geometry in a Higgs field $\Phi(w)$. A D2-brane probe perceives $\Phi(w)$ as an $\mathcal{N}=2$ deformation of its 3d $\mathcal{N}=4$ affine Dynkin quiver gauge theory via polynomial and monopole superpotential terms. By exploiting 3d mirror symmetry, we obtain an effective theory that correctly reproduces the quiver-collapsing mechanism known in the mathematical literature. We present several examples, including non-toric and non-resolvable cases.