The Potency of Nilpotence

The dynamics of $\mathcal{N}=1$ SUSY gauge theories with matter in adjoint and fundamental representations and the superpotentials given by Arnold's ADE singularities has been extensively studied in the literature. It was also conjectured that supersymmetric models with $W_{A_k}$, $W_{D_{k+2}}$ and $W_{E_7}$ superpotentials possess a dual description. In this paper we revisit the analysis of the moduli space of $A_k$ and $D_{k+2}$ models by considering the duality along nilpotent directions on the moduli space. While our analysis provides additional evidence for the duality conjecture in $W_{A_k}$ models, we show that the duality conjecture fails for the $W_{D_{k+2}}$ models.